27 Ground spaces and reduced densities of PEPS regions
Cutting a PEPS on a finite graph along the boundary of a region \(R\) writes the state as a sum over the bond configurations on the crossing edges of products of an open contraction on \(R\) and one on its complement. The open contractions on \(R\) span the local ground space \(\mathcal G_R\), and positive operators whose kernel is \(\mathcal G_R\) assemble into a frustration-free parent Hamiltonian of the state. The same cut writes the reduced density \(\rho _R\) as \(M_RM_R^\dagger \) for the coefficient matrix \(M_R\) across the cut. For regular \(G\)-injective tensors of Chapter 26 and a cut whose two sides are connected, the range of \(\rho _R\) is \(\mathcal G_R\). A cut with \(n+1\) crossing bonds then has entropy at most \(n\log |G|\), and the regular virtual boundary state has deficient rank, so it is not the exponential of a Hamiltonian on the full boundary space.
27.1 Ground spaces of finite PEPS regions
Let \(R\) be a region of a finite graph carrying PEPS site tensors. Write \(T_R(\mu ;\sigma )\) for the contraction over the internal bonds, with crossing-bond configuration \(\mu \) and physical configuration \(\sigma \) on \(R\). The local ground space is
This is the construction of [ 17 , Section IV.C.1 ] , local source lines 2003–2008. It applies to arbitrary finite graphs and site-dependent tensors.
For any linear subspace \(S\) of the physical space on \(R\),
The regional ground space is the linear span of these contraction columns. A linear subspace contains that span exactly when it contains every column.
If the virtual space on a crossing edge \(e\) has dimension \(D_e\), then \(\dim \mathcal G_R\leq \prod _{e\in \partial R}D_e\).
The contraction is a linear map from the boundary virtual space onto \(\mathcal G_R\), so its range has dimension at most that of its domain.
Let \(\psi \) be the closed PEPS vector. For each fixed complementary physical configuration \(\tau \), the vector \(\sigma \mapsto \psi (\sigma ,\tau )\) belongs to \(\mathcal G_R\). Consequently, if a linear operator \(h_R\) satisfies \(\mathcal G_R\subseteq \ker h_R\), then \(h_R\psi (\cdot ,\tau )=0\). This is the local contraction assertion underlying frustration freeness in [ 17 , Section IV.C.1 ] , local source lines 2008–2011.
If a global virtual configuration exists, every bond space has positive dimension. The exact contraction across the cut gives \(\psi (\sigma ,\tau )=\sum _\mu T_R(\mu ;\sigma )T_{R^c}(\mu ;\tau )\), which is a vector of (1). If no global virtual configuration exists, the defining sum for \(\psi \) is empty and the slice is zero. Applying \(h_R\) proves the final assertion.
27.2 Parent Hamiltonians on finite PEPS graphs
The construction of [ 17 , Section IV.C.1 ] , local source lines 2003–2011, applies to arbitrary finite graphs and site-dependent tensors. Different interaction terms may act on different regions.
A parent interaction on a region \(R\) is a positive semidefinite operator \(h_R\) on its physical space satisfying \(\ker h_R=\mathcal G_R\). It need not be an orthogonal projector.
The orthogonal projector onto \(\mathcal G_R^\perp \) is a regional parent interaction.
Orthogonal projections are positive semidefinite, and the kernel of the projector onto \(\mathcal G_R^\perp \) is \(\mathcal G_R^{\perp \perp }=\mathcal G_R\).
Let \((R_i)_{i\in I}\) be a finite family of regions, and let \(h_i\) be a parent interaction on \(R_i\). Its extension to the full graph and the parent Hamiltonian are
The corresponding space of regional boundary conditions is
The parent Hamiltonian is positive semidefinite and satisfies
Thus its kernel depends only on the regional PEPS spaces, rather than on the choice of positive interactions with those kernels.
Positivity is preserved by tensoring with the identity and by finite sums. For each fixed complementary configuration \(\tau \), (2) acts on the slice \(\phi (\cdot ,\tau )\) as \(h_i\). Finally, \(\langle \phi ,H\phi \rangle =\sum _i \langle \phi ,\widetilde h_i\phi \rangle \) is a sum of nonnegative numbers. If \(H\phi =0\), each summand is zero, and positivity gives \(\widetilde h_i\phi =0\). The converse follows by summing.
The closed PEPS vector \(\psi \) satisfies \(\widetilde h_i\psi =0\) for every \(i\in I\), and hence \(H\psi =0\). Whenever \(\psi \ne 0\), it is a ground state of this positive Hamiltonian, and it minimizes every local term simultaneously.
Every complementary slice of \(\psi \) belongs to the regional PEPS ground space. Therefore every extended interaction kills \(\psi \), and their sum does as well.
27.3 The one-dimensional specialization
Let \(n\geq 3\) and \(0{\lt}L{\lt}n\). Enumerate an arc \(R\) of \(L\) sites in cyclic order. A physical configuration on \(R\) is then a word \(\sigma =(i_1,\ldots ,i_L)\). There are exactly two crossing bonds, entering and leaving the arc. If a boundary coefficient is \(x(a,b)\) on their ordered bond values, its boundary matrix is \(X_{ba}=x(a,b)\). Write \(E_R\) for the corresponding ordering of physical coefficients.
Let \(A\) be a matrix product tensor of bond dimension \(D\), and let \(\Gamma _R\) be its open contraction on \(R\). Let \(m\) be the number of assignments to the cycle bonds wholly outside \(R\). Then
If \(D{\gt}0\), both scalar factors are nonzero, so the regional boundary map differs from the MPS boundary map only by an invertible scalar change of boundary coordinates.
Summing a regional contraction over all cycle bonds counts it once for each exterior assignment. At fixed incoming and outgoing bond values \((a,b)\), the same contraction equals \(D^{n-L-1}(A^{i_1}\cdots A^{i_L})_{ab}\). Sum against \(x(a,b)\) and use \(\sum _{a,b}x(a,b)M_{ab}=\operatorname {tr}(MX)\). Positive bond dimension permits cancellation of the exterior assignment count.
For a matrix product tensor \(A\) on a cycle of \(n\geq 3\) sites, and every arc \(R\) of length \(0{\lt}L{\lt}n\),
No injectivity assumption is needed. This identifies the one-dimensional and graph definitions in [ 17 , Section IV.C.1 ] , local source lines 1987–2011.
For \(D{\gt}0\), the two-bond boundary coefficients range over all matrices, and the nonzero scalar changes of boundary coordinates in (6) preserve the range. For \(D=0\), the nonempty arc has an empty boundary-configuration space, while the matrix boundary space has only the zero matrix. Both ground spaces are therefore zero.
27.4 Reduced densities of PEPS regions
Let \(\psi \) be the closed PEPS vector on a finite graph and let \(R\) be a region. Its unnormalized reduced density is \(\rho _R=\operatorname {tr}_{R^c}|\psi \rangle \langle \psi |\). If \(M_R(\sigma ,\tau )=\psi (\sigma ,\tau )\) is its coefficient matrix across the cut, then \(\rho _R=M_RM_R^\dagger \). This is the physical cut associated with the regional construction of [ 17 , Section IV.C.1 ] , local source lines 2003–2011.
For every finite PEPS graph, region \(R\), and pair of physical configurations \(\sigma ,\tau \) on \(R,R^c\), respectively, \(\psi (\sigma ,\tau )=\sum _\mu T_R(\mu ;\sigma )T_{R^c}(\mu ;\tau )\). This identity includes zero-dimensional virtual spaces.
If a global virtual configuration exists, every bond space has positive dimension and the exact contraction across the cut applies. Otherwise, the closed contraction is zero. If incident-edge configurations existed on both \(R\) and \(R^c\), choosing the label from either incident region at each edge would produce a global virtual configuration. Thus one regional incident configuration space is empty, and its open contraction vanishes. Both sides of the claimed identity are then zero.
The operator \(\rho _R\) is positive semidefinite, and \(\operatorname {ran}\rho _R=\operatorname {ran}M_R \subseteq \mathcal G_R\). No injectivity of the regional tensor is assumed.
The identity \(\rho _R=M_RM_R^\dagger \) gives positivity and \(\operatorname {ran}\rho _R\subseteq \operatorname {ran}M_R\). The two operators have the same rank, so these ranges are equal. Each column of \(M_R\) is a local slice of \(\psi \) and lies in \(\mathcal G_R\). Their span therefore lies in \(\mathcal G_R\).
Every regional parent interaction \(h_R\) satisfies \(h_R\rho _R=\rho _Rh_R=0\). This is the density-operator form of local frustration freeness in [ 17 , Section IV.C.1 ] , local source lines 2008–2011.
Since \(\operatorname {ran}\rho _R\subseteq \mathcal G_R=\ker h_R\), the product \(h_R\rho _R\) vanishes. Taking adjoints gives \(\rho _Rh_R=0\).
For a finite PEPS region with regular group labels, numbering its crossing bonds identifies the columns of its open-region matrix \(T_R\) with the open tensors defining \(\mathcal G_R\). Thus \(\operatorname {ran}T_R=\mathcal G_R\).
Numbering and converting between group labels and the chosen finite basis are bijections of boundary configurations. They preserve the span of the open-region tensors.
Let \(G\) be a finite group, and put regular \(G\)-injective tensors at the vertices of a finite connected graph. Suppose the subgraphs induced by \(R\) and \(R^c\) are connected. Then the reduced density of the closed PEPS vector satisfies \(\operatorname {ran}\rho _R=\mathcal G_R\). The same conclusion holds without connectedness of the ambient graph whenever the cut has a crossing edge. This is an auxiliary consequence of the connected-region construction in [ 235 , Lemma 5.2 ] and the regular boundary cut in its Theorem 6.9.
Choose one numbering for the crossing bonds on both sides. Let \(T_R,T_{R^c}\) denote the resulting open-region matrices. Both maps are injective on the invariant boundary space, and the coefficient matrix is \(M_R=T_RT_{R^c}^{\mathsf T}\). A left inverse \(L\) of \(T_{R^c}\) on that invariant space satisfies \(LT_{R^c}=P\), where \(P\) is its averaging projector. Since \(P^{\mathsf T}=P\) and \(T_RP=T_R\), one has \(M_RL^{\mathsf T}=T_R\). Hence \(\operatorname {ran}M_R=\operatorname {ran}T_R=\mathcal G_R\); the reduced density has this same range. In a connected ambient graph, the two nonempty induced regions have a crossing edge, so the required numbering exists.
Consider two regular \(G\)-injective PEPS tensor families on the same finite connected graph, with the same physical spaces. If their closed coefficient vectors agree, then \(\mathcal G_R(A)=\mathcal G_R(B)\) for every cut whose two induced subgraphs are connected. This is a necessary local relation toward a Fundamental Theorem for regular \(G\)-injective PEPS.
Equality of the closed coefficient vectors gives equality of their reduced densities. Each of the two regional ground spaces is the range of that common density.
Consider regular \(G\)-injective tensors on a finite graph and a connected region \(R\) with \(n+1\) crossing bonds. Then \(\dim \mathcal G_R=|G|^n\). If the complementary induced region is also connected, the physical reduced density has rank \(|G|^n\), and the closed PEPS vector is nonzero. These are auxiliary untwisted-cut consequences of the invariant boundary calculation in [ 235 , Corollary 6.10 ] .
The open-region map is injective on the invariant boundary space and has range \(\mathcal G_R\). The simultaneous regular action on \(n+1\) boundary labels has invariant dimension \(|G|^n\). If the complement is connected, \(\rho _R\) has range \(\mathcal G_R\) and therefore this same rank. The rank is positive; hence the closed vector cannot vanish.
Let two tensor families be \(G\)-injective and \(H\)-injective for the regular representations of finite groups \(G\) and \(H\), respectively, and represent the same closed PEPS vector on a finite graph. Suppose a cut has connected induced regions on both sides and at least two crossing bonds. Then \(|G|=|H|\). Equality of the group orders is asserted; the multiplication laws are not determined by this rank argument.
Write the number of crossing bonds as \(n+1\), with \(n{\gt}0\). The common physical reduced density has rank both \(|G|^n\) and \(|H|^n\). The map \(k\mapsto k^n\) is injective on the natural numbers, so \(|G|=|H|\).
For a nonzero closed PEPS vector, write \(\widehat\rho _R=\rho _R/\operatorname {tr}\rho _R\) for its normalized physical reduced density. The trace is positive for every region. The normalized density is positive semidefinite, has trace one, and has the same range and rank as \(\rho _R\).
The physical coefficient matrix \(M_R\) is nonzero because assembly identifies its entries with the coefficients of the closed vector. Thus \(\operatorname {tr}(M_RM_R^*){\gt}0\). Division by this trace preserves positivity, rank, and range and makes the trace equal to one.
Consider regular \(G\)-injective tensors on a finite graph and a cut whose two induced subgraphs are connected and whose boundary is nonempty. Then \(\operatorname {ran}\widehat\rho _R=\mathcal G_R\). In particular, for every local linear operator \(h\),
Every regional parent interaction annihilates this normalized density. Positivity of \(h\) is not needed for the displayed equivalence.
The closed vector is nonzero. Normalization preserves the range, which is \(\mathcal G_R\). The identity \(h\widehat\rho _R=0\) means precisely that this range is contained in the kernel of \(h\).
Let regular \(G\)-injective tensors define a PEPS on a finite graph. Suppose that both induced subgraphs of a cut are connected and that the cut has \(n+1\) crossing bonds. Then the closed vector is nonzero and its normalized physical reduced density satisfies
Here \(S\) is the von Neumann entropy and \(S_0\) is the zero-order Rényi entropy, with natural logarithms. Neither isometry nor flatness of the physical spectrum is assumed. This is an auxiliary connected regular-cut consequence of the boundary calculation in [ 235 , Corollary 6.10 ] .
Normalization preserves the invariant boundary rank \(|G|^n\). The von Neumann entropy of a positive trace-one matrix lies between zero and the logarithm of its rank. The zero-order Rényi entropy is that logarithm by definition.
27.5 Boundary support and Gibbs operators
The bulk–boundary conjecture in [ 15 ] asks whether a gap of the parent Hamiltonian is equivalent to a short-range Gibbs representation of the virtual boundary states. The following finite-dimensional results concern the support of such states. They do not assert locality or a uniform gap.
Let \(G\) be a finite group with \(|G|{\gt}1\), and let a regular virtual boundary have \(b\geq 1\) legs. Its normalized density \(\rho =P_G/|G|^{b-1}\) has rank \(|G|^{b-1}{\lt}|G|^b\). In particular, it is not invertible and cannot equal \(e^H\) for any finite matrix \(H\) on the full boundary space. More generally, if \(T\) is a regular \(G\)-injective open map, its virtual Gram operator \(T^\dagger T\) has the same rank and cannot be such an exponential.
The rank of \(\rho \) is the dimension of the invariant boundary space. The rank of \(T^\dagger T\) equals the rank of \(T\), which is also that dimension by \(G\)-injectivity. Since \(|G|{\gt}1\), both ranks are strictly smaller than the full boundary dimension. Every matrix exponential is invertible, its inverse being the exponential of the negative matrix.
In relative boundary coordinates the restriction of \(\rho \) to its invariant support is
This is a statement on the invariant support, whose dimension is \(|G|^{b-1}\). It imposes no interaction decomposition on the original \(b\) virtual legs.
The averaging projector fixes invariant vectors, so the density acts on this space by \(|G|^{-(b-1)}\). The exponential of the scalar diagonal matrix \(-\log (|G|^{b-1})I\) has precisely that scalar on each diagonal entry.
27.6 Physical changes of coordinates and regional ground spaces
Let \(A\) be a tensor family on a finite graph and let \(F_v:\mathbb {C}^d\to \mathbb {C}^e\) be a linear map at every vertex. Its physical deformation \(A^F\) keeps the virtual bond dimensions and has components
For a region \(R\), write \(F_R=\bigotimes _{v\in R}F_v\). When every \(F_v\) is invertible, \(F_R\) is invertible with inverse \(\bigotimes _{v\in R}F_v^{-1}\).
Let \(\Gamma _R^A\) denote the open-region map and \(\mathcal G_R(A)=\operatorname {im}\Gamma _R^A\). Then \(\Gamma _R^{A^F}=F_R\Gamma _R^A\) and \(\mathcal G_R(A^F)=F_R\mathcal G_R(A)\). If every \(F_v\) is invertible, restriction of \(F_R\) gives an isomorphism \(\mathcal G_R(A)\cong \mathcal G_R(A^F)\), and \(\ker \Gamma _R^{A^F}=\ker \Gamma _R^A\). In particular the regional ground spaces have equal dimensions. No injectivity assumption on \(A\) is required. The physical changes of coordinates in [ 235 , §6.1, accessible virtual coordinates ] satisfy these identities.
Each physical map acts independently of every internal virtual summation. Distributing the products over the finite physical sums gives \(\Gamma _R^{A^F}=F_R\Gamma _R^A\) on each boundary basis vector, hence on every boundary vector by linearity. Taking images gives the regional-space identity. The composition rule for product maps gives \(F_RF_R^{-1}=F_R^{-1}F_R=\mathbb {1}\). The restriction is therefore a linear isomorphism. Injectivity of \(F_R\) gives equality of the virtual kernels.
If the physical maps are invertible, the transport of a regional interaction \(h_R\) is \(h_R^F=(F_R^{-1})^\dagger h_RF_R^{-1}\).
If \(h_R\geq 0\) and \(\ker h_R=\mathcal G_R(A)\), then \(h_R^F\geq 0\) and \(\ker h_R^F=\mathcal G_R(A^F)\). The original interaction may be any positive semidefinite operator with the prescribed kernel.
Congruence preserves positive semidefiniteness. Since \((F_R^{-1})^\dagger \) is invertible, \(h_R^F\psi =0\) is equivalent to \(h_RF_R^{-1}\psi =0\). Thus \(\ker h_R^F=F_R\ker h_R =F_R\mathcal G_R(A)=\mathcal G_R(A^F)\).
27.7 Physical transport of the common parent ground space
For physical maps \(F_v:\mathbb {C}^d\to \mathbb {C}^e\) at the vertices of a finite graph, their product map \(F_V=\bigotimes _{v\in V}F_v\) acts by
If each \(F_v\) is invertible, then \(F_V\) is a linear isomorphism.
The closed PEPS vectors satisfy \(|\Psi (A^F)\rangle =F_V|\Psi (A)\rangle \).
Expand both coefficients as finite sums over virtual labels and physical configurations. Interchange the two sums and distribute the product of physical coefficients at the vertices. This gives the defining local physical sums of \(A^F\).
Fix a region \(R\subseteq V\) and write \(R^c=V\setminus R\). For a complementary physical configuration \(\beta \), let \(\psi _\beta (\alpha )=\psi (\alpha ,\beta )\). Every complementary output configuration \(\tau \) satisfies
Split each input configuration into its restrictions to \(R\) and \(R^c\). The coefficient product factors over these two sets. Interchanging the two finite sums gives the stated identity.
Let \(\{ R_i\} _{i\in I}\) be any family of regions, and let \(\mathcal G(A)\) be the space of full physical vectors whose slices on every \(R_i\) belong to \(\mathcal G_{R_i}(A)\). If each \(F_v\) is invertible, then \(\mathcal G(A^F)=F_V\mathcal G(A)\). Restriction of \(F_V\) is a linear isomorphism between these common ground spaces, so their dimensions agree. For a finite family of positive parent interactions \(h_i\), define \(h_i^F=(F_{R_i}^{-1})^\dagger h_iF_{R_i}^{-1}\). The corresponding full parent Hamiltonians satisfy \(\ker H^F=F_V\ker H\).
If every slice \(\psi _\beta \) lies in \(\mathcal G_{R_i}(A)\), then \(F_{R_i}\psi _\beta \) lies in \(\mathcal G_{R_i}(A^F)\). The slice identity and closure under finite linear combinations give \(F_V\mathcal G(A)\subseteq \mathcal G(A^F)\). Applying the inverse physical maps gives the reverse inclusion. Restriction of \(F_V\) is therefore a linear isomorphism. The full Hamiltonian kernel is the common regional ground space for every choice of positive interactions with the specified local kernels, giving the final identity.
27.8 Unitary physical transport of parent Hamiltonians
The matrix of the full product physical map is \((F_V)_{\tau \sigma }=\prod _{v\in V}(F_v)_{\tau _v\sigma _v}\).
Suppose the regional interactions \(h_i\) and \(k_i\) satisfy \(k_iF_{R_i}=F_{R_i}h_i\) for every region in a finite family. Their extensions to the full graph satisfy \(\widehat{k_i}F_V=F_V\widehat{h_i}\), and their full Hamiltonian sums satisfy \(KF_V=F_VH\). These identities require neither positivity of the interactions nor invertibility of the physical maps.
On a fixed complementary slice, a regional extension acts by its local interaction matrix. The slice of \(F_V\psi \) is a linear combination of \(F_{R_i}\psi _\beta \). Applying \(k_iF_{R_i}=F_{R_i}h_i\) to each summand proves the intertwining identity on every slice, hence on the full vector. Summing the identities gives the full Hamiltonian identity.
If every \(F_v\) is unitary, then every regional product \(F_R\) and the full product \(F_V\) are unitary. For any finite family of regional interaction matrices, their inverse-adjoint transports satisfy
Consequently \(H^F\) and \(H\) have equal characteristic polynomials and equal spectra, with the same eigenvalue multiplicities.
Product composition and adjoint identities give \(F_R^\dagger F_R=F_RF_R^\dagger =\mathbb {1}\) from the corresponding vertex identities, and likewise for \(F_V\). Thus \(F_R^{-1}=F_R^\dagger \), giving the local conjugation formula. The local identity \(h_i^FF_{R_i}=F_{R_i}h_i\) gives \(H^FF_V=F_VH\). Multiplication by \(F_V^\dagger \) gives the full conjugation formula. Equality of characteristic polynomials follows by exchanging the two factors of the rectangular product \((F_VH)F_V^\dagger \), using equality of the physical dimensions. The spectra are the roots of these characteristic polynomials.
27.9 The full-region ground space
The following statements concern the full vertex region in the construction of [ 17 , Section IV.C.1 ] , local source lines 2003–2011. A term on this region acts on the entire graph.
Let \(\psi \) be the closed PEPS vector on a finite graph. Identifying the full-region physical configurations with the configurations of the graph, every full-region open tensor is \(\psi \), and \(\mathcal G_V=\operatorname {span}\{ \psi \} \). If \(\psi =0\), this space is zero.
The full region has neither crossing bonds nor exterior edges. Thus its open contraction sums precisely the virtual labels of the closed PEPS contraction. The boundary-configuration space has one element, so its physical range is the asserted span.
If a collection of regions contains the entire vertex set, its common regional ground space is \(\operatorname {span}\{ \psi \} \). For a finite collection, every positive parent Hamiltonian with those regional kernels has this same kernel. It is one-dimensional when \(\psi \ne 0\).
The full-region condition forces a vector to belong to \(\operatorname {span}\{ \psi \} \). Conversely, every regional condition is satisfied by \(\psi \) and hence by its span. Apply the common-kernel characterization and the dimension formula for the span of a nonzero vector.
The canonical interaction \(P_R\) and its extension \(\widetilde P_R\) are orthogonal projections. Their spectra contain only zero and one. On the orthogonal complement of either kernel the corresponding projector is the identity, and hence has unit norm gap.
The canonical interaction is the orthogonal projector onto \(\mathcal G_R^\perp \). Tensoring with the identity and permuting the physical coordinates preserve idempotence and positivity. For a symmetric projection, its range is the orthogonal complement of its kernel, and it acts as the identity on its range. Idempotence restricts the spectrum to \(\{ 0,1\} \).
27.10 Nesting of regional PEPS ground spaces
The regional contraction with arbitrary boundary conditions of [ 17 , Section IV.C.1 ] , local source lines 2003–2011, implies a compatibility between nested regions. No injectivity or nonzero bond dimensions are needed.
Let \(R\subseteq S\), and write \(E_R,E_S\) for their incident edges. Then \(E_R\subseteq E_S\), and assignments on \(E_S\) split uniquely into assignments on \(E_R\) and \(E_S\setminus E_R\). For any coefficient \(f\) on the crossing-bond labels of \(R\), contraction against its open tensor satisfies
Restriction gives the decomposition of assignments; its inverse joins the two assignments on their disjoint edge sets. Expand the open tensor in (16), interchange the finite sums, and retain the unique boundary label \(\mu _R=\eta _R|_{\partial R}\).
For \(R\subseteq S\) and fixed physical labels \(\tau \) on \(R^c\), let \(Q_{R,S,\tau }\) send a vector \(\phi _S\) on \(S\) to the vector \(\sigma _R\mapsto \phi _S(\sigma _R,\tau |_{S\setminus R})\). Fixing all complementary labels of \(R\) factors as fixing the labels on \(S^c\) and then applying \(Q_{R,S,\tau }\).
If \(R\subseteq S\), then every \(\phi _S\in \mathcal G_S\) and every complementary physical configuration \(\tau \) satisfy \(Q_{R,S,\tau }\phi _S\in \mathcal G_R\).
It suffices to consider a boundary-labelled tensor \(\psi _{S,\mu _S}\). Separate the incident labels into \(\eta _R\) on \(E_R\) and \(\zeta \) on \(E_S\setminus E_R\). Tensor factors at vertices in \(S\setminus R\) and the boundary constraint on \(S\) depend on \(\eta _R\) only through its crossing labels \(\mu _R\). Equation (16) therefore gives
where each coefficient is the product of the tensor factors on \(S\setminus R\) when the assembled boundary labels agree with \(\mu _S\), and is zero otherwise. The right side belongs to \(\mathcal G_R\). Linear combinations give the assertion for every \(\phi _S\in \mathcal G_S\).
Let \((R_i)_{i\in I}\) and \((S_j)_{j\in J}\) be collections of regions such that each \(R_i\) is contained in some \(S_j\). Then the common regional ground space for \((S_j)\) is contained in that for \((R_i)\). In particular, for finite collections, any positive parent Hamiltonians built from these regions satisfy \(\ker H_S\subseteq \ker H_R\).
For each \(R_i\), choose a containing \(S_j\). A vector satisfying the \(S_j\) condition has every complementary slice in \(\mathcal G_{S_j}\). Apply the nesting theorem and the factorization of physical slices to obtain the \(R_i\) condition. The parent-kernel statement follows from the characterization by common regional conditions.
27.11 Comparison of finite PEPS parent Hamiltonians
The comparison of positive interactions with orthogonal projectors in [ 17 , Section IV.C.3 ] , local source line 2170, applies to the regional PEPS interactions of Section IV.C.1. All collections of regions below are finite.
Let \(h_R\) be a positive regional parent interaction, and let \(P_R\) be the orthogonal projector onto \(\mathcal G_R^\perp \). There are strictly positive constants \(\kappa _R,C_R\) such that
The kernel is \(\mathcal G_R\) in the physical Hilbert space. On its orthogonal complement, a positive operator in finite dimension has a positive lower bound. Its operator norm gives an upper bound. On the kernel all three operators vanish.
Extension by the identity on the complementary physical space is linear and preserves Loewner order. In particular, if two collections \((h_i)_{i\in I}\) and \((k_i)_{i\in I}\) on the same regions satisfy \(\kappa k_i\le h_i\le Ck_i\) for every \(i\), their sums satisfy \(\kappa H_k\le H_h\le CH_k\).
Tensoring a positive semidefinite difference with the identity preserves positivity. Reordering the physical coordinates also preserves positivity. Sum the resulting inequalities.
For a finite family of regional parent interactions \(h_i\), let \(H_P=\sum _i\widetilde P_i\) be the sum of the canonical regional projectors extended by the identity. There are \(\kappa ,C{\gt}0\) such that \(\kappa H_P\le H_h\le CH_P\). Consequently, for any two positive parents \(H_h,H_k\) on the same family of regions, there are \(a,b{\gt}0\) such that \(aH_k\le H_h\le bH_k\).
For a nonempty family take \(\kappa =\min _i\kappa _i\) and \(C=\max _i C_i\) in (18). The empty family gives zero Hamiltonians, so any positive constants suffice. If the respective comparisons with \(H_P\) use constants \(\kappa _h,C_h\) and \(\kappa _k,C_k\), take \(a=\kappa _h/C_k\) and \(b=C_h/\kappa _k\).
Let \(H_h,H_k\) be positive PEPS parents on the same finite family of regions. They have the same kernel \(\mathcal K\). Suppose \(H_h\ge \kappa H_k\), where \(\kappa {\gt}0\), and suppose that for every \(\phi \in \mathcal K^\perp \), \(\gamma \| \phi \| \le \| H_k\phi \| \), where \(\gamma \ge 0\). Then every \(\phi \in \mathcal K^\perp \) satisfies \(\kappa \gamma \| \phi \| \le \| H_h\phi \| \). Existence of a strictly positive bound of this form is therefore independent of the choice of positive regional interactions. The comparison constants may depend on the finite collection.
The common regional kernels determine \(\mathcal K\). Positivity and the spectral theorem convert the norm bound for \(H_k\) into \(\langle \phi ,H_k\phi \rangle \ge \gamma \| \phi \| ^2\) on \(\mathcal K^\perp \). Loewner comparison gives \(\langle \phi ,H_h\phi \rangle \ge \kappa \gamma \| \phi \| ^2\). Cauchy–Schwarz yields the asserted norm bound. Apply the two-sided comparison to obtain the final equivalence.
27.12 Finite-overlap gap criteria
We apply the projector expansion of [ 17 , Section IV.C.3 ] , local source lines 2170–2179, to the canonical regional parent interactions on a finite graph. The anticommutator estimate below is an explicit hypothesis. Deriving it from properties of a boundary state is a separate question.
Let \(R,S\) be disjoint vertex regions and let \(h_R,h_S\) be arbitrary linear operators on their respective physical spaces. Their extensions to the full physical space commute.
The extension of \(h_R\) sums only the physical indices in \(R\) and retains all complementary labels. Replacements on disjoint regions commute. Interchanging the two finite sums therefore identifies the two compositions.
Let \(P_i\) be the extended canonical parent projector for a region \(R_i\), and let \(H=\sum _i P_i\). Suppose that each region overlaps at most \(m\) other regions, where \(m\ge 1\), and that \(\varepsilon \ge 0\) satisfies
for every distinct overlapping pair. Then
If \(m\varepsilon {\lt}1\), then every \(\xi \perp \ker H\) satisfies
If another positive parent with the same regional kernels satisfies \(H'\ge \kappa H\) for some \(\kappa {\gt}0\), its norm gap is at least \(\kappa (1-m\varepsilon )\).
Expand \(H^2\). The diagonal terms sum to \(H\) since \(P_i^2=P_i\). A disjoint pair consists of commuting orthogonal projections, so its anticommutator is positive. Sum the assumed lower bounds over the remaining unordered pairs. Each projector occurs at most \(m\) times, giving the quadratic-form inequality. The spectral theorem for a positive finite-dimensional operator then gives the asserted norm bound on the complement of its kernel. The comparison assertion follows from the common kernel and the corresponding lower bound between positive operators.
Consider an arbitrary family of finite graphs, regional canonical PEPS parents, and physical and bond dimensions. If the overlap bound \(m\) and anticommutator bound \(\varepsilon \) hold throughout the family and satisfy \(m\varepsilon {\lt}1\), then every parent in the family has norm gap at least the same positive number \(1-m\varepsilon \).
Apply the preceding estimate to each member of the family. The resulting constant does not depend on the graph or its volume.
If the extended canonical parent projectors commute pairwise, their sum satisfies \(H^2\ge H\) and has norm gap at least one on \((\ker H)^\perp \). In particular, this holds for pairwise disjoint regions.
Every cross term in the projector expansion is nonnegative. Thus \(H^2\ge H\), and the preceding spectral argument applies.
Let \(R\) be a region of a finite graph with regular virtual bonds, and number its \(n+1\) boundary edges. Write \(T_R\) for the actual open-region contraction and \(U\) for the normalized invariant-boundary embedding. Define
Suppose the induced graph on \(R\) is connected and each site tensor is regular \(G\)-injective. If \(R\) has at least one boundary edge, then \(H_R\) is Hermitian and
The complement need not be connected. This gives a finite supported Gibbs representation; spatial decay of the interactions in \(H_R\) remains a separate condition in the boundary-Hamiltonian conjecture of [ 15 ] . The regular virtual-space restriction and the broader source scope are recorded in [ 96 ] .
Blocking a connected region preserves regular \(G\)-injectivity. The supported Gram operator is therefore positive definite. Applying its spectral logarithm and the invariant-boundary compression formula proves the identity.
27.13 Parent spaces determined by the physical ray
Suppose two finite-graph PEPS vectors satisfy \(|\Psi _B\rangle =c|\Psi _A\rangle \) with \(c\ne 0\). For every region \(R\), their unnormalized reduced densities satisfy \(\rho _R(B)=|c|^2\rho _R(A)\). Their reduced supports and ranks agree, and their normalized reduced densities are equal. The regular connected-cut restriction and the remaining bondwise classification are recorded in [ 96 ] .
The outer product scales by \(c\overline c\), and partial trace is linear. Multiplication by a nonzero scalar preserves the range and rank. Dividing by the trace cancels the same scalar. The zero-vector convention also respects this identity.
If two tensor families have equal regional spaces on every region in a prescribed collection, they have the same admissible positive parent interactions and the same canonical local projectors on those regions. Their common parent ground spaces agree. For a finite collection, the canonical sums of local projectors are equal as operators.
A parent interaction is characterized by positivity and its regional kernel, and the canonical projector is determined by that kernel. The common ground space is the intersection of the corresponding slice conditions. Summing equal lifted projectors gives equal canonical Hamiltonians.
Let a finite ambient graph be connected. Suppose two regular group-injective tensor families have proportional closed vectors with nonzero proportionality scalar. Their finite virtual groups may differ. Across any cut with both induced sides connected, their regional PEPS spaces agree. Their common parent ground spaces therefore agree for any family of such cuts.
Each regional PEPS space is the support of the corresponding physical reduced density. The rescaling lemma identifies these supports. Equality of the common ground spaces then follows from equality of all regional conditions.
Suppose two regular group-injective descriptions have proportional closed vectors with nonzero scalar. If a cut has connected induced sides and \(b\geq 2\) crossing edges, the two virtual groups \(G\) and \(H\) have equal order. This assertion does not identify their multiplication laws.
Their reduced ranks agree, while the regular cut-rank formula gives \(|G|^{b-1}=|H|^{b-1}\). Since \(b-1{\gt}0\), the power map on natural numbers is injective, and \(|G|=|H|\) follows.
27.14 Entanglement bounds on finite graphs
Let a PEPS on a finite graph have virtual dimension \(D_e\) on edge \(e\), and let \(R\) be any region. Its unnormalized reduced density satisfies
If the closed vector is nonzero, every virtual bond has positive dimension, and the normalized reduced density \(\widehat\rho _R\) satisfies
For a common crossing-bond dimension \(D\), this gives \(S(\widehat\rho _R)\leq |\partial R|\log D\). This is the finite-graph boundary-dimension argument underlying the PEPS area law in [ 17 , Section II.A.3 ] .
The reduced-density range lies in the span of the open-region tensors, indexed by the crossing-bond configurations. That span has dimension at most \(\prod _{e\in \partial R}D_e\). Normalization preserves the range when the vector is nonzero. The entropy of a trace-one positive operator is at most the logarithm of its rank. Monotonicity of the logarithm and the logarithm-of-product identity yield the entropy bound. If any bond dimension vanished, every virtual contraction would be an empty sum, contradicting nonvanishing.
If every physical site has dimension \(d\), then
For a nonzero closed PEPS, a cut whose crossing virtual bonds all have dimension one has zero entanglement entropy. This includes cuts with no crossing edges.
The reduced-density rank equals the rank of the physical coefficient matrix across the cut. Its row and column dimensions are \(d^{|R|}\) and \(d^{|R^c|}\). If all crossing dimensions are one, the preceding entropy upper bound is zero; nonnegativity gives equality.
Let regular \(G\)-injective tensors be placed at the vertices of the native square torus, with periods \(W,H\geq 3\). The tensors may vary with the vertex. Let \(R\) be a positive coordinate rectangle of side lengths \(w,h\), bounded by the coordinate intervals, with \(w{\lt}W\) and \(h{\lt}H\). For the ordinary untwisted closed vector,
Isometry and flatness of the physical spectrum are not assumed. Coordinate intervals may start at zero or end at their torus periods. This is an auxiliary regular rectangular-cut consequence of the boundary count in [ 235 , Corollary 6.10 ] . The regular representation and proper-rectangle restrictions are recorded in [ 96 ] .
Both the rectangle and its complementary induced graph are connected. The boundary has \(2w+2h\) bonds, hence \(2w+2h-1\) relative regular labels. The exact regular-cut rank and entropy statements give the asserted formulas and inequality.
Consider a nonzero PEPS on a finite square-lattice torus whose two periods are at least three. Let \(R\) be a coordinate rectangle of width \(w\) and height \(h\) contained in the chosen coordinate domain. Suppose each virtual bond crossing its boundary has dimension \(D\). The normalized reduced density matrix obeys
In particular a square of side length \(L\) obeys \(S(\rho _R)\le 4L\log D\), the PEPS area-law bound in [ 17 , Section II.A.3 ] . The tensors may vary from site to site. Empty rectangles and rectangles spanning a full coordinate period are included. We use the convention \(\log 0=0\); a nonzero PEPS permits \(D=0\) here only when no bond crosses the cut. The period restriction of the finite simple-torus model is recorded in [ 96 ] .
A coordinate rectangle has at most \(2w+2h\) crossing bonds. The general entropy bound gives the logarithm of the product of their dimensions. Since \(\log D\ge 0\) for natural-number dimensions, the boundary count yields the asserted inequality. When \(D=0\), nonvanishing forces an empty boundary, and the entropy is zero.
Let \(G\) be a finite group and let a four-leg tensor be \(G\)-injective for the regular representation. On a native square torus with both periods at least three, identify its four virtual labels with the top, right, down, and left incident bonds. The resulting virtual-to-physical map is \(G\)-injective on the actual incident graph coordinates. Its ordinary closed graph contraction equals the native closure with both closure elements equal to the identity. The restriction to regular virtual spaces and untwisted closures is recorded in [ 96 ] .
The four actual legs are a bijection. The resulting bijection of virtual configurations intertwines simultaneous regular translation. Changing variables in the finite contraction sum identifies the two site maps. Invariance and injectivity on invariant vectors are therefore preserved. Identity closure elements insert identity operators on every seam bond, so the closed contraction agrees with the ordinary graph contraction.
Place one regular \(G\)-injective four-leg tensor at every site of a native square torus whose periods are at least three. For its untwisted closed vector, the physical density of every site \(v\) satisfies
Express native \(G\)-injectivity on the actual incident bonds and apply the single-site support and rank statements.
Consider two native four-leg regular \(G\)-injective tensors with the same physical space on a square torus whose periods are at least three. If their untwisted closed coefficient vectors agree, their single-site ground spaces agree at every vertex. If the two descriptions instead use regular representations of different finite groups \(G\) and \(H\), equality of these untwisted closed vectors forces \(|G|=|H|\). These are necessary local relations toward a Fundamental Theorem; neither a group isomorphism nor a product of individual bond gauges is asserted.
Identity native closures are the ordinary graph contractions. The incident-bond descriptions remain \(G\)-injective, so the local ground-space and group-order rigidity statements apply.
Suppose the native square torus has periods \(W,H\geq 3\). Let \(R\) be the coordinate rectangle
with \(x_0+w\leq W\), \(y_0+h\leq H\), \(w{\lt}W\), and \(h{\lt}H\). Then there are \(2h\) horizontal and \(2w\) vertical crossing bonds, and hence
In particular a bounded square of side length \(L\) has \(4L\) crossing bonds. The coordinate intervals may start at zero or end at their torus periods; the formula includes the cyclic seam bonds in these cases. These are the actual lattice bonds used in the rectangular boundary construction of [ 235 , Theorem 6.9 ] . The period restriction comes from the finite simple-graph model; see [ 96 ] .
For a cyclic coordinate interval \([s,s+\ell )\) shorter than its period \(N\), membership changes between \(a\) and \(a+1\) at precisely two origins. Their coordinate values are \(s+\ell -1\) and \(s-1\), where the latter is read as \(N-1\) if \(s=0\). The two origins are distinct because \(0{\lt}\ell {\lt}N\). A horizontal bond crosses the rectangle exactly when its first coordinate is one of these two origins and its second coordinate lies in the vertical interval. This gives \(2h\) bonds. The vertical argument gives \(2w\) bonds. On a torus with both periods at least three, right and up bonds enumerate the graph edges bijectively, so these two counts add without multiplicity.
On a native square torus with periods \(W,H\geq 3\), every bounded coordinate rectangle of side lengths \(w,h\geq 0\) satisfies \(|\partial R|\leq 2w+2h\). No strict upper bound on either side length is required; empty rectangles and full coordinate bands are included.
A zero side length gives the empty rectangle. A side length equal to its torus period produces no crossing bond in that coordinate direction. In every other case, the horizontal crossing count is \(2h\) and the vertical crossing count is \(2w\). The two directional bounds add to the stated perimeter bound.
Suppose both periods of the native square torus are at least three. The graph is connected. For every vertex \(v\), the subgraphs induced by \(\{ v\} \) and its complement are connected, and the boundary of \(\{ v\} \) has exactly four bonds. The native top, right, down, and left legs enumerate these incident bonds bijectively. The regular virtual-space and period restrictions are recorded in [ 96 ] .
A singleton is the coordinate rectangle of side lengths one. Its complement is connected by the unused row and column construction. The same rectangular connectivity argument applied to both full coordinate intervals gives connectivity of the torus. The four native legs exhaust the incident bonds and are distinct: horizontal and vertical bonds cannot agree, and opposite bonds have different origins when the corresponding period exceeds two. Every incident bond has just one endpoint at \(v\), so these are precisely the crossing bonds of the singleton.
Let regular \(G\)-injective tensors of physical dimension \(d\) be placed at the vertices of the native square torus, with both periods at least three. The tensors may vary with the vertex. Their closed vector is nonzero. For every vertex \(v\),
The normalized density satisfies
No isometry or flat spectrum is assumed.
Apply the connected regular-cut statements to the singleton and its connected complement. Its four crossing bonds give three relative boundary labels. The density acts on a \(d\)-dimensional physical space, so its rank \(|G|^3\) is at most \(d\).
On a native square torus with both periods at least three, two regular \(G\)-injective descriptions of the same closed physical vector have the same single-site ground spaces at every vertex. If the two descriptions use regular representations of different finite groups \(G\) and \(H\), the equality of their closed vectors implies \(|G|=|H|\). No assertion about the multiplication laws or individual bond gauges follows from this rank argument.
Equal closed vectors give equal reduced densities. Their ranges recover the local ground spaces. The common single-site density has rank both \(|G|^3\) and \(|H|^3\), so the group orders agree.
Consider regular \(G\)-injective tensors on a native square torus with periods at least three. Let \(R\) be a coordinate rectangle of positive side lengths, bounded by the coordinate intervals, with each side length strictly smaller than the corresponding torus period. Then \(\operatorname {ran}\rho _R=\mathcal G_R\).
The rectangle is connected, and its complement is connected by paths along an unused row and an unused column. The torus is connected, so the exact support theorem applies.
Consider a finite graph tensor with regular group labels and bond insertions equal to the identity on every edge incident to \(R\). The inserted tensor has the same regional ground space \(\mathcal G_R\) as the original tensor. The strict interior restriction and the remaining global closure problem are recorded in [ 96 ] .
All tensor coefficients in the open contraction of \(R\) are unchanged. Thus the open-region matrices, and hence their physical ranges, agree.
Let a four-leg tensor with regular group labels be placed on a native square torus with both periods at least three. Let \(R\) be a coordinate rectangle whose starting coordinates are positive and whose ending coordinates are strictly below the torus periods. For every pair of closure elements \(g,h\), every physical slice of the native closure vector \(\Psi _{g,h}\) across \(R\) lies in the original regional ground space \(\mathcal G_R\). Consequently this holds for every coherent sum \(\Psi =\sum _i\mu _i\Psi _{g_i,h_i}\), and every regional parent interaction \(h_R\) for the original tensor satisfies
No isometry, \(G\)-injectivity, or commutativity of \(g_i,h_i\) is assumed. This is an interior local constraint; it does not assert the full torus ground-space classification of [ 235 , Theorem 5.5 ] .
The native closure is the graph contraction with its holonomy operators inserted at ordered seam endpoints. No seam bond is incident to the strict interior rectangle. Its open contraction therefore has the original regional ground space. Every slice of the closed inserted tensor belongs to that space. A subspace contains coherent sums of its vectors, so the assertion extends to \(\Psi \). The globally extended interaction vanishes precisely when its regional kernel contains every such slice.
Let a native four-leg regular \(G\)-injective tensor be placed on a square torus with both periods at least three. Let \(\Psi =\sum _i\mu _i\Psi _{g_i,h_i}\) be a nonzero coherent sum of closure vectors, where \(g_i h_i=h_i g_i\) for every \(i\). Let \(R\) be a positive coordinate rectangle whose starting coordinates are positive and whose ending coordinates are strictly below the torus periods. Then the physical coefficient matrix across \(R\) has range exactly the original regional ground space \(\mathcal G_R\). Equivalently, the actual reduced density of \(\Psi \) satisfies
This is an interior support consequence of [ 235 , Theorem 5.5 and Corollary 6.10 ] ; it does not classify the full torus parent-Hamiltonian ground space. The strict interior regular-sector scope and the remaining full parent-kernel classification are recorded in [ 96 ] .
Every complementary physical slice of \(\Psi \) lies in \(\mathcal G_R\), so the coefficient range is contained in that space. The simply connected regular Schmidt-rank theorem gives rank \(|G|^{|\partial R|-1}\) for the nonzero coherent sum. The connected open-region contraction has precisely the same dimension. Inclusion and equality of dimensions therefore give equality of the two spaces. The reduced density is the product of the coefficient matrix and its adjoint, which has the same range as the coefficient matrix.
27.15 A counterexample to bare inverse-adjoint parent transport
The exact-kernel assertion following the displayed parent-transport formula in [ 17 , Section IV.C.1 ] , local source lines 2031–2037, fails for rectangular injective site maps. The correction and its relation to genuine parent uniqueness are recorded in [ 68 ] .
Consider two vertices joined by a bond of dimension \(D=1\). At either endpoint place the site map \(A:\mathbb C\longrightarrow \mathbb C^2\), \(A(z)=(z,0)\), and choose the left inverse \(\ell (z_0,z_1)=z_0\). The product maps \(F=A\otimes A\) and \(L=\ell \otimes \ell \) satisfy \(LF=I\), so both the site and product maps are injective. The virtual Bell vector is the scalar \(1\). Its complement interaction is \(H=0\), which is positive and has exactly the Bell line as its kernel.
Direct multiplication gives \(\ell A=1\) and \((\ell \otimes \ell )(A\otimes A)=1\). The two-endpoint virtual space is one-dimensional, and the Bell vector is nonzero, so its span is the whole virtual space.
Bare inverse-adjoint transport gives \(L^\dagger HL=0\) on \(\mathbb C^2\otimes \mathbb C^2\). The vector \(e_1\otimes e_0\) belongs to its kernel, but lies outside \(\operatorname {ran}F\). The bare kernel has dimension four, whereas the intended product-site range has dimension one. Hence
The example uses a positive bond dimension and genuinely injective tensors. The corrected interaction with its physical reconstruction summand is treated separately.
The transported interaction is zero because \(H=0\). Every vector in \(\operatorname {ran}(A\otimes A)\) has zero coefficient at \((1,0)\), whereas \(e_1\otimes e_0\) has coefficient one there. Thus it is an additional physical kernel vector.
Let \(F_R\) be the product of injective site maps and let \(L_R\) be the product of their left inverses. For a positive virtual interaction \(H\), define
The second summand constrains a physical ground vector to the image of the product site map. This is a regional consequence of the inverse-adjoint deformation argument in [ 17 , Section IV.C.1 ] . For rectangular site maps, the bare printed interaction leaves unused physical directions in its kernel; the correction is recorded in [ 68 ] .
The transported interaction is positive and satisfies
Equivalently, \(\widetilde H_R\psi =0\) precisely when \(HL_R\psi =0\) and \(F_RL_R\psi =\psi \). In particular, if \(\ker H\) is the genuine virtual regional bond space, then \(\widetilde H_R\) is a parent interaction with kernel \(\mathcal G_R\). The site maps may be rectangular. The reconstruction penalty acts on the entire region \(R\); the assertion does not establish locality of a sum of edge interactions.
Both summands are positive. A vector in the kernel of their sum therefore lies in the kernel of each summand. Since \(L_RF_R=I\), the adjoint \(L_R^\dagger \) is injective, so the first condition is equivalent to \(HL_R\psi =0\). The kernel of the second summand is the kernel of \(I-F_RL_R\). Thus every physical ground vector is the image of a virtual ground vector, and conversely. The regional factorization \(T_R=F_RB_R\) identifies the transported virtual bond range with \(\mathcal G_R\).
Give each vertex of \(R\) an independent configuration of its incident virtual labels. Thus each internal edge has two labels, one at each endpoint. Let \(F_R=\bigotimes _{v\in R}A_v\) be the product site map. For injective sites let \(L_R\) be the product of their left inverses. Let \(B_R\) denote the virtual region map that pairs the two endpoint labels of every internal edge and leaves the crossing labels open. Its coefficients are sums of basis vectors indexed by the actual incident-edge assignments, with the boundary labels fixed.
For arbitrary site tensors the actual open-region map satisfies \(T_R=F_RB_R\), and \(\mathcal G_R=F_R(\operatorname {ran}B_R)\). If all site maps are injective, then
No nonzero-bond hypothesis is needed for these identities. These are the finite-graph virtual-coordinate identities underlying the parent argument in [ 17 , Section IV.C.1 ] .
The product site map takes a virtual basis vector to the product of its site tensor coefficients. Summing over compatible internal edge labels with the crossing labels fixed gives the actual open-region contraction, so \(T_R=F_RB_R\). Tensoring the site left inverses gives \(L_RF_R=I\), hence injectivity of \(F_R\) and \(L_RT_R=B_R\). The range and inverse-image formulas follow.
27.16 The virtual Bell-bond ground space
The virtual part of the injective-parent uniqueness argument in [ 17 , Section IV.C.1 ] , local source lines 2017–2044, concerns a product of independent maximally entangled bonds. We first establish this part without physical maps.
Let \(E\) be a finite set of bonds with dimensions \(D_e\ge 0\). Write \(\zeta _e=(a_e,b_e)\) for their endpoint labels and put
The common virtual ground space consists of the vectors whose slice on each bond, for every fixed configuration of all other bonds, belongs to \(\operatorname {span}\{ \omega _e\} \).
The common virtual ground space is \(\operatorname {span}\{ \Omega \} \). The vector \(\Omega \) is nonzero exactly when every \(D_e\) is positive; in this case the common ground space is one-dimensional. If a bond dimension vanishes, the entire virtual configuration space is empty and the vector space is zero. The assertion also includes an empty bond collection, whose product vector is the scalar vacuum.
The product vector satisfies each slice condition by separating its corresponding Bell factor. Conversely, the slice condition forces a coefficient to vanish whenever the two labels of some bond differ. On diagonal configurations, replacing one common endpoint label by another leaves the coefficient unchanged. Finitely many such replacements join any two diagonal configurations. Thus every common ground vector is a scalar multiple of \(\Omega \). Its coefficient at any diagonal configuration is one, proving nonvanishing when all dimensions are positive. A zero-dimensional bond makes the configuration space empty, giving the stated degenerate case.
Let the site tensors on a finite graph be injective. Suppose a collection of regions contains every vertex in some region and contains the two endpoints of every edge together in some region. Then its common parent space is
Every finite positive interaction family with the genuine regional kernels has this same zero-energy space. If the tensor is an injective PEPS tensor, so that all bond dimensions are positive, the closed vector is nonzero and the ground space is one-dimensional. The span formula also holds when a bond dimension vanishes, in which case the span can be zero.
This is the covering-region form of the virtual-pair argument in [ 17 , Section IV.C.1 ] . Vertex coverage is explicit, including for isolated vertices. The physical reconstruction correction is recorded in [ 68 ] .
Vertex coverage supplies the genuine site-image conditions, so every physical parent vector is the product-site image of a vector satisfying all actual virtual regional conditions. Edge coverage imposes the Bell condition on each separate virtual bond. Their common space is precisely the line spanned by the product Bell vector. Its image is the actual closed PEPS vector. Conversely the closed vector satisfies every regional condition. Positivity gives the Hamiltonian-kernel statement, and positive bond dimensions give nonvanishing and dimension one.
27.17 Nearest-neighbor parents of injective PEPS
For each edge \(e=\{ u,v\} \) of a finite graph, use the endpoint region \(R_e=\{ u,v\} \). The augmented family additionally contains the singleton \(\{ v\} \) for each isolated vertex \(v\). All regional spaces are the images of the actual open-boundary contractions. Isolated-site conditions are needed to exclude unused physical directions at vertices absent from every edge; see [ 68 ] .
Every edge is contained in its endpoint region. If every vertex has a neighbor, the endpoint regions also cover every vertex. On an arbitrary finite graph, the augmented family covers every vertex and every edge.
A vertex with a neighbor belongs to the region of their edge. An isolated vertex belongs to its singleton region.
Suppose every site tensor is injective. If every vertex has a neighbor, the common space for the edge-pair regions is exactly the span of the closed PEPS vector. For an arbitrary finite graph, the same conclusion holds for the augmented edge and isolated-site family. If the tensor is an injective PEPS tensor, so that every bond dimension is positive, each of these spaces is one-dimensional. These are the nearest-neighbor and isolated-site forms of the virtual-pair argument in [ 17 , Section IV.C.1 ] . Without positive bond dimensions the span equality still holds, and the PEPS span may be zero.
Apply the covering-region theorem to the two coverage statements. Positive bond dimensions make the virtual Bell product nonzero; injectivity preserves this property in the physical PEPS vector.
Choose any positive semidefinite interaction on each region, with kernel exactly its actual regional PEPS space, and sum their extensions to the full physical graph. Under the corresponding hypotheses of the preceding theorem, the Hamiltonian kernel is exactly the closed PEPS span. For an injective PEPS tensor, whose bond dimensions are positive, this kernel is one-dimensional. The edge-only conclusion requires that every vertex have a neighbor; the augmented conclusion includes the isolated-site singleton interactions explicitly.
The kernel of a finite sum of positive interactions is the intersection of their kernels, which is precisely the common regional parent space.
27.18 Regional supports of ordinary injective tensors
The region map obtained by summing all graph-edge labels is the actual open-region map multiplied by the number of assignments to edges wholly outside the region. Injectivity of the former implies injectivity of the latter. If every virtual bond is nonzero dimensional, the two maps have the same range.
Split the edge labels into those incident to the region and those wholly outside it. The exterior labels do not enter any site tensor, so they contribute their cardinality. An equality of actual images implies an equality of globally summed images. A nonzero scalar factor preserves a linear range.
If the actual open-region map of \(R^c\) is injective, then the unnormalized reduced density of the closed PEPS has support exactly equal to \(\mathcal G_R\). Neither injectivity of \(R\) nor connectedness of either side is needed.
Write the physical coefficient matrix as \(M=T_RS^\mathsf T\), where \(S\) is the complementary open map after matching the crossing labels. A left inverse of \(S\) gives a right inverse of its transpose. Consequently \(\operatorname {ran}M=\operatorname {ran}T_R\). The reduced density \(MM^\dagger \) has the same range as \(M\).
If the actual open maps of \(R\) and \(R^c\) are injective, then \(\operatorname {rank}\rho _R=\prod _{e\in \partial R}D_e\). If two coefficient vectors coincide and their complementary open maps are injective, their regional ground spaces on \(R\) coincide. These assertions need neither site injectivity nor connectedness.
The complementary map identifies the reduced support with \(\mathcal G_R\), and injectivity of the regional map gives its exact boundary dimension. Equal coefficient vectors have equal reduced densities, so the corresponding support spaces agree.
Let \(A\) be an injective PEPS tensor, so that every site tensor is injective and every virtual bond has positive dimension. For every finite region \(R\),
The region and its complement may be disconnected. The positive-bond convention is recorded in [ 67 ] . This is the injective Schmidt-span consequence used in [ 220 , Section 3 ] and the virtual-pair argument of [ 17 , Section IV.C.1 ] .
A contraction of injective sites with nonzero virtual bonds is injective. This applies both to \(R\) and to its complement. The complementary support theorem identifies the reduced support with \(\mathcal G_R\). Injectivity of the region map identifies its range dimension with the crossing-bond configuration dimension.
Fix all physical indices in \(R\) except the index at \(v\in R\). The resulting one-site coefficient vector is a linear slice of the regional vector.
Every one-site slice at \(v\in R\) of a vector in \(\mathcal G_R\) lies in the image of the site map \(A_v\). Consequently a collection of parent regions covering every vertex supplies all one-site image conditions. No injectivity or positive-bond hypothesis is needed. This is a consequence of the finite contractions in [ 17 , Section IV.C.1 ] .
Expand a regional vector as a sum over actual incident-edge assignments, with a coefficient depending on the crossing labels. For each assignment the selected site’s coefficient vector is a column of \(A_v\) multiplied by the product of all other site coefficients. Every summand belongs to the site image, hence so does the slice. Apply this to a parent region containing each vertex.
Suppose the site maps are injective and the parent regions cover every vertex. Their common physical ground space is exactly the image under the product site map of the common genuine virtual regional bond space. The kernel of any positive exact-kernel parent Hamiltonian with finitely many terms on these regions has the same description.
The regional conditions imply the site-image conditions by the preceding theorem. The product site inverse therefore reconstructs each physical vector and sends its actual regional conditions to the genuine virtual conditions. The product site map sends the virtual conditions back to the physical conditions. Positivity identifies the kernel of the parent sum with the common local space.
27.19 Vertex and bond coordinates
The virtual construction in [ 17 , Section IV.C.1 ] , local source lines 2017–2044, can be described either by independent labels at the vertices or by the two endpoint labels of each edge. These descriptions are equivalent on the original finite graph, including vertices incident to no edge.
Let every edge \(e\) of a finite graph have a dimension \(D_e\ge 0\). Independent virtual configurations at all vertices are in bijection with configurations \(\zeta _e=(a_e,b_e)\) of the two endpoints of each edge. Equality of an edge’s two incident vertex labels is precisely the condition \(a_e=b_e\). Such consistent configurations are exactly those obtained by assigning one common label to each bond.
Read the two endpoint labels of each edge. Conversely, recover each vertex’s incident label from its corresponding endpoint. An edge has distinct endpoints, so these operations are inverse. The consistency assertion then follows coordinate by coordinate.
In the full vertex region, sum the consistent virtual bond assignments in the original contraction. Under the endpoint-pair coordinate equivalence, the resulting coefficient vector is exactly
No injectivity, connectedness, or positivity of the bond dimensions is assumed.
The full region has no crossing edges, and its incident labels are exactly its global bond labels. An endpoint configuration therefore contributes once if it is consistent and zero times otherwise. This is the displayed product of diagonal indicators.
27.20 Internal Bell constraints from actual regional contractions
The virtual-pair argument of [ 17 , Section IV.C.1 ] , local source lines 2017–2044, uses the Bell condition on each original graph edge. Here this condition is derived from the actual regional virtual bond maps, before any physical injectivity assumption is imposed.
Let both endpoints of an edge \(e\) belong to a finite region \(R\). Every vector in the actual regional virtual bond range vanishes when the two half-edge labels of \(e\) differ. Its coefficients are unchanged when the same permutation of the bond alphabet is applied to both labels. The assertion includes arbitrary boundary coefficients and vanishing bond dimensions.
Each incident-edge assignment uses one common label at the two endpoints of \(e\), so no assignment contributes to a coefficient with unequal labels. A permutation of this common internal label is a bijection of the incident assignments, leaves every boundary label fixed, and permutes the two endpoint labels together. The resulting identities extend by linearity to the actual regional virtual map.
Let \(x\) be a global vector in independent virtual vertex coordinates. Suppose that, for every fixed complementary virtual configuration, its slice on \(R\) belongs to the actual regional virtual bond range. Reorganize \(x\) into its original edge-pair coordinates, obtaining \(y\). For every internal edge \(e\) of \(R\) and every fixed configuration \(\zeta \) of the other endpoint pairs, one has
Thus the corresponding one-bond slice belongs to the Bell line. In particular, the assertion applies to the two-vertex region formed by the endpoints of \(e\). No bond dimension is assumed positive.
Restrict a global configuration to \(R\) and its complement. The regional support condition gives vanishing off the diagonal. Jointly permuting the two endpoint labels leaves the complementary configuration fixed because both endpoints belong to \(R\). Transpositions therefore identify all diagonal coefficients. A coefficient function that vanishes off the diagonal and is constant on it is a scalar multiple of the Bell vector.
Let \((R_i)\) be a family of finite vertex regions such that the two endpoints of every edge lie together in some \(R_i\). If every complementary slice of \(x\) on every \(R_i\) belongs to its actual regional virtual bond range, then its edge-pair coefficient vector belongs to the common Bell-bond ground space.
For each edge choose a region containing both endpoints and apply the Bell-slice theorem. This gives every defining condition of the common Bell-bond space.
27.21 The physical image of the Bell product
The global virtual-pair construction of [ 17 , Section IV.C.1 ] , local source lines 2017–2044, sends the independent Bell product through the site tensor maps. The following identities retain the actual graph contraction and include isolated vertices and zero-dimensional bonds.
Let \(F\) be the product of all site tensor maps, expressed on its domain in the independent endpoint-pair coordinates of the edges. Its codomain is the physical coefficient space on the full vertex region.
Write \(\psi \) for the actual closed PEPS vector and \(\Omega \) for the product of virtual Bell vectors. Then, in full-region physical coordinates,
Neither identity requires site injectivity or positive bond dimensions.
The full-region virtual contraction is exactly \(\Omega \) in endpoint-pair coordinates. Applying the product site map gives the full-region open coefficient vector, which equals the closed state because no boundary edges remain. The common Bell space is the span of \(\Omega \), and linear maps preserve spans.
If every ordinary site tensor map is injective, then \(F\) is injective and
These assertions concern the global state and the full-region range. They do not yet identify the ground space of a parent built from smaller regions.
Tensor the site left inverses and transport them to the endpoint-pair coordinates. This gives an inverse on the image of \(F\). The inverse-image assertion follows from injectivity. By \(F\Omega =\psi \), nonvanishing is equivalent to that of \(\Omega \), which holds precisely when every bond dimension is positive.
27.22 The common one-site image space
Let \(A\) be a tensor on a finite graph, with physical space \(\mathcal P_v=\mathbb C^d\) at each vertex and independent virtual half-edge space \(\mathcal V_v\) at that vertex. Write \(A_v:\mathcal V_v\to \mathcal P_v\) for its site map and \(T_A=\bigotimes _v A_v\) for their product. The common one-site image space \(\mathcal K_A\) consists of those physical vectors whose one-site slices lie in \(\operatorname {ran}A_v\), for every vertex \(v\) and every choice of physical indices at all other vertices. If every \(A_v\) is injective, choose left inverses \(L_v\) and set \(P_v=A_vL_v\). These retractions need not be orthogonal. This is the independent site-image condition preceding the virtual bond constraints in [ 17 , Section IV.C.1 ] .
If every site map \(A_v\) is injective, then
More precisely, every \(\psi \in \mathcal K_A\) satisfies
This assertion holds for an arbitrary finite graph. It identifies the product site image, before imposing virtual bond constraints; it does not assert uniqueness of a PEPS parent ground state.
For each \(v\), the map \(P_v=A_vL_v\) fixes exactly \(\operatorname {ran}A_v\). Therefore its action at the vertex \(v\), with the identity at all other vertices, fixes every vector in \(\mathcal K_A\). Multiplying these commuting actions gives \(\bigotimes _vP_v=T_A(\bigotimes _vL_v)\), proving the reconstruction identity and the inclusion \(\mathcal K_A\subseteq \operatorname {ran}T_A\). Conversely, \(P_vA_v=A_v\) shows that every vector in \(\operatorname {ran}T_A\) satisfies every one-site image condition.
27.23 Regional conditions after the site inverses
Let \(R\) be any region of a finite graph and let \(F_v:\mathbb C^d\to \mathcal Q_v\) be local maps, whose output spaces may depend on \(v\). For an outside output configuration \(\tau \), the \(R\)-slice of \((\bigotimes _vF_v)\psi \) is
where \(\beta \) ranges over outside physical configurations and \(\psi _\beta \) denotes the corresponding physical \(R\)-slice. This is a finite contraction identity; it does not require injectivity or an inner product on the output spaces.
Expand the product map in physical configurations. Split each configuration into its coordinates in \(R\) and its coordinates outside \(R\), and factor the product of the local coefficients. Sum first over the coordinates in \(R\).
For any tensor on a finite graph and any family of regions \((R_i)_i\), let \(\mathcal K_{\mathrm{virt}}\) consist of the independent virtual vectors whose \(R_i\)-slices belong to the actual virtual bond range \(\mathcal B_{R_i}\), for every \(i\) and every outside virtual configuration. This definition requires no injectivity.
Suppose every site map is injective, with chosen left inverses \(L_v\), and write \(L_A=\bigotimes _vL_v\). For a region \(R\), let \(\mathcal B_R\) be the image of the actual virtual bond contraction map, with the crossing bond coordinates free. If every physical \(R\)-slice of \(\psi \) belongs to the regional PEPS space, then every virtual \(R\)-slice of \(L_A\psi \) belongs to \(\mathcal B_R\). The implication holds simultaneously for any family of regions. No reconstruction identity or virtual support condition is assumed. This is the site-inverse step of the virtual-pair argument in [ 17 , Section IV.C.1 ] ; the identification of the common virtual conditions with the globally paired bond space is a separate assertion.
The regional product inverse sends the regional physical space into \(\mathcal B_R\), since composing it with the actual regional open-boundary map gives the virtual bond contraction map. Apply the slicing identity with \(F_v=L_v\). Every summand lies in the subspace \(\mathcal B_R\), and hence so does their linear combination.
27.24 Exact transport of the common virtual conditions
Let \(A\) be a tensor on a finite graph, and let \(T_A=\bigotimes _vA_v\) be its product site map. For an arbitrary family of regions \((R_i)_i\), let \(\mathcal K_{\mathrm{virt}}\) be the common space of independent virtual vectors whose \(R_i\)-slices lie in the actual regional virtual bond range, for every \(i\) and every outside virtual configuration. Let \(\mathcal K_{\mathrm{phys}}\) be the common physical regional PEPS space, defined by all corresponding physical slice conditions. Then
This implication requires no injectivity of the site maps.
Split an independent virtual configuration into its coordinates inside and outside a region. A physical regional slice of \(T_Ax\) is a linear combination of the regional product site map applied to virtual regional slices of \(x\). The regional product site map sends the actual virtual bond range onto the actual physical regional PEPS space, so each summand satisfies the physical regional condition.
If every site map is injective, then
where \(\mathcal K_A\) is the common singleton site-image space. These singleton conditions are explicit, and no reconstruction condition is a hypothesis. This is the physical-to-virtual correspondence in the virtual-pair argument of [ 17 , Section IV.C.1 ] . Identifying \(\mathcal K_{\mathrm{virt}}\) with the global paired-bond space remains a separate assertion.
The forward inclusion follows from the preceding theorem and the singleton-image property of the product site map. Conversely, let \(\psi \in \mathcal K_A\cap \mathcal K_{\mathrm{phys}}\). Its product inverse \(x=L_A\psi \) belongs to \(\mathcal K_{\mathrm{virt}}\) by the regional inverse theorem. The singleton-image theorem gives \(T_Ax=\psi \).
27.24.1 Physical changes on the site images
Fix a finite simple graph with ordered vertices \(V\), site maps \(A_v:\mathcal V_v\to \mathbb C^d\), and linear maps \(F_v:\mathbb C^d\to \mathbb C^e\). Write \(B_v=F_vA_v\), \(\mathcal G_S(A)=\operatorname {im}\mathcal C_S(A)\) for the full open-region range, and \(\mathcal P_{\mathcal R}(A)\) for the common parent space of a family \(\mathcal R=(R_i)_{i\in \mathcal I}\). Set \(F_S=\bigotimes _{v\in S}F_v\) and \(F_{\mathrm{all}}=\bigotimes _{v\in V}F_v\). The following are consequences of the site inverse in [ 235 , Definition 5.1 ] and the regional parent construction. The virtual spaces and representations remain fixed.
If every \(F_v\) is injective on \(\operatorname {im}A_v\), there are linear maps \(L_v:\mathbb C^e\to \mathbb C^d\) such that \(L_vF_vx=x\) for \(x\in \operatorname {im}A_v\). For any such choice, \(L_vB_v=A_v\), and \(F_vL_vy=y\) for \(y\in \operatorname {im}B_v\).
The injection \(F_v:\operatorname {im}A_v\to \mathbb C^e\) has a linear left inverse. Compose it with the inclusion of \(\operatorname {im}A_v\) into \(\mathbb C^d\) to obtain \(L_v\). Applying \(L_vF_v\) to each column of \(A_v\) gives \(L_vB_v=A_v\). If \(y=B_vz=F_vA_vz\), then \(F_vL_vy=F_vA_vz=y\).
Suppose \(L_vF_v\) fixes \(\operatorname {im}A_v\) at every vertex. For any region \(S\) and \(\psi \in \mathcal G_S(A)\), \(L_SF_S\psi =\psi \). Consequently site-image injectivity of the \(F_v\) implies injectivity of \(F_S\) on \(\mathcal G_S(A)\) for every \(S\). All virtual boundary conditions are allowed.
Write \(\psi =\mathcal C_S(A)\theta \) for arbitrary boundary data \(\theta \). Physical deformation commutes with contraction, so
Apply \(L_S\) to an equality \(F_S\psi =F_S\phi \) to obtain injectivity. When only site-image injectivity is given, choose \(L_v\) by Theorem 27.24.1.1.
Let \(P_v:\mathbb C^d\to \mathbb C^d\) fix \(\operatorname {im}A_v\) pointwise. If every one-site slice of \(\psi \) at \(v\) belongs to \(\operatorname {im}A_v\), for all \(v\), then \((\bigotimes _vP_v)\psi =\psi \). In particular, if \(L_vF_v\) fixes \(\operatorname {im}A_v\) and every vertex belongs to some \(R_i\), then \(L_{\mathrm{all}}F_{\mathrm{all}}\psi =\psi \) for every \(\psi \in \mathcal P_{\mathcal R}(A)\).
The operator acting as \(P_v\) at one vertex and as the identity elsewhere fixes \(\psi \), because it acts on its one-site slices. Multiplying these operators gives the first assertion. Vertex coverage puts every one-site slice of a parent vector in the appropriate site image. Take \(P_v=L_vF_v\) and use \(L_{\mathrm{all}}F_{\mathrm{all}}=\bigotimes _v(L_vF_v)\).
Every physical product map satisfies \(F_{\mathrm{all}}\mathcal P_{\mathcal R}(A) \subseteq \mathcal P_{\mathcal R}(B)\), without injectivity or coverage assumptions. If each \(F_v\) is injective on \(\operatorname {im}A_v\) and the regions cover all vertices, its restriction is a linear isomorphism
An inverse is \(L_{\mathrm{all}}\) for any site inverses from Theorem 27.24.1.1.
The identity \(\mathcal G_S(B)=F_S\mathcal G_S(A)\) sends each regional constraint to the corresponding constraint for \(B\). In a fixed exterior slice, the physical maps on exterior vertices only form linear combinations of source slices; hence the forward inclusion holds for arbitrary \(F_v\). Apply it also to \(L_v\), using \(LB=A\). Since \(LF\) fixes the source site images and \(FL\) fixes the target site images, Theorem 27.24.1.3 gives both inverse identities on the parent spaces. This proves (22) and the dimension equality.
Assume the site-image injectivity and vertex coverage of Theorem 27.24.1.4, and suppose \(\mathcal I\) is finite. Let \(H_i,K_i\) be positive semidefinite regional interactions with \(\ker H_i=\mathcal G_{R_i}(A)\) and \(\ker K_i=\mathcal G_{R_i}(B)\). If \(H\) and \(K\) are the sums of their placements on the full graph, then \(F_{\mathrm{all}}(\ker H)=\ker K\).
Positivity identifies the kernel of each sum with the intersection of its local kernels. Thus \(\ker H=\mathcal P_{\mathcal R}(A)\) and \(\ker K=\mathcal P_{\mathcal R}(B)\), to which (22) applies.
Let \(G\) be finite, \(\rho \) a complex representation on \(\mathcal W\), and \(T:\mathcal W\to \mathcal P\) and \(S:\mathcal W\to \mathcal Q\) be \(G\)-injective for \(\rho \). There is a linear \(F:\mathcal P\to \mathcal Q\) such that \(FT=S\) and \(F\) is injective on \(\operatorname {im}T\). In particular, two tensor families \(A,B\) on the same virtual bond spaces, with site maps \(G\)-injective for the same \(\rho _v\), admit such maps \(F_v\) with \(B_v=F_vA_v\). Their ambient physical dimensions may differ.
Choose \(L\) with \(LT=\Pi _\rho \), where \(\Pi _\rho =|G|^{-1}\sum _g\rho (g)\). Set \(F=SL\). Invariance of \(S\) gives \(FT=S\Pi _\rho =S\). Since both \(T\) and \(FT=S\) are \(G\)-injective, Theorem 26.4.1 gives injectivity of \(F\) on \(\operatorname {im}T\). Apply this argument at every vertex and evaluate on virtual basis vectors to obtain equality of the tensor coefficients.
Let \(A,B\) have the same virtual bond spaces and let their site maps be \(G\)-injective for the same representations \(\rho _v\) of a finite group. If the regions \(R_i\) cover every vertex, there are rectangular physical maps \(F_v\) such that, for every region \(S\),
The second restriction is injective, and the parent spaces have equal dimensions. If \(B=FA\) is already specified and both site families are \(G\)-injective for \(\rho _v\), the second equality holds for that \(F\). For a finite region family and arbitrary positive regional interactions with these exact local kernels, the same comparison gives \(F_{\mathrm{all}}(\ker H)=\ker K\), injectivity on \(\ker H\), and \(\dim \ker K=\dim \ker H\).
Theorem 27.24.1.6 supplies \(B=FA\) and site-image injectivity. Regional contraction commutes with \(F\), giving the first equality. Apply Theorem 27.24.1.4 to the common parent spaces. For a prescribed \(F\), its site-image injectivity follows directly from \(G\)-injectivity of \(A\) and \(FA\). Finally, positivity identifies each Hamiltonian kernel with its common parent space.
Consider the actual canonical regular contraction on any finite graph region \(R\). Choose group elements \(r_e\) on its bonds, with \(r_e=1\) on every crossing bond. Simultaneously multiply both exposed half-edge labels of each internal bond \(e\) on the right by \(r_e\). Every canonical regional column remains unchanged. In particular this holds for one internal bond, with both its endpoints in \(R\). The same statement holds in the presence of arbitrary inserted left group multiplications.
Let the physical coefficients be \(a_v\), and suppose that local inverse coefficients \(F_v\) satisfy
where \(\Pi _v\) is the regular averaging projector. Then every original regional ground vector exposed by the product of the \(F_v\) has this internal right invariance. Consequently, every complementary slice of an arbitrary global vector satisfying the original regional parent condition has this invariance after exposure. Such inverse coefficients exist for regular \(G\)-injective tensors.
No \(G\)-isometry, flat-connection reconstruction, or global closure-span representation is assumed. This is a local identity of the contractions appearing in the accessible coordinate argument of [ 235 ] . This auxiliary regular bond identity does not establish global closure spanning; see [ 96 ] .
The regular averaging kernel is unchanged when its two configurations are multiplied on the right, leg by leg, by the same labels: common left translation commutes with these right multiplications. Change each summed shared bond label \(\eta _e\) to \(\eta _e r_e\). This is a bijection of the summation variables. It fixes the crossing condition, and associativity gives the same identity in the presence of inserted left group multiplications. The canonical column equality follows.
Apply the stated inverse coefficients to an actual original regional column. The result is its canonical column, so it inherits the identity. Linearity extends it to the entire original regional ground space. A regional parent constraint places every complementary physical slice in that space.
Consider a finite graph with site-dependent regular \(G\)-injective tensors. Let \(R\) induce a connected graph, and insert regular bond operators \(u_e\) at their ordered heads. If an internal closed walk has nonidentity holonomy, the original regional ground space and the inserted regional ground space have zero intersection. More generally, a nonidentity spanning-tree cycle residual already forces this separation.
Every regional ground vector, exposed by a local \(G\)-injective inverse, vanishes at half-edge coordinates whose cycle labels are not all the identity. In particular, this holds for every complementary slice of an arbitrary vector satisfying an original regional parent constraint. No inserted-bond representation of that vector is assumed.
Consequently, if a nonzero inserted global PEPS vector is annihilated by an original regional parent term on \(R\), every internal closed-walk holonomy in \(R\) is the identity. If a vertex gauge removes all internal inserted operators, the original and inserted regional ground spaces coincide: crossing operators only relabel the boundary condition. This last assertion requires only coefficient invariance. No \(G\)-isometry is assumed. This is a necessary local implication in the closure argument of [ 235 , Theorem 5.5 ] ; it does not assert that every parent-kernel vector admits an inserted-bond description. This necessary local holonomy condition does not give a global closure representation; see [ 96 ] .
Apply a local \(G\)-injective inverse at every vertex of \(R\). This exposes the canonical regular half-edge coordinates, and applying the original local coefficient maps restores each regional ground vector. In spanning-tree coordinates, the uninserted contraction is supported on identity cycle labels. The inserted contraction is supported on simultaneous conjugates of its cycle residuals. A nonidentity residual makes these supports disjoint. Since the local coordinate map has a left inverse on the regional ground spaces, their physical intersection is also zero.
A common nonzero regional vector therefore forces all residuals to be the identity. Evaluating the actual closed-walk word at these residuals gives identity holonomy. Finally choose a nonzero physical slice of the inserted global vector. It belongs to the inserted regional ground space by contraction, and to the original one by the original parent constraint.
Consider a finite graph with regular \(G\)-injective tensors, a connected region \(R\), and a closed walk \(p\) within \(R\). There is a physical linear operator \(Q\) on \(R\) which fixes the original regional ground space. On a regional ground vector with inserted bonds \(u\), it acts as the identity if the holonomy around \(p\) is the identity, and as zero otherwise.
Extended by the identity outside \(R\), the same operator fixes every global vector satisfying the original regional parent condition and has the corresponding scalar action on every actual inserted closed PEPS. No \(G\)-isometry is assumed. This is a linear operator; Hermiticity and positivity are not asserted. It is an auxiliary consequence of the closure and accessible-coordinate calculations in [ 235 , Theorem 5.5 ] . These auxiliary regular filters do not establish global closure spanning; see [ 96 ] .
Choose a spanning tree of \(R\). The exposed cycle labels of an inserted block are simultaneous conjugates of its cycle residuals. Evaluation of the closed-walk word commutes with simultaneous conjugation. Thus the diagonal detector of identity word value acts on each inserted block column by the stated scalar. Transport this detector through a local \(G\)-injective inverse and the original physical maps. These maps recover every regional ground vector, so the scalar action is preserved. An uninserted closed walk has identity holonomy. Apply the regional statements to each complementary physical slice to obtain the global statements.
Let both native torus periods be at least three, and let the tensor be regular \(G\)-injective. Let \(\mathcal C\) be the span of all native pair-conjugacy-class closure vectors, including noncommuting labels, and let \(\mathcal C_{\mathrm{comm}}\) be the commuting closure span. Let \(H\) be a finite sum of positive original parent interactions on connected regions whose actual closed-cell realizations are simply connected. Assume that this family contains the plaquette at the intersection of the two closure seams. Then
Its dimension is the number of commuting pair-conjugacy classes, equivalently the centralizer sum. This equality applies to arbitrary coherent sums of closure vectors and requires no \(G\)-isometry. It proves the final closure constraint within the finite closure span in the regular case of [ 235 , Theorem 5.5 ] . The separate assertion that every parent-kernel vector lies in \(\mathcal C\) remains unproved. The restriction to the closure-span intersection and the remaining full-kernel identification are recorded in [ 96 ] .
At the seam-intersection plaquette, identity holonomy is equivalent to commutativity of the two closure labels. The identity-holonomy filter therefore fixes every commuting closure and removes every noncommuting closure. Its image of \(\mathcal C\) is contained in \(\mathcal C_{\mathrm{comm}}\). Positivity of the parent sum forces each of its local constraints separately, so every vector in \(\ker H\) is fixed by the corner filter. Such a vector in \(\mathcal C\) thus belongs to \(\mathcal C_{\mathrm{comm}}\). The reverse inclusion is the local parent inclusion for commuting closures.
Let both torus periods be at least three, and let the native tensor be regular \(G\)-injective. The original parent condition on the plaquette at the intersection of the two seams forces the labels \(g,h\) of a closure vector to commute. Conversely, every commuting closure satisfies the original parent condition on every connected region whose actual closed-cell realization is simply connected. These regions may meet either seam.
For any finite family of such regions, let \(H\) be a sum of positive parent interactions with the original regional kernels. Then the span of the commuting closure vectors is contained in \(\ker H\), and
where \(c_C\) represents \(C\). No \(G\)-isometry is required. This proves the inclusion and the resulting lower bound for the regular case of [ 235 , Theorems 5.5 and 5.9 ] . The assertion that every parent-kernel vector lies in the closure span remains separate. The regular native-torus restriction and the missing reverse global spanning inclusion are recorded in [ 96 ] .
Regular \(G\)-injectivity makes every native closure vector nonzero. The local parent constraint therefore forces trivial plaquette holonomy. At the intersection of the two seams, this is precisely the commutation relation for the closure labels.
A simply connected closed-cell region admits an integer lift. For commuting labels, the lift defines a vertex gauge which removes every internal closure operator. The remaining operators merely relabel the virtual boundary, so the original and inserted regional ground spaces agree. Every physical slice of the closed inserted contraction thus satisfies the original regional parent condition. The kernel of a positive sum is the intersection of the local kernels. Finally use linear independence of the commuting pair-conjugacy classes and their centralizer count.
Let a native four-leg regular \(G\)-injective tensor be placed on a square torus with both periods at least three. Insert arbitrary regular group operators on its ordered bonds, and suppose that the resulting physical PEPS vector is nonzero. If an original plaquette parent term annihilates this vector, the holonomy around that plaquette is the identity.
If the sum of all original plaquette parent terms annihilates it, every elementary plaquette has identity holonomy. Equivalently, writing \(A_v\) for transport to the right and \(B_v\) for transport upward, one has
at every vertex. In particular, a nonzero one-bond flux state with nonidentity flux is not annihilated by the original parent term surrounding its endpoint.
No \(G\)-isometry is assumed. This is a necessary local step in [ 235 , Theorem 5.5 ] ; it does not claim that arbitrary parent-kernel vectors have inserted-bond descriptions, or that flatness alone gives the full torus ground-space classification. The regular native-torus scope and the remaining global representation step are recorded in [ 96 ] .
Each native plaquette is the support of a four-step closed walk, so its induced graph is connected. Apply local regular parent flatness to the induced walk and then identify its holonomy with the original native plaquette holonomy. Positivity of every regional parent term implies that a vector killed by their sum is killed by each term separately.
The ordered plaquette holonomy is \(B_v^{-1}A_{v+(0,1)}^{-1}B_{v+(1,0)}A_v\). It is the identity exactly when the two elementary square transports agree. The explicit one-bond insertion has its prescribed group element as holonomy, giving the exclusion of nonidentity flux.
For a region \(R\) in a finite simple graph and an arbitrary alphabet \(X\), regroup the physical incidence labels as one label on each crossing bond, one tail label on each internal bond, and one head label on each internal bond. This is a coordinate bijection using the ordered endpoints and retaining every incidence at the boundary. It is the incidence part of the regional coordinates in [ 235 , Section 6.2 ] and of the bond-endpoint argument in [ 235 , Section 7 ] .
The two internal labels are precisely the physical labels at the ordered tail and head. On a crossing bond, the single retained physical label is the tail label when the tail lies in \(R\), and the head label when the head lies in \(R\).
Each internal edge has exactly two incidences in \(R\), whereas each crossing edge has exactly one. Read these incidences in the coordinate bijection.
Let \(R\) be a region in a finite simple graph. Incident virtual labels split into internal-bond labels and crossing-bond labels. For a finite virtual alphabet \(X\), site coefficients \(A_v\), prescribed crossing labels \(\theta \), and physical configuration \(\sigma \), define the actual open coefficient by summing the product of the site coefficients over incident assignments whose crossing restriction equals \(\theta \). The physical alphabet may depend on \(v\). These are the finite contraction coordinates used in [ 235 , Sections 6.2–6.3 and 7 ] .
The two factors of the incident-label bijection read exactly the original internal and crossing labels. Prescribing \(\theta \) therefore gives
with each internal assignment counted once. When the physical alphabet is uniformly numbered, enumerating \(X\) into finite indices gives exactly the original numbered open-region coefficient of the same site tensors. No boundary-state equality or Gram identity is a hypothesis.
Partition incident edges into internal and crossing edges and transport their labels along the resulting bijection. The prescribed crossing condition removes the crossing-label sum. For numbered coefficients, reindex each incident virtual label by the same finite enumeration; the site products and boundary condition are unchanged.
For a region \(R\) in a finite simple graph, a product over its incidences equals the product of both endpoint factors on each internal edge, multiplied by the product of the single retained endpoint factor on each crossing edge. This is the regional incidence decomposition underlying [ 235 , Sections 6.2 and 7 ] .
Extend incidence factors by \(1\) outside \(R\), regroup the ambient incidence product by edges, and separate internal edges from crossing edges.
Let \(G\) be finite, \(U:G\to \operatorname {Mat}_X(\mathbb C)\) a representation, \(W\) a virtual weight, and \(R\) a region of a finite simple graph with ordered endpoints. Write \(\kappa =|G|\), and prescribe crossing virtual labels \(\theta _e\). The physical labels are \((\beta _{\partial },\beta _t,\beta _h)\) in crossing, internal-tail, and internal-head coordinates. For \(q:R\to G\), set
Define
These are the regional endpoint factors from the contraction calculation in [ 235 , Sections 6.2 and 7 ] ; the site average is normalized as described in [ 96 ] .
Suppose \(W\) commutes with every \(U(g)\). The actual open contraction of the averaging sites dressed by \(W\), evaluated at a physical incidence configuration \(\alpha \), equals \(H_R(\theta ,\beta )\), where \(\beta \) is the incidence regrouping of \(\alpha \). Thus regrouping by the coordinate bijection gives exactly the displayed bond formula. The case \(W=I\) gives the ordinary averaging-site coefficients. If the local physical incidence configurations have uniform finite enumerations, the same formula is exactly the numbered open-region coefficient of those enumerated site tensors. The crossing labels are arbitrary throughout. These finite-simple-graph coefficient identities do not assert an equivalence of all open-boundary state spaces.
Fix the crossing labels and sum only over internal virtual labels. Expand the independent site averages and regroup their incidence products. Each internal label is summed once, giving \((U(q_h)W)(WU(q_t^{-1}))=W^2U(q_hq_t^{-1})\). Crossing labels are fixed, so their single endpoint factors remain. The incidence bijection reads the physical endpoint labels literally. Finally, finite enumerations reindex the same coefficient sum.
27.24.2 Matching sectors in open weighted regions
For a finite simple graph with edge set \(E\) and ordered endpoints, write \(E_R\) for edges internal to \(R\) and \(\partial E_R\) for edges crossing its boundary. An internal edge \(f=(t,h)\) is indexed in head–tail order. The open contractions below retain arbitrary virtual boundary labels. They give support consequences of the bond transformation in [ 235 , Section 7 ] for the regional parent spaces.
For finite coordinate sets \(X_i\), let \(X=\coprod _iX_i\). The matching-sector coordinate inclusion is \(j:\coprod _i(X_i\times X_i)\hookrightarrow X\times X\), given by \(j(i,a,b)=((i,a),(i,b))\). Its matrix is the matching-sector bond inclusion \(E_X\).
Let \(j:B\hookrightarrow A\) be an injection with \(B\) finite, and let \(E_j(a,b)=\delta _{a,j(b)}\). Then \((E_jx)(j(b))=x(b)\) and \((E_jx)(a)=0\) outside \(j(B)\). Consequently
For any finite set \(E\), the product matrix \(\bigotimes _{f\in E}E_j\) is the coordinate inclusion induced by \(j^E:B^E\hookrightarrow A^E\). In particular, a vector in \(\operatorname {im}\bigotimes _{f\in E}E_X\) vanishes whenever one endpoint pair has different sector labels.
At \(a=j(b)\), the sum \(\sum _c\delta _{a,j(c)}x(c)\) has only the term \(c=b\); outside \(j(B)\) every summand is zero. Conversely, if \(\psi \) vanishes outside \(j(B)\), take \(x(b)=\psi (j(b))\) to prove (31). The product of coordinate indicators is the indicator of equality of the full configurations. For a mismatched bond, each summand in the product inclusion has a zero factor at that bond.
Let \(X\) and \(G\) be finite, \(G\) a group, \(U:G\to \operatorname {Mat}_X(\mathbb C)\) a representation, and \(W\in \operatorname {Mat}_X(\mathbb C)\). Fix virtual labels \(\theta :\partial E_R\to X\) and physical labels \(\sigma :\partial E_R\to X\). For \(q:R\to G\), define
For a regional physical vector \(\psi \), its internal slice \(\mathcal S_{R,\sigma }\psi \) fixes these physical crossing labels and regards the remaining indices as \(\beta :E_R\to X\times X\), with \(\beta _f=(\beta _{h,f},\beta _{t,f})\). If incident endpoint configurations at every vertex are enumerated by bijections \(e_v:X^{\operatorname {Inc}(v)}\cong \{ 0,\ldots ,p-1\} \), the same slice in numbered physical coordinates first uses these bijections to assemble each site index.
Suppose \(W\) commutes with \(U(g)\) for every \(g\in G\). Let \(A\) be the weighted averaging-site family and let \(\mathcal C_R(A)|\theta \rangle \) denote its actual open contraction with fixed virtual boundary labels \(\theta \). Then
Regroup the open contraction into crossing endpoints and internal head–tail pairs. The internal virtual sum gives \((U(q_h)W)(WU(q_t^{-1}))=W^2U(q_hq_t^{-1})\). Each crossing edge contributes only its retained endpoint factor from (35). Collecting these factors and the site-average normalization yields (36).
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\end{tenkz}](tenkz_svg/tenkz-463214026a457209.svg)
Let \(I\) be finite, \(d_i{\gt}0\), and let \(D_i:G\to \operatorname {Mat}_{d_i}(\mathbb C)\) be representations. Put \(X=\coprod _i\{ 0,\ldots ,d_i-1\} \), \(U(g)=\bigoplus _iD_i(g)\) and \(W=\bigoplus _i d_i^{1/4}I_{d_i}\). Let \(A\) be the weighted averaging-site family, and set \(\psi =\mathcal S_{R,\sigma }\mathcal C_R(A)|\theta \rangle \). For arbitrary \(R,\theta ,\sigma \),
where \(F\) is the full endpoint multiplicity-restoring map with multiplicity \(d_i\) in sector \(i\), and \(U'(g)=\bigoplus _i(D_i(g)\otimes I_{d_i})\). The coefficients \(c_R\) still use the original \(U,W,\theta ,\sigma \).
The block-scalar matrix \(W\) commutes with \(U(g)\), and \(W^2U(g)=\bigoplus _i\sqrt{d_i}\, D_i(g)\). Apply coherent multiplicity restoration to (36), with coherent coefficients \(c_R(\theta ,\sigma ;q)\) and bond labels \(q_hq_t^{-1}\). It gives both product support and (38). Only internal bonds are acted on, so the crossing factors remain those in (35).
Use the representations and positive dimensions of Theorem 27.24.2.5, and suppose each site’s incident endpoint configurations have a bijection \(e_v:X^{\operatorname {Inc}(v)}\cong \{ 0,\ldots ,p-1\} \). Let \(A\) be the resulting weighted averaging tensor in these numbered physical coordinates. For any region \(R\), any crossing physical labels \(\sigma \), and every \(\psi \in \mathcal G_R(A)\),
Thus \(\psi \) vanishes at every physical configuration whose two endpoint labels on some internal edge belong to different sectors.
The open contractions \(\mathcal C_R(A)|\theta \rangle \) span \(\mathcal G_R(A)\). Reindexing their virtual and physical labels identifies their slices with those of Theorem 27.24.2.5. The slice is linear and the product inclusion range is a subspace, so support extends to every vector in \(\mathcal G_R(A)\). At a given physical configuration, choose its own crossing labels for \(\sigma \) and regroup the remaining endpoints. A mismatched internal sector then forces a zero coefficient by Theorem 27.24.2.2.
Let \(A\) and the endpoint enumerations be as in Theorem 27.24.2.6. Suppose that for every edge \(f=(t,h)\) there is a parent region \(R_i\) containing both \(t\) and \(h\). If \(\psi \in \mathcal P_{\mathcal R}(A)\), then \(\psi \) vanishes whenever one physical bond has different endpoint sector labels. Equivalently, regrouping the physical endpoint coordinates by bonds gives a vector \(\widetilde\psi \) satisfying
Neither irreducibility nor completeness of the block representations is required.
Choose a region containing both endpoints of a mismatched bond. Fix the exterior physical configuration. The resulting regional slice belongs to \(\mathcal G_{R_i}(A)\) by the parent condition, so Theorem 27.24.2.6 makes its coefficient zero. Reassembling the regional and exterior indices gives the original coefficient of \(\psi \). The image of the coordinate inclusion \(j^E\) consists exactly of configurations with matching sectors on every bond: for each matched pair, select its shared sector and its two internal labels. Equation (31) therefore turns the vanishing condition into (40).
27.25 Regional support of physical bond transformations
The bond transformation of [ 235 , Section 7 ] acts on head–tail endpoint pairs. We describe its regional blocks, including the one-ended factors on crossing edges, and derive the product support of regional parent spaces. Throughout the graph statements, \(\Gamma \) is a finite simple graph with ordered vertices. For a region \(R\), write \(E_R\) for its internal edges, \(\partial E_R\) for its crossing edges, and \(E_{R^c}\) for its exterior edges. An edge \(f=(t,h)\) has \(t{\lt}h\).
27.25.1 Products, coordinate slices, and multiplicity maps
Let \(B,E\) be finite sets, with matrices \(F_b:\mathbb {C}^X\to \mathbb {C}^Y\) and \(K_f:\mathbb {C}^P\to \mathbb {C}^Q\). Their mixed product has coefficients
When \(X,P\) are finite, its action is \((\mathcal M(F,K)\psi )(\tau ,\beta ) =\sum _{\sigma ,\alpha }\mathcal M(F,K)_{(\tau ,\beta ),(\sigma ,\alpha )} \psi (\sigma ,\alpha )\).
The adjoint satisfies \(\mathcal M(F,K)^\dagger =\mathcal M(F^\dagger ,K^\dagger )\). Assume \(X,P\) are finite. For a finite index set \(A\), weights \(w_a\), and factor vectors \(x_{b,a}\in \mathbb {C}^X\), \(y_{f,a}\in \mathbb {C}^P\), one has
Let \(S,X,Y\) be finite and \(F:\mathbb {C}^X\to \mathbb {C}^Y\) a matrix. For \(\psi \in \mathbb {C}^{Y^S}\), define \(\psi _{v,\tau }(s)=\psi (\tau [v\mapsto s])\). Then
In the forward implication \(Y\) need not be finite. For any matrix \(P:\mathbb {C}^Y\to \mathbb {C}^Y\) with \(Y\) finite, if \(P\psi _{v,\tau }=\psi _{v,\tau }\) for every \(v,\tau \), then \(P^{\otimes S}\psi =\psi \); idempotence of \(P\) is unnecessary. The assertions allow empty sets and impose no rank condition on \(F\).
A slice of \(F^{\otimes S}\chi \) is a linear combination of columns of \(F\), with coefficients obtained by contracting all other factors. For the converse choose a linear right inverse of \(F\) on its image and extend it to \(L:\mathbb {C}^Y\to \mathbb {C}^X\). Thus \(FL\) fixes every slice. Applying \(FL\) at one coordinate fixes \(\psi \). Induction over finite subsets of \(S\) shows that applying it at all coordinates also fixes \(\psi \). Consequently \(F^{\otimes S}(L^{\otimes S}\psi )=(FL)^{\otimes S}\psi =\psi \). The same induction proves the assertion about an arbitrary \(P\).
Let \(I\), the sets \(X_i\), and the multiplicity sets \(M_i\) be finite. Put \(X=\coprod _iX_i\), \(Y=\coprod _i(X_i\times M_i)\), and \(m_i=|M_i|\). For \(y=(i,a,u)\in Y\), define \(b(y)=(i,a)\) and \(c(y)=(i,u)\). Set
Whenever the first case occurs, \(M_i\) contains \(u\), so \(m_i{\gt}0\).
Let \(F=E_YJE_X^\dagger :\mathbb {C}^{X\times X}\to \mathbb {C}^{Y\times Y}\) be the full endpoint multiplicity map, where \(J|i,a,b\rangle =m_i^{-1/2}\sum _{u\in M_i}|i,(a,u),(b,u)\rangle \) and an empty sum is zero. Its coefficient and action formulas are
Also \(a(y,z)=a(z,y)\). These formulas require neither nonempty multiplicity sets nor support conditions on the input vectors. More generally, for a coordinate inclusion \(j:B\hookrightarrow A\) with \(A\) finite, its inclusion matrix satisfies \((E_j^\dagger \phi )(i)=\phi (j(i))\).
The adjoint of a coordinate inclusion selects its included coordinates. For equal sectors, the multiplicity map retains equal copy labels and multiplies by \(m_i^{-1/2}\); for unequal sectors its coefficient is zero. This proves the formula for \(F\phi \). Evaluating it on coordinate vectors gives (47); the product of its coordinate indicators selects the single input configuration in (49). Equal copy labels have equal sectors and hence equal normalizations, proving symmetry.
Assume every \(M_i\) is nonempty. Let \(E_X\) be the matching-sector inclusion and let \(\mathcal F=F^{\otimes E}\) for a finite set \(E\). Then \(FF^\dagger F=F\). For \(\psi \in \operatorname {im}E_X^{\otimes E}\) and \(\phi \in \operatorname {im}\mathcal F\),
Since \(F^\dagger F=E_XE_X^\dagger \) and \(E_X^\dagger E_X=I\), the factorization \(F=E_YJE_X^\dagger \) gives \(FF^\dagger F=F\). Products respect multiplication. Write \(\psi =E_X^{\otimes E}\chi \) and \(\phi =\mathcal F\eta \), respectively, and cancel the corresponding local factors.
27.25.2 One-ended boundary transformations
For weights \(\lambda _i\in \mathbb {C}\) and a fixed exterior endpoint \(z\in Y\), define \(B_z^\lambda :\mathbb {C}^X\to \mathbb {C}^Y\). For \(y=(i,a,u)\), put
Write \(B_z=B_z^1\) for unit weights.
For arbitrary matrices \(T_i\) on \(X_i\), write \(M=\bigoplus _iT_i\), \(M^{[m]}=\bigoplus _i(T_i\otimes I_{M_i})\), and \(M_\lambda =\bigoplus _i\lambda _iT_i\). For \(x\in X\) and \(y,z\in Y\),
The entries of \(B_z\) are real, hence \(B_z^\dagger =B_z^{\mathsf T}\). A block scalar may be read from either index: \((M_\lambda )_{x,r}=\lambda _{i(x)}M_{x,r} =\lambda _{i(r)}M_{x,r}\).
The block-diagonal and identity factors force the sector and copy coordinates in the sum. The remaining matrix entry is that of \(T_i\), multiplied by \(\lambda _i m_i^{-1/2}\). Transposing the block matrices exchanges the two contraction formulas. The action formula follows by summing the single surviving input coordinate in (52). Its unit-weight coefficient is real. Off-block entries vanish; on a block, the row and column sector labels agree.
Suppose \(\lambda _i\ne 0\) for every \(i\). With the preceding notation,
Independently of these weights, if every \(M_i\) is nonempty then
for arbitrary matrices \(T_i\) on \(X_i\), where \(\operatorname {vec}(T)(x,r)=T_{x,r}\). In particular, for representations \(D_i\) of dimensions \(d_i\), \(D=\bigoplus _iD_i\) and \(W=\bigoplus _i d_i^{1/4}I_{d_i}\) satisfy \(WD(g)=D(g)W=\bigoplus _i d_i^{1/4}D_i(g)\).
Apply (54) with unit weights and use \(B_z^\dagger =B_z^{\mathsf T}\). Reading the block weight from the fixed \(y\) index permits cancellation of \(\lambda _{i(y)}\). The repeated matrix-entry vector in (58) is \(F\) applied to the square-root-weighted block vector. Apply \(F^\dagger F=E_XE_X^\dagger \); the block-diagonal input lies in the matching-sector range. The fourth-root identities follow by multiplying block-scalar matrices on the left and on the right.
27.25.3 Physical coordinates and regional operator blocks
Choose bijections \(e_v:X^{\operatorname {Inc}(v)}\to \{ 0,\ldots ,p-1\} \). Decoding a numbered configuration and grouping its endpoint labels gives a bijection \(j_e:\{ 0,\ldots ,p-1\} ^V\to (X\times X)^E\). The pair on \(f=(t,h)\) is \((x_{h,f},x_{t,f})\). Its coefficient-space isomorphism is \(J_e\psi (\beta )=\psi (j_e^{-1}\beta )\). Regionally the same regrouping gives
The inverse assigns each crossing label to its inside incidence, and each internal pair to its head and tail incidences, before applying the corresponding \(e_v\). If \(\sigma \) is a numbered exterior configuration, let \(\gamma ^e_R(\sigma )_f\) be its endpoint label outside \(R\): the head label when \(t\in R\), and the tail label when \(h\in R\).
Let \(e^X,e^Y\) enumerate incident configurations of \(X,Y\), respectively, and let \(F:\mathbb {C}^{X\times X}\to \mathbb {C}^{Y\times Y}\). Define the matrix \(N_F\) between the numbered physical spaces by
The matrix \(N_F\) is the matrix of \(F^{\otimes E}\) with rows relabelled by \(j_{e^Y}\) and columns by \(j_{e^X}\). Thus \(J_{e^Y}N_F=F^{\otimes E}J_{e^X}\) and \(N_F^\dagger =N_{F^\dagger }\), where the latter operator exchanges the two enumerations.
The product coefficients agree by their definitions. Reindex the matrix-vector sum by the bijection \(j_{e^X}\) to obtain the intertwining identity. Conjugating each factor in (60) proves the adjoint identity.
For a matrix \(M\) between global numbered physical spaces, define its regional block by \(M_R[\tau ,\sigma ]_{y,x}=M_{y\cup \tau ,x\cup \sigma }\), where \(x,y\) are configurations on \(R\), and \(\sigma ,\tau \) are configurations on \(R^c\). Write \(S_{R,\sigma }\psi (x)=\psi (x\cup \sigma )\) for an exterior slice.
For every exterior output configuration \(\tau \),
Let \(A,B\) be tensors on the same graph, possibly with different physical dimensions, and let \(\mathcal R=(R_i)_{i\in \iota }\) be any region family. If \(M_{R_i}[\tau ,\sigma ]\mathcal G_{R_i}(A) \subseteq \mathcal G_{R_i}(B)\) for every \(i,\tau ,\sigma \), then \(M\mathcal P_{\mathcal R}(A)\subseteq \mathcal P_{\mathcal R}(B)\). Here \(\mathcal P_{\mathcal R}(A)\) consists of vectors whose every exterior slice on \(R_i\) belongs to \(\mathcal G_{R_i}(A)\).
Split the global input-coordinate sum into its regional and exterior coordinates to obtain (61). Taking the adjoint exchanges input and output configurations. If \(\psi \in \mathcal P_{\mathcal R}(A)\), every summand in (61) lies in \(\mathcal G_{R_i}(B)\) by the assumed block inclusion. This space is closed under sums, proving all target parent constraints.
Fix exterior configurations \(\tau ,\sigma \), and put \(\gamma _Y=\gamma _R^{e^Y}(\tau )\), \(\gamma _X=\gamma _R^{e^X}(\sigma )\). The matrix on the inside endpoint of a crossing edge is
Decode the exterior endpoint labels as \(\tau _{h,f},\tau _{t,f}\) and \(\sigma _{h,f},\sigma _{t,f}\). Define the exterior scalar and, for the multiplicity map, its crossing-indicator factor by
For any weights \(w_f\), the edge partition gives \(\prod _{f\in E}w_f=(\prod _{f\in E_{R^c}}w_f) (\prod _{f\in \partial E_R}w_f)(\prod _{f\in E_R}w_f)\). Accordingly, the blocks of the global bond operator satisfy
Every edge has zero, one, or two endpoints in \(R\). Separate the product in (60) into these three cases. The first contributes \(c_{R,F}\), the second gives (65), and the third retains \(F\) on its head–tail pair. Regional regrouping places these factors in the two coordinate families of \(\mathcal M\).
For the multiplicity map \(F\) and any \(z,y\in Y\), \(r,x\in X\),
Thus \(C_f=\delta _{b(\gamma _Y(f)),\gamma _X(f)}B_{\gamma _Y(f)}\) in either orientation, and
27.25.4 Actual open contractions and their product support
Let \(G\) and \(X\) be finite, \(U:G\to \operatorname {Mat}_X(\mathbb {C})\) a representation, and \(W\) a matrix on \(X\). For \(q:R\to G\), define
With virtual boundary labels \(\theta \in X^{\partial E_R}\), put
This is the coherent open bond vector in crossing and internal head–tail coordinates.
Let \(a_v^{U,W}(\eta ,\xi )\) be the dressed averaging site. In a chosen incident enumeration \(e\), the numbered tensor is \(A(U,W,e)_v(\eta ,s)=a_v^{U,W}(\eta ,e_v^{-1}(s))\), with virtual labels numbered by a bijection \(X\cong \{ 0,\ldots ,|X|-1\} \). Dressing by \(W=I\) gives the unweighted averaging site. Define the open bond space by \(\mathcal H_R(U,W)=\operatorname {span}_{\mathbb {C}} \{ H_R^{U,W}(\theta ):\theta \in X^{\partial E_R}\} \).
Suppose \(WU(g)=U(g)W\) for every \(g\). Then
In particular, \(J_{e,R}\phi \in \mathcal H_R(U,W)\) if and only if \(\phi \in \mathcal G_R(A(U,W,e))\). More generally, for any site family \(a_v(\eta ,\xi )\), numbering its virtual and physical labels leaves every internal slice of each open column unchanged: with the same \(\theta \) and fixed crossing physical labels \(\sigma \), the numbered and incidence contractions give the same function of internal head–tail pairs.
Decode the numbered labels and apply the actual open contraction formula. The inverse regional coordinate map puts each crossing, head, and tail label back at its original incidence, proving the column equality in (78). Open columns span \(\mathcal G_R\), and their boundary numberings are bijective, so their images span exactly \(\mathcal H_R\). Injectivity of \(J_{e,R}\) gives the membership equivalence. The final assertion follows directly by decoding the virtual boundary labels in the finite contraction sum and then fixing the crossing physical coordinates; it needs no representation or commutation hypothesis.
Let \(A\) be any tensor with incident enumeration \(e\) by a finite endpoint set \(X\), and let \(F:\mathbb {C}^Z\to \mathbb {C}^{X\times X}\) with \(Z\) finite. Suppose every edge is internal to some region \(R_i\). Assume that, for every \(i\), every crossing physical configuration \(\sigma \), and every \(\phi \in \mathcal G_{R_i}(A)\),
Then \(J_e\psi \in \operatorname {im}F^{\otimes E}\) for every \(\psi \in \mathcal P_{\mathcal R}(A)\). The coordinate identity used here is the following. Fix global bond labels \(\beta \) and an internal edge \(f\in E_R\). Let \(\sigma \) be their crossing labels inside \(R\), and let \(\tau \) be their numbered exterior configuration. For every endpoint pair \(s\),
Updating an internal edge changes only its two inside incidences; it leaves all crossing and exterior incidences fixed. This proves (80). For each edge \(f\), choose a covering region \(R_i\). The parent condition places \(S_{R_i,\tau }\psi \) in \(\mathcal G_{R_i}(A)\), so (79) and the forward implication of (43) put its \(f\)-slice in \(\operatorname {im}F\). By (80) this is the global \(f\)-slice. Apply the reverse implication of (43) to all edges.
Let \(U:G\to \operatorname {Mat}_X(\mathbb {C})\) be a representation of a finite group, let \(W\) commute with every \(U(g)\), and let \(F:\mathbb {C}^Z\to \mathbb {C}^{X\times X}\), with \(X,Z\) finite. Suppose \(\operatorname {vec}(W^2U(g))\in \operatorname {im}F\) for every \(g\). Every internal slice of every open column, with arbitrary virtual boundary labels, belongs to \(\operatorname {im}F^{\otimes E_R}\). The same holds for every \(\phi \in \mathcal G_R(A(U,W,e))\). If every edge is internal to a member of \(\mathcal R\), then
Choose \(x_g\) with \(Fx_g=\operatorname {vec}(W^2U(g))\). For each internal slice use the preimage
where \(c_R\) retains all boundary factors and the averaging normalization. Applying \(F^{\otimes E_R}\) term by term gives the weighted slice formula. Numbered and incidence slices agree, and the regional range is spanned by open columns, so the inclusion extends by linearity. Theorem 27.25.4.4 then proves (81).
Let \(d_i{\gt}0\) and let \(D_i:G\to \operatorname {Mat}_{d_i}(\mathbb {C})\) be representations of a finite group. Put \(U'(g)=\bigoplus _i(D_i(g)\otimes I_{d_i})\), and let \(F\) restore multiplicity \(d_i\) on the full endpoint pairs. Every internal slice of each actual open column for the unweighted averaging tensor of \(U'\) lies in \(\operatorname {im}F^{\otimes E_R}\). This also holds for each vector in the numbered regional range \(\mathcal G_R(A(U',I,e))\). For any edge-covering family \(\mathcal R\),
Neither irreducibility nor unitarity of the \(D_i\) is assumed.
Expand each target slice with its own boundary coefficients \(c_R^{U',I}(\theta ,\sigma ;q)\). Coherent multiplicity restoration sends the internal factors \(\operatorname {vec}(\bigoplus _i\sqrt{d_i}\, D_i(q_hq_t^{-1}))\) to \(\operatorname {vec}(U'(q_hq_t^{-1}))\), while retaining these same coefficients. This supplies a preimage for each target open slice. Linear combinations give all regional vectors; edge coverage and Theorem 27.25.4.4 give (83).
27.26 Positive multiplicities and complete canonical parent spaces
Fix a finite group \(G\), a finite ordered simple graph, and a finite family of matrix representations \(D_i:G\to \operatorname {Mat}_{d_i}(\mathbb {C})\). Write \(X=\coprod _i\{ 0,\ldots ,d_i-1\} \) and \(X_m=\coprod _i(\{ 0,\ldots ,d_i-1\} \times \{ 0,\ldots ,m_i-1\} )\). An edge is written \(f=(t,h)\); physical bond coordinates are ordered head–tail. The construction extends the dimension multiplicities in [ 235 , Section 7 ] to arbitrary positive integers \(m_i\). Irreducibility of \(D_i\) is not required.
27.26.1 Copy weights and arbitrary open boundaries
Define \(D(g)=\bigoplus _iD_i(g)\), \(D^{[m]}(g)=\bigoplus _iD_i(g)\otimes I_{m_i}\), and \(W_m=\bigoplus _i m_i^{1/4}I_{d_i}\). Taking \(m_i=d_i\) gives dimension multiplicity restoration and the fourth-root dimension weight. Let \(A_m\) be the actual averaging tensor for \(D\) with virtual weight \(W_m\), and let \(B_m\) be the unweighted averaging tensor for \(D^{[m]}\). Their site averages use the same factor \(|G|^{-1}\).
For every \(g\in G\),
Multiply within each diagonal block. Its scalar weight commutes with every matrix, and \((m_i^{1/4})^2=\sqrt{m_i}\).
Let \(F_m:\mathbb {C}^{X\times X}\to \mathbb {C}^{X_m\times X_m}\) be the supported normalized copy map. For \(y=(i,a,c)\), write \(\pi y=(i,a)\), and set \(\alpha _m(y,z)=m_i^{-1/2}\) when \(y,z\) have sector \(i\) and the same copy label, and zero otherwise. Its one-ended map with fixed exterior output \(z\) is \(B_z(y,x)=\alpha _m(y,z)\delta _{\pi y,x}\). For a region \(R\), let \(E_R\) be its internal edges and \(\partial E_R\) its crossing edges. Given \(\gamma :\partial E_R\to X_m\), define
The numbered regional matrix is obtained by regrouping the site labels into these crossing endpoints and internal head–tail pairs. The dimension-multiplicity versions are obtained by setting \(m=d\).
Assume \(m_i{\gt}0\). Fix \(q:R\to G\). Denote the internal bond factors of \(A_m,B_m\) by \(a_f(q),b_f(q)\), and the crossing factors with virtual label \(\theta \) by \(a_f(q;\theta ),b_f(q;\theta )\). For an internal edge, \(F_ma_f(q)=b_f(q)\) and \(F_m^\dagger b_f(q)=a_f(q)\). For a crossing edge and exterior output \(z\in X_m\),
where \(w_y=m_i^{1/4}\) for \(y\) in sector \(i\). These formulas hold for both crossing orientations; each crossing retains exactly one fourth-root weight.
The internal factor is \(W_m^2D(q_hq_t^{-1})\). Equation (85) supplies \(\sqrt{m_i}\) in each block, canceled by the normalized copy map. Its adjoint sums \(m_i\) equal-copy terms with coefficient \(m_i^{-1/2}\), recovering the same factor. At a crossing, use \(W_mD(q_t^{-1})\) when the tail lies in \(R\) and \(D(q_h)W_m\) when the head lies in \(R\). The sparse coefficient of \(B_z\) fixes the base coordinate and copy label; moving the remaining \(w_y\) to the virtual boundary coefficient gives (87). Conjugating the same sparse coefficient and dividing by the nonzero \(w_y\) gives (88).
Assume \(m_i{\gt}0\), and write \(a_R(\theta ),b_R(\eta )\) for the open contractions of \(A_m,B_m\) in regional bond coordinates. For arbitrary virtual labels \(\theta :\partial E_R\to X\) and \(\eta :\partial E_R\to X_m\),
Linear extension therefore transports arbitrary, possibly entangled, virtual boundary coefficients. These identities also give the dimension-multiplicity formulas when \(m=d\).
Expand each open contraction as \(|G|^{-|R|}\) times a sum over \(q:R\to G\) of the product of its crossing and internal factors. Apply Theorem 27.26.1.4 to each factor. Distributing the crossing sums gives (89); collecting their scalar coefficients gives (90). The transformation never restricts the original virtual boundary.
If \(m_i{\gt}0\), then for every region and every fixed exterior output \(\gamma \), the regional matrix and its adjoint satisfy
The statements hold both in bond coordinates and after the supplied enumerations of the site physical spaces, including \(m=d\).
27.26.2 Global supported inverses
Suppose \(m_i{\gt}0\) and let \(M_m\) be the actual product \(\bigotimes _{f\in E}F_m\) in the supplied numbered site coordinates. For any family \(\mathcal R=(R_i)_i\),
No coverage assumption on the regions is needed for these inclusions.
Fix a region and the input and output exterior configurations. The corresponding block of \(M_m\) is a scalar times \(M_{R,\gamma }\): exterior edges give scalar factors, and crossing edges additionally enforce equality of the exterior base labels. Every such block carries the source regional range into the target range by (91). Each target regional slice is a sum of these block images of source slices, and so satisfies the target condition. The adjoint block is the conjugate scalar times \(M_{R,\gamma }^\dagger \), proving the reverse inclusion.
Assume \(m_i{\gt}0\) and every edge has both endpoints in some \(R_i\). For \(\psi \in \mathcal P_{\mathcal R}(A_m)\) and \(\phi \in \mathcal P_{\mathcal R}(B_m)\),
In particular, \(M_m\) is injective on the source parent space. These identities concern every parent vector, including for \(m=d\).
The source internal factors \(\bigoplus _i\sqrt{m_i}D_i(g)\) lie in the matching-sector inclusion range. The target internal factors \(\bigoplus _iD_i(g)\otimes I_{m_i}\) lie in \(\operatorname {im}F_m\). Fixing crossing physical labels therefore puts every internal slice of a regional generator in the corresponding product range. Linear extension gives the same property on its regional range. For each edge, choose a parent region containing both endpoints; its regional constraint places every global one-bond slice in the required range. The product-range criterion then gives source matching-sector support and target product-image support. On these supports the product partial-isometry identities are exactly (93). Applying \(M_m^\dagger \) to \(M_m\psi =M_m\psi '\) proves injectivity.
If \(m_i{\gt}0\) and every edge is internal to some parent region, restriction of the actual product map gives
Its inverse is the restriction of \(M_m^\dagger \). For any finite region family and positive interactions whose local kernels are these exact regional ranges, \(M_m(\ker H)=\ker K\). Taking \(m=d\) gives canonical dimension-multiplicity restoration.
27.26.3 Removing weights and changing multiplicities
For an incident physical configuration \(\sigma \) at \(v\), put \(c_v(\sigma )=\prod _{f\in \operatorname {Inc}(v)}m_{i(\sigma _f)}^{1/4}\). When \(m_i{\gt}0\), define the invertible diagonal physical filter \(C_v=\operatorname {diag}(c_v)\) in the numbered site coordinates, and let \(C=\bigotimes _vC_v\).
Let \(A_0\) be the unweighted averaging tensor for \(D\). Every virtual column satisfies \((A_m)_v=C_v(A_0)_v\). If \(m_i{\gt}0\), then every \(c_v(\sigma )\) is nonzero and, for any region family without a coverage assumption,
On each incident leg, the block-scalar weight can be read from the physical sector label. Taking the product of these weights gives \(c_v(\sigma )\), independently of the virtual column. Positive \(m_i\) make the diagonal matrices invertible. Physical deformation commutes with every open contraction. Applying \(C\) and \(C^{-1}\) to regional slices proves (95).
Assume all \(m_i{\gt}0\), and that every edge is internal to some \(R_i\). Define \(T_m=M_mC\). Its restriction is the composite isomorphism from the unweighted single-copy parent space to that of \(D^{[m]}\). For any two positive multiplicity families \(m,n\), define their parent-space comparison by \(T_nT_m^{-1}\), where the inverse is taken on the parent space.
For arbitrary representations \(D_i\), positive \(m_i\), supplied common incident enumerations, and internal coverage of every edge,
Consequently any two positive multiplicity families have isomorphic full canonical parent spaces. For a finite region family and arbitrary positive local interactions with the corresponding exact regional kernels, \(T_m(\ker H)=\ker K\).
First apply \(C\) using (95), then apply \(M_m\) using (94). Both restrictions are isomorphisms, so their composition preserves dimensions. Reverse the comparison for \(m\) and follow it by the comparison for \(n\). The kernel statement again follows from positivity and the exact local-kernel hypotheses.
27.27 Fourier coordinates for regional parent spaces
Let \(G\) be finite, and let \(U,L\) be representations on finite coordinate sets \(X,Y\). Suppose \(Q:\mathbb {C}^X\to \mathbb {C}^Y\) satisfies \(Q^\dagger Q=I\) and \(L(g)=QU(g)Q^\dagger \). All graph edges retain their ordered tail–head orientation. The physical bond coordinate map is \(B(Q)=Q\otimes \overline Q\) in head–tail order, as in [ 235 , Section 7 ] . Every open contraction retains its full virtual boundary coefficient.
27.27.1 Regional coordinate changes
The identity \(L(g)=QU(g)Q^\dagger \) alone implies \(QQ^\dagger =I\). If also \(Q^\dagger Q=I\), then \(U(g)=Q^\dagger L(g)Q\).
Evaluate the first identity at \(g=1\), since \(U(1)=L(1)=I\). For the second, multiply it on the left by \(Q^\dagger \) and on the right by \(Q\), and cancel \(Q^\dagger Q\).
For a crossing edge \(f=(t,h)\) of a region \(R\), put \(Q_f=\overline Q\) when \(t\in R\) and \(Q_f=Q\) when \(h\in R\). On its physical crossing endpoints and internal head–tail pairs, define
The virtual boundary matrix is \(Q_{\partial R}=\bigotimes _{f\in \partial E_R}Q_f\).
For finite products with compatible index spaces, multiplying product matrices multiplies the matrices at each factor, and a product of identity matrices is the identity. In particular, \(B(P)B(Q)=B(PQ)\) and \(B(I)=I\). If \(QQ^\dagger =I\), then \(Q_R(Q^\dagger )_R=I\).
In matrix multiplication, the sum over an intermediate product configuration factors into the product of the individual sums. Conjugation commutes with multiplication, proving the bond formula. For a tail crossing the two factors are \(\overline Q\) and \(\overline{Q^\dagger }\), with product \(\overline{QQ^\dagger }=I\); at a head they are \(Q,Q^\dagger \). The internal product is \(B(QQ^\dagger )=I\), proving the regional cancellation identity.
Let \(a_R^U(\theta )\) and \(a_R^L(\eta )\) be the actual unweighted open averaging contractions in regional bond coordinates. If \(Q^\dagger Q=I\) and \(L(g)=QU(g)Q^\dagger \), then
Here \(c\) is an arbitrary virtual boundary coefficient vector. For each internal edge the individual factor is transformed by \(B(Q)\), using only \(L(g)=QU(g)Q^\dagger \). At each crossing the individual factor obeys the corresponding one-ended sum over \(\eta _f\) in (98).
For an internal edge, vectorization of \(L(q_hq_t^{-1})=QU(q_hq_t^{-1})Q^\dagger \) gives its factor identity. At a head crossing use \(L(g)Q=QU(g)\); at a tail crossing use \(Q^\dagger L(g)=U(g)Q^\dagger \), with \(g=q_t^{-1}\), and take coefficients in tail order. These identities leave the oriented virtual boundary matrix \(Q_f\). Expand the site averages and distribute the product of crossing sums to obtain (98). Summing against \(c(\theta )\) and interchanging finite sums proves (99).
Under the coordinate hypotheses above, \(Q_R\mathcal G_R(U)=\mathcal G_R(L)\) in regional bond coordinates, for every \(R\). In particular the forward range inclusion holds.
27.27.2 Exterior blocks and complete kernels
For fixed exterior crossing configurations \(\gamma _X,\gamma _Y\), define
The exterior endpoint has the complementary orientation to the retained regional endpoint.
Let \(\mathcal Q\) be \(\bigotimes _{f\in E}B(Q)\) in numbered site coordinates. For fixed input and output exterior configurations, its regional matrix block is the product of the scalar from wholly exterior bonds, the coefficient (102), and \(Q_R\) in numbered regional coordinates. The same factorization holds for the mixed product of its crossing and internal matrices before numbering. Hence every block maps \(\mathcal G_R(U)\) into \(\mathcal G_R(L)\), and \(\mathcal Q\mathcal P_{\mathcal R}(U) \subseteq \mathcal P_{\mathcal R}(L)\) for any region family.
Fixing one endpoint in \(B(Q)_{(y_h,y_t),(x_h,x_t)}=Q_{y_h,x_h}\overline{Q_{y_t,x_t}}\) extracts exactly the exterior scalar and leaves \(Q_f\) on the regional endpoint. Multiplying over edges gives the block formula. Theorem 27.27.1.5 gives each local range inclusion. Every output slice is a sum over exterior input configurations of these block images, so all parent constraints are preserved.
Suppose \(Q^\dagger Q=I\) and \(L(g)=QU(g)Q^\dagger \). For supplied incident enumerations and any family of regions,
Their dimensions agree. The global operation associated to \(Q^\dagger \) is its inverse on the entire physical space: \(QQ^\dagger =I\) suffices for the right-inverse identity, and \(Q^\dagger Q=I\) suffices for injectivity. For a finite region family, arbitrary positive local interactions whose kernels are the respective regional ranges satisfy \(\mathcal Q(\ker H)=\ker K\). No vertex or edge coverage assumption is needed.
Regroup the numbered physical coordinates by edges. Composition of the global products reduces to \(B(Q)B(Q^\dagger )=B(QQ^\dagger )=I\) at each edge. Reversing the order proves injectivity when \(Q^\dagger Q=I\). Theorem 27.27.1.1 supplies both cancellations and the representation identity for \(Q^\dagger \). Apply Theorem 27.27.2.2 in both directions and restrict the inverse operations to the parent spaces. Positivity identifies the Hamiltonian kernels with those spaces.
Let \(d_i{\gt}0\), and suppose an isometric matrix \(Q:\mathbb {C}^{\coprod _i\{ 0,\ldots ,d_i-1\} ^2}\to \mathbb {C}^G\) satisfies \(L(g)=QD^{[d]}(g)Q^\dagger \). Given incident enumerations at all three stages, define \(T=\mathcal Q M_d\). When every edge is internal to some parent region, its restriction is the composition of the copy-restoration and Fourier parent-space isomorphisms.
With the hypotheses of Definition 27.27.2.4, let \(A_d\) be the averaging tensor for \(D=\bigoplus _iD_i\) with weights \(d_i^{1/4}\), and let \(A_{\mathrm{reg}}\) be the unweighted averaging tensor for \(L\) in group coordinates. Then
For a finite region family and arbitrary positive local interactions with these exact regional kernels, \(T(\ker H)=\ker K\).
Copy restoration identifies the full weighted single-copy parent space with the unweighted \(D^{[d]}\) parent space. The Fourier isomorphism then identifies that entire space with the regular parent space. Compose the two maps, use preservation of dimension, and identify each positive Hamiltonian kernel with its parent space.
27.28 Semi-regular representations and physical parent kernels
Let \(G\) be finite. A semi-regular representation contains every irreducible representation of \(G\). The comparison below derives the irreducible blocks and their positive multiplicities from the original unitary representation, using the representation decomposition and bond transformations of [ 235 , Sections 4.1 and 7 ] . The graph is finite and simple, its vertices are ordered, and each edge \(f=(t,h)\) has the resulting tail–head orientation. Write \(\mathcal G_R(A)=\operatorname {im}\mathcal C_R(A)\) for the range of the open contraction with arbitrary virtual boundary coefficients, and \(\mathcal P_{\mathcal R}(A)\) for the common regional parent space.
27.28.1 Irreducible coordinates and the regular representation
Let \(U:G\to \operatorname {Mat}_X(\mathbb {C})\) be unitary, with \(X\) finite. There are a finite index set \(I\), positive integers \(d_i,m_i\), unitary irreducible representations \(D_i\) of dimensions \(d_i\) with pairwise distinct characters, and a matrix \(Q\) such that
If \(U\) is semi-regular, then \(D=\bigoplus _iD_i\) is unitary and semi-regular. The dimensions, multiplicities, and coordinate matrix are conclusions, rather than additional assumptions on \(U\).
Choose the orthonormal irreducible rows of Theorem 26.9.73. The matrices of the action on one row of each component give \(D_i\); the common action on its \(m_i\) rows gives \(D_i\otimes I_{m_i}\) after exchanging the row and matrix indices. The change between these orthonormal coordinates and the original basis is \(Q\), giving (105). The character multiplicity equals the number of rows. Apply Theorem 27.28.1.2 for semi-regularity; the direct sum of the unitary \(D_i\) is unitary.
Suppose \(U\) is semi-regular and \(U(g)=Q(\bigoplus _iD_i(g)\otimes I_{m_i})Q^\dagger \). Then \(\bigoplus _iD_i\) is semi-regular. This implication does not require \(Q\) to be isometric, or the \(D_i\) to be irreducible.
Semi-regularity is equivalent to linear independence of the group operators. If \(\sum _gc_gD(g)=0\), each diagonal block gives \(\sum _gc_gD_i(g)=0\). Tensoring with \(I_{m_i}\) and conjugating by \(Q\) yields \(\sum _gc_gU(g)=0\). Hence every \(c_g\) is zero.
For any finite family of representations \(D_i\) of dimensions \(d_i\), put \(D=\bigoplus _iD_i\) and \(D^{[d]}=\bigoplus _iD_i\otimes I_{d_i}\). Then \(\chi _D=\sum _i\chi _{D_i}\) and \(\chi _{D^{[d]}}=\sum _id_i\chi _{D_i}\). If each \(D_i\) is irreducible and \(D\) is semi-regular, the occurring irreducible characters are exactly \(\{ \chi _{D_i}:i\in I\} \). If these characters are also pairwise distinct, then
Traces add under direct sums and multiply under tensor products. Character orthogonality shows that an irreducible character absent from all the summands has zero multiplicity. Conversely, semi-regularity includes every irreducible character of the regular representation. Its character expansion is therefore \(\sum _id_i\chi _{D_i}\), with no repetitions. Evaluating at the identity gives the dimension formula.
For \(a,b\in \{ 0,\ldots ,d_i-1\} \) and \(g\in G\), define
For arbitrary \(D_i\), \(TL(g)=D^{[d]}(g)T\), where \(L\) is the left regular representation. If \(D\) is semi-regular, \(T\) is injective. If the \(D_i\) are also irreducible with pairwise distinct characters, \(T\) is invertible and \(D^{[d]}\) and \(L\) are equivalent: there are matrices \(S,R\) with \(RS=SR=I\) and \(L(g)=SD^{[d]}(g)R\). If the \(D_i\) are unitary, one may take \(Q=\widehat T^\dagger \), giving
Multiplicativity of \(D_i\) gives \(TL(g)=D^{[d]}(g)T\). If \(Tc=0\), all matrix coefficients of \(\sum _gc_gD(g)\) vanish; semi-regularity forces \(c=0\). The dimension identity in (109) makes \(T\) bijective. For unitary blocks, direct multiplication gives
Equal dimensions give the other inverse identity. The row weights commute with \(D^{[d]}\), so the intertwining identity also holds for \(\widehat T\), proving (111).
27.28.2 The oriented incident action
Let \(U:G\to \operatorname {Mat}_X(\mathbb {C})\) be a representation. On the incident coordinate space \(\mathcal V_v=\mathbb {C}^{X^{\operatorname {Inc}(v)}}\), put
Thus a head carries \(U(g)\) and a tail carries \(U(g^{-1})^{\mathsf T}\). A numbering \(\{ 0,\ldots ,|X|-1\} \cong X\) gives a coefficient-space isomorphism \(J_v\) from numbered virtual labels to \(\mathcal V_v\). Define the numbered action by \(\rho _v^{\mathrm{num}}=J_v^{-1}\rho _vJ_v\). A physical enumeration \(e_v:X^{\operatorname {Inc}(v)}\cong \{ 0,\ldots ,p-1\} \) similarly gives \(E_v:\mathcal V_v\cong \mathbb {C}^p\).
The matrices in (115) satisfy \(\rho _v(1)=I\) and \(\rho _v(gh)=\rho _v(g)\rho _v(h)\), and act on coefficient functions by matrix multiplication. The unweighted averaging site has local map \(P_v=|G|^{-1}\sum _g\rho _v(g)\). It is \(G\)-injective for \(\rho _v\). In numbered coordinates its local map is \(A_v=E_vP_vJ_v\) and is \(G\)-injective for \(\rho _v^{\mathrm{num}}\). No unitarity or semi-regularity assumption is required here.
Products of independent incident indices separate into products of matrix products. At a tail, inversion and transposition each reverse the order, giving multiplicativity. Expanding the averaging site yields \(P_v\). Averaging gives \(P_v\rho _v(g)=P_v\) and fixes every invariant vector, hence is injective on the invariant subspace. Conjugation by \(J_v\) transports invariance, and \(E_v\) is injective; therefore \(E_vP_vJ_v\) has the same two \(G\)-injectivity properties.
27.28.3 Equivalence of the complete parent spaces
If \(|\operatorname {Inc}(v)|=k\) at every vertex and \(X\) is finite, choose the finite-set enumeration \(e^X_v:X^{\operatorname {Inc}(v)}\cong \{ 0,\ldots ,|X|^k-1\} \). Write \(A(U,e)\) for the unweighted canonical averaging tensor in a supplied enumeration \(e\), and put \(A_{\mathrm{reg}}=A(L,e^G)\).
Suppose one unitary semi-regular representation \(U\) is used on all edges, \(|\operatorname {Inc}(v)|=k\) for every vertex, and every edge has both endpoints in some region \(R_i\). For any common physical enumeration \(e\) of the incident \(X\)-labels, there is a linear isomorphism
The dimensions are equal. For a finite region family, the kernels of any positive local parent Hamiltonians with these exact regional ranges are also linearly isomorphic.
Derive \(D_i,d_i,m_i,Q\) from Theorem 27.28.1.1. Transport canonical parents from \(U\) to \(D^{[m]}\) by \(Q^\dagger \). Theorem 27.26.3.4 changes the positive copy family \(m_i\) to \(d_i\). The normalized Fourier matrix then transports \(D^{[d]}\) to \(L\) by Theorem 27.27.2.3. Constant incidence supplies the intermediate common alphabets, and edge coverage supplies the supported inverse for the copy change. Compose these three isomorphisms. Positivity identifies the kernel of each parent Hamiltonian with its common regional parent space.
Let \(A_v:\mathcal V_v^{\mathrm{num}}\to \mathbb {C}^p\) be \(G\)-injective for \(\rho _v^{\mathrm{num}}\), and let \(e\) enumerate the canonical physical incident space, possibly with another dimension \(q\). If every vertex belongs to a region \(R_i\), then \(\mathcal P_{\mathcal R}(A)\cong \mathcal P_{\mathcal R}(A(U,e))\). Neither constant incidence nor semi-regularity is needed for this fixed-representation comparison, provided \(e\) is supplied.
The canonical averaging tensor is \(G\)-injective for the same incident action. The fixed-representation comparison supplies physical maps \(F_v\) that are injective on its site images and satisfy \(A_v=F_vA(U,e)_v\). Vertex coverage makes the restriction of \(\bigotimes _vF_v\) an isomorphism of the parent spaces. Take its inverse.
Let \(U:G\to \operatorname {Mat}_X(\mathbb {C})\) be one unitary semi-regular representation used uniformly on the edges of the finite ordered simple graph. Suppose \(|\operatorname {Inc}(v)|=k\) at every vertex and \(A_v:\mathcal V_v^{\mathrm{num}}\to \mathbb {C}^p\) is \(G\)-injective for the actual oriented action \(\rho _v^{\mathrm{num}}\). Assume that every vertex is in some \(R_i\), and that every edge has both endpoints in some \(R_i\). Then
If the region family is finite, let \(H_i,K_i\ge 0\) satisfy \(\ker H_i=\mathcal G_{R_i}(A)\) and \(\ker K_i=\mathcal G_{R_i}(A_{\mathrm{reg}})\). Their sums over the corresponding region placements obey \(\ker H\cong \ker K\). The physical dimension \(p\) need not equal \(|G|^k\).
Use the common incident enumeration \(e^X\) and compose
The first comparison uses vertex coverage; the second uses edge coverage, constant incidence, and semi-regularity. Dimensions are preserved by linear isomorphisms. For positive interactions, \(\langle \psi ,H\psi \rangle =0\) holds exactly when every regional term annihilates \(\psi \); hence \(\ker H=\mathcal P_{\mathcal R}(A)\) and likewise for \(K\).
27.29 Open contractions with native edge orientations
Let \(\Gamma =(V,E)\) be a finite simple graph with ordered vertices. Write an ordered edge as \(e=(t,h)\), with \(t{\lt}h\), and let \(o:E\to \{ 0,1\} \) specify its orientation: \(o_e=0\) means \(t\to h\) and \(o_e=1\) means \(h\to t\). The ordering of vertices is used only for coordinates. The physical pair on an edge is always ordered as \((x_h,x_t)\), even when the native arrow is reversed. For \(R\subseteq V\), denote its internal, crossing, and exterior edges by \(E_R\), \(\partial E_R\), and \(E_{R^c}\). For a region family \(\mathcal R\), write \(\mathcal P_{\mathcal R}(A)\) for the physical vectors whose every slice on each \(R\in \mathcal R\) belongs to the regional contraction range of \(A\). These conventions extend the bond transformations of [ 235 , Section 7 ] to the rightward and downward arrows of a torus.
27.29.1 Averaging tensors and unrestricted regional boundaries
An incidence \((v,e)\) is a native tail when \(v=t\) and \(o_e=0\), or when \(v=h\) and \(o_e=1\). Thus the ordered first incidence is a native tail exactly when \(o_e=0\), and the ordered second incidence exactly when \(o_e=1\).
Let \(G\) be finite, \(U:G\to \mathrm{GL}(\mathbb {C}^X)\) a representation, and \(W\) a matrix on \(\mathbb {C}^X\). For virtual and physical incidence labels \(\eta ,\sigma :\operatorname {Inc}(v)\to X\), set
A bijection \(e_v:X^{\operatorname {Inc}(v)}\to \{ 0,\ldots ,p-1\} \) numbers the physical indices. Its coefficient at \(s\) is \(A^{U,W,o}_v(\eta ,e_v^{-1}(s))\). Fix any numbering of \(X\) on the virtual bonds. If every \(o_e=0\), this is the ordered dressed averaging tensor. No symmetry of \(W\) is required. The normalization is the group-average normalization of [ 96 ] .
Fix \(q:R\to G\). On an internal edge \(e=(t,h)\) define
For a crossing edge, let \(v\) be its retained vertex, \(\theta \) its virtual boundary label, and \(x\) its retained physical label. Put \(B_e(q;\theta ,x)=(WU_{q_v^{-1}})_{\theta ,x}\) when \(v\) is the native tail, and \(B_e(q;\theta ,x)=(U_{q_v}W)_{x,\theta }\) otherwise. For arbitrary \(\theta \in X^{\partial E_R}\), define
Here \(z_e=(x_h,x_t)\) and \(x\) lists the retained crossing endpoints. Setting \(o=0\) recovers each ordered boundary factor, internal factor, open vector, and open span.
Assume \(WU_g=U_gW\) for every \(g\). Let \(\mathcal C_R\) be the open contraction of the tensors (121), and let \(J_R\) regroup its physical incidences into crossing labels and ordered internal pairs. Then
These identities hold before or after numbering physical and virtual labels. In particular, \(\psi \in \operatorname {im}\mathcal C_R\) exactly when \(J_R\psi \in \mathcal S_R(U,W,o)\). The boundary coefficients may be arbitrary linear combinations of all \(\theta \), including entangled boundary vectors.
Expand the normalized group average separately at every vertex, obtaining the factor \(|G|^{-|R|}\) and the sum over \(q:R\to G\). Regroup the incidence product by internal and crossing edges. Summation over one internal virtual index gives either \((U_{q_h}W)(WU_{q_t^{-1}})\) or \((U_{q_t}W)(WU_{q_h^{-1}})\) with the physical indices interchanged. Commutation with \(W\) gives (124). Each crossing edge retains its original \(\theta _e\), giving (125); commuting the scalar products also gives the internal-first form of the same sum. The bijections of incidence and numbered labels preserve every finite sum. Finally, the regional range is the span of all its boundary columns, proving (128).
On \(\mathbb {C}^{X^{\operatorname {Inc}(v)}}\), define
Matrix multiplication gives \(\rho _v^o(1)=1\) and \(\rho _v^o(gh)=\rho _v^o(g)\rho _v^o(h)\). At a tail this follows from \((U_{(gh)^{-1}})^T=(U_{g^{-1}})^T(U_{h^{-1}})^T\). Thus these matrices act as a representation by multiplication on coefficient vectors. If \(E_v\) changes numbered virtual coefficients to incidence coefficients, the numbered representation is \(E_v^{-1}\rho _v^o(g)E_v\).
The local coefficient map of \(A^{U,1,o}_v\) is \(\Pi _v=|G|^{-1}\sum _g\rho _v^o(g)\), and is \(G\)-injective. With physical numbering \(P_v\) and virtual numbering \(E_v\), its local map is \(P_v\Pi _v E_v\) and is \(G\)-injective for \(E_v^{-1}\rho _v^o E_v\). This assertion needs neither unitarity nor semi-regularity of \(U\).
27.29.2 Internal support and regional blocks
Let \(F:\mathbb {C}^Z\to \mathbb {C}^{X\times X}\). Assume both coefficient vectors of \(W^2U_g\) and \((W^2U_g)^T\) belong to \(\operatorname {im}F\) for every \(g\). Then each \(I_e(q)\) belongs to \(\operatorname {im}F\). Assume also \(WU_g=U_gW\). Fixing the physical crossing labels \(x\) in an open vector leaves
This holds for every vector in the regional range after taking its internal physical slice. If a region family \(\mathcal R\) contains both endpoints of every edge in some member, every vector in its parent space \(\mathcal P_{\mathcal R}(A^{U,W,o})\) lies, after regrouping, in \(\operatorname {im}F^{\otimes E}\).
Each orientation selects one of the two assumed matrix orders. Choose a preimage under \(F\) for every \(I_e(q)\) and take their product, with the scalar coefficient in (132). Its image under \(F^{\otimes E_R}\) is the displayed sum. Span closure proves the assertion for every regional vector. For a global parent vector, choose a region containing the endpoints of a prescribed edge. Every one-edge slice then belongs to \(\operatorname {im}F\). The product-range criterion gives the full \(E\)-fold range.
For \(F_e:\mathbb {C}^{X\times X}\to \mathbb {C}^{Y\times Y}\), define \(M(F)=\bigotimes _{e\in E}F_e\), expressed in the numbered site coordinates through the incidence-to-bond bijections. Thus \(M(F)_{\tau ,\sigma }=\prod _eF_e((\tau _h,\tau _t),(\sigma _h,\sigma _t))\) in bond coordinates. Fix exterior endpoint labels \(\gamma _Y,\gamma _X\). On a crossing edge put
For fixed exterior physical configurations \(\tau ,\sigma \), let \(c_R(F;\tau ,\sigma )\) be the product of their matrix coefficients on all \(e\in E_{R^c}\).
The block obtained by fixing exterior output \(\tau \) and input \(\sigma \) satisfies
where \(\gamma _Y,\gamma _X\) are the crossing endpoint labels determined by \(\tau ,\sigma \), and \(J_{X,R},J_{Y,R}\) regroup regional coordinates.
With compatible finite endpoint alphabets, \(M(F)^\dagger =M(F^\dagger )\), \(M(F)M(L)=M((F_eL_e)_e)\), and \(M((1)_e)=1\). In particular, \(F_eL_e=1\) on every edge implies \(M(F)M(L)=1\).
Conjugation distributes over each coefficient product. For composition, the sum over the intermediate site configuration is, under the bond-coordinate bijection, a sum over independent intermediate edge labels. It factors into the products \(F_eL_e\). The product of edgewise Kronecker deltas is the delta of the complete configuration, proving the identity assertion.
27.30 Coordinate changes with native edge orientations
Continue with the incidence and bond coordinates of Section 27.29. The matrix on a reversed edge is the entrywise conjugate of the matrix on an unreversed edge. This conjugation follows from exchanging the two physical endpoints; it does not require real representation matrices.
For a matrix \(Q:\mathbb {C}^X\to \mathbb {C}^Y\), put \(T(Q)=Q\otimes \overline Q\) on ordered head–tail pairs, and let \(S_X,S_Y\) exchange their two coordinates. Then \(T(\overline Q)=S_YT(Q)S_X\) and, for every matrix \(M\) on \(\mathbb {C}^X\),
Here \(\operatorname {vec}(M)(a,b)=M_{a,b}\).
The coefficient of \(T(Q)\) is \(Q_{a,c}\overline{Q_{b,d}}\). Exchanging both endpoint pairs gives \(\overline{Q_{a,c}}Q_{b,d}\), which is \(T(\overline Q)_{(a,b),(c,d)}\). Summing this coefficient against \(M_{d,c}\) gives \((QMQ^\dagger )_{b,a}\).
Set \(Q_e=Q\) if \(o_e=0\) and \(Q_e=\overline Q\) if \(o_e=1\). The internal bond matrix is \(T_e(Q)=Q_e\otimes \overline{Q_e}\). On a crossing edge \(e=(t,h)\), set \(B_{e,R}(Q)=\overline{Q_e}\) when \(t\in R\), and \(B_{e,R}(Q)=Q_e\) when \(h\in R\). Define
The corresponding global matrix in numbered site coordinates is \(M_o(Q)=M((T_e(Q))_e)\).
If \(AB=1\), then \(\overline A\, \overline B=1\). Consequently \(QQ^\dagger =1\) implies \(T_R^o(Q)T_R^o(Q^\dagger )=1\) for every region and orientation.
Take entrywise conjugates of the finite sums defining \(AB\). Apply the resulting identity to each \(Q_e\), its conjugate on a retained tail, and the two factors of every internal pair. Multiplication of the independent boundary and internal tensor factors proves the regional assertion.
Suppose \(Q^\dagger Q=1\) and \(U,L\) are representations satisfying \(L_g=QU_gQ^\dagger \). For \(W=1\), internal factors obey \(T_e(Q)I_e^U(q)=I_e^L(q)\), while crossing factors obey
The internal identity only uses \(L_g=QU_gQ^\dagger \). Put \(K_R(Q)_{a,\theta } =\prod _{e\in \partial E_R}B_{e,R}(Q)_{a_e,\theta _e}\). Then
At a native head, use \(L_gQ=QU_g\); at a native tail use \(Q^\dagger L_g=U_gQ^\dagger \), with conjugates as specified by \(B_{e,R}(Q)\). This gives (139). Internal factors follow by conjugating the relative representation matrix, using (137) on reversed edges. Apply these identities inside (125) and factor the independent boundary sums to obtain (140). Linearity gives (141) and the forward range inclusion. At \(g=1\), the representation identity gives \(1=L_1=QQ^\dagger \). The reverse coordinate change \(Q^\dagger \) therefore satisfies the same hypotheses, and cancellation proves (142).
For fixed crossing exterior labels \(\gamma _Y,\gamma _X\), put
The mixed product of crossing blocks and internal coordinate matrices is \(b_R(Q;\gamma _Y,\gamma _X)T_R^o(Q)\). Under the hypotheses of Theorem 27.30.4, every exterior block of \(M_o(Q)\) sends the regional range for \(A^{U,1,o}\) into that for \(A^{L,1,o}\). Thus, for every region family \(\mathcal R\),
No coverage condition is needed for this inclusion.
A crossing coefficient of \(Q_e\otimes \overline{Q_e}\) factors into its fixed exterior entry and its retained endpoint entry. Taking their product gives (145). By (136), each global block is \(c_Rb_RJ_{Y,R}^{-1}T_R^o(Q)J_{X,R}\). Equations (128) and (142) show that this block preserves the required regional range. Each output slice of a global vector is a sum of such blocks applied to its input slices, proving (146).
If \(QQ^\dagger =1\), then \(M_o(Q)M_o(Q^\dagger )=1\) on the full target physical space. If \(Q^\dagger Q=1\), then \(M_o(Q)\) is injective on the full source physical space.
Each edge factor and its inverse cancel. The conjugated factors on reversed edges cancel as well. Their global product cancels by Theorem 27.29.2.4. Applying the same identity to \(Q^\dagger \) supplies a left inverse under \(Q^\dagger Q=1\) and hence injectivity.
If \(Q^\dagger Q=1\) and \(L_g=QU_gQ^\dagger \) for representations \(U,L\), then for every \(\mathcal R\),
These parent spaces have equal dimension. For a finite region family, let \(H_U,H_L\) be sums of positive regional interactions whose kernels are exactly the respective regional contraction ranges. Then \(M_o(Q)\ker H_U=\ker H_L\).
Evaluating the intertwining identity at \(1\) gives \(QQ^\dagger =1\). Apply the block inclusion in both directions and cancel the global coordinate matrices. The restriction of \(M_o(Q)\) is then bijective, which gives equal dimension. Positivity identifies each Hamiltonian kernel with the intersection of its regional parent conditions, giving the kernel assertion.
Under the hypotheses of Theorem 27.30.8, define \(E_Q^o:\mathcal P_{\mathcal R}(A^{U,1,o}) \simeq \mathcal P_{\mathcal R}(A^{L,1,o})\) to be the restriction of \(M_o(Q)\). Its inverse is the restriction of \(M_o(Q^\dagger )\).
27.31 Oriented multiplicity and physical transport
Let \(D=\bigoplus _iD_i\) and \(D^{(m)}=\bigoplus _i(D_i\otimes 1_{m_i})\), where the \(D_i\) are representations on \(\mathbb {C}^{d_i}\) and \(m_i{\gt}0\). The multiplicities \(m_i\) need not equal \(d_i\). Put \(W_m=\bigoplus _i m_i^{1/4}1_{d_i}\), \(A_m=A^{D,W_m,o}\), and \(B_m=A^{D^{(m)},1,o}\). The single-copy and repeated endpoint alphabets are \(X=\coprod _i\{ 0,\ldots ,d_i-1\} \) and \(Y=\coprod _i(\{ 0,\ldots ,d_i-1\} \times \{ 0,\ldots ,m_i-1\} )\). Write \(\pi (i,a,u)=(i,a)\) and \(\alpha (y,z)=m_i^{-1/2}\) when \(y=(i,a,u),z=(i,b,u)\) share a sector and copy, and zero otherwise. The bond map \(F_m\) sends a matching-sector pair \((i,a;i,b)\) to \(m_i^{-1/2}\sum _u(i,a,u;i,b,u)\) and vanishes on different-sector pairs. These are the copy maps of [ 235 , Section 7 ] , with arbitrary positive multiplicities.
27.31.1 Copy maps on open oriented regions
For any matrices \(M_i\), set \(M=\bigoplus _i\sqrt{m_i}M_i\) and \(N=\bigoplus _i(M_i\otimes 1_{m_i})\). Then \(F_m\operatorname {vec}(M^T)=\operatorname {vec}(N^T)\) and \(F_m^\dagger \operatorname {vec}(N^T)=\operatorname {vec}(M^T)\). More generally, the same statements hold for arbitrary finite nonempty copy-label sets of cardinality \(m_i\).
Apply the ordinary restoration and adjoint formulas to \(M_i^T\). Transposition commutes with block sums and scalar multiplication, and \((M_i\otimes 1_{m_i})^T=M_i^T\otimes 1_{m_i}\).
Write \(I_e^A,I_e^B\) and \(B_e^A,B_e^B\) for the internal and crossing factors of \(A_m,B_m\). On an internal edge, \(F_mI_e^A(q)=I_e^B(q)\) and \(F_m^\dagger I_e^B(q)=I_e^A(q)\). Let \(B_z(y,x)=\alpha (y,z)\delta _{\pi y,x}\) be the one-ended copy matrix for fixed exterior label \(z\in Y\), and put \(w_y=m_i^{1/4}\) for \(y\) in sector \(i\). On a crossing edge,
These formulas hold for either native orientation and either retained endpoint. For exterior crossing labels \(\gamma \), set \(F_{m,R}(\gamma )=\bigotimes _{e\in \partial E_R}B_{\gamma _e} \otimes \bigotimes _{e\in E_R}F_m\). The complete open vectors satisfy
Since \(W_m^2D_g=\bigoplus _i\sqrt{m_i}D_i(g)\), the ordinary bond identities give the internal claim on unreversed edges; Theorem 27.31.1.1 gives it on reversed edges. On a crossing edge the sparse coefficient \(\alpha (y,z)\delta _{\pi y,x}\) retains exactly one factor \(w_y\). Taking its adjoint gives the factor \(w_y^{-1}\alpha (y,z)\), which proves (148) and (149). Substitute into (125) and distribute the independent boundary sums. This gives (150) and (151).
For every \(R,\gamma \), \(F_{m,R}(\gamma )\mathcal S_R(D,W_m,o) \subseteq \mathcal S_R(D^{(m)},1,o)\) and \(F_{m,R}(\gamma )^\dagger \mathcal S_R(D^{(m)},1,o) \subseteq \mathcal S_R(D,W_m,o)\). The corresponding numbered matrices give the same inclusions between the actual regional contraction ranges of \(A_m\) and \(B_m\).
Let \(M_m=M((F_m)_e)\). For any region family \(\mathcal R\), \(M_m\mathcal P_{\mathcal R}(A_m)\subseteq \mathcal P_{\mathcal R}(B_m)\) and \(M_m^\dagger \mathcal P_{\mathcal R}(B_m) \subseteq \mathcal P_{\mathcal R}(A_m)\). If each edge is internal to some member of \(\mathcal R\), then
In particular, \(M_m\) is injective on the source parent space.
Each regional block of \(M_m\) is a scalar times the numbered \(F_{m,R}(\gamma )\); the adjoint block has the conjugate scalar. The regional inclusions therefore give the two global inclusions. The source factors \(\bigoplus _i\sqrt{m_i}D_i(g)\) and their transposes lie in the matching-sector subspace. The target factors \(\bigoplus _i(D_i(g)\otimes 1_{m_i})\) and their transposes lie in \(\operatorname {im}F_m\). Theorem 27.29.2.1 derives these product supports for every parent vector from edge coverage. On the source support \(F_m^\dagger F_m\) is the identity; on the target support \(F_mF_m^\dagger \) is the identity. Taking products proves (152) and (153).
If \(m_i{\gt}0\) and \(\mathcal R\) contains both endpoints of every edge in some member, then \(M_m\mathcal P_{\mathcal R}(A_m)=\mathcal P_{\mathcal R}(B_m)\) and these spaces have equal dimension. For a finite region family and positive parent Hamiltonians \(H_A,H_B\) with the respective genuine regional kernels, \(M_m\ker H_A=\ker H_B\).
For \(\phi \in \mathcal P_{\mathcal R}(B_m)\), its preimage \(M_m^\dagger \phi \) belongs to the source parent space and satisfies \(M_mM_m^\dagger \phi =\phi \). The source identity proves injectivity. The kernel statement follows from positivity and the exact local kernel conditions.
Define \(E_m^o:\mathcal P_{\mathcal R}(A_m) \simeq \mathcal P_{\mathcal R}(B_m)\) as the restriction of \(M_m\) under the hypotheses of Theorem 27.31.1.5. Its inverse is \(M_m^\dagger \) restricted to the target parent space.
27.31.2 Removing weights and comparing arbitrary physical tensors
For a physical incidence configuration \(\sigma \), let \(i(\sigma _e)\) be its sector and put \(S_v(\sigma )=\prod _{e\ni v}m_{i(\sigma _e)}^{1/4}\). Then \(A_m(\eta ,\sigma )=S_v(\sigma )A^{D,1,o}_v(\eta ,\sigma )\). This coefficient identity also holds for zero multiplicities. If every \(m_i{\gt}0\), the diagonal physical maps \(S_v\) are invertible, the numbered tensors satisfy \(A_m=S\cdot A^{D,1,o}\), and for every region family
Each \(D_i(g)\) preserves its sector. A diagonal factor \(m_i^{1/4}\) can therefore be moved to the physical index at either end of an incidence, regardless of \(o_e\). Factor the product of these scalars out of the group average. Positive multiplicities make every diagonal entry nonzero. Invertible physical transport gives (154).
Restricting \(\bigotimes _vS_v\) defines an equivalence \(S_m^o\) from \(\mathcal P_{\mathcal R}(A^{D,1,o})\) to \(\mathcal P_{\mathcal R}(A_m)\) for every region family. With edge coverage, put \(L_m^o=E_m^oS_m^o\). This is an equivalence from the unweighted single-copy parent space to \(\mathcal P_{\mathcal R}(A^{D^{(m)},1,o})\). For two positive multiplicities \(m,n\), the equivalence between their parent spaces is \(L_n^o(L_m^o)^{-1}\).
Let \(U\) be an arbitrary unitary semi-regular representation of a finite group, used on every edge. Suppose every vertex has degree \(k\), and every edge is internal to some region of \(\mathcal R\). Then the oriented canonical parent spaces for \(U\) and the left regular representation \(L\) are linearly equivalent:
In particular they have equal dimension. For a finite region family, any positive parent Hamiltonians with these exact local contraction kernels have linearly equivalent full kernels.
The decomposition theorem supplies inequivalent irreducibles \(D_i\), dimensions \(d_i{\gt}0\), actual multiplicities \(m_i{\gt}0\), and coordinates \(Q\) such that \(U_g=QD^{(m)}_gQ^\dagger \), \(Q^\dagger Q=1\). Semi-regularity supplies every irreducible sector, so normalized Fourier coordinates give \(L_g=FD^{(d)}_gF^\dagger \) with \(F^\dagger F=1\). Constant degree supplies physical numberings of all intermediate incidence alphabets. The equivalence is the composition
Here all parent spaces use \(\mathcal R\). Dimensions are preserved by each equivalence; the positive-parent common-kernel theorem gives the Hamiltonian assertion.
Let \(a_v\) be vertex-dependent physical tensors which are \(G\)-injective for \(\rho _v^o\), expressed in their virtual numbering. If every vertex belongs to some region of \(\mathcal R\), then \(\mathcal P_{\mathcal R}(a)\simeq \mathcal P_{\mathcal R}(A^{U,1,o})\). The physical dimensions of \(a\) and the averaging tensor may differ. Neither semi-regularity nor unitarity of \(U\) is required here.
The averaging tensor is \(G\)-injective by Theorem 27.29.1.6. The two local maps have the same kernel, namely the complement of the invariant subspace killed by the group average. The fixed-representation physical comparison constructs local maps \(F_v\) with \(a_v=F_vA^{U,1,o}_v\), injective on the local image. Vertex coverage makes their product injective on the parent space and identifies its image with \(\mathcal P_{\mathcal R}(a)\). Invert that equivalence.
Suppose \(U\) is unitary and semi-regular, the graph has constant degree \(k\), and \(a_v\) is \(G\)-injective for its oriented incident representation at every vertex. Suppose \(\mathcal R\) covers all vertices and contains both endpoints of every edge in some member. Then \(\mathcal P_{\mathcal R}(a)\simeq \mathcal P_{\mathcal R}(A^{L,1,o})\), with equal dimensions. For a finite region family, arbitrary positive regional interactions with these genuine local kernels have equivalent full Hamiltonian kernels. The tensors \(a_v\) may vary with \(v\); the representation \(U\) is uniform on the edges.
Use vertex coverage to compare \(a\) with \(A^{U,1,o}\), then use edge coverage and (155) to compare the latter with \(A^{L,1,o}\). Compose the equivalences. The dimension and positive-parent kernel assertions follow. The positive filters \(S_v\) need not be unitary, so this assertion concerns ground spaces and kernels, without a spectral comparison.
27.32 Full native semi-regular torus parent spaces
Let \(V=\mathbb Z/m\mathbb Z\times \mathbb Z/n\mathbb Z\), with \(m,n\ge 3\), and join nearest neighbors to form the native simple-graph torus. Each site has legs ordered as top, right, down, left. The native arrows point right and down, so top and left are incoming. Throughout this section one matrix representation \(U\) is used on all edges and one tensor \(a\) is repeated at every vertex. The final dimension and spanning results are the corresponding uniform native-torus statements of [ 235 , Theorems 5.7 and 5.9 ] . The restrictions to a uniform representation, a repeated tensor, and periods at least three are retained; the link-dependent and smaller-period cases are discussed in [ 96 , 173 ] . The oriented graph comparison in Theorem 27.31.2.5, in contrast, allows vertex-dependent physical tensors.
27.32.1 Native geometry and the local invariant subspace
For the horizontal edge from \(v=(x,y)\) to \((x+1,y)\) the native tail is \(v\). For the vertical edge joining \(v\) to \((x,y+1)\), the native tail is \((x,y+1)\), including wraparound edges. If \(e=(t,h)\) is its ordered-edge description, define \(o_e=1\) exactly when \(t\) is not this native tail.
Every native tail is an endpoint of its edge. Every vertex has four distinct incident edges, bijectively indexed by \(0,1,2,3\) in top, right, down, left order. At \(v\), the native-tail incidences are precisely indices \(1,2\). Equivalently, the graph native-tail predicate for \(o\) holds exactly when the chosen torus tail equals \(v\).
The horizontal and vertical edge bijection identifies the selected tails with actual endpoints. The four-leg incidence bijection gives degree four. Evaluating the tail on its four legs gives \(v\) on right and down; on top and left it gives a neighboring vertex, different from \(v\). The definition of \(o_e\) then identifies this test with the ordered-graph native-tail predicate.
The bijection \(j_v:X^{\operatorname {Inc}(v)}\to X^4\) sends \(\eta \) to \((\eta _{\mathrm{top}},\eta _{\mathrm{right}}, \eta _{\mathrm{down}},\eta _{\mathrm{left}})\). A native tensor \(a:X^4\times \{ 0,\ldots ,d-1\} \to \mathbb {C}\) defines the graph site \(a_v(\eta ,s)=a(j_v\eta ,s)\). Its native linear map is \(\mathcal A x(s)=\sum _{\xi \in X^4}a(\xi ,s)x(\xi )\).
Define the native four-leg representation by
The site coefficient \(a(\eta _t,\eta _r,\eta _b,\eta _l;s)\) has the following native leg orientation; the transverse physical index is \(s\).
![\begin{tenkz}[rows={wire,wire,wire}, cols=3, bonds=none, frame=plane]
\tn[at=(2,2), name=nativeSite, physical=up,
ports={90:virtual,0:virtual,270:virtual,180:virtual},
label pos=sw]{$a$}
\tnwire[dir=from, name=nativeTop]{nativeSite.90}{open n}
\tnwire[dir=to, name=nativeRight]{nativeSite.0}{open e}
\tnwire[dir=to, name=nativeDown]{nativeSite.270}{open s}
\tnwire[dir=from, name=nativeLeft]{nativeSite.180}{open w}
\tnmark[form=label, label pos=n]{on nativeTop 1}{$\eta_t$}
\tnmark[form=label, label pos=e]{on nativeRight 1}{$\eta_r$}
\tnmark[form=label, label pos=s]{on nativeDown 1}{$\eta_b$}
\tnmark[form=label, label pos=w]{on nativeLeft 1}{$\eta_l$}
\tnmark[form=label, label pos=w]{on leg-n-2-2 1}{$s$}
\end{tenkz}](tenkz_svg/tenkz-9171420522c321b5.svg)
Then \(\rho _v^o(g)_{\sigma ,\eta } =\rho _\square (g)_{j_v\sigma ,j_v\eta }\). For coefficient vectors, let \(J_vx=x\circ j_v^{-1}\). The corresponding identities are \(J_v\rho _v^o(g)=\rho _\square (g)J_v\) and \(\mathcal A_v=\mathcal A J_v\). Hence \(G\)-injectivity of \(\mathcal A\) for \(\rho _\square \) implies \(G\)-injectivity of every \(\mathcal A_v\) for \(\rho _v^o\). In numbered virtual coordinates the local map is \(\mathcal A_vE_v\) and is \(G\)-injective for \(E_v^{-1}\rho _v^oE_v\).
The outgoing indices \(1,2\) contribute \(U_{g^{-1}}^T\); indices \(0,3\) contribute \(U_g\). Reindexing the four incidence factors gives (157). Reindexing the finite coefficient sum by \(j_v\) gives both intertwining identities. Since \(J_v\) is invertible, invariance and injectivity on invariant vectors pass from \(\mathcal A\) to \(\mathcal A_v\). Reindex once more by \(E_v\) to obtain the numbered assertion.
For every finite group representation \(U\) on a finite alphabet \(X\), there are a finite physical dimension \(p\) and a native tensor \(b:X^4\times \{ 0,\ldots ,p-1\} \to \mathbb {C}\) which is \(G\)-injective for \(\rho _\square \).
Let \(\Pi =|G|^{-1}\sum _g\rho _\square (g)\) and enumerate the coefficient basis of \(\mathbb {C}^{X^4}\) by a linear isomorphism \(E\) to \(\mathbb {C}^p\), with \(p=|X|^4\). Take \(b\) to be the matrix entries of \(E\Pi \). The average satisfies \(\Pi \rho _\square (g)=\Pi \) and is the identity on invariant vectors. Composing with \(E\) therefore gives the required invariance and restricted injectivity.
27.32.2 Commuting closures satisfy the original plaquette constraints
On any finite simple graph, let \(a_v\) be arbitrary site tensors with finite virtual alphabet \(X\), and insert a matrix \(K_e\) on each ordered bond. Suppose \(K_e=1\) whenever both endpoints lie in \(R\). For every fixed exterior physical configuration \(\tau \), the resulting vector \(\Psi _K(\, \cdot \, ,\tau )\) on \(R\) belongs to the original regional contraction range \(\operatorname {im}\mathcal C_R\). No invertibility, group symmetry, or positivity of the inserted matrices is required.
Split endpoint assignments into internal labels and pairs \(\zeta \) on all noninternal edges. The identity matrices on internal edges force their two endpoint labels to agree. For fixed \(\zeta \), the remaining coefficient is
Here \(\theta (\zeta )\) selects the retained endpoint on every crossing edge. Each summand is a boundary column of the original regional map, so the sum belongs to its range.
Let \(O^h_v\) act on the native rightward bond from \(v\), and \(O^v_v\) on the downward bond from \((x,y+1)\) to \(v=(x,y)\). For an ordered edge \(e=(t,h)\) put
Thus a transpose is inserted exactly when the native arrow and ordered-edge arrow disagree.
Reindex all native top/right/down/left labels by the incidence bijections \(j_v\), then regroup them into ordered endpoint pairs. The horizontal and vertical bond coefficients become, respectively,
Equations (162) and (165) give exactly these entries in both possible vertex orders. The site factors are unchanged, so the reindexed finite sums agree.
For \(v=(x,y)\) let \(R_v=v+\{ (0,0),(1,0),(1,1),(0,1)\} \). Place the horizontal closure seam at \(x+2\) and the vertical closure seam at \(y+2\). For any \(g,h\in G\), the corresponding inserted graph matrices are the identity on every edge internal to \(R_v\). This assertion includes \(m=3\) or \(n=3\).
The first coordinates in \(R_v\) are \(x,x+1\), whereas \(x+2\) differs from both modulo \(m\ge 3\). The analogous statement holds for \(y,y+1,y+2\) modulo \(n\). An internal rightward edge cannot end at the horizontal seam, and an internal upward edge cannot end at the vertical seam. Their insertion matrices are therefore \(1\); any transpose required by ordered coordinates leaves \(1\) unchanged.
Let \(U\) be any representation on finite \(X\), and suppose \(\mathcal A\rho _\square (k)=\mathcal A\) for every \(k\in G\). If \(gh=hg\), every physical slice of the native closure \(\lvert a;(g,h)\rangle \) on \(R_v\) lies in \(\operatorname {im}\mathcal C_{R_v}\). Consequently the closure belongs to the full plaquette parent space. If \(P_v\ge 0\) and \(\ker P_v=\operatorname {im}\mathcal C_{R_v}\), then every embedded \(P_v\) annihilates it, and so does \(H=\sum _v(P_v\otimes 1_{R_v^c})\). In particular,
This implication uses only local virtual invariance, without semi-regularity or \(G\)-injectivity.
Local invariance and \(gh=hg\) move both closure seams to \(x+2,y+2\) without changing the vector. Theorem 27.32.2.4 makes every internal insertion on \(R_v\) the identity. After the native-to-ordered contraction identity, formula (159) expresses each physical slice as a sum of unrestricted regional boundary columns. For the four-site plaquette, collecting equal boundary labels gives the contraction below. All four sites carry \(a\).
![\begin{tenkz}[lattice={2x2}, frame=plane, physical=up] % chktex 29
\tn[at=(1,1), name=nativePlaquette11,
ports={180:virtual:$\theta_{l,1}$,0:virtual,
90:virtual:$\theta_{t,0}$,270:virtual}, label pos=sw]{}
\tn[at=(1,2), name=nativePlaquette12,
ports={180:virtual,0:virtual:$\theta_{r,1}$,
90:virtual:$\theta_{t,1}$,270:virtual}, label pos=sw]{}
\tn[at=(2,1), name=nativePlaquette21,
ports={180:virtual:$\theta_{l,0}$,0:virtual,
90:virtual,270:virtual:$\theta_{b,0}$}, label pos=sw]{}
\tn[at=(2,2), name=nativePlaquette22,
ports={180:virtual,0:virtual:$\theta_{r,0}$,
90:virtual,270:virtual:$\theta_{b,1}$}, label pos=sw]{}
\end{tenkz}](tenkz_svg/tenkz-0efc093fdc997327.svg)
The coefficients \(c_\tau \) include the exterior contraction and every crossing insertion. Each local term therefore annihilates every closure slice. Summing the terms gives \(H\lvert a;(g,h)\rangle =0\); linearity proves (167).
27.32.3 Dimension and exhaustion of the full kernel
Let \(G\) be finite, let \(U\) be one unitary semi-regular representation on finite \(X\), and let \(a\) be one \(G\)-injective native tensor for \(\rho _\square \), repeated at every vertex of a torus with \(m,n\ge 3\). For each native plaquette \(R_v\), let \(P_v\ge 0\) have kernel \(\operatorname {im}\mathcal C_{R_v}\) and set \(H=\sum _v(P_v\otimes 1_{R_v^c})\). If \(\mathcal K_G=\{ (g,h)\in G^2:gh=hg\} /G\), with simultaneous conjugation, then
The block decomposition and its positive multiplicities are consequences of \(U\); no expansion of a parent vector into closures is assumed. This is the stated uniform-torus case of [ 235 , Theorem 5.9 ] .
Native incidence coordinates make every graph tensor \(G\)-injective for the actual rightward/downward orientation. The torus has degree four. Every vertex belongs to its plaquette, and every edge has both endpoints in a native plaquette. Theorem 27.31.2.5 therefore identifies the entire parent space of \(a\) with that of the oriented regular averaging tensor. Independently construct a native \(G\)-injective site \(b\) for the regular representation using Theorem 27.32.1.5. Fixed-representation comparison identifies its parent space with the same oriented regular canonical space. The already established regular native-torus theorem gives dimension \(|\mathcal K_G|\) for \(b\). Thus
Finally, positivity and the genuine local kernel conditions give \(\ker H=\mathcal P_{\{ R_v\} }(a)\).
Under the hypotheses of Theorem 27.32.3.1,
Thus the actual commuting closure vectors exhaust the entire positive plaquette-parent kernel for one repeated \(G\)-injective tensor and one uniform unitary semi-regular representation on native tori of periods at least three. This is the corresponding case of [ 235 , Theorems 5.7 and 5.9 ] .
Let \(\mathcal V\) be the span on the right of (170). Invariance gives \(\mathcal V\subseteq \ker H\) by (167). Closures in the same simultaneous-conjugacy class coincide, and semi-regular \(G\)-injectivity makes the vectors from distinct commuting classes linearly independent. Hence \(\dim \mathcal V=|\mathcal K_G|\) by Theorem 26.1.6.10. Equation (168) gives \(\dim \ker H=|\mathcal K_G|\). Inclusion and equal finite dimension imply \(\mathcal V=\ker H\). Taking canonical positive plaquette interactions identifies the same span with the full intersection of regional range conditions.
27.32.4 Labelled bonds and the four cuts
Let \(w,\ell \ge 0\) and \(\Lambda =\mathbb Z_w\times \mathbb Z_\ell \). A bond is a pair \(e=(v,d)\) with \(d\in \{ \mathrm H,\mathrm V\} \). Its tail \(s(e)\) and head \(t(e)\) are
The direction is part of the bond label. At period two, the two bonds connecting the same vertices are distinct; at period one, the head and tail incidences of a loop are distinct.
Suppose \(w,\ell {\gt}0\). For \(c\in \mathbb Z_w\) and \(r\in \mathbb Z_\ell \), set
A cut endpoint is a pair \((e,\varepsilon )\) with \(e\in E_{c,r}\) and \(\varepsilon \in \{ -,+\} \), recording the tail or the head.
When \(w=\ell =2\), there are eight labelled bonds. Each \(E_{c,r}\) contains four bonds and has eight distinct endpoints. Every bond \(e=((x,y),d)\) is uncut in \(E_{x,y}\).
The bond set has \(2\cdot 2\cdot 2=8\) elements. For each of the four choices of \((c,r)\), exactly two horizontal and two vertical bonds satisfy (172). Each has two endpoints. A bond anchored at \((x,y)\) is outside \(E_{x,y}\) because \(x+1\ne x\) and \(y+1\ne y\) in \(\mathbb Z_2\).
For \(w,\ell {\gt}0\), the incidences at \(v=(x,y)\) are in bijection with \(\{ \mathrm H,\mathrm V\} \times \{ -,+\} \). The inverse bijection is
Thus every vertex has four incident endpoint indices, including when a period equals one.
27.32.5 Flat group labels on the actual bonds
For a group \(G\), an assignment \(p:E\to G\) has horizontal and vertical components \(a_{x,y}=p((x,y),\mathrm H)\) and \(b_{x,y}=p((x,y),\mathrm V)\). We write \(p=(a,b)\) for these components.
A vertex assignment \(q:\Lambda \to G\) acts by
Pointwise multiplication of vertex assignments satisfies \(1\cdot p=p\), \(q\cdot (r\cdot p)=(qr)\cdot p\), and \(q^{-1}\cdot (q\cdot p)=p\).
The assignment \(p=(a,b)\) is flat if every plaquette satisfies
For every \(q\), the assignment \(q\cdot p\) is flat if and only if \(p\) is flat.
Bond flatness of \((a,b)\) is equivalent to flatness of the rightward and upward transports \((a,b^{-1})\): \(b_{x+1,y}^{-1}a_{x,y}=a_{x,y+1}b_{x,y}^{-1}\).
Multiply (175) on the left by \(b_{x+1,y}^{-1}\) and on the right by \(b_{x,y}^{-1}\), and reverse the resulting equality. Multiplication by the inverse factors gives the converse.
Given \(g,h\in G\), the standard closure assignment \(\kappa (g,h)\) is
These are the group labels of the two closures in [ 235 , Theorem 5.5 and Definition 5.6 ] .
The assignment \(\kappa (g,h)\) is flat if and only if \(gh=hg\).
The relative labels of \(q\) are \(q\cdot 1\), namely \(p_e=q_{t(e)}q_{s(e)}^{-1}\). They are flat, since the identity assignment is flat and vertex gauges preserve flatness.
Suppose \(w,\ell {\gt}0\). For \(0\le x{\lt}w\) and \(0\le y{\lt}\ell \), put \(A_x=a_{x-1,0}\cdots a_{0,0}\) and \(B_{x,y}=b_{x,y-1}^{-1}\cdots b_{x,0}^{-1}\), with empty products equal to \(1\). Define \(q_{x,y}=(B_{x,y}A_x)^{-1}\).
If \(p\) is flat, then the inverse rooted transport satisfies \(q\cdot p=\kappa (B_{0,\ell }^{-1},A_w)\).
Apply the two rooted-transport identities of Theorem 26.1.4.150 to \((a,b^{-1})\). The horizontal identity is the claimed horizontal component. Invert the vertical identity to obtain the downward component.
For every \(w,\ell {\gt}0\), a bond assignment is flat if and only if \(p=q\cdot \kappa (g,h)\) for some vertex assignment \(q\) and commuting \(g,h\). Equivalently, a flat assignment admits \(q\) with \(q\cdot p=\kappa (g,h)\). This includes periods one and two.
The based holonomies \(B_{0,\ell }\) and \(A_w\) commute, hence so do \(g=B_{0,\ell }^{-1}\) and \(h=A_w\). Theorem 27.32.5.10 gives \(q\cdot p=\kappa (g,h)\); applying \(q^{-1}\) gives the other form. Conversely, commuting closure labels are flat and remain flat under a vertex gauge.
Suppose \(G\) is finite and \(w,\ell {\gt}0\). For any function \(f\) on bond assignments with values in a commutative additive monoid, \(\sum _r f(r\cdot (q\cdot p))=\sum _r f(r\cdot p)\).
Substitute \(r\cdot (q\cdot p)=(rq)\cdot p\) and reindex the sum by the bijection \(r\mapsto rq\) of vertex assignments.
27.32.6 Moving the commuting seams
Define \(\kappa _{c,r}(g,h)\) by replacing \(x+1=0\) and \(y+1=0\) in (180) by \(x+1=c\) and \(y+1=r\). In particular, \(\kappa _{0,0}(g,h)=\kappa (g,h)\).
For \(w,\ell {\gt}0\), write \(s_{k,c}(x)=k^{-1}\) when the least nonnegative representatives satisfy \(x{\lt}c\), and \(s_{k,c}(x)=1\) otherwise. Set \(q_{c,r}(x,y)=s_{g^{-1},r}(y)s_{h,c}(x)\).
If \(gh=hg\) and \(w,\ell {\gt}0\), then
In the first two lines, a one-coordinate function denotes its pullback to \(\Lambda \). The first line needs only \(w{\gt}0\) and the second only \(\ell {\gt}0\). The last line in particular supplies a vertex gauge taking the standard closure to any chosen pair of seams. It is the group-valued form of the string movement in [ 235 , Section 5 ] .
The one-dimensional seam identity gives the changed horizontal component. Every value of \(s_{h,c}\) commutes with \(g\), so conjugation leaves the vertical component fixed. The inverse-label seam identity gives the vertical movement, while commutation fixes the horizontal component. Compose the horizontal gauge and then the vertical gauge; their pointwise product is \(q_{c,r}\).
For finite \(G\), positive periods, commuting \(g,h\), and any function \(f\) into a commutative additive monoid, \(\sum _q f(q\cdot \kappa _{c,r}(g,h)) =\sum _q f(q\cdot \kappa (g,h))\).
Replace \(\kappa _{c,r}(g,h)\) using (181) and reindex the orbit sum by multiplication with \(q_{c,r}\).
On individual bonds, write \(\kappa _e(g,h)\) and \(\kappa _{c,r;e}(g,h)\) for the assignments whose horizontal and vertical components are \(\kappa (g,h)\) and \(\kappa _{c,r}(g,h)\).
If \(e\notin E_{c,r}\), then \(\kappa _{c,r;e}(g,h)=1\).
In each direction, being outside the cut makes the seam condition false.
For positive periods and commuting \(g,h\), there exists \(q:\Lambda \to G\) such that \(q_{t(e)}\kappa _e(g,h)q_{s(e)}^{-1} =\kappa _{c,r;e}(g,h)\) for every bond \(e\).
Take the gauge of Theorem 27.32.6.3 and read its horizontal or vertical equality according to the direction of \(e\).
For positive periods, every flat assignment \(p:E\to G\) admits a vertex assignment \(q\) and commuting \(g,h\) with \(q_{t(e)}p_eq_{s(e)}^{-1}=\kappa _e(g,h)\) for every \(e\).
Classify the horizontal and vertical components, then evaluate the resulting pair of equalities in the direction of each bond.
Let \(w=\ell =2\). Suppose \(p_e=q_{t(e)}q_{s(e)}^{-1}\) for every \(e\notin E_{c,r}\). Then \(b_{c+1,r}a_{c,r+1}=a_{c,r}b_{c,r}\).
The four bonds in this equality are uncut. Substitution gives
27.32.7 Contractions with independently sized bonds
Let \(\mathcal V\) and \(\mathcal E\) be finite sets of vertices and labelled oriented bonds, with tail and head maps \(t,h:\mathcal E\to \mathcal V\). Each bond \(e\) has its own finite alphabet \(X_e\), and each vertex \(v\) has its own physical alphabet \(Y_v\). Parallel bonds remain distinct; a bond with \(t(e)=h(e)\) has two distinct incidences. These conventions express the linkwise matching in [ 235 , Definition 5.1 ] .
The incidence set is \(\mathcal E\times \{ -,+\} \), where \(-\) is the tail and \(+\) the head. Put \(I_v=\{ (e,\epsilon ):v(e,\epsilon )=v\} \) and \(\mathcal X_v=\prod _{(e,\epsilon )\in I_v}X_e\). An incidence configuration \(\beta \) assigns \(\beta _{e,\epsilon }\in X_e\) independently. Restriction to \(I_v\) defines a bijection \(R:\prod _{e,\epsilon }X_e\to \prod _v\mathcal X_v\) with \((R\beta )_v(e,\epsilon )=\beta _{e,\epsilon }\); its inverse reads the coordinate in the unique incident vertex. The bond-pair bijection is \(S\beta =(\beta _{e,+},\beta _{e,-})_{e\in \mathcal E}\).
For factors in any commutative monoid, \(\prod _v\prod _{p\in I_v}f(p)=\prod _p f(p) =\prod _e f(e,-)f(e,+)\). For functions \(f_e:X_e^2\to R\) into a commutative semiring,
The sets \(I_v\) partition the incidence set. Regroup first by this partition and then by the two signs of each bond. For the sum, change coordinates by \(S\) and apply finite distributivity.
Let \(A_v(\eta ,s)\) be the coefficient of a tensor with virtual configuration \(\eta \in \mathcal X_v\) and physical index \(s\in Y_v\). For matrices \(B_e\in \operatorname {Mat}_{X_e}(\mathbb {C})\) define \(w_B(\beta )=\prod _e B_e(\beta _{e,+},\beta _{e,-})\) and
Thus every matrix has its head index as row and tail index as column. The second expression is the change of coordinates by \(R\).
For \(C\subseteq \mathcal E\), let \(\partial C=C\times \{ -,+\} \) and \(\mathcal X_{\partial C}=\prod _{(e,\epsilon )\in \partial C}X_e\). Write \(r_C\beta =\beta |_{\partial C}\) and \(w_C(\beta )=\prod _{e\notin C}\delta _{\beta _{e,+},\beta _{e,-}}\). The linear cut map is
Its range is \(\mathcal S_{A,C}=\operatorname {im}\mathcal C_{A,C}\). Equivalently, \(\psi \in \mathcal S_{A,C}\) if and only if one \(M\) satisfies \(\mathcal C_{A,C}(M)(\sigma )=\psi (\sigma )\) for every \(\sigma \). The function \(M\) may correlate all cut indices.
Let \(\delta _\eta \) denote a coordinate vector on \(\mathcal X_{\partial C}\). Then
A matrix family determines the particular boundary \(M_B(\eta )=\prod _{e\in C}B_e(\eta _{e,+},\eta _{e,-})\). Its value at \(r_C\beta \) is the product of these same cut-bond factors, and \(\mathcal C_{A,C}(M_B)=\mathcal N_A(B^C)\), where \(B^C_e=B_e\) on \(C\) and \(B^C_e=I\) elsewhere. Define \(E^\eta _e\) to be the matrix unit \(|\eta _{e,+}\rangle \langle \eta _{e,-}|\) on \(C\), and \(I\) elsewhere. Then \(M_{E^\eta }=\delta _\eta \) and \(\mathcal C_{A,C}(\delta _\eta )=\mathcal N_A(E^\eta )\).
Expand \(M=\sum _\eta M(\eta )\delta _\eta \) and use linearity. In the contraction with \(M_B\), the factors on \(C\) are the entries of \(B_e\), and the remaining factors are identity entries. Products of matrix-unit entries equal one exactly when all cut coordinates equal \(\eta \), proving the last assertions without factoring a general \(M\).
For finite input alphabets \(J_v\), arbitrary output alphabets \(K_v\), and matrices \(F_v: \mathbb {C}^{J_v}\to \mathbb {C}^{K_v}\), put \((\bigotimes _v F_v)_{\tau ,\sigma }=\prod _v F_v(\tau _v,\sigma _v)\). The associated linear map \(\mathcal F\) satisfies
Products compose coordinatewise: \((\bigotimes _vF_v)(\bigotimes _vL_v) =\bigotimes _v(F_vL_v)\), at both the matrix and linear-map levels. Products of identities are identities. The product with \(P_v\) at one coordinate and identities elsewhere acts by \(\psi (\tau )\mapsto \sum _sP_v(\tau _v,s)\psi (\tau [v\leftarrow s])\). If each \(P_v\) fixes every slice \(s\mapsto \psi (\tau [v\leftarrow s])\), then \(\bigotimes _vP_v\) fixes \(\psi \); the \(P_v\) need not be projections.
Finite distributivity separates the intermediate sum in a matrix product into one sum at each coordinate. Identity entries force equality of the remaining coordinates. For the fixed-vector assertion, induct on the finite set of coordinates at which \(P_v\) replaces the identity, using the one-coordinate formula and coordinatewise composition.
Every one-coordinate slice of \(\mathcal F\chi \) lies in \(\operatorname {im}F_v\). If all input and output alphabets are finite, the converse holds:
Empty alphabets, an empty vertex set, zero matrices, and different local ranks are allowed.
For the forward implication, expand a slice as a sum of columns of \(F_v\), with coefficients obtained by contracting all other coordinates. For each \(v\), choose a linear right inverse on \(\operatorname {im}F_v\) and extend it to a map \(L_v\) on the output space. Thus \(F_vL_v\) fixes every slice. The preceding theorem gives \((\bigotimes _vF_vL_v)\psi =\psi \), so \((\bigotimes _vL_v)\psi \) is a preimage.
27.32.8 Matched representations and a common local inverse
Retain the incidence coordinates of Definition 27.32.7.1. Let \(G\) be a finite group and let each bond carry a representation \(U_e:G\to \operatorname {GL}(\mathbb {C}^{X_e})\). The same \(U_e\) occurs at both ends of that bond, with the dual action at its tail. This is the matching convention of [ 235 , Definition 5.1 ] .
The representation \(\rho _v\) acts by the matrix
It satisfies \(W_v(1)=I\) and \(W_v(gh)=W_v(g)W_v(h)\), so \(\rho _v(g)x=W_v(g)x\). The local coefficient map is \(T_vx(s)=\sum _{\eta \in \mathcal X_v}A_v(\eta ,s)x(\eta )\). The identity and multiplication laws follow coordinatewise, since \(g\mapsto U_e(g^{-1})^{\mathsf T}\) is also a representation.
For any representation \(\rho _v\) on \(\mathbb {C}^{\mathcal X_v}\), let \(P_v=|G|^{-1}\sum _g\rho _v(g)\) and define \(A_v^0(\eta ,s)=(P_v)_{s,\eta }\). Its local coefficient map is exactly \(P_v\). For the incident representation, \(A_v^0(\eta ,s)=|G|^{-1}\sum _gW_v(g)_{s,\eta }\).
Every \(A_v^0\) is \(G\)-injective for its representation \(\rho _v\). In particular this holds for independently chosen incident bond actions.
Right multiplication by \(\rho _v(g)\) permutes the group average, so \(P_v\rho _v(g)=P_v\). On the invariant subspace \(P_v\) is the identity, hence is injective there.
Local sums expand independently, and local scalars multiply: \(\mathcal N_{\sum _i A^i}(B)=\sum _{q:\mathcal V\to J} \mathcal N_{(A^{q_v}_v)_v}(B)\) and \(\mathcal N_{(c_vA_v)_v}(B)=(\prod _vc_v)\mathcal N_A(B)\). For fixed vertex labels \(q\), contracting the local matrices \(W_v(q_v)\) gives the product of the entries of \(U_e(q_{h(e)})B_eU_e(q_{t(e)}^{-1})\). Consequently, with \(n=|\mathcal V|\),
No semi-regularity or unitarity is required. This is the independent-bond version of the projector contraction in [ 235 , Theorem 5.9 ] .
Expand each local average, collecting the factor \(|G|^{-n}\). Regroup the incident matrix factors by bonds. The independent sum over the two indices of each bond is the displayed three-matrix product, by (183).
For any \(q:\mathcal V\to G\) and arbitrary matrices \(B_e\), set \(B_e^q=U_e(q_{h(e)})B_eU_e(q_{t(e)}^{-1})\). Then \(\mathcal N_{A^0}(B^q)=\mathcal N_{A^0}(B)\). In particular the group-label change \(p_e\mapsto q_{h(e)}p_eq_{t(e)}^{-1}\) preserves the canonical network. This is the virtual symmetry of [ 235 , Definition 5.1 ] .
In (190), change the vertex summation variable from \(r\) to \(rq\). The inverse at a tail reverses the product in precisely the required order.
Given finite \(Y_v\) and matrices \(F_v:\mathbb {C}^{Y_v}\to \mathbb {C}^{Z_v}\), define \((FA)_v(\eta ,s)=\sum _tF_v(s,t)A_v(\eta ,t)\). Its local coefficient map is \(F_vT_v\). Equality of local coefficient maps at every site implies equality of tensor families, by evaluating those maps on every coordinate vector.
Let \(\mathcal F=\bigotimes _vF_v\). For every joint boundary \(M\) and every bond-matrix family \(B\), \(\mathcal F\mathcal C_{A,C}(M)=\mathcal C_{FA,C}(M)\) and \(\mathcal F\mathcal N_A(B)=\mathcal N_{FA}(B)\).
Expand the product physical map, interchange the finite physical and incidence sums, and distribute the product of the local sums. The factors \(M(r_C\beta )w_C(\beta )\), or \(w_B(\beta )\), remain unchanged.
Suppose the physical alphabets are finite and every \(T_v\) is \(G\)-injective for a representation \(\rho _v\) on its virtual space. There are \(L_v\) with \(L_vT_v=P_v\); hence \(LA=A^0\). Invariance alone gives \(T_vP_v=T_v\) and \(TA^0=A\). Put \(\mathcal L=\bigotimes _vL_v\) and \(\mathcal T=\bigotimes _vT_v\). For every cut \(C\) and every joint boundary \(M\),
Therefore, for any nonempty family \((C_i)_{i\in J}\) of cuts,
Choose each \(L_v\) by the left-inverse characterization of \(G\)-injectivity. Average the invariance equations to get \(T_vP_v=T_v\). The physical-contraction identities yield (191) with the same \(M\). If \(\psi \) lies in every original cut range, then \(\mathcal L\psi \) lies in every canonical range. Choose one cut in the nonempty family to conclude \(\mathcal T\mathcal L\psi =\psi \). Conversely, \(\mathcal T\) carries each canonical range into the corresponding original range.
The identity tensor has coefficients \(I_v(\eta ,s)=\delta _{\eta ,s}\), so \(\mathcal N_I(B)(\sigma )=w_B(R^{-1}\sigma )\). Define the product invariant projection \(\mathcal P=\bigotimes _vP_v\).
For any representations \(\rho _v\), any joint cut boundary \(M\), and any matrices \(B_e\), \(\mathcal P\mathcal C_{A^0,C}(M)=\mathcal C_{A^0,C}(M)\) and \(\mathcal P\mathcal N_I(B)=\mathcal N_{A^0}(B)\).
For the first equality apply canonical recovery to the \(G\)-injective tensors \(A^0\). For the second, applying \(P_v\) to the physical leg of \(I_v\) gives exactly \(A_v^0\); commute these local changes through the network.
27.32.9 Coherent coefficients forced by complementary cuts
Use the independent incidence spaces and incident representations of Definitions 27.32.7.1 and 27.32.8.1. This section derives a bond-product expansion from actual cut membership before extracting its coefficients. The trace duality is [ 235 , Lemma 4.6 ] ; its use in the four-block comparison is part of [ 235 , Theorem 5.5 ] .
The bijection \(J=SR^{-1}\) sends a site configuration to its head–tail bond pairs. It induces the coefficient permutation \((\mathcal R\psi )(\beta )=\psi (J^{-1}\beta )\). For a matrix family \(B\), let \(b_B(\sigma )=\prod _eB_e((J\sigma )_e^+,(J\sigma )_e^-)\), so \((\mathcal Rb_B)(\beta )=\prod _eB_e(\beta _e^+,\beta _e^-)\). For group labels \(p:\mathcal E\to G\), abbreviate \(b_p=b_{(U_e(p_e))_e}\).
Given linear functionals \(K_e\) on the bond matrices, define
With the source’s trace-dual functionals \(\ell _{e,g}(B)=\operatorname{tr}[U_e(g^{-1})B\Delta _e]\), write \(c_p(\psi )=(\bigotimes _e\ell _{e,p_e})(\psi )\) for this product functional.
For arbitrary \(K_e\) and \(B_e\), \(K(b_B)=\prod _eK_e(B_e)\). If every \(U_e\) is semi-regular, then
Thus two coherent sums in the \(b_p\) are equal if and only if their complete coefficient functions are equal.
Distribute (193) over the bond pairs. The \(e\)-th sum is \(K_e(B_e)\) by expansion in matrix units. Trace duality gives \(\ell _{e,p_e}(U_e(q_e))=\delta _{p_e,q_e}\); their product is \(\delta _{p,q}\). Apply linearity to the entire coherent sum, then compare each coefficient.
Let \(F_e:\mathbb {C}^G\to \mathbb {C}^{X_e\times X_e}\) have columns \(F_e((a,b),g)=U_e(g)_{a,b}\). Each representation matrix is therefore a vector in \(\operatorname {im}F_e\). The slice at bond \(e\) and remaining configuration \(\tau \) is \(S_{e,\tau }\psi (s)=(\mathcal R\psi )(\tau [e\leftarrow s])\). Put \(B_e(q)=U_e(q_{h(e)})B_eU_e(q_{t(e)}^{-1})\).
A coordinate slice of \(\prod _w f_w(\tau _w)\) is \((\prod _{w\ne e}f_w(\tau _w))f_e\). With \(n=|\mathcal V|\), the canonical contraction and its regrouping obey
These identities require neither semi-regularity nor restrictions on \(B\).
Separate the \(e\)-th factor for the slice identity. Regroup (190) by \(J\), obtaining the first two formulas. Apply the product-functional identity to each term to obtain the third formula.
If \(B_e=I\), then every \(S_{e,\tau }\mathcal N_{A^0}(B)\) lies in \(\operatorname {im}F_e\). Consequently, if \(e\notin C\) and \(\psi \in \mathcal S_{A^0,C}\), then \(S_{e,\tau }\psi \in \operatorname {im}F_e\) for every \(\tau \), even when the cut boundary is arbitrarily correlated.
In a summand of (195), an identity insertion gives \(B_e(q)=U_e(q_{h(e)}q_{t(e)}^{-1})\). Every slice is a scalar multiple of this representation matrix; sums retain membership in \(\operatorname {im}F_e\). Expand a general boundary in fixed configurations. Each resulting matrix-unit network still has \(B_e=I\) when \(e\notin C\).
Let \((C_i)_{i\in J}\) be any family of cuts such that every bond is uncut in at least one \(C_i\). If \(\psi \in \mathcal S_{A^0,C_i}\) for every \(i\), then \(\mathcal R\psi \in \operatorname {im}(\bigotimes _eF_e)\). In particular there are coefficients \(a_p\) with \(\psi =\sum _pa_pb_p\). If all \(U_e\) are semi-regular, this becomes
For each bond, choose a cut leaving it uncut. The preceding theorem places every slice in \(\operatorname {im}F_e\). The product-range criterion supplies a preimage under \(\bigotimes _eF_e\), whose entries are the coefficients \(a_p\). Apply \(c_p\) to the whole expansion to identify \(a_p=c_p(\psi )\) in the semi-regular case.
Assume every \(U_e\) is semi-regular. If \(B_e=I\) outside \(C\) and no \(q:\mathcal V\to G\) satisfies \(p_e=q_{h(e)}q_{t(e)}^{-1}\) simultaneously for all \(e\notin C\), then \(c_p(\mathcal N_{A^0}(B))=0\). The same vanishing holds for every \(\psi \in \mathcal S_{A^0,C}\). Equivalently,
Fix a vertex assignment \(q\) in the extracted sum of (195). Incompatibility supplies an uncut bond with \(p_e\ne q_{h(e)}q_{t(e)}^{-1}\); its trace-dual factor is zero. Thus every summand vanishes. For an arbitrary boundary, expand into fixed-coordinate columns and use their matrix-unit realization. The implication is the contrapositive of this vanishing.
27.32.10 The four-block closure property
The four blocks in [ 235 , Theorem 5.5 ] occupy \(\Lambda =(\mathbb Z/2\mathbb Z)^2\). There are eight labelled bonds, including two distinct bonds between each horizontal or vertical pair of blocks. Every bond \(e\) has its own finite virtual alphabet \(X_e\) and semi-regular representation \(U_e\); every block has its own finite physical alphabet \(Y_v\). The head acts by \(U_e\) and the tail by \(U_e^{-\mathsf T}\), so the two incident tensors use the same representation on their shared bond.
Let \(C_{c,r}\) be the cut consisting of horizontal bonds with head column \(c\) and vertical bonds with tail row \(r\). Its virtual domain is \(\mathcal X_{\partial C_{c,r}}\), with eight independent incidence indices. Define
Here \(p^{g,h}\) is \(h\) on the horizontal seam, \(g\) on the vertical seam, and \(1\) elsewhere. Membership in \(\mathcal I_A\) means membership in all four displayed ranges. Every bond is uncut for at least one of these four choices: its own base-vertex cut leaves it uncut.
Let \(G\) be finite. Assign a semi-regular representation \(U_e\) to each of the eight bonds, and let the four tensors \(A_v\) be \(G\)-injective for their actual incident representations. Then
This is [ 235 , Theorem 5.5 ] . The bond and physical dimensions may differ independently. Each boundary \(M\) is one arbitrary tensor on all eight open incidences. Algebraic semi-regularity suffices; the unitary convention of the source is a special case.
Write \(A^0\) for the four incident averaging tensors and \(\mathcal T=\bigotimes _vT_v\) for the product of the original local coefficient maps. A common product of genuine \(G\)-injective inverses gives \(\mathcal I_A=\mathcal T(\mathcal I_{A^0})\); the same physical map gives \(\mathcal T(\mathcal Z_{A^0})=\mathcal Z_A\). For the canonical tensors, every commuting seam can be moved to any of the four cuts, giving explicit boundary witnesses. Conversely, uncut-bond slices give a coherent expansion of every \(\psi \in \mathcal I_{A^0}\). Trace-dual extraction forces each nonzero coefficient to satisfy every plaquette equation. A vertex gauge sends every such flat assignment to commuting seams, and the product invariant projection turns each bare bond product into its actual closure. Hence \(\mathcal I_{A^0}=\mathcal Z_{A^0}\), proving the result.
For arbitrary local physical maps and \(\mathcal F=\bigotimes _vF_v\), \(\mathcal F\zeta _A(g,h)=\zeta _{FA}(g,h)\) and \(\mathcal F(\mathcal Z_A)=\mathcal Z_{FA}\). For \(G\)-injective \(A\), let \(\mathcal T=\bigotimes _vT_v\) and let \(A^0\) be the incident averaging tensors. Then \(\mathcal I_A=\mathcal T(\mathcal I_{A^0})\). Invariance of \(A\) alone gives \(\mathcal T(\mathcal Z_{A^0})=\mathcal Z_A\).
Commute physical maps through the network defining \(\zeta _A\) and take spans. Apply (192) to the four cuts, choosing \((0,0)\) as a member of this nonempty family. Finally, \(T_vP_v=T_v\) identifies the recovered tensors with \(A_v\) themselves.
On the four-block torus, the canonical tensors are \(A_v^0(\eta ,s)=(P_v)_{s,\eta }\) and their product projection is \(\mathcal P=\bigotimes _{v\in \Lambda }P_v\). The physical alphabet of the canonical tensor at \(v\) is precisely its own \(\mathcal X_v\).
If \(gh=hg\), then \(\zeta _{A^0}(g,h)\in \mathcal S_{A^0,C_{c,r}}\) for every \((c,r)\). In particular \(\mathcal Z_{A^0}\subseteq \mathcal I_{A^0}\). This inclusion requires no semi-regularity.
Move the seams to \((c,r)\) by the group-valued vertex gauge, producing labels \(p^{g,h;c,r}\) equal to \(1\) off \(C_{c,r}\). An explicit boundary is \(M(\eta )=\prod _{e\in C_{c,r}} U_e(p^{g,h;c,r}_e)_{\eta _{e,+},\eta _{e,-}}\). The matrix-boundary contraction and vertex-gauge invariance identify its image with \(\zeta _{A^0}(g,h)\). Take spans and intersect the four ranges.
For every bond assignment \(p\), one has \(\mathcal P b_p=\mathcal N_{A^0}((U_e(p_e))_e)\). If \(p\) is flat, there exist \(g,h\) with \(gh=hg\) and \(\mathcal P b_p=\zeta _{A^0}(g,h)\); thus \(\mathcal P b_p\in \mathcal Z_{A^0}\).
The bare product \(b_p\) is the identity-site network, so product averaging proves the first equality. Flat labelled bonds are vertex-gauge equivalent to two commuting seams. Canonical vertex-gauge invariance therefore gives the asserted closure.
Suppose every \(U_e\) is semi-regular. If \(\psi \in \mathcal I_{A^0}\) and \(p\) is not flat, then \(c_p(\psi )=0\).
Choose a plaquette where flatness fails, and use the corresponding cut. If \(c_p(\psi )\ne 0\), (197) supplies one vertex assignment realizing all four uncut labels simultaneously. Its relative labels satisfy that plaquette equation, a contradiction.
If all eight bond representations are semi-regular, then \(\mathcal I_{A^0}\subseteq \mathcal Z_{A^0}\) and in fact \(\mathcal I_{A^0}=\mathcal Z_{A^0}\).
Let \(\psi \in \mathcal I_{A^0}\). Membership in the \((0,0)\) cut gives \(\mathcal P\psi =\psi \). Every bond is uncut in at least one of the four cuts, so (196) and linearity give
Nonflat terms vanish by coefficient extraction; each remaining projected term is a commuting closure. The forward inclusion proves equality.
27.32.11 Independent four-block closure classes
Retain the four blocks, eight separately represented bonds, and actual cut spaces of Definition 27.32.10.1. Write \(p^{g,h}\) for the standard seam labels and \(A^0\) for the four incident averaging tensors. Let \(K\) be the number of simultaneous conjugacy classes of commuting pairs in \(G\). The independence argument of [ 235 , Theorem 5.9 ] gives an exact dimension for this four-block intersection. The identification with a microscopic parent kernel is a separate geometric assertion.
For any \(x,g,h\in G\), one has \(p^{xgx^{-1},xhx^{-1}}_e=xp^{g,h}_ex^{-1}\) on every bond. Given pairs \((a,b),(g,h)\) and vertex labels \(q\), the identities
hold on all eight bonds if and only if \(q\) is constant, with value \(x=q_{(0,0)}\), and \(a=xgx^{-1}\) and \(b=xhx^{-1}\). Commutation is unnecessary here.
Both standard closures equal \(1\) off their seams. On those bonds the asserted identities give equality of neighboring vertex labels. Nonseam connectivity implies \(q_v=q_{(0,0)}\) at all four vertices. Reading the vertical bond based at \((0,-1)\) and the horizontal bond based at \((-1,0)\) gives \(a=xgx^{-1}\) and \(b=xhx^{-1}\). Conversely, these equalities give the seam identities, while constant conjugation fixes the identity labels on the other bonds.
Suppose \(G\) is finite and every \(U_e\) is semi-regular. Let \(E_{a,b}=c_{p^{a,b}}\) be the product of the actual eight bond trace-dual functionals. Then
Here \(C_G(g,h)=\{ x\in G:xg=gx,\ xh=hx\} \) is the common centralizer. In particular, the coefficient is zero for different simultaneous conjugacy classes. These formulas hold even for noncommuting pairs.
Expanding the four normalized local averages and applying the eight trace-dual functionals gives
Theorem 27.32.11.1 identifies the surviving labels with constant functions \(q_v=x\), proving (203). On the diagonal the surviving \(x\) are precisely the common centralizer. It contains the identity, so (204) is nonzero.
For a finite group and locally invariant tensors \(A_v\), one has \(\zeta _A(xgx^{-1},xhx^{-1})=\zeta _A(g,h)\) for all \(x,g,h\). Thus closures depend only on their simultaneous conjugacy class. Neither semi-regularity nor injectivity is needed for this assertion.
A constant vertex gauge transforms the two canonical seam labels by simultaneous conjugation and leaves the canonical network unchanged. Writing \(T_v\) for the original site maps, local invariance gives \(T_vP_v=T_v\). Hence \(\zeta _A(g,h)=(\bigotimes _vT_v)\zeta _{A^0}(g,h)\), which transports the equality to the original physical tensors.
For a simultaneous conjugacy class \(C=[(g,h)]\), define \(\zeta _A(C)=\zeta _A(g,h)\). This does not depend on the representative. The set of closures indexed by commuting pairs equals the set indexed by commuting pair-conjugacy classes.
Suppose each of the eight actual bond representations is semi-regular and every physical tensor is \(G\)-injective for its own incident action. Then the vectors \(\zeta _A(C)\) are linearly independent as \(C\) ranges over all pair-conjugacy classes. In particular, the commuting classes give a linearly independent family. The four physical alphabets and all eight virtual alphabets may differ independently.
First take the canonical tensors. Apply \(E_{g,h}\), for a representative of any fixed class \(C\), to a relation \(\sum _{C'}\lambda _{C'} \zeta _{A^0}(C')=0\). The off-class coefficients vanish, and the remaining equation is \(\lambda _C|G|^{-4}|C_G(g,h)|=0\). Thus every \(\lambda _C=0\). For the actual tensors, choose their genuine local inverses with \(L_vT_v=P_v\). Physical contraction gives \((\bigotimes _vL_v)\zeta _A(C)=\zeta _{A^0}(C)\) for every class \(C\). Applying this same linear map to a relation proves the result for \(A\). The inverse and scalar extraction are depicted below. Each vertical leg denotes a complete configuration, retaining its independently typed component indices; \(\mathcal L=\bigotimes _vL_v\).
![\begin{tenkz}[rows={bra,op,ket}, cols=1, bonds=none]
\tn[at=(1,1), name=extractActual, size=l,
ports={270:physical}]{E_{a,b}}
\tn[at=(2,1), name=inverseActual, skin=mpo, size=l,
ports={90:physical,270:physical}]{\mathcal L}
\tn[at=(3,1), name=closureActual, size=l,
ports={90:physical}]{\zeta_A(g,h)}
\tnwire[name=extractedIncidences]{extractActual.270}{inverseActual.90}
\tnmark[form=label, label pos=e]{on extractedIncidences .5}{$\sigma$}
\tnwire[name=originalPhysical]{inverseActual.270}{closureActual.90}
\tnmark[form=label, label pos=e]{on originalPhysical .5}{$s$}
\end{tenkz}](tenkz_svg/tenkz-7674446d58c2770e.svg)
\(=\) ![\begin{tenkz}[rows={bra,ket}, cols=1, bonds=none]
\tn[at=(1,1), name=extractCanonical, size=l,
ports={270:physical}]{E_{a,b}}
\tn[at=(2,1), name=closureCanonical, size=l,
ports={90:physical}]{\zeta_{A^0}(g,h)}
\tnwire[name=canonicalIncidences]{extractCanonical.270}{closureCanonical.90}
\tnmark[form=label, label pos=e]{on canonicalIncidences .5}{$\sigma$}
\end{tenkz}](tenkz_svg/tenkz-62102f7d64dd3566.svg)
\(\displaystyle =|G|^{-4} \sum _{x\in G}\mathbf1_{a=xgx^{-1},\, b=xhx^{-1}}\). Under the hypotheses of Theorem 27.32.11.5, the commuting class-labelled closures form a basis of \(\mathcal I_A=\mathcal Z_A\). Both spaces have complex dimension \(K\), the number of simultaneous conjugacy classes of commuting pairs in \(G\).
The four-cut closure theorem gives \(\mathcal I_A=\mathcal Z_A=\operatorname{span}_{\mathbb {C}}\{ \zeta _A(C):C\text{ commuting}\} \). The stated family is linearly independent, so the dimension of its span is its cardinality \(K\).
The four boundary contractions.
Put \(A=(0,1)\), \(B=(1,1)\), \(C=(0,0)\) and \(D=(1,0)\). Write \(H_{xy}=((x,y),\mathrm H)\) and \(V_{xy}=((x,y),\mathrm V)\). The four contractions in [ 235 , Theorem 5.5 ] are \(\alpha (M)=\mathcal C_{0,0}(M)\), \(\beta (N)=\mathcal C_{1,0}(N)\), \(\gamma (P)=\mathcal C_{0,1}(P)\) and \(\delta (Q)=\mathcal C_{1,1}(Q)\). Each central tensor below is one arbitrary joint tensor on eight endpoints; the four outer wires are the four uncut identity pairings. A superscript \(+\) identifies a head endpoint, and \(-\) a tail endpoint. Thus two wires bearing the same bond label but different superscripts are independent arguments of the boundary tensor, not an identity pairing. The uncontracted physical legs are \(i_A,i_B,i_C,i_D\). The drawings move the boundary tensor to the centre without changing any incidence. Their displayed geometry does not identify parallel bonds.
\(\alpha (M)=\mathcal C_{0,0}(M)\).
![\begin{tenkz}[rows={wire,wire}, cols=2, bonds=none]
\tn[at=(2,2), name=fourCutMA, skin=box,
ports={0:virtual,270:virtual,330:virtual,300:virtual,135:physical:$i_A$}]{A}
\tn[at=4 e of fourCutMA, name=fourCutMB, skin=box,
ports={180:virtual,270:virtual,210:virtual,240:virtual,45:physical:$i_B$}]{B}
\tn[at=4 s of fourCutMA, name=fourCutMC, skin=box,
ports={0:virtual,90:virtual,30:virtual,60:virtual,225:physical:$i_C$}]{C}
\tn[at=4 e of fourCutMC, name=fourCutMD, skin=box,
ports={180:virtual,90:virtual,150:virtual,120:virtual,315:physical:$i_D$}]{D}
\tn[at=2 e of 2 s of fourCutMA, name=fourCutMBoundary, skin=ring,
ports={30:virtual,60:virtual,120:virtual,150:virtual,210:virtual,240:virtual,300:virtual,330:virtual}]{M}
\tnwire[name=fourCutMAphysical, via={1 nw of fourCutMA}]{fourCutMA.135}{open nw}
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\tnwire[name=fourCutMAVend]{fourCutMA.300}{fourCutMBoundary.150}
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\tnmark[form=label, label pos=n]{on fourCutMBHend 0.55}{$H_{11}^{-}$}
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\tnmark[form=label, label pos=s]{on fourCutMBVend 0.55}{$V_{11}^{+}$}
\tnwire[name=fourCutMCHend]{fourCutMC.30}{fourCutMBoundary.240}
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\tnwire[name=fourCutMCVend]{fourCutMC.60}{fourCutMBoundary.210}
\tnmark[form=label, label pos=n]{on fourCutMCVend 0.55}{$V_{01}^{-}$}
\tnwire[name=fourCutMDHend]{fourCutMD.150}{fourCutMBoundary.300}
\tnmark[form=label, label pos=s]{on fourCutMDHend 0.55}{$H_{10}^{-}$}
\tnwire[name=fourCutMDVend]{fourCutMD.120}{fourCutMBoundary.330}
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\tnmark[form=label, label pos=w]{on fourCutMUncut2 0.5}{$V_{00}$}
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\(\beta (N)=\mathcal C_{1,0}(N)\).
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\tnwire[name=fourCutNDHend]{fourCutND.150}{fourCutNBoundary.300}
\tnmark[form=label, label pos=s]{on fourCutNDHend 0.55}{$H_{00}^{+}$}
\tnwire[name=fourCutNDVend]{fourCutND.120}{fourCutNBoundary.330}
\tnmark[form=label, label pos=n]{on fourCutNDVend 0.55}{$V_{11}^{-}$}
\tnwire[name=fourCutNUncut0]{fourCutNA.0}{fourCutNB.180}
\tnmark[form=label, label pos=n]{on fourCutNUncut0 0.5}{$H_{11}$}
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\tnmark[form=label, label pos=w]{on fourCutNUncut2 0.5}{$V_{00}$}
\tnwire[name=fourCutNUncut3]{fourCutNB.270}{fourCutND.90}
\tnmark[form=label, label pos=e]{on fourCutNUncut3 0.5}{$V_{10}$}
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\(\gamma (P)=\mathcal C_{0,1}(P)\).
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ports={180:virtual,270:virtual,210:virtual,240:virtual,45:physical:$i_B$}]{B}
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\tnmark[form=label, label pos=s]{on fourCutPCHend 0.55}{$H_{10}^{+}$}
\tnwire[name=fourCutPCVend]{fourCutPC.60}{fourCutPBoundary.210}
\tnmark[form=label, label pos=n]{on fourCutPCVend 0.55}{$V_{00}^{+}$}
\tnwire[name=fourCutPDHend]{fourCutPD.150}{fourCutPBoundary.300}
\tnmark[form=label, label pos=s]{on fourCutPDHend 0.55}{$H_{10}^{-}$}
\tnwire[name=fourCutPDVend]{fourCutPD.120}{fourCutPBoundary.330}
\tnmark[form=label, label pos=n]{on fourCutPDVend 0.55}{$V_{10}^{+}$}
\tnwire[name=fourCutPUncut0]{fourCutPA.0}{fourCutPB.180}
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\(\delta (Q)=\mathcal C_{1,1}(Q)\).
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ports={0:virtual,270:virtual,330:virtual,300:virtual,135:physical:$i_A$}]{A}
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\end{tenkz}](tenkz_svg/tenkz-b7cd83f027e35c51.svg)
27.32.12 Four cut spaces with common coordinate alphabets
The following constructions give the common-coordinate specialization of the four boundary spaces in [ 235 , Theorem 5.5 ] , whose full statement is Theorem 27.32.10.2. Let \(X=\mathbb {Z}/w\mathbb {Z}\times \mathbb {Z}/\ell \mathbb {Z}\), where \(w,\ell {\gt}0\), and write \(e=(1,0)\) and \(n=(0,1)\). Horizontal bonds run from \(v\) to \(v+e\); vertical bonds run from \(v+n\) to \(v\). Let \(V\) be a finite virtual alphabet and \(P\) a physical alphabet. All eight bonds remain distinct when \(w=\ell =2\). Write \(\mathcal N_A(O^{\mathrm h},O^{\mathrm v})\) for the physical vector obtained from the bond contraction of Definition 26.1.4.1, with independent site tensors \(A_v:V^4\times P\to \mathbb {C}\).
The boundary configurations for the cut \((c,r)\) form \(\mathcal E=(V\times V)^{\mathbb {Z}/\ell \mathbb {Z}}\times (V\times V)^{\mathbb {Z}/w\mathbb {Z}}\). Each pair is ordered tail, head. For independent bond-end labels \(\beta :X\to V^4\), put
For an arbitrary joint boundary tensor \(M:\mathcal E\to \mathbb {C}\), define the linear map \(\mathcal C^A_{c,r}\) by
Put \(\mathcal S^A_{c,r}=\operatorname {ran}\mathcal C^A_{c,r}\). Thus \(\psi \in \mathcal S^A_{c,r}\) precisely when some \(M\) satisfies \((\mathcal C^A_{c,r}M)(\sigma )=\psi (\sigma )\) for every \(\sigma \). On \(X=(\mathbb {Z}/2\mathbb {Z})^2\), put \(\mathcal S^A=\bigcap _{c,r}\mathcal S^A_{c,r}\); membership means that each cut admits such a boundary tensor, which may depend on the cut. Here \(|\mathcal E|=|V|^8\).
The boundary occurs linearly in (211). The range descriptions follow by equality of all coefficients, and the intersection description follows by imposing all four range conditions. The two rows and two columns each contribute two independent endpoints, giving \(|V|^{2\cdot 2+2\cdot 2}=|V|^8\) configurations.
For bond matrices \(O^{\mathrm h}_v,O^{\mathrm v}_v\), define
Its value at \(R_{c,r}(\beta )\) is the product of the corresponding \(O^{\mathrm h}_v(\beta _v^2,\beta _v^1)\) and \(O^{\mathrm v}_v(\beta _v^4,\beta _v^3)\) on cut bonds, with factor one on every other bond. For fixed \(\eta \in \mathcal E\), let \(E^{\mathrm h}_{\eta ,v}=E_{\eta ^{\mathrm h,2}_{v_y}, \eta ^{\mathrm h,1}_{v_y}}\) and \(E^{\mathrm v}_{\eta ,v}=E_{\eta ^{\mathrm v,2}_{v_x}, \eta ^{\mathrm v,1}_{v_x}}\) be matrix units. Then \(M^{E_\eta }_{c,r}(\theta )=\delta _{\eta ,\theta }\).
Substitution of (206) selects the designated seam bonds. Every matrix unit fixes its two endpoints; their product is one exactly when all endpoint labels agree with \(\eta \).
Write \(\widehat O^{\mathrm h,c}_v=O^{\mathrm h}_v\) on \(v_x+1=c\) and \(\mathbb {1}\) elsewhere, and define \(\widehat O^{\mathrm v,r}\) analogously. Then
In particular, \(\mathcal C^A_{c,r}\delta _\eta \) is the network with \(E^{\mathrm h}_{\eta ,v}\) and \(E^{\mathrm v}_{\eta ,v}\) on cut bonds and identities elsewhere.
Multiplying the restricted product boundary by \(W_{c,r}\) gives exactly the bond weights in (213). Expand \(M=\sum _\eta M(\eta )\delta _\eta \) and use linearity. Each fixed-boundary vector already belongs to the range, proving the reverse span inclusion. The matrix-unit identity gives the final assertion.
Let \(G\) be a group and assign representations \(U^{\mathrm h}_v\) and \(U^{\mathrm v}_v\) on \(\mathbb {C}^V\) to the individual bonds. Their action on the top, right, down, left coordinates of site \(v\) is
The coefficient action is multiplication by this matrix. The bond family is unitary when every \(U^d_v(k)\) is unitary, and, for finite \(G\), is semi-regular when every individual representation \(U^d_v\) is semi-regular. These are separate conditions. For any representation \(\rho \) on \(\mathbb {C}^{V^4}\) of finite \(G\), put \(\Pi _\rho =|G|^{-1}\sum _k\rho (k)\) and \(A^0_\rho (t,r,b,l;s)=(\Pi _\rho )_{s,(t,r,b,l)}\). Its site map is exactly \(\Pi _\rho \). Write \(A^0_v=A^0_{\rho _v}\) for the matched family. Constant bond representations recover the uniform four-leg action and its averaging tensor. The matching convention is that of [ 235 , Definition 5.1 ] : each link carries its own representation, used at both ends.
The inverse transpose reverses the order twice, so (214) is a representation. The stated coefficient action and the site map of \(A^0_\rho \) follow by reading their matrix entries. Constant bond families give the uniform formulas by substitution.
For \(g,h\in G\), define
When every bond representation is \(U\), these are the site-dependent uniform closures and their span. If also every site tensor equals \(A\), the closure is \(\mathcal M(A\mid g,h)\) of Definition 26.1.4.96.
Suppose every bond carries \(U\) and each site map obeys \(\mathcal P(A_v)\rho (k)=\mathcal P(A_v)\) for every \(k\in G\). If \(gh=hg\), then for every cut \((c,r)\),
Consequently \(\mathcal Z_A\subseteq \mathcal S^A\) on the two-by-two torus.
Put \(B(t,r,b,l;(v,s))=A_v(t,r,b,l;s)\), with \((v,s)\) as its physical label. Its invariance holds separately at each physical label. Apply Theorem 26.1.4.99 to \(B\) and the configuration \(v\mapsto (v,\sigma _v)\). The product-boundary identity gives the explicit witness in (220); taking spans proves the inclusion.
For \(x,c\in \mathbb {Z}/m\mathbb {Z}\), with \(m{\gt}0\), use representatives \(\bar x,\bar c\in \{ 0,\ldots ,m-1\} \) and put \(s_{k,c}(x)=k^{-1}\) if \(\bar x{\lt}\bar c\), and \(s_{k,c}(x)=1\) otherwise.
Write \(K^c_k(x)=k\) if \(x+1=c\), and \(K^c_k(x)=1\) otherwise. For every \(k\in G\),
If \(kl=lk\), then \(s_{k,c}(x)\) commutes with \(l\).
Check separately the wrap \(x+1=0\), the crossing \(x+1=c\), and the two intervals between them. The adjacent factors cancel except at the seam crossings, proving (221). Invert that formula with \(k^{-1}\) in place of \(k\) to obtain (222). Each \(s_{k,c}(x)\) is either \(1\) or \(k^{-1}\), proving commutation.
Suppose \(\mathcal P(A_v)\rho _v(k)=\mathcal P(A_v)\) for every \(v,k\). For \(gh=hg\) and every \((c,r)\),
On the two-by-two torus, \(\mathcal Z_A\subseteq \mathcal S^A\). This is the first inclusion in the proof of [ 235 , Theorem 5.5 ] , with a common coordinate alphabet.
Apply the matched vertex-gauge identity first with \(q_v=s_{h,c}(v_x)\) and then with \(q_v=s_{g^{-1},r}(v_y)\). Formulas (221) and (222) move the two seams. Commutation with the transverse closure label leaves its matrices unchanged. Each bond uses its own representation of these group identities. The product boundary gives the stated range witness; linearity then gives the span inclusion.
27.32.13 Physical inverses and matched projector contractions
For a finite site set \(I\) and matrices \(F_v:\mathbb {C}^P\to \mathbb {C}^Q\), put
The map formula assumes \(P\) finite; the matrix formula does not. A constant family gives \(F^{\otimes I}\). Taking \(I=X\) gives the site-dependent physical map on the torus.
For compatible matrices \(F_v,L_v\) with finite intermediate alphabets, \(\mathcal T_F\mathcal T_L=\mathcal T_{(F_vL_v)_v}\) as matrices; the same equality holds as maps when the input alphabet is finite. A product of identity matrices is the identity matrix and, for finite alphabets, the identity map. If \(P\) is finite, \(B\) is a matrix on \(\mathbb {C}^P\), and \(F_v=B\) at a single site \(v\) and \(F_w=\mathbb {1}\) elsewhere, then
In particular, if matrices \(B_v\) fix every slice \(s\mapsto \psi (\tau [v\mapsto s])\) at their respective sites, then \(\mathcal T_B\psi =\psi \).
Specialize the product identities to constant input and output alphabets. Finite distributivity factors the intermediate-coordinate sum into \(F_vL_v\) at every site. Identity entries force agreement at all unchanged coordinates, giving (225). Apply the single-coordinate maps successively over the finite site set; every step fixes \(\psi \), and their product is \(\mathcal T_B\).
Let \(I,P\) be finite. Every one-coordinate slice of \(\mathcal T_F\chi \) lies in the corresponding range \(\operatorname {ran}F_v\). If also \(Q\) is finite, then
Empty alphabets and an empty site set are allowed, as are zero maps and maps of different ranks.
Apply Theorem 27.32.7.8 with constant alphabets. Explicitly, a slice of a product image is a linear combination of the columns of \(F_v\). Conversely, choose \(L_v\) with \(F_vL_v\) equal to the identity on \(\operatorname {ran}F_v\). The slice conditions imply \(\mathcal T_{FL}\psi =\psi \), so \(\psi =\mathcal T_F(\mathcal T_L\psi )\).
Let the input physical alphabet be finite, and set
For every boundary \(M\) and every pair of bond-operator families,
The closure identities include uniform bond representations.
Expand the physical and virtual sums and exchange their order. The physical sum factors into \(\sum _sF_v(q,s)A_v(\cdot ;s)\) at each site, leaving every boundary coefficient and bond operator untouched. Substitute the closure operators for the third equality, and take the image of the span of commuting closures for the fourth.
Let \(G\) be finite and let \(\rho _v\) be arbitrary representations on \(\mathbb {C}^{V^4}\). Put \(R_v=\mathcal P(A_v)\). If \(R_v\rho _v(k)=R_v\) for all \(v,k\), then \(R_v\Pi _{\rho _v}=R_v\), so \(R_vA^0_{\rho _v}=A_v\) as tensors and
If every \(A_v\) is \(G\)-injective and the physical alphabet is finite, there exist matrices \(L_v\) with \(L_vR_v=\Pi _{\rho _v}\), hence \(L_vA_v=A^0_{\rho _v}\) and, simultaneously for all \(c,r,M\),
On the two-by-two torus this gives \(\mathcal S^A=\mathcal T_R(\mathcal S^{A^0})\). For the matched actions, invariance alone also gives \(\mathcal T_R(\mathcal Z_{A^0})=\mathcal Z_A\). These are common-alphabet local-inverse identities supporting [ 235 , Theorem 5.5 ] .
Average \(R_v\rho _v(k)=R_v\) to obtain \(R_v\Pi _{\rho _v}=R_v\). Theorem 26.1.2 supplies \(L_v\) independently at each site. Apply (228) to these two local identities. If \(\psi \in \mathcal S^A\), the same vector \(\mathcal T_L\psi \) lies in every canonical cut range by (230); recovering it using one cut gives \(\mathcal T_R\mathcal T_L\psi =\psi \). Conversely, (229) maps each canonical range into the corresponding physical range. Finally, physical transport of the closure span and \(R_vA^0_v=A_v\) give the asserted closure recovery.
For a finite family \(A^i_v\) and scalars \(z_v\), the cut contraction obeys
For finite \(G\), the canonical tensor has coefficients \(A^0_{\rho _v}(c;s)=|G|^{-1}\sum _{k\in G}\rho _v(k)_{s,c}\). Consequently, if \(B^q_v(c;s)=\rho _v(q_v)_{s,c}\), then
This includes the matched matrices (214). On the two-by-two torus the normalization is \(|G|^{-4}\) and there are four independent group labels.
Distribute the finite sums independently at every site and exchange them with the bond-end sum. Site scalars factor out as their product. The averaging formula gives \(|G|^{-1}\) at each site and the stated sum of matrix entries. Throughout, the same function \(M\) multiplies each bond-end configuration.
For four matrices \(T,R,B,L\) on \(\mathbb {C}^V\), define \(K=T\otimes R^{\mathsf T}\otimes B^{\mathsf T}\otimes L\) and, for a row tensor \(a:V^4\to \mathbb {C}\), define its dressed row to be \(aK\). In coordinates,
For the fixed row \(a(c)=\delta _{s,c}\), the dressed row is \(K_{s,c'}\).
The entries follow from the tensor-product matrix and transpose formulas. The fixed row removes every summand except \(c=s\).
For arbitrary site rows \(a_v\) and matrix families \(O^{\mathrm h},O^{\mathrm v},T,R,B,L\), put \(K_v=T_v\otimes R_v^{\mathsf T}\otimes B_v^{\mathsf T}\otimes L_v\). Then the scalar bond contractions satisfy
Expand each three-matrix bond product and each dressed site row. The bijection \(\beta \mapsto (s_v(\beta ))_v\) from (206) regroups the intermediate endpoint indices by site. Reindex the factors attached to \(L_{v+e}\) and \(B_{v+n}\) by translation. The scalar summands on both sides then coincide, with the same original bond matrices remaining between them.
Suppose \(\mathcal P(A_v)\rho _v(k)=\mathcal P(A_v)\) for every \(v,k\). Each physical row of \(A_v\) therefore satisfies \(A_v^{s}\rho _v(k)=A_v^{s}\). For every \(q:X\to G\), put
Then \(\mathcal N_A(O^{\mathrm h,q},O^{\mathrm v,q}) =\mathcal N_A(O^{\mathrm h},O^{\mathrm v})\).
Test the site-map invariance on each virtual basis vector to obtain the row identity. In (232), choose \(T_v=U^{\mathrm v}_v(q_v)\), \(R_v=U^{\mathrm h}_v(q_v^{-1})\), \(B_v=U^{\mathrm v}_{v-n}(q_v^{-1})\), and \(L_v=U^{\mathrm h}_{v-e}(q_v)\). These four factors form \(\rho _v(q_v)\), which each physical row absorbs.
Let \(G\) be finite and use the bond matrices (233) and (234). Then
Here \(\sigma _v=(t_v,r_v,b_v,l_v)\), \(\mathcal R\) is physical bond regrouping, and \(O^q_{(v,d)}=O^{d,q}_v\) defines the single bond-indexed family. No semi-regularity or unitarity is required.
Expand each local average as a sum of its matched matrices. Finite multilinearity produces \(|G|^{-|X|}\sum _q\). For each \(q\), apply (232) to the fixed rows \(\delta _{\sigma _v}\); the remaining identity-site contraction is the product of its bond entries. This gives (235). Substitute the inverse physical regrouping formulas and combine the horizontal and vertical products to obtain (236).
27.32.14 Bond support and common-alphabet closure reconstruction
Throughout this subsection \(G\) and \(V\) are finite, and the torus periods are positive. The site tensors \(A^0_v\) are the matched averaging tensors of Definition 27.32.12.4. Let \(\mathcal B=X\times \{ \mathrm h,\mathrm v\} \) denote the actual bonds.
A bond labelling is \(p=(p^{\mathrm h},p^{\mathrm v})\in G^X\times G^X\). The bijection to \(G^{\mathcal B}\) sends \(p\) to \((v,d)\mapsto p^d_v\); its inverse restricts to the two directions. For any finite label alphabet \(G\), there are \(|G|^8\) such labellings on the two-by-two torus. Define the bare product vector
For each bond \(f\), let \(U_f\) denote its representation and put \(\delta _{f,k}(M)=\operatorname{tr}(U_f(k^{-1})M\Delta _{U_f})\), using the trace-dual operator of Theorem 26.1.4.115. The linear coefficient functional is
For unitary representations, \(U_f(k^{-1})=U_f(k)^\dagger \), recovering the pairing in [ 235 , Lemma 4.6 ] .
Suppose every individual bond representation is semi-regular. Then
The family \((B_p)_p\) is linearly independent. In particular, \(\sum _pa_pB_p=\sum _pb_pB_p\) if and only if \(a_p=b_p\) for every \(p\).
Pair the product vector bond by bond. Trace duality gives \(E_p(B_q)=\prod _f\delta _{p_f,q_f}=\delta _{p,q}\), since all labelled bonds are distinct. Linearity extracts a specified coefficient from any finite coherent sum. Applying every \(E_p\) to a zero sum proves linear independence and applying them to an equality proves coefficient uniqueness.
For arbitrary \(a,b,c,d\) in a group, put \(h_1=b^{-1}a\), \(h_2=d^{-1}c\), \(v_1=c^{-1}a\), \(v_2=d^{-1}b\). Then \(v_2h_1=h_2v_1\). More generally, any four labels satisfying \(v_2h_1=h_2v_1\), \(h_1=h_2\), and \(v_1=v_2\) obey \(v_1h_1=h_1v_1\). These are the group identities in the proof of [ 235 , Theorem 5.5 ] .
Both \((d^{-1}b)(b^{-1}a)\) and \((d^{-1}c)(c^{-1}a)\) equal \(d^{-1}a\). Substituting equal horizontal and vertical labels into the compatibility equation gives commutation.
Read \(U_{(v,\mathrm h)}=U^{\mathrm h}_v\) and \(U_{(v,\mathrm v)}=U^{\mathrm v}_v\). Define \(F_f:\mathbb {C}^G\to \mathbb {C}^{V^2}\) by \(F_f((i,j),k)=U_f(k)_{i,j}\). Thus \(\operatorname {ran}F_f\) is the span of the representation matrices, and each vector \((i,j)\mapsto U_f(k)_{i,j}\) is its column \(F_f\delta _k\). For \(\tau :\mathcal B\to V^2\), the physical bond slice is the linear map \(S_{f,\tau }\psi (s)=(\mathcal R\psi )(\tau [f\mapsto s])\).
Matrix multiplication by the coordinate vector \(\delta _k\) gives exactly the indicated representation column. Evaluation after the inverse bond regrouping is linear in \(\psi \).
For any finite set \(I\), functions \(f_i:Q\to \mathbb {C}\), \(j\in I\), and \(\tau :I\to Q\),
If the inserted bond operator \(O_f\) equals \(\mathbb {1}\), then every slice \(S_{f,\tau }\mathcal N_{A^0}(O)\) lies in \(\operatorname {ran}F_f\). The other bond operators are arbitrary.
If \(\psi \in \mathcal S^{A^0}_{c,r}\) and \(f\) is uncut, then \(S_{f,\tau }\psi \in \operatorname {ran}F_f\) for every \(\tau \). Here uncut means \(v_x+1\ne c\) for \(f=(v,\mathrm h)\) or \(v_y+1\ne r\) for \(f=(v,\mathrm v)\). On \(X=(\mathbb {Z}/2\mathbb {Z})^2\), every \(\psi \in \mathcal S^{A^0}\) satisfies
for suitable coefficients. If all bond representations are semi-regular, these coefficients are uniquely determined and
Expand each boundary in fixed endpoint tensors. Each such cut vector has matrix units on cut bonds and identities on uncut bonds, so Theorem 27.32.14.5 gives the slice condition; linearity extends it to the entire cut range. A bond based at \(v\) is uncut in the cut \((v_x,v_y)\) on the two-by-two torus. Thus every bond satisfies the slice condition, and (226) gives a vector \(a\) with \(\mathcal R\psi =\mathcal T_Fa\). Expand this product map and undo regrouping to obtain the first sum in (241); the bond-label bijection gives the second. Only after this existence argument, apply each \(E_p\) and use (239) to obtain (242).
For arbitrary inserted bond matrices, with \(O^q_f\) as in (233) and (234),
This identity requires no semi-regularity assumption.
Apply the linear functional \(E_p\) to (235). Move it through the finite sum and normalization, and factor the pairing of each product into its individual bond pairings.
Assume \(X=(\mathbb {Z}/2\mathbb {Z})^2\) and that every bond representation is semi-regular. Fix a cut \((c,r)\) and a labelling \(p\) for which
If \(O^{\mathrm h}_v=\mathbb {1}\) when \(v_x+1\ne c\) and \(O^{\mathrm v}_v=\mathbb {1}\) when \(v_y+1\ne r\), then \(E_p(\mathcal N_{A^0}(O))=0\). Consequently \(E_p(\psi )=0\) for every \(\psi \in \mathcal S^{A^0}_{c,r}\). If \(\psi \in \mathcal S^{A^0}\) and \(p\) is not flat, then \(E_p(\psi )=0\).
A nonzero summand in (243) has no zero bond pairing. Trace duality on its uncut bonds forces \(p^{\mathrm h}_v=q_{v+e}q_v^{-1}\) and \(p^{\mathrm v}_v=q_vq_{v+n}^{-1}\) there. The four uncut edges then give
contradicting (244). Thus every summand vanishes. Apply this conclusion to the matrix-unit networks spanning the cut range. Finally, a nonflat labelling fails the plaquette equation at some \((c,r)\), whose cut membership gives the result.
For arbitrary representations \(\rho _v\) of finite \(G\) on \(\mathbb {C}^{V^4}\), define \(\mathcal Q_\rho =\mathcal T_{(\Pi _{\rho _v})_v}\). Each averaging tensor is invariant: \(\mathcal P(A^0_{\rho _v})\rho _v(k)=\mathcal P(A^0_{\rho _v})\) for all \(k\).
Right multiplication by \(k\) permutes the summands of \(\Pi _{\rho _v}=|G|^{-1}\sum _g\rho _v(g)\), so \(\Pi _{\rho _v}\rho _v(k)=\Pi _{\rho _v}\).
For every boundary \(M\), \(\mathcal Q_\rho \mathcal C^{A^0}_{c,r}M=\mathcal C^{A^0}_{c,r}M\). Therefore \(\mathcal Q_\rho \psi =\psi \) for every \(\psi \in \mathcal S^{A^0}_{c,r}\). For any bond representations \(U^{\mathrm h},U^{\mathrm v}\) and labelling \(p\),
The actions \(\rho _v\) need not be the matched actions for these bond representations.
Apply the recovery identity (229) with \(A_v=A^0_{\rho _v}\); its invariance proves the fixed-vector assertion. Express \(B_p\) as a bond network whose local tensors have coefficients \(\delta _{(t,r,b,l),s}\). Applying \(\Pi _{\rho _v}\) to such an identity tensor produces \(A^0_{\rho _v}\). Physical transport through the contraction gives (245).
Use the matched actions \(\rho _v\). For every flat bond labelling \(p\) on a torus with positive periods, there exist \(g,h\in G\) with \(gh=hg\) such that \(\mathcal Q_\rho B_p=\mathcal M_{A^0}(g,h)\). In particular, \(\mathcal Q_\rho B_p\in \mathcal Z_{A^0}\).
Theorem 27.32.5.11 gives a vertex gauge \(q\) sending \(p\) to two seams labelled by commuting \(g,h\). Formula (245) expresses \(\mathcal Q_\rho B_p\) as the corresponding actual averaging network. Its invariant sites permit the matched vertex-gauge transformation. Applying each bond’s representation to the transformed labels turns its matrices into \(H^0(h)\) and \(V^0(g)\), giving \(\mathcal M_{A^0}(g,h)\). This vector is a generator of the stated span.
On the two-by-two torus, assume every individual bond representation is semi-regular. Then \(\mathcal S^{A^0}\subseteq \mathcal Z_{A^0}\) and, together with the opposite inclusion, \(\mathcal S^{A^0}=\mathcal Z_{A^0}\).
For \(\psi \in \mathcal S^{A^0}\), the cut \((0,0)\) gives \(\mathcal Q_\rho \psi =\psi \). Apply this projector to the derived expansion (242):
The nonflat terms have zero extracted coefficient, while every flat term is a commuting closure. The averaging sites are invariant, so Theorem 27.32.12.9 gives \(\mathcal Z_{A^0}\subseteq \mathcal S^{A^0}\).
Let \(G,V,P\) be finite and \(X=(\mathbb {Z}/2\mathbb {Z})^2\). Assign an arbitrary semi-regular representation on \(\mathbb {C}^V\) to each of the eight bonds, and let each independent site tensor \(A_v:V^4\times P\to \mathbb {C}\) be \(G\)-injective for its matched action \(\rho _v\). Then
This is the common-coordinate specialization of [ 235 , Theorem 5.5 ] ; the independently sized bond and physical spaces are treated in Theorem 27.32.10.2.
Put \(R_v=\mathcal P(A_v)\). The common local inverse and the canonical equality give
The last equality is closure recovery by the original invariant site maps. Each cut retains its arbitrary joint boundary throughout.
Let \(R\) be a connected finite region with \(b{\gt}0\) crossing bonds, and suppose its original tensors are \(G\)-injective for the regular incident-edge action. The actual open contraction \(A_R\) has \(\dim \operatorname {im}A_R=|G|^{b-1}\).
The contracted block is \(G\)-injective on its invariant boundary space. Its restriction to that space is an isomorphism onto its physical image. The invariant regular boundary space has dimension \(|G|^{b-1}\).
Under the preceding hypotheses, let \(M:\mathbb C^B\to \mathcal P\) and \(L:\mathcal P\to \mathcal H_R\) be arbitrary linear maps between finite dimensional spaces. If \(|B|{\lt}|G|^{b-1}\), then \(\operatorname {im}A_R\not\subseteq L(\operatorname {im}M)\). In particular, a block with \(m\) boundary coordinates per leg cannot cover the complete regular boundary image when \(m^b{\lt}|G|^{b-1}\). This is a dimension comparison relevant to the closed bondwise equivalence in [ 235 , Section 7 ] ; the source does not assert such containment for arbitrary open boundary tensors.
Containment would give \(|G|^{b-1}\leq \operatorname {rank}(LM)\leq \operatorname {rank}M\leq |B|\), contrary to the strict inequality.
On a native square torus with both periods at least three, take \(R=\{ 0,1\} \times \{ 0\} \) and a group of order six. Suppose the original incident-edge tensors are locally regular \(G\)-injective. The six actual crossing bonds have a numbering for which, for every block \(M:(\mathbb C^4)^{\otimes 6}\to \mathcal P\) and every linear physical map \(L:\mathcal P\to \mathcal H_R\), the image of \(LM\) does not contain the complete regular boundary image.
The positive bounded rectangle is connected. Its actual perimeter gives six crossing bonds, and \(4^6=4096{\lt}7776=6^5\). Apply the dimension obstruction.
27.32.15 Partial averaging and unrestricted exterior bonds
Let \(\mathcal V\) and \(\mathcal E\) be finite sets of vertices and labelled oriented bonds, with tail and head maps \(t,h:\mathcal E\to \mathcal V\). Each bond has its own finite virtual alphabet \(X_e\), and each vertex its own physical alphabet \(Y_v\). Parallel bonds and self-bonds are allowed; the two incidences of a self-bond remain distinct. Write \(v(e,-)=t(e)\), \(v(e,+)=h(e)\), \(I_v=\{ (e,\epsilon ):v(e,\epsilon )=v\} \), and \(\mathcal X_v=\prod _{(e,\epsilon )\in I_v}X_e\). An endpoint configuration \(\beta \) specifies all \(\beta _{e,\epsilon }\). The bijection \(R\) groups these endpoints by vertex, \((R\beta )_v=\beta |_{I_v}\), while \(S\) groups them into head–tail pairs, \((S\beta )_e=(\beta _{e,+},\beta _{e,-})\).
The tensor \(A_v(\eta ,s)\) defines \(T_vx(s)=\sum _{\eta \in \mathcal X_v}A_v(\eta ,s)x(\eta )\). Its network contraction with matrices \(B_e\) is
For a cut \(C\subseteq \mathcal E\), put \(\mathcal X_{\partial C}=\prod _{e\in C,\epsilon \in \{ -,+\} }X_e\), \(r_C\beta =\beta |_{C\times \{ -,+\} }\), and \(w_C(\beta )=\prod _{e\notin C}\delta _{\beta _{e,+},\beta _{e,-}}\). The cut map and its full range are
The boundary \(M:\mathcal X_{\partial C}\to \mathbb {C}\) is arbitrary. Expanding it in coordinate vectors shows that \(\mathcal S_{A,C}\) is spanned by networks with matrix units on \(C\) and identities elsewhere.
When the physical alphabets are \(\mathcal X_v\), put \(J=SR^{-1}\) and \((\mathcal R\psi )(\gamma )=\psi (J^{-1}\gamma )\). Define \(b_B(\sigma )=\prod _eB_e((J\sigma )_e^+,(J\sigma )_e^-)\) and the bond slice \(S_{e,\tau }\psi (s)=(\mathcal R\psi )(\tau [e\leftarrow s])\). Let \(G\) be a finite group and let \(U_e:G\to \operatorname {GL}(\mathbb {C}^{X_e})\) be a representation on each bond, matched at its two ends. Write \(F_e((a,b),g)=U_e(g)_{a,b}\), so \(\operatorname {im}F_e=\operatorname{span}_{\mathbb {C}}\{ U_e(g):g\in G\} \) in matrix coordinates. The incident representation \(\rho _v\) acts by
For any representations on these local spaces, use \(P_v=|G|^{-1}\sum _g\rho _v(g)\), \(A_v^0(\eta ,s)=(P_v)_{s,\eta }\), and \(\mathcal P=\bigotimes _vP_v\).
Products of finite-coordinate maps are taken in coefficient coordinates: \((\bigotimes _vF_v)x(\sigma )=\sum _\eta \prod _vF_v(\sigma _v,\eta _v)x(\eta )\). They compose coordinatewise. In finite dimensions their range consists exactly of vectors whose every one-coordinate slice lies in the corresponding local range. To see the reverse implication, choose \(L_v\) with \(F_vL_v\) the identity on \(\operatorname {im}F_v\). Each single-coordinate action \(F_vL_v\) fixes the vector, hence so does their product, giving the preimage \((\bigotimes _vL_v)\psi \). The same coordinatewise argument shows that a product of maps fixes a vector whenever each map fixes every slice at its coordinate.
These support and reconstruction statements provide the open-boundary steps of [ 235 , Theorem 5.4 ] .
Choose a group endomorphism \(f_v:G\to G\) at each vertex and define \(\rho _v^f(g)=\rho _v(f_v(g))\). Its action is \(\rho _v^f(g)x=W_v(f_v(g))x\). When \(f_v\) is the identity, \(\rho _v^f=\rho _v\); when \(f_v(g)=1\) for every \(g\), the representation \(\rho _v^f\) is trivial. These equalities follow by evaluating the composed representation at each group element.
The partially averaged tensor and its local map are
Thus the local map of \(A_v^f\) is \(P_v^f\). At an identity endomorphism this is the canonical incident averaging tensor. At a trivial endomorphism \(P_v^f=I\) and \(A_v^f(\eta ,s)=\delta _{s,\eta }\). Indeed every vector is invariant for a trivial representation, so averaging fixes it. In particular, one may average at core vertices while retaining identity tensors at exterior vertices.
At every vertex, \(P_v^f\) is \(G\)-injective for \(\rho _v^f\).
Right multiplication permutes the group average, giving \(P_v^f\rho _v^f(g)=P_v^f\). Every invariant vector is fixed by every summand, so \(P_v^f\) is the identity on \((\mathbb {C}^{\mathcal X_v})^{\rho _v^f}\) and is injective there.
For any matrix family \(B_e\), put \(B_e^f(q)=U_e(f_{h(e)}(q_{h(e)}))B_e U_e(f_{t(e)}(q_{t(e)})^{-1})\). With \(n=|\mathcal V|\), one has
All vertex labels are summed together; neither the inserted matrices nor the resulting coefficient function are required to factor further.
Expand (251) independently at every vertex, obtaining the scalar \(|G|^{-n}\) and one assignment \(q\) of group labels. Contracting the fixed incident actions gives \(B_e^f(q)\) on each bond. Applying the site-to-bond permutation gives the last formula, and the definition of \(b_B\) gives the vector equality.
For every \(q:\mathcal V\to G\) and every family \(B_e\), \(\mathcal N_{A^f}(B^f(q))=\mathcal N_{A^f}(B)\). In particular the replacement \(p_e\mapsto f_{h(e)}(q_{h(e)})p_e f_{t(e)}(q_{t(e)})^{-1}\) preserves the network with bond matrices \(U_e(p_e)\).
In (252), right multiplication \(r\mapsto rq\) is a bijection of vertex assignments. The identities \(f_v(r_vq_v)=f_v(r_v)f_v(q_v)\) and \((ab)^{-1}=b^{-1}a^{-1}\) identify the transformed summands. For representation-matrix insertions, use multiplicativity of \(U_e\).
If \(B_e=I\), every bond slice \(S_{e,\tau }\mathcal N_{A^f}(B)\) belongs to \(\operatorname {im}F_e=\operatorname{span}_{\mathbb {C}}\{ U_e(g):g\in G\} \). Consequently, if \(e\notin C\) and \(\psi \in \mathcal S_{A^f,C}\), then \(S_{e,\tau }\psi \in \operatorname {im}F_e\) for every \(\tau \). The boundary defining \(\psi \) may correlate every exposed incidence.
For an identity insertion, \(B_e^f(q)=U_e(f_{h(e)}(q_{h(e)})f_{t(e)}(q_{t(e)})^{-1})\). Fixing all other bond coordinates in each term of (252) leaves a scalar multiple of this matrix, hence a vector in \(\operatorname {im}F_e\). Linearity preserves this membership through the sum over \(q\). Finally expand an arbitrary cut boundary in coordinate vectors. Their networks have matrix units on \(C\) and identities outside \(C\), so the first assertion applies to every column of the cut map.
Fix a set \(C\) of exterior bonds. The label set at bond \(e\) is \(\Lambda _e=G\sqcup (X_e\times X_e)\). Write its two kinds of labels as \([g]\) and \([a,b]\). Define matrices \(Q_e^C\) by
Let \(\widetilde F_e^C:\mathbb {C}^{\Lambda _e}\to \mathbb {C}^{X_e\times X_e}\) be the matrix whose column at \(p\) is \(Q_e^C(p)\), read in head-row, tail-column order.
If \(e\notin C\), then \(\operatorname {im}F_e\subseteq \operatorname {im}\widetilde F_e^C\). If \(e\in C\), then \(\operatorname {im}\widetilde F_e^C=\mathbb {C}^{X_e\times X_e}\).
In the first case, extend any coefficient function on \(G\) by zero on \(X_e\times X_e\). The resulting column combination is unchanged. In the second case, for a matrix \(x\) take coefficient zero at every \([g]\) and coefficient \(x_{a,b}\) at \([a,b]\); its image is \(\sum _{a,b}x_{a,b}E_{a,b}=x\).
Suppose \(S_{e,\tau }\psi \in \operatorname {im}F_e\) for every \(e\notin C\) and every remaining bond configuration \(\tau \). Then there is a function \(c:\prod _e\Lambda _e\to \mathbb {C}\) such that
No condition is imposed on slices at exterior bonds, and \(c\) may correlate all interior and exterior labels.
Every slice of \(\mathcal R\psi \) belongs to \(\operatorname {im}\widetilde F_e^C\): use the hypothesis for interior bonds and the full-range assertion for exterior bonds. The product-range criterion gives \(\mathcal R\psi =(\bigotimes _e\widetilde F_e^C)c\). Expand this product map and undo \(\mathcal R\) to obtain (258).
27.32.16 Open cut intersections from removable internal labels
For arbitrary local tensors \(A\), if \(B_e=I\) whenever \(e\notin C\), then \(\mathcal N_A(B)\in \mathcal S_{A,C}\).
Take the boundary \(M_B(\eta )=\prod _{e\in C}B_e(\eta _{e,+},\eta _{e,-})\). Its contraction equals \(\mathcal N_A(B)\) because every remaining bond already carries the identity.
If \(C\subseteq C'\), then \(\mathcal S_{A,C}\subseteq \mathcal S_{A,C'}\) for arbitrary \(A\).
Expand \(\mathcal S_{A,C}\) in fixed-boundary columns. Each column is a network of matrix units on \(C\) and identities elsewhere. It therefore has identity insertions outside \(C'\) and belongs to \(\mathcal S_{A,C'}\). Taking linear spans proves the inclusion.
Suppose every \(Y_v\) is finite and let \(F_v:\mathbb {C}^{Y_v}\to \mathbb {C}^{Z_v}\). Then \((\bigotimes _v F_v)(\mathcal S_{A,C})=\mathcal S_{FA,C}\). The output alphabets \(Z_v\) need not be finite.
The maps satisfy \((\bigotimes _vF_v)\mathcal C_{A,C}=\mathcal C_{FA,C}\) for every boundary argument. Taking ranges gives the equality.
For any family of representations \(\rho _v\) on \(\mathbb {C}^{\mathcal X_v}\), let \(A^0\) be its averaging tensors and \(\mathcal P=\bigotimes _v P_v\). For every matrix family \(B\), \(\mathcal P b_B=\mathcal N_{A^0}(B)\).
Identity tensors give \(b_B=\mathcal N_I(B)\). Applying the product averaging projection replaces those identity tensors by \(A^0\).
Suppose that every \(p:\mathcal E\to G\) admits \(q:\mathcal V\to G\) such that
With \(\mathcal P^f=\bigotimes _vP_v^f\), every label family \(a\in \prod _e\Lambda _e\) satisfies \(\mathcal P^f b_{(Q_e^C(a_e))_e}\in \mathcal S_{A^f,C}\).
Read \(p_e=g\) from a label \([g]\) and \(p_e=1\) from a matrix-unit label. Averaging the bond product gives its network with tensors \(A^f\). Choose \(q\) from (259) and apply its vertex gauge. On every interior bond the transformed matrix is \(U_e(f_{h(e)}(q_{h(e)})p_ef_{t(e)}(q_{t(e)})^{-1})=I\). The exterior matrices remain unrestricted. The network therefore belongs to \(\mathcal S_{A^f,C}\).
Let \((C_i)_{i\in I}\) be a nonempty family of cuts and let \(C_0\) be an exterior cut contained in every \(C_i\). Suppose every \(e\notin C_0\) is uncut in at least one \(C_i\), and suppose (259) holds with \(C=C_0\). Then
Let \(\psi \) belong to every cut range. For each interior bond choose a cut leaving it uncut. Its slices then have representation support, so (258) gives \(\psi =\sum _a c(a)b_{Q(a)}\) with exterior cut \(C_0\). Choose one cut from the nonempty family. Its local maps are \(P_v^f\), and \((P_v^f)^2=P_v^f\), so applying their product leaves its contraction unchanged: \(\mathcal P^f\psi =\psi \). Hence
by removal of internal labels term by term. Conversely, \(C_0\subseteq C_i\) implies \(\mathcal S_{A^f,C_0}\subseteq \mathcal S_{A^f,C_i}\) for each \(i\).
Suppose the physical alphabets are finite and every \(T_v\) is \(G\)-injective for \(\rho _v^f\). Let the nonempty family \((C_i)_{i\in I}\) and the exterior cut \(C_0\) satisfy the containment, uncut-covering, and label-removal hypotheses of Theorem 27.32.16.6. Then
Set \(\mathcal T=\bigotimes _vT_v\). Choose \(L_v\) with \(L_vT_v=P_v^f\) and put \(\mathcal L=\bigotimes _vL_v\). Physical maps commute with cut contraction, so \(\mathcal L\mathcal C_{A,C}(M)=\mathcal C_{A^f,C}(M)\). Invariance gives \(T_vP_v^f=T_v\), hence also \(\mathcal T\mathcal C_{A^f,C}(M)=\mathcal C_{A,C}(M)\). Thus \(\mathcal L\) carries the original intersection into the canonical intersection; choosing one cut in the nonempty family shows \(\mathcal T\mathcal L\psi =\psi \) there. The reverse image inclusion follows from the second contraction identity. Therefore
The middle equality is (260); the last is the physical-image identity.
27.32.17 Lifted regional contractions and physical support
The constructions in this subsection and the reconstruction below require no group action. Retain finite incidence and virtual alphabets, but allow arbitrary physical alphabets unless finiteness is explicitly assumed. A region \(\Omega \subseteq \mathcal V\) retains all physical coordinates outside \(\Omega \) as part of one joint boundary.
Put \(Y_{\Omega ^c}=\prod _{v\notin \Omega }Y_v\) and write \(\sigma _{\Omega ^c}=(\sigma _v)_{v\notin \Omega }\) for the outside restriction of a physical configuration. If \(v\in \Omega \), then \((\sigma [v\leftarrow s])_{\Omega ^c}=\sigma _{\Omega ^c}\), because none of the restricted coordinates has changed.
For an arbitrary joint function \(M:\mathcal X_{\partial C}\times Y_{\Omega ^c}\to \mathbb {C}\), define
This expression is linear in \(M\). Its range is \(\widehat{\mathcal S}_{A,\Omega ,C} =\operatorname {im}\widehat\Gamma _{A,\Omega ,C}\). Only tensors in \(\Omega \) are contracted; the outside physical configuration and every exposed virtual index may be correlated in \(M\).
If \(v\in \Omega \) and \(\psi \in \widehat{\mathcal S}_{A,\Omega ,C}\), then every physical slice \(s\mapsto \psi (\sigma [v\leftarrow s])\) belongs to \(\operatorname {im}T_v\). In particular this holds for \(\psi =\widehat\Gamma _{A,\Omega ,C}(M)\) and every joint \(M\).
Changing \(\sigma _v\) leaves \(\sigma _{\Omega ^c}\) fixed. Separate the factor at \(v\) in (264) to obtain
Each vector \(s\mapsto A_v((R\beta )_v,s)\) is a column of \(T_v\). The finite sum therefore lies in its range. A boundary realizing any vector in the lifted range gives the general assertion.
Assume the physical alphabets are finite. Put \(\mathcal T=\bigotimes _vT_v\) and \(\mathcal H_A=\operatorname {im}\mathcal T\). Let \((\Omega _i,C_i)_{i\in I}\) be any family such that each vertex belongs to some \(\Omega _i\) or has \(\operatorname {im}T_v=\mathbb {C}^{Y_v}\). Then
Thus only vertices with nonsurjective local maps require regional coverage.
For a covered vertex choose a region containing it and apply its local-slice conclusion. At an uncovered vertex every slice belongs to the full local range. The product-range criterion then places the whole vector in \(\operatorname {im}\mathcal T\).
Say that \(C\) exposes the outside of \(\Omega \) when every incidence \((e,\epsilon )\) at a vertex outside \(\Omega \) has \(e\in C\). Under this condition a cut configuration \(\kappa \) determines \(o_v(\kappa )\in \mathcal X_v\) at every \(v\notin \Omega \), by \(o_v(\kappa )(e,\epsilon )=\kappa _{e,\epsilon }\). In particular \(o_v(r_C\beta )=(R\beta )_v\).
If \(C\) exposes the outside of \(\Omega \), then \(\mathcal S_{A,C}\subseteq \widehat{\mathcal S}_{A,\Omega ,C}\). For every cut \(C\), independently of this exposure condition, \(\mathcal S_{A,C}\subseteq \operatorname {im}(\bigotimes _vT_v)\). These containments do not require finite physical alphabets.
For a boundary \(M\) of the full cut, define \(\widetilde M(\kappa ,\tau ) =M(\kappa )\prod _{v\notin \Omega }A_v(o_v(\kappa ),\tau _v)\). Substituting \(o_v(r_C\beta )=(R\beta )_v\) and splitting the vertex product into \(\Omega \) and its complement gives \(\widehat\Gamma _{A,\Omega ,C}(\widetilde M) =\mathcal C_{A,C}(M)\). For the second inclusion take the virtual coefficient function \(x(\eta )=M(r_CR^{-1}\eta )w_C(R^{-1}\eta )\). Changing the incidence sum to site coordinates shows \((\bigotimes _vT_v)x=\mathcal C_{A,C}(M)\).
27.32.18 Reconstructing full cuts from lifted vectors
Given \(\kappa \in \mathcal X_{\partial C}\) and outside local configurations \(\xi \in \prod _{v\notin \Omega }\mathcal X_v\), define \(\kappa [\Omega ^c\leftarrow \xi ]\) by retaining \(\kappa _{e,\epsilon }\) at incidences whose vertex is in \(\Omega \), and replacing it by \(\xi _{v(e,\epsilon )}(e,\epsilon )\) at every other cut incidence. This replacement is defined whether or not \(C\) exposes every outside incidence.
For \(\tau \in Y_{\Omega ^c}\), write \(\sigma [\Omega ^c\leftarrow \tau ]\) for the configuration equal to \(\sigma _v\) on \(\Omega \) and \(\tau _v\) outside \(\Omega \). Its outside restriction is \(\tau \), and \(\sigma [\Omega ^c\leftarrow \sigma _{\Omega ^c}]=\sigma \). Both identities follow by evaluating the inside and outside coordinates.
Suppose the physical alphabets are finite and \(C\) exposes the outside of \(\Omega \). Choose arbitrary maps \(L_v:\mathbb {C}^{Y_v}\to \mathbb {C}^{\mathcal X_v}\) at the outside vertices. For a lifted boundary \(M\), define the full cut boundary
The old outside virtual labels \(\xi \) are summed independently of the new labels \(o_v(\kappa )\) used by \(L_v\).
Under the hypotheses of Definition 27.32.18.3, put \(\psi =\widehat\Gamma _{A,\Omega ,C}(M)\) and \(\psi _\sigma (\tau )=\psi (\sigma [\Omega ^c\leftarrow \tau ])\). Then
Expand the left side as a sum over endpoint labels \(\beta \), old outside virtual labels \(\xi \), and outside physical labels \(\tau \). Exchange the outside part of \(\beta \) with \(\xi \); this is an involution of the two configuration sets. It keeps every inside local configuration unchanged and replaces each outside local configuration by \(\xi _v\). Every uncut bond has both ends in \(\Omega \), so \(w_C(\beta )\) is unchanged by this exchange. The argument of \(M\) becomes \(r_C\beta \).
Split the tensor product over vertices into inside and outside factors. Summing each new outside virtual index gives
Interchanging the remaining finite sums yields \(\sum _\tau \prod _{v\notin \Omega }(T_vL_v)_{\sigma _v,\tau _v} \psi (\sigma [\Omega ^c\leftarrow \tau ])\), which is (268).
Suppose every physical alphabet is finite and \(C\) exposes the outside of \(\Omega \). With \(\mathcal H_A=\operatorname {im}(\bigotimes _vT_v)\),
In particular every lifted regional vector in product physical support is a full cut contraction. No injectivity or nonempty-alphabet assumption is needed.
The full cut range lies in both spaces on the right by Theorem 27.32.17.6. Conversely, let \(\psi =\widehat\Gamma _{A,\Omega ,C}(M)\in \mathcal H_A\). For each outside vertex choose a right inverse of \(T_v\) on its range and extend it to a linear map \(L_v\) on \(\mathbb {C}^{Y_v}\). Thus \(T_vL_vq=q\) for \(q\in \operatorname {im}T_v\). The product-range criterion places every one-coordinate slice of \(\psi \) in the corresponding local range. The same holds for the outside-coordinate function \(\psi _\sigma \), since changing an outside coordinate commutes with replacement of the outside configuration. Consequently \((\bigotimes _{v\notin \Omega }T_vL_v)\psi _\sigma =\psi _\sigma \), by the fixed-slice product identity. Substituting in (268) gives \(\mathcal C_{A,C}(K_{L,M})(\sigma ) =\psi _\sigma (\sigma _{\Omega ^c})=\psi (\sigma )\). This constructs a single full cut boundary for \(\psi \).
27.32.19 Intersections of lifted regional ranges
Suppose the physical alphabets are finite. Let \((\Omega _i,C_i)_{i\in I}\) be a family such that each \(C_i\) exposes the outside of \(\Omega _i\). Suppose every vertex lies in some \(\Omega _i\) or its local map \(T_v\) is surjective. Then
In particular, if all omitted local maps are surjective, then \(\widehat{\mathcal S}_{A,\Omega ,C}=\mathcal S_{A,C}\). For two regions \(\Omega ,\Xi \) with exposed cuts \(C,E\), it suffices that every vertex belongs to \(\Omega \cup \Xi \) or has a surjective local map; in that case
The boundary tensors witnessing the two memberships are independent.
A vector in every lifted range belongs to \(\mathcal H_A\) by the covering argument. For each \(i\), reconstruction then places it in \(\mathcal S_{A,C_i}\). The reverse inclusion follows from the inclusion of every full cut range in its lifted range. The one-region and two-region assertions are the corresponding specializations of this family equality.
Let \(G\) be finite and let \(\rho _v\) be representations on the virtual spaces. Suppose every physical alphabet is finite and \(T_v\) is \(G\)-injective for \(\rho _v\). Let \((\Omega _i,C_i)_{i\in I}\) be a nonempty family satisfying the exposure and coverage hypotheses of Theorem 27.32.19.1. Put \(\mathcal T=\bigotimes _vT_v\). Then
Here \(A^0\) consists of the averaging tensors of the \(\rho _v\).
Apply (271) to replace the lifted intersection by the intersection of full cut ranges. Choose \(L_vT_v=P_v\) and set \(\mathcal L=\bigotimes _vL_v\). The local identities \(L_vT_v=P_v\) and \(T_vP_v=T_v\) give, for every cut \(C\) and the same boundary \(M\), \(\mathcal L\mathcal C_{A,C}(M)=\mathcal C_{A^0,C}(M)\) and \(\mathcal T\mathcal C_{A^0,C}(M)=\mathcal C_{A,C}(M)\). Thus \(\mathcal L\) sends every vector in the full intersection to the canonical intersection. Choosing one cut gives its recovery under \(\mathcal T\), while the second identity gives the reverse inclusion. This proves (273).
Under the hypotheses of Theorem 27.32.19.2, there is one family \(L_v:\mathbb {C}^{Y_v}\to \mathbb {C}^{\mathcal X_v}\), with \(\mathcal L=\bigotimes _vL_v\), such that for every cut \(C\) and every joint boundary \(M\),
For every \(\psi \in \bigcap _i\widehat{\mathcal S}_{A,\Omega _i,C_i}\), the same family satisfies
By Theorem 26.1.2, choose \(L_v\) with \(L_vT_v=P_v\). Applying their product to a cut contraction replaces each local map \(T_v\) by \(P_v\) and retains the same joint boundary, proving (274) for all cuts at once. By (271), the vector \(\psi \) has a full boundary \(M_i\) at each cut \(C_i\). Applying (274) at every \(i\) proves canonical membership. Choose any one \(i\) from the nonempty family and recover its original contraction using \(T_vP_v=T_v\) at every vertex. This proves \(\mathcal T\mathcal L\psi =\psi \).
27.32.20 The open intersection for three four-legged tensors
Let \(V=\{ 0,\ldots ,10\} \) and \(E=\{ 0,\ldots ,9\} \), with directed edges
Put \(T=\{ 0,1,2\} \), \(R=\{ 0,1\} \), \(S=\{ 1,2\} \), \(C_0=E\setminus \{ 0,1\} \), \(C_L=E\setminus \{ 0\} \) and \(C_R=E\setminus \{ 1\} \). Each \(v\in T\) has four incidences and \(|C_0|=8\). Give each edge its own finite alphabet \(X_e\) and write \(\mathcal X_v=\prod _{p\in I_v}X_{e(p)}\) for its local virtual configurations. No equality between different \(X_e\) is imposed.
For representations \(U_e:G\to \operatorname {GL}(\mathbb {C}^{X_e})\), put \(f_v(g)=g\) on \(T\) and \(f_v(g)=1\) outside \(T\). The local action is \(\rho _v^f(g)=\rho _v(f_v(g))\), where \(\rho _v\) is the incident representation. Thus an edge acts by \(U_e(g)\) at its head and \(U_e(g^{-1})^{\mathsf T}\) at its tail; exterior vertices act trivially. An incoming exterior action \(U_e(g)\) is equivalently outgoing for \(U_e^\vee (g)=U_e(g^{-1})^{\mathsf T}\), since \(U_e^\vee (g^{-1})^{\mathsf T}=U_e(g)\). Semi-regularity is preserved by this change of orientation, by Theorem 26.1.13.
Every \(p:E\to G\) admits \(q:V\to G\) such that \(f_{h(e)}(q_{h(e)})p_ef_{t(e)}(q_{t(e)})^{-1}=1\) for \(e\notin C_0\).
Take \(q_0=1\), \(q_1=p_0^{-1}\), \(q_2=p_0^{-1}p_1^{-1}\) and \(q_v=1\) at the exterior vertices. The only identities required are \(p_0^{-1}p_0=1\) and \(p_0^{-1}p_1^{-1}p_1p_0=1\).
Let \(G\) be finite and let \(A_v:\mathcal X_v\times Y_v\to \mathbb {C}\) have finite physical alphabets. If each coefficient map \(T_v\) is \(G\)-injective for \(\rho _v^f\), then
This assertion does not require semi-regularity or unitarity of \(U_e\).
Both cuts contain \(C_0\); edge \(0\) is uncut in \(C_L\) and edge \(1\) in \(C_R\). Theorem 27.32.20.3 supplies the vertex labels required by Theorem 27.32.16.7.
Given physical alphabets \(Y_v\) and tensors \(A_v\) only for \(v\in T\), extend them to \(V\) by
The corresponding coordinate identifications give \(\widetilde T_v=T_v\) at core vertices and \(\widetilde T_v=I_{\mathbb {C}^{\mathcal X_v}}\) outside the core. In particular every exterior local map is surjective.
If the three \(T_v\), \(v\in T\), are \(G\)-injective for \(\rho _v\), all eleven \(\widetilde T_v\) are \(G\)-injective for \(\rho _v^f\).
At a core vertex the representation and coefficient map are unchanged, up to the coordinate identifications in Definition 27.32.20.5. At an exterior vertex the representation is trivial and the coefficient map is the identity, which is injective on the entire virtual space.
A semi-regular matrix representation on \(\mathbb {C}^X\), for finite \(X\), has \(X\ne \varnothing \). Hence independently semi-regular \(U_e\) supply a choice of one coordinate in each \(X_e\).
Semi-regularity includes the trivial irreducible representation, so there is a nonzero invariant vector. If \(X\) were empty, \(\mathbb {C}^X\) would be the zero vector space. Apply this argument to each edge.
For any alphabets \(Y_v\) on \(V\), write \(\Sigma _T=\prod _{v\in T}Y_v\) and \(\Sigma _{T\setminus Q}=\prod _{v\in T\setminus Q}Y_v\). Restriction is \(r_T(\sigma )=\sigma |_T\); constant extension of vectors is \(J_T\psi (\sigma )=\psi (r_T\sigma )\). For \(\tau \in \prod _{v\notin T}Y_v\), let \(a_T(s,\tau )\) equal \(s\) on \(T\) and \(\tau \) elsewhere. Then \(r_Ta_T(s,\tau )=s\).
If an exterior physical configuration \(\tau \) exists, \(J_T\) is injective.
Evaluate \(J_T\psi =J_T\chi \) at \(a_T(s,\tau )\) to obtain \(\psi (s)=\chi (s)\).
Let \(Q\subseteq T\) and \(C\subseteq E\). With endpoint coordinates \(\beta \in \prod _{p\in E\times \{ -,+\} }X_{e(p)}\), define
Set \(\mathcal K_{A,Q,C}=\operatorname {range}K_{A,Q,C}\). Here \(r_C\) retains both endpoint coordinates of each cut edge and \(w_C(\beta )=\prod _{e\notin C}\delta _{\beta _{e,+},\beta _{e,-}}\). The arbitrary function \(M\) may correlate the whole doubled cut with every omitted core physical coordinate.
Extend a boundary \(M\) constantly in the exterior physical coordinates, obtaining \(M^\uparrow \). Then \(J_TK_{A,Q,C}(M)=\widehat\Gamma _{A,Q,C}(M^\uparrow )\). Conversely, fix an exterior configuration \(\tau \) and restrict a graph boundary \(N\) to \(\tau \), obtaining \(N_\tau \). Then
Each contraction uses tensors only in \(Q\subseteq T\). Inserting \(M^\uparrow \) or fixing \(\tau \) therefore changes only the outside argument of the boundary, term by term in (283). These two constructions give the opposite range inclusions.
Partition the endpoints as \(I_T\sqcup I_{V\setminus T}\) according to their incident vertex. There are twelve core and eight exterior endpoints. Every tail belongs to \(T\); a head belongs to \(T\) exactly on edges \(0,1\). Write \(\eta \) and \(\xi \) for configurations on these two endpoint sets, and \(a(\eta ,\xi )\) for their union. The actual open boundary is \(\Theta =\prod _{e\in C_0}X_e\), with \(r_0\eta =(\eta _{e,-})_{e\in C_0}\). A doubled exterior cut is assembled by \(b(\theta ,\xi )_{e,-}=\theta _e\) and \(b(\theta ,\xi )_{e,+}=\xi _{e,+}\). Thus \(r_{C_0}a(\eta ,\xi )=b(r_0\eta ,\xi )\), its exterior restriction recovers \(\xi \), and its local core configurations are \(\eta |_{I_v}\).
Put \(w_0(\eta )=\delta _{\eta _{0,+},\eta _{0,-}} \delta _{\eta _{1,+},\eta _{1,-}}\). It equals \(w_{C_0}(a(\eta ,\xi ))\) for every exterior configuration. For an arbitrary joint boundary \(X:\Theta \to \mathbb {C}\), define
Set \(\mathcal O_A=\operatorname {range}\zeta _A\). The domain of \(X\) has precisely eight virtual factors; the output has precisely the three physical factors \(\mathbb {C}^{Y_0}\otimes \mathbb {C}^{Y_1}\otimes \mathbb {C}^{Y_2}\).
The product of omitted core physical alphabets for \(Q=T\) has its unique empty configuration \(*\). For a doubled-cut boundary \(M\), set \(m(M)(\theta )=\sum _\xi M(b(\theta ,\xi ),*)\). Given one exterior configuration \(\xi _0\), embed an arbitrary \(X\) by \(j_{\xi _0}(X)(\kappa ,*)=X((\kappa _{e,-})_{e\in C_0}) \delta _{\kappa |_{I_{V\setminus T}},\xi _0}\). Then \(m(j_{\xi _0}X)=X\).
Endpoint summation separates as \(\sum _\beta F(\beta )=\sum _\eta \sum _\xi F(a(\eta ,\xi ))\). Consequently \(K_{A,T,C_0}(M)=\zeta _A(m(M))\). If an exterior configuration \(\xi _0\) exists, then \(\mathcal K_{A,T,C_0}=\mathcal O_A\).
Split the independent endpoint product into its core and exterior factors. Only \(M\) depends on \(\xi \), so its sum is \(m(M)\); the internal weight and all three tensors retain precisely the factors in (285). Conversely use \(j_{\xi _0}X\), whose marginal is \(X\).
For \(Q\subseteq T\) put \(I_Q=\{ p:v(p)\in Q\} \) and \(B_{Q,C}=\{ p\in I_Q:e(p)\in C\} \). Regional configurations lie in \(\prod _{p\in I_Q}X_{e(p)}\) and their minimal boundary is \(\Theta _{Q,C}=\prod _{p\in B_{Q,C}}X_{e(p)}\). Denote restriction to \(B_{Q,C}\) by \(r_{Q,C}\) and restriction to \(I_v\) by \(\eta \mapsto \eta _v\). The two relevant boundary incidence sets satisfy \(|B_{R,C_L}|=|B_{S,C_R}|=6\).
Assemble regional labels \(\eta \) with complementary labels \(\xi \) into \(a_Q(\eta ,\xi )\). Assemble a doubled cut \(b_{Q,C}(\theta ,\xi )\) by taking \(\theta \) at its regional endpoints and \(\xi \) elsewhere. Then \(r_Ca_Q(\eta ,\xi )=b_{Q,C}(r_{Q,C}\eta ,\xi )\); restriction recovers \(\theta \), and local configurations in \(Q\) recover \(\eta _v\). If every uncut edge has both endpoints in \(Q\), all complementary endpoint labels are on the cut, and their restriction recovers \(\xi \).
Assume \(Q\subseteq T\) and every uncut edge has both endpoints in \(Q\). For \(\eta \in \prod _{p\in I_Q}X_{e(p)}\) let \(w_{Q,C}(\eta )=\prod _{e\notin C}\delta _{\eta _{e,+},\eta _{e,-}}\). This equals \(w_C(a_Q(\eta ,\xi ))\), independently of \(\xi \). For an arbitrary \(M:\Theta _{Q,C}\times \Sigma _{T\setminus Q}\to \mathbb {C}\), put
Set \(\mathcal G_{A,Q,C}=\operatorname {range}\Gamma _{A,Q,C}\). For \(Q=R,S\), the boundary has six virtual legs and one omitted physical leg, all of which may be correlated.
Under the same condition on uncut edges, define
Any fixed complementary configuration \(\xi _0\) gives \(m_{Q,C}(j_{\xi _0}M)=M\), preserving all virtual–physical correlations.
For \(Q\subseteq T\) with all uncut edges internal,
If a complementary configuration \(\xi _0\) exists, then \(\mathcal K_{A,Q,C}=\mathcal G_{A,Q,C}\).
Split the endpoint sum according to membership in \(I_Q\). All tensor factors and uncut identity pairings depend only on \(\eta \); summing \(\xi \) gives (287). Every minimal boundary is recovered by its embedding \(j_{\xi _0}\).
Let \(G\) be finite, let every \(U_e\) be semi-regular, and let the three maps \(T_v:\mathbb {C}^{\mathcal X_v}\to \mathbb {C}^{Y_v}\) be \(G\)-injective for their incident representations, with finite, independently chosen \(Y_v\). Then
Choose exterior virtual and physical configurations using Theorem 27.32.20.7. Identity exterior tensors are surjective, and \(R\cup S=T\). Thus the lifted reconstruction theorem and (277) give
Inverse images under \(J_T\) preserve intersections. Apply (284) to all three terms, then Theorem 27.32.20.15 to the whole core.
Let \(G\) be finite. Give the ten bonds independently chosen finite virtual alphabets and matched semi-regular representations \(U_e\). Let \(A_0,A_1,A_2\) be \(G\)-injective four-legged tensors for the incident actions, with independently chosen finite physical alphabets. Write \(\mathbf x=(x_2,x_3,x_4)\), \(\mathbf y=(x_5,x_6)\) and \(\mathbf z=(x_7,x_8,x_9)\). Then
Here \(a\in X_0\), \(b\in X_1\), and \(x_e\in X_e\). The functions \(M,N,X\) are arbitrary joint tensors on their displayed arguments. The equality is in the three-factor physical space; neither an auxiliary physical factor nor a factorization of a boundary is assumed. This is the intersection property of [ 235 , Theorem 5.4 ] , allowing the endpoint tensors to differ as well. Its displayed \(A\)–\(B\)–\(A\) case is obtained by identifying the two endpoint tensors and their alphabets. In the incidence diagram for \(\zeta (X)\), the eight legs carry \(x_2,\ldots ,x_9\) in clockwise order along the upper boundary of \(X\), starting at its left horizontal leg.
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Semi-regularity supplies one complementary virtual configuration for each region. Theorem 27.32.20.19 therefore identifies the two six-leg ranges with \(\mathcal K_{\widetilde A,R,C_L}\) and \(\mathcal K_{\widetilde A,S,C_R}\), without restricting their joint boundaries. Apply (288). The resulting range is the exact eight-leg map in (285), whose coordinate expansion is \(\zeta (X)\) in (290).
Let \(A\) be regular \(G\)-injective on a native torus with both periods at least three, with the same tensor repeated at every site. For each native plaquette \(R_v\), let \(P_v\geq 0\) have kernel equal to the image of the regional PEPS map, and put \(H=\sum _v(P_v\otimes \mathbf1_{R_v^c})\). Every vector \(\psi \in \ker H\), including zero, admits coefficients \(\mu _{g,h}\in \mathbb C\) such that
This is the coefficient form of [ 235 , Theorem 5.7 ] in the stated finite regular-torus setting.
The parent kernel equals the span of the commuting closures. Since the indexing set of commuting pairs is finite, membership in this span gives (291).
Under the hypotheses of 27.32.20.22, suppose \(A\) is \(G\)-isometric. Let \(R\) have connected occupied nearest-neighbour graph and simply connected closed-cell realization, and let \(b\) count its crossing bonds. For \(\psi \in \ker H\setminus \{ 0\} \), let \(M_\psi (\sigma ,\tau )=\psi (\sigma \cup \tau )\) be its physical coefficient matrix across \(R\mid R^c\). There is one density matrix \(\rho _R\), independent of \(\psi \), such that \(\operatorname {tr}(M_\psi M_\psi ^\dagger ){\gt}0\) and
The Rényi equality holds for every finite real \(\alpha \geq 0\). This is [ 235 , Theorem 6.9 ] for all actual parent ground states in the stated geometry.
Insert (291) into \(M_\psi \). Restricting an arbitrary global configuration to \(R\) and \(R^c\) and reassembling it recovers that configuration, so \(M_\psi =0\) would imply \(\psi =0\). The common-density theorem for nonzero commuting-closure superpositions therefore applies.
Under the hypotheses of 27.32.20.23, let \(\psi ,\phi \in \ker H\setminus \{ 0\} \). Write \(\widehat M_\psi \) and \(\widehat M_\phi \) for their normalized physical coefficient matrices. Every operator \(O\) on \(R\) has equal normalized expectation, and some physical unitary \(U\) on \(R^c\) satisfies
In particular, the complement of the nonwrapping rectangle of sites with both coordinates nonzero is \(\{ (x,y):x=0\lor y=0\} \). A unitary on these two width-one stripes relates any two normalized nonzero parent ground states. These are the conclusions of [ 235 , Theorem 6.7 and Corollary 6.8 ] in the stated scope.
Expand both ground vectors using (291). Their cuts are nonzero by restriction and reassembly. Apply the local-equivalence and stripe-unitary theorems to these derived expansions.
Under the hypotheses of 27.32.20.22, let \(R\) have connected occupied graph and simply connected closed-cell realization. Let \(b\) count its crossing bonds and let \(\psi \in \ker H\setminus \{ 0\} \). Its normalized reduced density \(\rho _R=M_\psi M_\psi ^\dagger /\operatorname {tr}(M_\psi M_\psi ^\dagger )\) has positive denominator, is positive semidefinite with trace one, and satisfies
Only \(G\)-injectivity is required. This is [ 235 , Corollary 6.10 ] for all actual parent ground states in the stated regular native-torus scope.
Substitute (291) into the physical coefficient matrix and apply the rank and zero-order entropy theorem to the resulting nonzero closure superposition.
27.32.21 Microscopic parent constraints and native seam boundaries
The parent-Hamiltonian argument in [ 235 ] uses the regional intersection and closure statements of Theorems 5.4 and 5.5. The auxiliary results below pass directly from microscopic plaquette constraints to boundary tensors on the original lattice.
Let \(\Gamma =(V,E)\) be a finite simple graph with ordered vertices. An edge \(e\) has tail \(t(e){\lt}h(e)\) and alphabet \(X_e=\{ 0,\ldots ,D_e-1\} \). Each site has its own tensor \(A_v\), with common physical codomain \(\mathbb C^d\) until the final theorem. For \(R\subseteq V\), define \(C_R=\{ e:\{ t(e),h(e)\} \not\subseteq R\} \). Replacing an incident edge by its unique endpoint at \(v\) identifies the original tensor coordinates with the independent endpoint coordinates. Thus contracting identity bonds gives exactly the original PEPS vector, and a diagonal endpoint assignment restricts to its original crossing labels. Write \(\mathcal S_{A,C}\) for the range of the actual cut contraction against one arbitrary joint boundary on the two incidences of each \(e\in C\).
Suppose \(D_e{\gt}0\) for every bond and that the finite family of regions \((R_i)_i\) covers \(V\). Let \(H_i\geq 0\) have kernel equal to the genuine open-region range on \(R_i\), and let \(H=\sum _i H_i\otimes I_{V\setminus R_i}\). Every \(\psi \in \ker H\) belongs to \(\bigcap _i\mathcal S_{A,C_{R_i}}\). Neither injectivity nor a common bond dimension is required.
Positivity gives every regional slice in the corresponding open-region range. Select its virtual boundary jointly with the omitted physical coordinates. Close both endpoints of each additional cut bond by an identity matrix. The endpoint sum is exactly the original graph sum; its exterior factor is \(\prod _{e\cap R_i=\varnothing }D_e\), which is nonzero and can be divided out. The resulting vector lies in the literal lifted regional range. Coverage forces every physical site slice into its tensor image. Independent local linear retractions then absorb the omitted physical factors into the same joint virtual boundary.
Let \(G\) be finite. Give each bond a matching semi-regular representation \(U_e\), and suppose each \(A_v\) is \(G\)-injective for its incident action. Suppose also that the covering region family is nonempty and covers every edge internally. Every physical parent vector admits canonical coordinates \(\xi \) recovered by \(\psi =(\bigotimes _v A_v)\xi \), with
Here \(b_p\) is the product of the matrices \(U_e(p_e)\) and \(c_p\) is the product trace-dual coefficient. One family of local inverses supplies \(\xi \) and all its regional cut conditions simultaneously.
Semi-regularity gives a nonzero invariant vector on each bond, so every bond alphabet is nonempty. Apply the genuine local inverses to the boundaries obtained in Theorem 27.32.21.2. Every uncut bond has representation-matrix support. Edge coverage and the product-range characterization give one coherent bond expansion. The trace-dual identities extract its coefficients. A nonzero extracted coefficient of a regional cut must have one vertex-label realization on all its internal bonds. Applying the original invariant tensors recovers \(\psi \).
Let \(V=\mathbb Z_w\times \mathbb Z_h\) with \(w,h\geq 3\). The cut \(C_{c,r}\) consists exactly of the horizontal bonds from \((x,y)\) to \((x+1,y)\) with \(x+1=c\), and the vertical bonds from \((x,y)\) to \((x,y+1)\) with \(y+1=r\).
On this torus, let every \(U_e\) be semi-regular and every \(A_v\) be independently \(G\)-injective. Let \(H\) be any positive microscopic plaquette parent with the exact open-region kernels. Then \(\ker H\subseteq \mathcal S_{A,C_{c,r}}\) for every \(c,r\). The boundary correlates all exposed incidences, and the physical vector retains every original site.
The plaquettes cover all vertices and edges. For each nonzero coefficient in (298), regional vertex labels telescope around every plaquette, giving identity holonomy. Translate the tree gauge to the chosen origin \((c,r)\); it sends all labels outside \(C_{c,r}\) to the identity. Applying the product of the local averages therefore turns each supported bond product into an actual cut contraction. The coherent sum lies in the same cut range. Finally the original site maps recover the physical vector.
Allow arbitrary finite physical alphabets \(Y_v\). Embed their tagged disjoint union in a common alphabet and let \(E_v:\mathbb C^{Y_v}\to \mathbb C^{\sum _w|Y_w|}\) be coordinate zero extension, with coordinate restriction \(R_vE_v=I\). Form the ordinary graph tensors \(B_v=E_vA_v\) and their positive plaquette parent \(H_B\). Every vector \(\Psi \) in the full ambient kernel has a unique original physical representative \(\psi \) satisfying
No restriction to the site images is imposed on \(\Psi \) beforehand.
Zero extension is injective and preserves \(G\)-injectivity. Each cut contraction commutes with the product of the physical extensions and restrictions. Vertex-covering parent constraints supply an actual cut representation of every ambient kernel vector; hence it lies in the product extension range. The product left inverse gives recovery and uniqueness. Apply Theorem 27.32.21.5 to \(B\) and restrict its boundary witnesses to the original independent physical spaces.
One joint boundary on the original seam.
Fix \(C=C_{c,r}\) on the native torus with periods \(w,h\geq 3\). For each edge, let \(t(e){\lt}h(e)\) be its ordered graph orientation and \(X_e=\{ 0,\ldots ,D_e-1\} \). Write \(\alpha ^-=(\alpha ^-_e)_{e\in C}\) and \(\alpha ^+=(\alpha ^+_e)_{e\in C}\), each in \(\prod _{e\in C}X_e\). These are independent tuples at the tail and head incidences, respectively; they are not identified across a cut bond. For \(z\in \prod _{e\notin C}X_e\), let \(\eta _v(\alpha ^-,\alpha ^+,z)\) put \(\alpha ^-_e\) at \(t(e)\) and \(\alpha ^+_e\) at \(h(e)\) when \(e\in C\), and put the same \(z_e\) at both incidences when \(e\notin C\). Define
The box \(\mathcal K_{A,C}\) contains every microscopic site and exactly the uncut identity contractions. Its one physical leg denotes the entire tuple \(\sigma =(\sigma _v)_{v\in V}\in \prod _vY_v\).
![\begin{tenkz}[rows={wire}, cols=2, bonds=none]
\tn[at=(1,1), name=parentSeamM, skin=ring,
ports={30:virtual,330:virtual}]{M}
\tn[at=2.5 e of parentSeamM, name=parentSeamK, skin=box,
ports={150:virtual,210:virtual,0:physical:$\sigma$}]{\mathcal K_{A,C}}
\tnwire[name=parentSeamTail]{parentSeamM.30}{parentSeamK.150}
\tnwire[name=parentSeamHead]{parentSeamM.330}{parentSeamK.210}
\tnmark[form=label, label pos=n]{on parentSeamTail 0.5}{$\alpha^-$}
\tnmark[form=label, label pos=s]{on parentSeamHead 0.5}{$\alpha^+$}
\end{tenkz}](tenkz_svg/tenkz-1ffaca20d93c81e2.svg)
For the independently \(G\)-injective tensors and matching semi-regular bond representations of Theorem 27.32.21.5, every vector in the positive microscopic plaquette-parent kernel equals \(\mathcal C_{A,C}(M)\) for some such joint \(M\), at every choice of \(c,r\).
Padding leaves the joint boundary unchanged.
Put \(d=\sum _v|Y_v|\) and choose the tagged coordinate inclusion \(\iota _v:Y_v\hookrightarrow \{ 0,\ldots ,d-1\} \). Zero extension and restriction have coefficients \((E_v)_{j,i}=\delta _{j,\iota _v(i)}\) and \((R_v)_{i,j}=\delta _{\iota _v(i),j}\). Write \(E=\bigotimes _vE_v\), \(R=\bigotimes _vR_v\) and \(B_v=E_vA_v\). The physical tuples below are \(i=(i_v)\in \prod _vY_v\) and \(j=(j_v)\in \{ 0,\ldots ,d-1\} ^{V}\). For every joint \(M\), padding commutes with (301).
![\begin{tenkz}[rows={wire}, cols=3, bonds=none]
\tn[at=(1,1), name=parentPadM, skin=ring,
ports={30:virtual,330:virtual}]{M}
\tn[at=2 e of parentPadM, name=parentPadA, skin=box,
ports={150:virtual,210:virtual,0:physical}]{\mathcal K_{A,C}}
\tn[at=2 e of parentPadA, name=parentPadE, skin=box,
ports={180:physical,0:physical:$j$}]{E}
\tnwire[name=parentPadTail]{parentPadM.30}{parentPadA.150}
\tnwire[name=parentPadHead]{parentPadM.330}{parentPadA.210}
\tnwire[name=parentPadPhysical]{parentPadA.0}{parentPadE.180}
\tnmark[form=label, label pos=n]{on parentPadTail 0.5}{$\alpha^-$}
\tnmark[form=label, label pos=s]{on parentPadHead 0.5}{$\alpha^+$}
\tnmark[form=label, label pos=n]{on parentPadPhysical 0.5}{$i$}
\end{tenkz}](tenkz_svg/tenkz-fc5d6b7d513a047a.svg)
\(=\) ![\begin{tenkz}[rows={wire}, cols=2, bonds=none]
\tn[at=(1,1), name=parentPadSameM, skin=ring,
ports={30:virtual,330:virtual}]{M}
\tn[at=2 e of parentPadSameM, name=parentPadB, skin=box,
ports={150:virtual,210:virtual,0:physical:$j$}]{\mathcal K_{B,C}}
\tnwire[name=parentPadSameTail]{parentPadSameM.30}{parentPadB.150}
\tnwire[name=parentPadSameHead]{parentPadSameM.330}{parentPadB.210}
\tnmark[form=label, label pos=n]{on parentPadSameTail 0.5}{$\alpha^-$}
\tnmark[form=label, label pos=s]{on parentPadSameHead 0.5}{$\alpha^+$}
\end{tenkz}](tenkz_svg/tenkz-81ead7c1001eec1f.svg)
In particular, \(R\mathcal C_{B,C}(M)=\mathcal C_{A,C}(M)\) with exactly the same boundary coefficients.
Recovery on the full ambient parent kernel.
The identity \(R_vE_v=I\) implies \(RE\psi =\psi \) for every original physical vector \(\psi \); thus \(E\) is injective.
![\begin{tenkz}[rows={wire}, cols=3, bonds=none]
\tn[at=(1,1), name=parentLeftPsi, skin=box,
ports={0:physical}]{\psi}
\tn[at=2 e of parentLeftPsi, name=parentLeftE, skin=box,
ports={180:physical,0:physical}]{E}
\tn[at=2 e of parentLeftE, name=parentLeftR, skin=box,
ports={180:physical,0:physical:$i$}]{R}
\tnwire[name=parentLeftOriginal]{parentLeftPsi.0}{parentLeftE.180}
\tnwire[name=parentLeftAmbient]{parentLeftE.0}{parentLeftR.180}
\tnmark[form=label, label pos=n]{on parentLeftOriginal 0.5}{$i'$}
\tnmark[form=label, label pos=n]{on parentLeftAmbient 0.5}{$j$}
\end{tenkz}](tenkz_svg/tenkz-02eb8d064ca1a812.svg)
\(=\) ![\begin{tenkz}[rows={wire}, cols=1, bonds=none]
\tn[at=(1,1), name=parentLeftRecovered, skin=box,
ports={0:physical:$i$}]{\psi}
\end{tenkz}](tenkz_svg/tenkz-ba967e8870419345.svg)
Conversely, let \(\Psi \in \ker H_B\) for the positive microscopic plaquette parent in Theorem 27.32.21.6. The parent constraints imply \(ER\Psi =\Psi \) without an assumed local-image condition.
![\begin{tenkz}[rows={wire}, cols=3, bonds=none]
\tn[at=(1,1), name=parentKernelPsi, skin=box,
ports={0:physical}]{\Psi}
\tn[at=2 e of parentKernelPsi, name=parentKernelR, skin=box,
ports={180:physical,0:physical}]{R}
\tn[at=2 e of parentKernelR, name=parentKernelE, skin=box,
ports={180:physical,0:physical:$j$}]{E}
\tnwire[name=parentKernelAmbient]{parentKernelPsi.0}{parentKernelR.180}
\tnwire[name=parentKernelOriginal]{parentKernelR.0}{parentKernelE.180}
\tnmark[form=label, label pos=n]{on parentKernelAmbient 0.5}{$j'$}
\tnmark[form=label, label pos=n]{on parentKernelOriginal 0.5}{$i$}
\end{tenkz}](tenkz_svg/tenkz-aa917016edb9ebf1.svg)
\(=\) ![\begin{tenkz}[rows={wire}, cols=1, bonds=none]
\tn[at=(1,1), name=parentKernelRecovered, skin=box,
ports={0:physical:$j$}]{\Psi}
\end{tenkz}](tenkz_svg/tenkz-a829a1626e8c950c.svg)
Thus the unique original representative is \(\psi =R\Psi \), and every native seam boundary for \(\Psi \) restricts to one for \(\psi \). The map \(ER\) is the coordinate projection onto \(\operatorname {ran}E\); padding is not assumed surjective onto the ambient physical space.
27.32.22 Nonuniform native parent classification
Let \(\Lambda =\mathbb Z_w\times \mathbb Z_h\) with \(w,h\geq 3\), and let \(E\) be the ordered edges of its native torus graph. Each edge \(e\) has its own finite virtual alphabet \(D_e\) and representation \(U_e\) of a finite group \(G\). Each vertex \(v\) has its own physical alphabet \(Y_v\) and tensor \(A_v\). The conclusions below give the native-simple-torus form of [ 235 , Theorems 5.7 and 5.9 ] , with no requirement that the representations, dimensions, or tensors be uniform.
Let \(\mathcal N_A(B)\) denote the actual contraction with bond matrices \(B_e\), acting from the tail index to the head index. Write \(p^{g,k}_e\) for the native closure labels: the ordered horizontal wrapping edges carry \(k^{-1}\), the ordered vertical wrapping edges carry \(g\), and other edges carry \(1\). Define
The local virtual alphabet is the product of the incident \(D_e\). The canonical tensor at \(v\) is the averaging projector \(P_v\) for its incident representation; write \(A^0_v=P_v\) and \(P=\bigotimes _vP_v\).
Given independent physical maps \(F_v\), finite physical alphabets give \(\zeta _{FA}(g,k)=(\bigotimes _vF_v)\zeta _A(g,k)\) and \(\mathcal Z_{FA}=(\bigotimes _vF_v)\mathcal Z_A\). If the original site maps \(T_v\) are invariant, \(T_vP_v=T_v\), so \(\mathcal Z_A=(\bigotimes _vT_v)\mathcal Z_{A^0}\).
Interchange the finite physical sums with the bond contraction. The identity \(T_vP_v=T_v\) then recovers every original site coefficient.
For the next regional-parent statements, take the physical alphabets to be a common finite alphabet. Independent physical alphabets are treated by Theorem 27.32.22.10.
On any finite ordered graph, suppose \(B_e=1\) for every edge internal to a region \(R\). Every exterior physical slice of \(\mathcal N_A(B)\) lies in the actual open-region range \(\mathcal S_{A,R}\). Moreover, invariant tensors satisfy
Neither assertion assumes semi-regularity or injectivity.
Split the endpoint indices into internal and noninternal edges. The internal identities identify their two endpoints, leaving the open-region contraction. Summing the noninternal indices supplies its boundary coefficient. For (303), first average the site representations and change the vertex labels in the finite gauge sum; then use \(T_vP_v=T_v\).
If each \(A_v\) is invariant, every commuting closure lies in every microscopic plaquette parent space. For the plaquette based at \(v=(x,y)\), the seams at \(x+2\) and \(y+2\) meet no internal edge. Consequently \(\mathcal Z_A\subseteq \bigcap _v\mathcal S_{A,R_v}^{\mathrm{ext}}\), where \(R_v=\{ x,x+1\} \times \{ y,y+1\} \) and the superscript denotes all exterior physical slices.
The native rightward transport is \(k\) on its seam; upward transport is \(g^{-1}\) on its seam. Their square holonomy is identity when \(gk=kg\). The translated tree gauge moves both seams to \(x+2,y+2\). Since neither coordinate equals either coordinate of the plaquette, all internal bond matrices become identities. Apply local regional support and then take linear combinations.
Suppose every \(U_e\) is semi-regular and every \(A_v\) is \(G\)-injective. Then \(\bigcap _v\mathcal S_{A,R_v}^{\mathrm{ext}}\subseteq \mathcal Z_A\). More precisely, for every flat bond assignment \(p\) there is a commuting pair \((g,k)\) such that \(PB(p)=\zeta _{A^0}(g,k)\), where \(B(p)\) is the product of the actual represented bond entries.
For a parent vector \(\psi \), the common local inverses give \(\xi \) with \(P\xi =\xi \), \((\bigotimes _vT_v)\xi =\psi \), and \(\xi =\sum _pc_pB(p)\). Regional trace-dual support makes every \(p\) with \(c_p\ne 0\) a vertex coboundary on each plaquette, hence flat. Its tree gauge leaves precisely two commuting seam holonomies. Gauge invariance of the actual average gives
For closure extraction and independence, allow independent physical alphabets \(Y_v\) again. These assertions concern the literal dependent contractions and do not require a new parent definition.
Local invariance gives \(\zeta _A(xgx^{-1},xkx^{-1})=\zeta _A(g,k)\). If all \(U_e\) are semi-regular, the native product trace-dual functional \(E_{a,b}\) satisfies
These equalities do not require \(g\) and \(k\) to commute.
The trace-dual pairings convert the local average into a sum over vertex labels \(q\), supported on \(p^{a,b}_e=q_{\mathrm{head}(e)}p^{g,k}_eq_{\mathrm{tail}(e)}^{-1}\). Off the seams these equations force neighboring \(q_v\) to coincide; nonseam connectivity therefore gives \(q_v=x\) everywhere. The two wrapping edges give \(a=xgx^{-1}\) and \(b=xkx^{-1}\), proving (305). On the diagonal the allowed \(x\) form the common centralizer, which contains identity. Constant gauge transformations and physical recovery prove the conjugacy invariance for the original tensors.
For a simultaneous-conjugacy class \(C=[(g,k)]\), put \(\zeta _A(C)=\zeta _A(g,k)\). This is well-defined. The set of closures indexed by commuting pairs equals the set indexed by commuting pair-conjugacy classes.
Suppose every \(U_e\) is semi-regular, all \(Y_v\) are finite, and every \(A_v\) is \(G\)-injective. The actual closure vectors belonging to distinct pair-conjugacy classes are linearly independent. In particular, \(\dim \mathcal Z_A=K\), where \(K\) is the number of simultaneous-conjugacy classes of commuting pairs.
Each functional in (305) vanishes on the other classes and is nonzero on its own class. Thus the canonical class vectors are independent. The product of genuine local \(G\)-injective inverses takes each original closure to its canonical closure, transferring independence. Restrict to commuting classes and count this independent spanning family.
Take the physical alphabets to be a common finite alphabet. For each plaquette choose any positive interaction \(H_v\) with kernel exactly its actual open-region range, and extend it by identity to the other sites. Under the preceding semi-regularity and \(G\)-injectivity assumptions, the genuine microscopic Hamiltonian \(H=\sum _vH_v^{\mathrm{ext}}\) satisfies \(\ker H=\mathcal Z_A\) and \(\dim \ker H=K\). This includes the canonical orthogonal-complement interactions of [ 235 , Theorems 5.7 and 5.9 ] .
Positivity identifies \(\ker H\) with the intersection of the local parent spaces. The two inclusions identify that intersection with \(\mathcal Z_A\), whose dimension was calculated above.
Now allow arbitrary finite \(Y_v\). Let \(E_v\) be the zero extension of \(Y_v\) into the common tagged alphabet \(\coprod _vY_v\), and let \(R_v\) restrict to its tagged coordinates. Put \(E=\bigotimes _vE_v\), \(R=\bigotimes _vR_v\), and \(\widetilde A_v=E_vA_v\). Every positive microscopic parent \(\widetilde H\) of these padded tensors has full ambient kernel
Every ambient kernel vector \(\Psi \) therefore has a unique original representative \(\psi \in \mathcal Z_A\) with \(E\psi =\Psi \), and \(\dim \ker \widetilde H=K\). No support restriction is imposed on \(\Psi \).
Zero extension preserves \(G\)-injectivity and commutes with every native closure, so \(\mathcal Z_{\widetilde A}=E\mathcal Z_A\). Apply the native kernel theorem to the existing padded graph tensor. Since \(RE=1\) on the full original physical space, \(E\) is injective. The kernel equality makes it surjective onto the full ambient kernel, where \(ER=1\). These are the asserted inverse maps, and the dimension is preserved.
27.33 Orthogonalization of nested cylinders
Let \(V\) be a finite set of sites, let \(\mathcal H_v\) be finite-dimensional coordinate Hilbert spaces, and write \(\mathcal H_R=\bigotimes _{v\in R}\mathcal H_v\). The empty tensor product is \(\mathbb {C}\). We use the canonical regrouping \(\mathcal H_V=\mathcal H_R\otimes \mathcal H_{V\setminus R}\). The following construction is the orthogonalization step in the proof of the adaptive patch constraint in the September 24, 2026 polynomial-PEPS manuscript. It applies to arbitrary nested regions and inside subspaces.
For \(S\subseteq \mathcal H_R\), its cylinder is \(\mathcal C_R(S)=S\otimes \mathcal H_{V\setminus R}\). Equivalently, a vector belongs to \(\mathcal C_R(S)\) precisely when every slice at a fixed complementary coordinate belongs to \(S\). The inside rank of the specified pair \((R,S)\) is \(\dim S\); this is distinct from the dimension of the cylinder. In particular, \(\mathcal C_R(0)=0\).
Let \(A\) act on \(\mathcal H_R\), and let \(P_S\) be the orthogonal projector onto \(S\subseteq \mathcal H_R\). Then
Each complementary slice of \((A\otimes I)x\) is \(A\) applied to the corresponding slice of \(x\). Conversely, choose a preimage under \(A\) for each complementary slice and join these slices into one vector. This proves (308). The identity extension of \(P_S\) is self-adjoint and idempotent, with the required range, proving (309).
For a finite family of subspaces \(S_i\subseteq \mathcal H_R\), a linear operator \(A\) on \(\mathcal H_R\), and \(R\subseteq U\subseteq V\),
Decompose each complementary slice to prove the sum identity for two subspaces, then use finite induction. For images, use \(AS=\operatorname {ran}(AP_S)\) and (308). For containing regions, extend \(P_S\) first to \(U\) and then to \(V\); the composite extension is \(P_S\otimes I_{V\setminus R}\).
Let \(R_0\subseteq \cdots \subseteq R_{n-1}\) and \(S_j\subseteq \mathcal H_{R_j}\). Set \(\mathcal C_j=\mathcal C_{R_j}(S_j)\). For \(0\leq j{\lt}n\), define
When \(n{\gt}0\), we have \(W_0=0\) and \(\widetilde S_0=S_0\).
For \(0\leq j{\lt}n\), write \(\mathcal E_{j-1}=\sum _{i{\lt}j}\mathcal C_i\). Then
If \(S_j\subseteq W_j\), then \(\widetilde S_j=0\).
Extend the defining sum for \(W_j\) from \(R_j\) to \(V\). Equations (310) and (312) identify its cylinder with \(\sum _{i{\lt}j}\mathcal C_i\). By (311), the cylinder of \((I-P_{W_j})S_j\) is the image of \(\mathcal C_j\) under \((I-P_{W_j})\otimes I_{V\setminus R_j}\). Equation (309) identifies this operator with \(I_V-P_{\mathcal E_{j-1}}\). Finally, if \(s\in S_j\subseteq W_j\), then \(P_{W_j}s=s\), so \((I-P_{W_j})s=0\).
The spaces \(\widetilde{\mathcal C}_j\) are pairwise orthogonal. For every integer \(k\geq 0\), with summation indices restricted to \(0\leq j{\lt}n\),
Each inside rank satisfies \(\dim \widetilde S_j\leq \dim S_j\), and hence \(\sum _j\dim \widetilde S_j\leq \sum _j\dim S_j\).
By (317), every vector in the new cylinder is obtained by subtracting its projection onto the earlier span. For \(x\in \mathcal C_j\), decompose \(x=P_{\mathcal E_{j-1}}x+(I-P_{\mathcal E_{j-1}})x\). This gives
The new range is orthogonal to the earlier span because \(P_{\mathcal E_{j-1}}(I-P_{\mathcal E_{j-1}})=0\). Induction on the index gives equality of the original and new prefix spans. Every earlier new cylinder lies in the earlier original span, proving pairwise orthogonality. Finally, apply the image-dimension inequality to (314) inside \(\mathcal H_{R_j}\) and sum over \(j\).
Put \(\mathcal E=\sum _{j{\lt}n}\mathcal C_j\). Its orthogonal projector is
Distinct summands have product zero. For any finite family of regions, even without nesting, if every \(R_j\subseteq U\), the projector onto \(\sum _j\mathcal C_j\) is supported on \(U\): there is an orthogonal projector \(B\) on \(\mathcal H_U\) whose identity extension is that projector.
In \(\mathbb {C}^2\), let \(W=\operatorname{span}\{ e_1\} \) and \(S=\operatorname{span}\{ e_1+e_2\} \). Then \((I-P_W)S=\operatorname{span}\{ e_2\} \), whereas \(S\cap W^\perp =0\). The compression \((I-P_W)P_S(I-P_W)=\tfrac 12 P_{\operatorname{span}\{ e_2\} }\) is not idempotent, so it is not the projector onto the projected image. The diagram records the restrictions of the two orthogonal projections. The generator \(e_1+e_2\) maps to \(e_1\) and \(e_2\), respectively.
![\begin{tikzcd}[column sep=huge,row sep=large]
W=\spn\{e_1\} \arrow[d,"I-P_W"'] &
S=\spn\{e_1+e_2\} \arrow[l,"P_W"'] \arrow[d,"I-P_W"] \\
\{0\} & (I-P_W)S=\spn\{e_2\} \arrow[l,"P_W"]
\end{tikzcd}](tenkz_svg/tenkz-79b90ecfd09aefa8.svg)
27.34 A uniform width bound for angular reset scales
Fix real constants \(C_3,C_4,K\) and a positive constant \(c\). The angular information argument bounds the reset width at scale \(s\) by
All powers in this section are real powers. The estimates below are used only for \(L\ge 2\) and \(s\ge 1\), where their bases are positive. They formalize the uniform scale argument following equation info-reset-scale-cost in the September 24, 2026 polynomial-PEPS manuscript. The source has nonnegative coefficients and exponent; the analytic bounds allow arbitrary fixed real values.
There is a real \(S\ge 1\) such that, for every \(s\ge S\),
For every fixed real \(K\), \((\log x)^K/x\) tends to zero as \(x\) tends to infinity. Substituting \(x=es\) and using \(\log (es)=1+\log s\) proves the same limit for \((1+\log s)^K/s\). Multiplication by the fixed coefficient and the positive tolerance \(c/2\) give one threshold valid for all larger scales.
Fix \(D{\gt}0\) with \(D\ge 4C_4/c\). There is a real \(L_0\ge 2\) such that, for every real \(L\ge L_0\), one has \(e^{\max (K,1)}\le D\log L\), and for every real \(s\ge D\log L\),
In particular one can choose \(D=\max \{ 1,4C_4/c\} \) before choosing \(L_0\). Once the coefficients and exponent are fixed, both choices are independent of \(L\) and \(s\).
Take \(S\) from Theorem 27.34.1. Set \(T=\max \{ S,e^{\max (K,1)}\} \) and \(L_0=\max \{ 2,e^{T/D}\} \). Then \(D\log L\ge T\) for all \(L\ge L_0\). Then every allowed \(s\) is at least \(S\), so the polylogarithmic term is at most \(cs/2\). The slope inequality and \(\log L\ge 0\) give \(C_4\log L\le (c/4)D\log L\le cs/4\). Adding proves the first inequality. The second is strict because \(c{\gt}0\) and \(s\ge 1\).
With the same fixed constants, any width function bounded above by \(B(L,s)\) is strictly smaller than \(cs\) throughout the stated range.
If \(w(L,s)\le B(L,s)\), then (323) gives \(w(L,s)\le B(L,s)\le \frac{3c}{4}s{\lt}cs\).
This establishes the numerical clearance needed at every annular layer. The physical reset theorem, geometric core separation, entropy estimates, and annular induction remain separate obligations.
27.35 Tag encodings with regional zero resets
Let \(V\) be a finite set of sites, with finite coordinate sets \(A_v\), and choose a configuration \(z\in \prod _{v\in V}A_v\). Write \(\mathcal H_R=\mathbb {C}^{\prod _{v\in R}A_v}\) and \(|0_R\rangle =\bigotimes _{v\in R}|z_v\rangle \). For nested regions \(R_0\subseteq \cdots \subseteq R_{n-1}\) and subspaces \(S_j\subseteq \mathcal H_{R_j}\), let \(\widetilde S_j\) be the projected images of Definition 27.33.4, and put \(\mathcal E=\sum _{j{\lt}n}\mathcal C_{R_j}(S_j)\). The tag and zero-reset construction is the linear map appearing in the adaptive hole encoding of the September 24, 2026 polynomial-PEPS manuscript.
For a finite coordinate set \(A\), a coordinate \(a_0\in A\), and an arbitrary vector \(v\in \mathbb {C}^A\), define \(Z_{a_0}(v)=|a_0\rangle \langle v|\). Its entry at \((a,b)\) is \(\overline{v(b)}\) if \(a=a_0\), and zero otherwise.
For any \(v\in \mathbb {C}^A\) and any orthonormal basis \((b_k)_{k\in B}\) of a subspace \(S\subseteq \mathbb {C}^A\),
and
The reset identity does not require \(v\) to be normalized.
The only nonzero output coordinate of \(Z_{a_0}(v)\) is \(a_0\). Its contribution to the Gram entry at \((a,b)\) is \(v(a)\overline{v(b)}\). Expansion in an orthonormal basis gives \(P_S\psi =\sum _k b_k\langle b_k,\psi \rangle \).
Set \(d_j=\dim \widetilde S_j\) and choose an orthonormal basis \((v_{j\ell })_{0\leq \ell {\lt}d_j}\) of each \(\widetilde S_j\). Define the tag set \(T=\{ (j,\ell ):0\leq j{\lt}n,\ 0\leq \ell {\lt}d_j\} \) and the tag space \(\mathcal J=\mathbb {C}^T\). The encoding \(K\colon \mathcal H_V\to \mathcal J\otimes \mathcal H_V\) is
In particular, the output retains one raw register at every site. For input and output configurations \(y,x\in \prod _{v\in V}A_v\), its matrix coefficient is
The reset region in this formula is \(R_j\), determined by the tag. At a fixed tag \(t=(j,\ell )\), the following tensor product depicts its summand.
![\begin{tenkz}[rows={wire,wire,wire}, cols=2, bonds=none]
\providecommand{\tnlHeReset}{R_j}
\providecommand{\tnlHeSites}{V}
\tn[at=(1,1), name=holeInsideBra, skin=box,
ports={180:physical:$y\vert_{\tnlHeReset}$}]{\bra{v_{j\ell}}}
\tn[at=3 e of holeInsideBra, name=holeResetKet, skin=box,
ports={0:physical:$x\vert_{\tnlHeReset}$}]{\ket{0_{\tnlHeReset}}}
\tn[at=2 s of holeResetKet, name=holeTagKet, skin=box,
ports={0:physical:$t$}]{\ket{j,\ell}}
\tn[at=4 s of holeInsideBra, name=holeComplement, skin=box,
ports={180:physical:$y\vert_{\tnlHeSites\setminus \tnlHeReset}$,
0:physical:$x\vert_{\tnlHeSites\setminus \tnlHeReset}$}]{I_{\tnlHeSites\setminus \tnlHeReset}}
\end{tenkz}](tenkz_svg/tenkz-daa8af54e2921924.svg)
Identifying coordinate vectors with their Euclidean Hilbert space preserves the dimension of every inside subspace. The tag dimension satisfies
The tags form a disjoint union of sets of cardinality \(d_j\). Sum the inside-dimension inequalities \(\dim \widetilde S_j\leq \dim S_j\).
The encoding has Gram operator
Orthogonality of the tag vectors removes all products between distinct summands of (326). For each remaining summand, \(\langle 0_{R_j} | 0_{R_j} \rangle =1\), so its Gram operator is \(|v_{j\ell }\rangle \! \langle v_{j\ell }|\otimes I_{V\setminus R_j}\). Summing over the orthonormal basis of \(\widetilde S_j\) gives \(P_{\widetilde S_j}\otimes I_{V\setminus R_j}\). Theorem 27.33.7 identifies the sum over \(j\) with \(P_{\mathcal E}\).
For every \(\psi \in \mathcal H_V\),
Thus \(\| K\| \leq 1\).
Since \(P_{\mathcal E}\) is an orthogonal projector, the \(C^*\)-identity gives \(\| K\| ^2=\| K^\dagger K\| =\| P_{\mathcal E}\| \leq 1\). Also,
Taking nonnegative square roots gives the norm equality. The operator-norm bound gives \(\| K\psi \| \leq \| K\| \| \psi \| \leq \| \psi \| \).
For the encoding \(K\) of Definition 27.35.3, every \(\psi \in \mathcal E\) satisfies \(\| K\psi \| =\| \psi \| \). This follows from the Gram identity in the one-hole encoding construction.
Since \(\psi \in \mathcal E\), its orthogonal projection satisfies \(P_{\mathcal E}\psi =\psi \). Hence \(\| K\psi \| =\| P_{\mathcal E}\psi \| =\| \psi \| \) by Theorem 27.35.6.
With a singleton tag set \(\{ \ast \} \), define \(K_{\mathrm{id}}\psi =|\ast \rangle \otimes \psi \). This is the encoding used when an outer square has empty physical sample.
The identity encoding satisfies \(K_{\mathrm{id}}^\dagger K_{\mathrm{id}}=I\). For every \(\psi \in \mathcal H_V\), \(\| K_{\mathrm{id}}\psi \| =\| \psi \| \).
The sum over the singleton tag has one term, and the raw matrix is \(I\). Thus the Gram matrix is \(I^\dagger I=I\). Equality of Gram matrices gives equality of the vector norms.
If \(n=0\) or every \(S_j=0\), then \(T\) is empty and \(K=0\).
An empty family has no branch index. If every \(S_j=0\), the tag dimension bound gives \(|T|\leq \sum _{j{\lt}n}\dim (0)=0\). Hence \(T\) is empty. In either case there are no tag coordinates, so the matrix is zero.
If \(n=0\), the tag set \(T\) is empty and \(K=0\); also \(\mathcal E=0\), in agreement with (331). More generally, a zero-dimensional projected image contributes no tags. For an empty region \(R_j\), the inside configuration set is a singleton and \(\mathcal H_{R_j}=\mathbb {C}\); the reset condition is automatic and the complement is all of \(V\). The separate choice \(K_{\mathrm{id}}\) has one tag and retains every raw configuration.
27.36 Regional embeddings in uniform physical dimension
Let \(V\) be finite, let \(q\in \mathbb {N}\), and set \(A_q=\{ 0,\ldots ,q-1\} \), with \(A_0=\varnothing \). For \(S\subseteq V\), write \(X_S=A_q^S\) and \(\mathcal H_S=\mathbb {C}^{X_S}\). Fix a region \(R\subseteq V\), its inclusion \(j_R:R\hookrightarrow V\), and an arbitrary matrix \(H\in \operatorname {End}(\mathcal H_R)\). The restriction map identifies \(X_V\) with \(X_R\times X_{V\setminus R}\); let \(L_R(H)\) denote \(H\otimes I_{V\setminus R}\) in the \(X_V\) coordinates. For a site inclusion \(j\), let \(E_j(H)\) act by \(H\) on its image and by the identity on the remaining sites.
For every \(\sigma ,\tau \in X_V\),
Since \(j_R(R)=R\), agreement outside the image of \(j_R\) is precisely equality of the restrictions to \(V\setminus R\). The complementary identity matrix contributes the Kronecker delta \(\delta _{\sigma |_{V\setminus R},\tau |_{V\setminus R}}\). Multiplying it by \(H_{\sigma |_R,\tau |_R}\) gives (334) for both operators.
Let \(\widetilde X_V=\prod _{v\in V}A_q\) and \(\theta _q:X_V\to \widetilde X_V\) be the bijection \(\theta _q(\sigma )=(\sigma (v))_{v\in V}\). Write \(D_R(H)\) for \(H\otimes I_{V\setminus R}\) after splitting this indexed product into its regional and complementary factors. For every \(\alpha ,\beta \in \widetilde X_V\),
The restrictions of \(\theta _q^{-1}(\alpha )\) and \(\theta _q^{-1}(\beta )\) select the same coordinates as the two indexed product factors. Both sides are \(H\) at the regional coordinates when the complementary coordinates agree, and zero otherwise.
If \(R=\varnothing \), then \(X_R\) has one element and \(L_R([c])=cI\). If \(R=V\), there is no complementary agreement condition. If \(q=1\), all configuration sets have one element, so every regional operator extends as a scalar identity. If \(q=0\) and \(V\ne \varnothing \), then \(X_V=\varnothing \) and the resulting operator is zero. If \(V=\varnothing \), then \(X_V\) still has one element, including when \(q=0\), and its scalar entry is retained.
Take \(V=\{ 0,1\} \), \(R=\{ 0\} \), \(q=2\), and \(H=\bigl(\begin{smallmatrix} 0 & i \\ 2+i & 0 \end{smallmatrix}\bigr)\). Formula (334) gives \(L_R(H)_{(0,0),(1,0)}=i\), \(L_R(H)_{(1,0),(0,0)}=2+i\), and \(L_R(H)_{(0,0),(1,1)}=0\).
27.37 Finite-domain area laws and polynomial PEPS approximation
An entropy bound for every cut and an approximation of the entire vector are different assertions. The first concerns arbitrary finite induced lattice domains. The second asks for one compatible tensor representation on the original open square. We state the two models separately, following [ 247 , 248 ] .
Let \(\Lambda \subset \mathbb Z^2\) be finite. Join sites at coordinate distance one, and let \(d_\Lambda \) be distance in this induced graph, with value \(+\infty \) between distinct components. For \(X\subseteq \Lambda \) and an integer \(r\ge 0\), the closed graph neighborhood
consists of the sites joined to some site of \(X\) by an induced walk of length at most \(r\). At every site put \(\mathbb C^q\), \(q\ge 1\). In configuration coordinates the physical Hilbert space is
Restriction to \(A\) and \(\Lambda \setminus A\) identifies the configuration set with the product of the two regional configuration sets. The configuration set of a region \(A\) has \(q^{|A|}\) elements. Empty domains, holes, and disconnected components are permitted.
Fix an integer \(R\ge 0\) and \(J{\gt}0\). For each nonempty \(X\subseteq \Lambda \) with \(\operatorname {diam}_{d_\Lambda }X\le R\), assign one Hermitian operator \(h_X\) supported on \(X\), with \(\| h_X\| \le J\). The assigned operator may be zero, and acts as the identity on the complementary tensor factors. Define
The norm is the Euclidean operator norm. Boundary edges are unordered and counted once; empty and full cuts have empty boundary.
A pair has induced distance at most \(R\) if and only if an induced walk of length at most \(R\) joins it. Every singleton is an admissible support at every range, and at range zero the admissible supports are exactly the singletons. Adjacent pairs are admissible at range one. The nearest-neighbor graph on \(\{ 0,\ldots ,w-1\} \times \{ 0,\ldots ,h-1\} \) is the induced graph of that integer rectangle, with the corresponding identification of configurations and range-one pair supports.
The extended graph distance is the least walk length, or infinity when no walk exists. A zero-length walk has equal endpoints. A nonempty support of diameter zero therefore has one element. The coordinate embedding preserves horizontal and vertical nearest neighbors and is bijective onto the integer rectangle. Restricting configurations along this bijection gives the last identification.
For a unit vector \(\Omega \), a real \(E_0\), and \(\Delta {\gt}0\), the eigenvector and full-system projector-gap conditions are
For \(A\subseteq \Lambda \), put
All logarithms are natural, with \(0\log 0=0\). The reduced matrix is Hermitian, including when its spectrum contains zero. No condition is imposed on the gap of a Hamiltonian restricted to a smaller region. The uniform boundary assertion requires a single finite constant for each \(q,R,J,\Delta \), chosen before every domain, Hamiltonian, ground vector, and cut.
Let \(H\) be a complex matrix indexed by a finite set, let \(E_0\) be real, and let \(\Omega \) be a unit vector satisfying (343) and (344) with \(\Delta {\gt}0\). Then the eigenspace of \(H\) at \(E_0\) is \(\operatorname {span}_{\mathbb C}\{ \Omega \} \). This is an elementary consequence of the global gap, giving the uniqueness assertion in [ 247 , Proposition 4.5 ] .
For an eigenvector \(\psi \) at \(E_0\), the gap inequality gives \(0\ge \Delta \| \psi -\langle \Omega ,\psi \rangle \Omega \| ^2\). Since \(\Delta {\gt}0\), \(\psi =\langle \Omega ,\psi \rangle \Omega \). Conversely, \(H\Omega =E_0\Omega \) puts every scalar multiple of \(\Omega \) in this eigenspace.
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\ge 0\) be an integer, and let \(\Omega \in \mathcal H_\Lambda \). For every \(A\subseteq \Lambda \), the regional reduction defined in Definition 27.37.4 satisfies
If \(\| \Omega \| =1\), then for every \(A\subseteq \Lambda \),
If \(\| \Omega \| =1\), the empty region and the full region also satisfy
No Hamiltonian or spectral-gap hypothesis is required. The identities include empty domains and regions and use the one-dimensional empty tensor product. The site dimension \(q\) may here be zero, in which case \(\mathcal H_\Lambda \) is the zero space for nonempty \(\Lambda \); the standing assumption \(q\ge 1\) of Definition 27.37.1 is not needed. These are the density and entropy conventions of [ 247 , Section 2 ] .
The pure-state matrix \(|\Omega \rangle \langle \Omega |\) is positive semidefinite. Restriction of configurations to \(A\) and its complement changes only the coordinates, and partial trace preserves positivity and trace. Thus \(\operatorname {Tr}\rho _A=\operatorname {Tr}|\Omega \rangle \langle \Omega | =\| \Omega \| ^2\). For a unit vector, the reduced matrix is positive with trace one. Its eigenvalues lie in \([0,1]\), so each term \(-\lambda \log \lambda \) in its entropy is nonnegative, with \(0\log 0=0\).
For the empty region, the reduced matrix acts on a one-dimensional space. For the full region, write \(\rho _\Lambda =MM^*\) using the coefficient matrix of the vector under the configuration splitting. The complement has one configuration, so \(M\) has one column and \(\rho _\Lambda \) has rank at most one. A positive trace-one matrix of rank at most one has one eigenvalue equal to one and all remaining eigenvalues zero. Its entropy therefore vanishes in both cases.
For fixed integers \(q\ge 1\), \(R\ge 0\) and reals \(J,\Delta {\gt}0\), there is a finite constant \(C=C(q,R,J,\Delta )\) such that every finite induced domain with the above interaction bounds and a unique unit ground vector, up to phase, satisfying the full-system projector gap obeys \(S_\Omega (A)\le C|\partial _\Lambda A|\) for every \(A\subseteq \Lambda \). The same constant applies to all domains, interaction families, and cuts. This is Theorem 1.1 of [ 247 ] .
For disjoint \(B,C\subseteq \Lambda \) write \(I_\Omega (B:C)=S_\Omega (B)+S_\Omega (C)-S_\Omega (B\cup C)\), and let \(Z\) be the set of endpoints of the edges in \(\partial _\Lambda A\). Empty and full cuts have zero entropy, and so does every cut with \(\partial _\Lambda A=\varnothing \). Put \(b=|\partial _\Lambda A|{\gt}0\).
The global gap first gives an exponent below volume: for a box \(Q\) of side at most \(r\), sufficiently far from \(Z\),
Spectral filtering replaces \(H\) by positive quasi-local constraints that annihilate \(\Omega \) and keep a global gap. A replica argument then compares two bounds on a filtered norm while a finite randomized scan moves the partition through a collar. Repeating the comparison improves the exponent in (353) and gives, for a template \(T\) at scale \(n\) and \(L_n=\lfloor n^{1-\varepsilon }\rfloor \),
Here \(\varepsilon \), \(\alpha \) and \(\beta \) are the exponents of Theorem 27.37.31. Products of the positive constraints amplify this bound. For every \(M\ge 1\) there is a radius \(r\le C_Mn^{(1-\varepsilon )/\alpha }\) with \(L_n+2r=o(n^\beta )\) such that
The planar construction partitions \(A\) into a residual set \(D\) and an ordered family of pieces \(X_i\subseteq A\cap T_i\) with labels \(f_i\in \{ 0,1\} \), templates \(T_i\) at scales \(n_i\), and
Put \(S_i=A\cap (T_i)_{L_{n_i}}\) and \(J_i=(\Lambda \setminus A)\cup \bigcup _{j{\lt}i,\, f_j=f_i}X_j\), and let \(r_i\) be the radius in (355) at scale \(n_i\), with the scale threshold large enough that \(L_{n_i}+2r_i{\lt}n_i^\beta \). A walk of length at most \(2r_i\) starting in \(S_i\) stays within ambient distance \(n_i^\beta \) of \(T_i\). By (357) it meets no earlier piece of the same family, and it cannot leave \(A\), since its first step out of \(A\) would end at a point of \(Z\). Hence \(J_i\cap N_{2r_i}^\Lambda (S_i)=\varnothing \). With \(M=100\), (355) and monotonicity of mutual information give
For one family \(f\), with \(A_f=\bigcup _{f_i=f}X_i\) and \(E_f=\sum _{f_i=f}\eta _i\), the chain rule, purity and subadditivity give
Adding (359) for \(f=0\) and \(f=1\) cancels every piece entropy. With (356),
Every threshold and constant above depends only on \(q,R,J,\Delta \), and is chosen before the domain and the cut.
On the original open \(L\times L\) graph, let
where terms are supported on their designated sites or edges, and only the total \(H\) is assumed Hermitian. For a unit vector \(\Omega \), a real \(E_0\), and \(\Delta {\gt}0\), the gapped ground-vector condition for this \(H\) is
A PEPS approximation with parameters \(C,c\) is a tensor family on those same edges with positive dimensions \(D_e\) satisfying \(D_e\le CL^c\) for every edge \(e\), and a nonzero contracted vector \(\Phi \) such that, for some \(\theta \in \mathbb R\),
The norm in (365) is the Euclidean norm of the entire vector. The local tensor maps need not be isometries and the bond dimensions need not be equal. The uniform approximation assertion requires \(C,c{\gt}0\) to be chosen for fixed \(q\ge 2\), \(J,\Delta {\gt}0\) before all \(L\ge 2\), Hamiltonians, and ground vectors.
Let \(\mathcal I_L\) be the disjoint union of the vertices of the open square and its forward nearest-neighbor edges. Every undirected edge has exactly one such orientation. The original term indexed by a vertex is its site term, and the term indexed by a forward edge is the term on the corresponding undirected edge.
The support \(S_i\) is \(\{ v\} \) for the vertex label \(i=v\), and \(\{ v,w\} \) for the forward edge label \(i=(v,w)\).
For each original term \(A_i\), put
This is the Hermitian replacement of the local terms in [ 248 , Theorem 1.1 ] . Only the original total Hamiltonian is assumed Hermitian.
Each \(A_i^{\mathrm h}\) is supported on \(S_i\), is Hermitian, and satisfies \(\| A_i^{\mathrm h}\| \le J\). These conclusions hold for every \(L,q\ge 0\) and every real interaction bound allowed by the original model, including empty grids and zero local dimension.
The supported operators form an algebra closed under adjoints, and hence under real linear combinations. Thus the Hermitian part has the same support. Its adjoint equals itself, and \(\| A_i^{\mathrm h}\| \le \| A_i\| \le J\).
For the original Hermitian Hamiltonian \(H\),
The bijection from forward edges to undirected edges gives \(\sum _i A_i=H\). Taking Hermitian parts commutes with this finite sum, and the Hermitian part of \(H\) is \(H\) itself.
Let \(L\ge 0\) be an integer, and identify the original square-grid sites with \(\Lambda =\{ 0,\ldots ,L-1\} ^2\) by the coordinate bijection \(\sigma \). For an original site or forward-edge label \(i\), put
These are the designated supports used to view the original square Hamiltonian as a finite-domain interaction family in [ 248 , 247 ] .
For every original site or forward-edge label \(i\), the set \(X_i\) is a nonempty support of induced graph diameter at most one in \(\Lambda \).
If \(i=v\) is a site label, then \(X_i=\{ \sigma (v)\} \) is a singleton, and every singleton is an admissible support at every range. If \(i=(v,w)\) is a forward-edge label, then \(v\) and \(w\) are adjacent in the square grid, so \(X_i=\{ \sigma (v),\sigma (w)\} \) is a pair of adjacent sites of the integer rectangle. Adjacent pairs are admissible at range one.
Distinct original interaction labels have distinct supports \(X_i\).
Site supports are singletons, whereas every edge has two distinct endpoints. Two equal unordered edge supports have equal ordered endpoints or reversed ordered endpoints. The latter case is excluded because both edges have the prescribed forward orientation.
For an integer \(q\ge 0\), let \(U\) be the unitary induced by the site bijection \(\sigma \) on the corresponding configuration spaces. The integer-domain vector corresponding to an original vector \(\Omega \) is \(U\Omega \). The local dimension and every physical tensor factor remain the same.
For every original vector \(\Omega \),
The coordinate identification permutes an orthonormal basis.
The Hermitian part \(A_i^{\mathrm h}\) of every original interaction belongs to the algebra of operators supported on its designated site or edge \(S_i\).
This algebra is closed under adjoints and real linear combinations. Apply these closure properties to \((A_i+A_i^*)/2\).
Let \(H=\sum _i A_i\) be an original Hermitian square Hamiltonian with interaction bound \(J\ge 0\). At every designated support set
and put \(h_X=0\) at every other admissible range-one support. This defines a Hermitian interaction family on \(\Lambda \), with the same bound \(\| h_X\| \le J\). Neither a nonempty lattice nor a positive local dimension is required.
The finite-domain interaction has total Hamiltonian
The support map is injective and every unused support contributes zero. Thus the sum is \(\sum _i U A_i^{\mathrm h}U^*\). Conjugation is linear and \(\sum _i A_i^{\mathrm h}=H\).
Let \(E_0,\Delta \in \mathbb R\). If an original unit vector \(\Omega \) satisfies
then \(U\Omega \) is a unit vector satisfying
The conclusion holds for any real lower-gap parameter \(\Delta \). In particular, a positive original full-system gap is preserved.
Conjugate the eigenvector equation and the positive semidefinite gap defect by the same unitary \(U\), and use norm preservation.
Let \(\sigma :V\to W\) be a bijection of finite site sets, let \(q\ge 0\), and let \(U_\sigma \) be the unitary that relabels the tensor factors of \((\mathbb C^q)^{\otimes V}\) along \(\sigma \). For every vector \(\Omega \) and every \(A\subseteq V\),
No normalization of \(\Omega \) is required.
Relabeling the sites of \(\sigma (A)\) and of its complement identifies the reduced matrix of \(U_\sigma \Omega \) on \(\sigma (A)\) with the reduced matrix of \(\Omega \) on \(A\), conjugated by a permutation of configurations. A permutation leaves the spectrum, and hence the entropy, unchanged.
For every set \(A\) of sites of the original grid, the number of original edges with exactly one endpoint in \(A\) equals \(|\partial _\Lambda \sigma (A)|\) in the integer rectangle \(\Lambda \).
The coordinate bijection \(\sigma \) is a graph isomorphism onto the induced graph of \(\Lambda \). It therefore maps the crossing edges of \(A\) bijectively onto those of \(\sigma (A)\).
The square-grid entropy area law is the assertion that for every integer \(q\ge 2\) and all reals \(J,\Delta {\gt}0\) there is a constant \(C_A=C_A(q,J,\Delta )\ge 0\) with the following property. For every \(L\), every Hamiltonian of the original model with interaction bound \(J\), and every unit vector \(\Omega \) and real \(E_0\) satisfying \(H\Omega =E_0\Omega \) and \(H-E_0I\ge \Delta (I-|\Omega \rangle \langle \Omega |)\), every set \(A\) of sites of the grid satisfies
where \(\partial _\Lambda A\) is the set of grid edges with exactly one endpoint in \(A\). The constant is chosen before \(L\), \(H\), \(\Omega \) and \(A\).
Apply the conditional specialization below to the uniform finite-domain area law.
Assume the uniform finite-domain area-law assertion: for fixed \(q\ge 1\), \(R\ge 0\) and \(J,\Delta {\gt}0\), one constant \(C(q,R,J,\Delta )\ge 0\) bounds \(S_\Omega (A)\) by \(C|\partial _\Lambda A|\) for every finite induced domain, every interaction family of range \(R\) and bound \(J\), every gapped unit ground vector, and every cut. Then the square-grid entropy area law holds, with \(C_A(q,J,\Delta )=C(q,1,J,\Delta )\).
Since \(q\ge 2\), the assumption applies with \(R=1\). Place \(H\) on the integer rectangle \(\Lambda \) through its Hermitian interaction parts. The vector \(U\Omega \) is a unit ground vector of the resulting Hamiltonian with the same full-system gap \(\Delta \). Apply the assumed bound to the cut \(\sigma (A)\). Its entropy equals \(S_\Omega (A)\), and its crossing edges correspond bijectively to the original crossing edges of \(A\).
Fix an integer \(q\ge 2\) and reals \(J,\Delta {\gt}0\). There are \(C,c{\gt}0\), depending only on these parameters, such that for every \(L\ge 2\), every Hamiltonian in the preceding model with a unique unit ground vector, up to phase, satisfying (364) has a nonzero PEPS on the original open grid with \(D_e\le CL^c\) for every edge \(e\) and
This is Theorem 1.1 of [ 248 ] .
Write \(\Lambda _L\) for the sites of the grid and \(\mathcal H_X\) for the tensor product of the site spaces in \(X\subseteq \Lambda _L\). The physical registers are distributed among parties indexed by dyadic squares, starting from one party holding \(\Omega \) and ending with one party per site. The area-law and collar estimates make the mutual information across tapered buffers smaller than any fixed inverse power of \(L\), which lets an owner split its purification approximately. Near a square of logarithmic size, the ground vector is projected with high accuracy onto a sum of mutually orthogonal subspaces
with nested square samples \(X_j\) and orthonormal vectors \(v_{jk}\in \mathcal H_{X_j}\). Each party takes part in boundedly many operations with other parties during its whole lifetime, and each such operation is a sum of polynomially many products of private contractions and bipartite vectors or covectors.
Telescoping the accumulated errors of these contractions, the final vector \(\Psi \) and the final reference vector \(\Omega \otimes \Xi \) satisfy
Density compression replaces the bipartite resources by polynomial sums of product operators and gives an operator network \(\sigma \) on the party graph with polynomial link dimensions and
If \(\eta {\lt}1\), write \(E=\sigma -|\Omega \rangle \langle \Omega |\) and \(a_z=\langle \Omega |z\rangle \) for product basis vectors \(|z\rangle \). Since \(\sum _z\| E|z\rangle \| ^2\le \eta ^2\) and \(\sum _z|a_z|^2=1\), some \(z\) has \(a_z\ne 0\) and \(\| E|z\rangle \| \le \eta |a_z|\). The column \(v=\sigma |z\rangle \) is then nonzero, has the same virtual dimensions, and satisfies
Routing each link horizontally and then vertically through the anchors of the dyadic blocks has edge congestion at most a constant \(\chi \), also after folding the padded grid into the \(L\times L\) grid. If every link has dimension at most \(C_1L^k\), grouping the links crossing each grid edge gives
For \(L\ge L_0\) the error \(2CL^{-5}\) is at most \(L^{-1}\), so (371) gives (366). Each of the finitely many sizes \(2\le L{\lt}L_0\) has an exact representation with bond dimensions bounded independently of the Hamiltonian, which is absorbed by enlarging the prefactor in (372).
Fix \(D_0{\gt}0\) and a sufficiently large positive numerical constant \(C_{\rm tpl}\) before choosing templates. A template with positive integer parameters \(n,s_0\) is a nonempty finite \(T\subset \mathbb Z^2\) with
Each \(P_a\) is a closed convex rectangle or triangle of sup-norm diameter at most \(s_0\), with side slopes in \(\{ 0,\infty ,1,-1\} \). Triangles have noncollinear vertices and rectangles have nonzero sides; this ordinary reading of the source’s polygons is recorded in [ 76 ] . The union may be disconnected. Geometric estimates require a uniform lower bound on \(C_{\rm tpl}\), chosen independently of the template. Separation from a fixed cut set \(Z\) means \(\operatorname {dist}_\infty (T,Z){\gt}4D_0s_0\). For the fixed cut \(A\subseteq \Lambda \) of (342), put
In particular, \(T_0=T\). These are ambient dilations, not the induced-graph neighborhoods (338). This is Definition 9.3 of [ 247 ] .
A finite region \(A\) is partitioned into a residual set \(D\) and pairwise disjoint pieces \(X_i\). Each piece has a family label in \(\{ 0,1\} \), and one finite order induces the order within each family. For a piece \(X_i\) in family \(f\), the earlier same-family region is \(\bigcup _{j{\lt}i,\, f(j)=f}X_j\). No size, separation, or information bound is part of this partition data; those are additional mathematical assertions.
The final box error exponent is \(e_*=2\cdot 10^{-6}\), and the scanner choices are \((\mu ,\nu ,\ell _{\rm sc},\omega ,\kappa )=(1/4,1/1000,1/100000,1/20000,1/1000000)\). The amplification uses \(\varepsilon =10^{-5}\) and \(\alpha =p/(p+1)\). Set \(\beta =1-2\cdot 10^{-6}\), \(\delta _0=2\cdot 10^{-7}\), \(\zeta =1-\delta _0/2\), \(\rho _0=(1+\beta )/2\), and \(\ell _{\rm geo}=10^{-5}\). Then
The choice \(p=10^6\) also satisfies \(p/(p+1){\gt}1-10^{-6}\). The scanner exponent \(\ell _{\rm sc}\) and the geometric exponent \(\ell _{\rm geo}\) are separate choices, which happen to have the same value.
Substitute the stated rational values and compare integers after multiplying by positive denominators. Each gap is strictly positive.
Fix \(q,R,J,\Delta \) and the template constants \(D_0,C_{\rm tpl}\). For disjoint \(B,C\subseteq \Lambda \) write \(I_\Omega (B:C)=S_\Omega (B)+S_\Omega (C)-S_\Omega (B\cup C)\). The sublinear collar bound for these constants is the assertion that there are \(C\) and \(N\) with the following property. Let \(n\ge N\), let \(\Omega \) be the gapped ground vector of an admissible Hamiltonian on a finite induced domain \(\Lambda \), let \(A\subseteq \Lambda \), and let \(T\) be a template with parameters \((n,s_0)\) separated from the endpoints of the cut edges. Put \(L=\lfloor n^{1-\varepsilon }\rfloor \), \(X=A\cap T\) and \(S_0=A\cap T_L\), and suppose \(L\le s_0\). Then
and each of the cuts \(X\), \(S_0\) and \(S_0\setminus X\) has at most \(Cn\) crossing edges. This is Proposition 9.6 of [ 247 ] .
Fix \(q,R,J,\Delta \). The amplification bound is the assertion that for every \(C_0{\gt}0\) and every integer \(M\ge 1\) there are \(C_M\) and \(N\) with the following property. Let \(n\ge \max (2,N)\), let \(\Omega \) be the gapped ground vector of an admissible Hamiltonian on a finite induced domain \(\Lambda \), let \(A\subseteq \Lambda \) and let \(T\subset \mathbb Z^2\) be finite. Put \(L=\lfloor n^{1-\varepsilon }\rfloor \), \(X=A\cap T\), \(S_0=A\cap T_L\) and \(Y=S_0\setminus X\). Suppose that \(|S_0|\le C_0n^2\), that each of the cuts \(X\), \(Y\), \(S_0\) has at most \(C_0n\) crossing edges, and that \(I_\Omega (X:\Lambda \setminus S_0)\le C_0n^{1-\varepsilon }\). Then there is an integer \(r\le C_Mn^{(1-\varepsilon )/\alpha }\) such that
If moreover \(\operatorname {dist}_\infty (T,Z){\gt}L+n^\beta \), where \(Z\) is the set of endpoints of the cut edges, then \(N_{2r}^\Lambda (S_0)\subseteq A\). This is Proposition 10.2 of [ 247 ] ; its further assertion \(L+2r=o(n^\beta )\) follows from the radius bound by Theorem 27.114.3.
Fix the template constants \(D_0,C_{\rm tpl}\). The two-family decomposition for these constants is the assertion that for every \(n_*\ge 1\) there is \(C\) with the following property. For every finite induced domain \(\Lambda \) and every \(A\subseteq \Lambda \) with \(b=|\partial _\Lambda A|{\gt}0\), with \(Z\) the set of endpoints of the cut edges, there is an ordered two-family partition of \(A\) into a residual set \(D\) and nonempty pieces \(X_i\) with labels \(f_i\), together with templates \(T_i\) with parameters \((n_i,s_i)\), such that \(|D|\le Cb\), \(X_i\subseteq A\cap T_i\), each \(T_i\) is separated from \(Z\), \(\lfloor n_i^{1-\ell _{\rm geo}}\rfloor \le s_i\), \(n_i\ge n_*\),
and \(\sum _in_i^{-100}\le Cb\). This is Proposition 11.2 of [ 247 ] .
Suppose that for all \(q\ge 1\), \(R\ge 0\) and \(J,\Delta {\gt}0\) there are template constants \(D_0,C_{\rm tpl}\) for which the sublinear collar bound and the two-family decomposition hold, and that the amplification bound holds for all such \(q,R,J,\Delta \). Then the conclusion of Theorem 27.37.7 holds: for each \(q,R,J,\Delta \) there is one \(C\ge 0\) such that \(S_\Omega (A)\le C|\partial _\Lambda A|\) for every finite induced domain, every admissible Hamiltonian with a gapped ground vector \(\Omega \), and every \(A\subseteq \Lambda \). This proves Theorem 1.1 of [ 247 ] only conditionally on the three propositions.
If \(|\partial _\Lambda A|=0\), the entropy vanishes by Theorem 27.116.4. Otherwise choose the collar constant \(C_1\), apply amplification with \(C_0=\max (C_1,1)\) and \(M=100\), and choose the scale threshold so that every admissible radius satisfies \(L_n+2r{\lt}n^\beta \). Apply the two-family decomposition with \(n_*\) at least these thresholds. For each piece, the collar bound supplies the hypotheses of amplification for \(T_i\). Since \(L_{n_i}{\lt}n_i^\beta \), the cut clearance \(\operatorname {dist}_\infty (T_i,Z){\gt}2n_i^\beta \) gives \(N_{2r_i}^\Lambda (S_i)\subseteq A\). A walk of length at most \(2r_i\) from \(S_i\) moves at most \(2r_i\) in sup-norm, so every point of \(N_{2r_i}^\Lambda (S_i)\) lies within \(L_{n_i}+2r_i{\lt}n_i^\beta \) of \(T_i\) and hence outside every earlier piece of the same family. Thus \(J_i=(\Lambda \setminus A)\cup \bigcup _{j{\lt}i,\, f_j=f_i}X_j\) avoids \(N_{2r_i}^\Lambda (S_i)\), and amplification applied to \(X_i\subseteq A\cap T_i\) gives \(I_\Omega (X_i:J_i)\le C_{100}n_i^{-100}\). The boundary estimate from partition bounds, with \(\sum _in_i^{-100}\le Cb\) and \(|D|\le Cb\), gives the linear bound with a constant chosen before the domain and the cut.
The proof of Proposition 9.6 of [ 247 ] combines the scanner estimates with three steps that use no estimate of the scan: an exact entropy identity at each leaf, the transfer from the truncated to the original ground state, and the cut and size assertions of the proposition. The results below prove these steps; the information inequality itself is not proved here.
Let \(\psi \) be a vector of a finite tensor product over a site set \(V\), and let \(X,U,Y,W\subseteq V\). Put
Let \(X,U,Y,W\) be pairwise disjoint sets covering \(V\). For every vector \(\psi \),
If a set \(F\) lies in \(W\), then \(2I(X:F)\le 2S(X)+g_{\rm pre}-g_{\rm post}\). Consequently, for finitely many leaves with weights \(p_i\ge 0\) summing to one, each with its own partition \(X,U_i,Y_i,W_i\) of \(V\) and with \(F\subseteq W_i\) for every \(i\),
These are the leaf entropy identity and the lower bound after the leaf average in the proof of Proposition 9.6 of [ 247 ] .
Purity gives \(S(X\cup W)=S(U\cup Y)\), \(S(X\cup Y\cup W)=S(U)\) and \(S(Y\cup W)=S(X\cup U)\), and the identity follows by expanding both mutual informations. Strong subadditivity makes mutual information monotone in each argument, and both \(W\) and \(Y\cup W\) contain \(F\). Multiply the bound at each leaf by \(p_i\) and sum.
Let \(\Omega ,\widetilde\Omega \) be unit vectors at trace distance \(\delta \), and let \(X,J\) be disjoint. Then
In particular, if \(\delta \le C_1n^{-500}\) and \(\log \dim \mathcal H_X\le C_2n^2\) for an integer \(n\ge 1\), then \(I_\Omega (X:J)\le I_{\widetilde\Omega }(X:J)+8\sqrt{C_1}(1+C_2)n^{-248}\). The dimension of \(J\) does not enter.
Let \(T\) be a template with parameters \((n,s_0)\) separated from the cut, with \(C_{\rm tpl}\ge 24\) and \(D_0\ge 1\), and let \(L\le s_0\). Put \(X=A\cap T\) and \(S_0=A\cap T_L\). Then the cuts of \(X\) and \(S_0\) have at most \(4n\) edges, the cut of \(S_0\setminus X\) has at most \(8n\) edges, and \(|S_0|\le 10n^2\).
The cut bounds are those of Theorem 27.53.1.3 with \(j=L\), since \(S_0\setminus X=A\cap (T_L\setminus T)\). The set \(S_0\) lies in \(T_L\), and \(|T_L|\le |T|+nL\le 9ns_0+nL\) by the area and depth-layer bounds. The scale condition \(C_{\rm tpl}N_0(s_0+1)\le n\) gives \(L\le s_0\le n\).
Let \(C_{\rm tpl}\ge 24\) and \(D_0\ge 1\). Suppose there are \(C\) and \(N\) such that, in the setting of Definition 27.37.32, \(I_\Omega (X:\Lambda \setminus S_0)\le Cn^{1-\varepsilon }\) whenever \(n\ge N\). Then the sublinear collar bound holds for \(D_0,C_{\rm tpl}\). This assumes the information inequality of Proposition 9.6 of [ 247 ] ; it is not a proof of that proposition.
Take the constant \(\max (C,10)\); the remaining assertions are those of Theorem 27.37.39.
27.37.1 Exact nearest-neighbor crossing sums
Let \(\Lambda \subset \mathbb Z^2\) be finite and let \(E_\Lambda \) be its unordered nearest-neighbor edges. Index supports by the disjoint union \(I=\Lambda \sqcup E_\Lambda \), assigning \(S_v=\{ v\} \) to a site and \(S_{\{ x,y\} }=\{ x,y\} \) to an edge. For \(A\subseteq \Lambda \), put \(C(A)=\{ i\in I:S_i\cap A\ne \varnothing ,\ S_i\setminus A\ne \varnothing \} \).
A singleton cannot meet both sides of a cut in any finite set. If \(\{ x,y\} \) meets both sides of \(A\), then \(\{ x,y\} \cap A=\{ v\} \) for exactly one endpoint \(v\). For an alphabet of cardinality \(q\ge 0\), the set of configurations on this intersection has cardinality \(q\).
If \(x\in A\), the crossing condition forces \(y\notin A\); otherwise it forces \(y\in A\). A function on a singleton is determined by its value at the unique point, including when the alphabet is empty.
For every \(A\subseteq \Lambda \), no on-site index belongs to \(C(A)\), and an edge index belongs to \(C(A)\) exactly when the edge lies in \(\partial _\Lambda A\). Consequently \(|C(A)|=|\partial _\Lambda A|\). For every \(i\in C(A)\) and every integer \(q\ge 0\), the configuration set on \(S_i\cap A\) has cardinality \(q\).
Choose any linear order on \(\Lambda \) and write each edge in increasing endpoint order. Forgetting this order maps crossing edge indices to unordered boundary edges. Equality of unordered pairs either identifies both endpoints or swaps them; the swap contradicts their strict order. Every boundary edge has an increasing ordering, so this map is bijective. The configuration count follows from the unique inside endpoint.
Let \(m,q\ge 0\) be integers, \(A_j\subseteq \Lambda \) for \(0\le j{\lt}m\), and let \(a_j,J\in \mathbb R\). If \(d_{ij}\) denotes the cardinality of the configuration set on \(S_i\cap A_j\) for an alphabet of size \(q\), then
This is the combinatorial identity used in the nearest-neighbor matrix-unit estimate of [ 248 , Section 4 ] .
Every summand in the inner sum has \(d_{ij}=q\), and there are exactly \(|\partial _\Lambda A_j|\) such summands. Substitute these identities and factor out \(q^2J\).
27.38 Regional states under relabelling
Let \(e:\Lambda \to \Lambda '\) be a bijection of finite domains, and let \(A\subseteq \Lambda \), \(A'\subseteq \Lambda '\) satisfy \(e(A)=A'\). It induces bijections \(e_A:A\to A'\) and \(e_B:\Lambda \setminus A\to \Lambda '\setminus A'\). For any physical vector \(\Omega \), its relabelled coefficients are
Regional configurations transform by the corresponding restricted bijections. The relabelling concerns tensor factors; preservation of lattice adjacency is unnecessary for this definition.
For configurations \(\sigma ,\tau :A\to \{ 0,\ldots ,q-1\} \), write \(\rho _A^\Omega (\sigma ,\tau )\) for the \((\sigma ,\tau )\) matrix entry of the reduction \(\rho _A\) of Definition 27.37.4 formed from the vector \(\Omega \). For every vector and region-preserving site bijection as above, and all \(\sigma ',\tau ':A'\to \{ 0,\ldots ,q-1\} \),
These identities hold without normalization or spectral assumptions, including empty regions and complements. They express the finite tensor-factor conventions of [ 247 , Section 2 ] in the induced-domain coordinates.
The three identities are the special case of a general covariance statement for finite tensor products: under a bijection of tensor factors combined with bijections of the local bases, the transported vector has the same norm, its regional state is the transported regional state, and its regional entropy is unchanged. Here all local dimensions equal \(q\) and every local basis bijection is the identity. The computation for this case is as follows. For \(\sigma :A\to \{ 0,\ldots ,q-1\} \) and \(\upsilon :\Lambda \setminus A\to \{ 0,\ldots ,q-1\} \), write \(\sigma \sqcup \upsilon \) for the configuration on \(\Lambda \) that restricts to \(\sigma \) and \(\upsilon \). Since \(\sigma '\mapsto \sigma '\circ e\) is a bijection of configuration sets,
which is (386). Because \(e\) maps \(A\) onto \(A'\) and \(\Lambda \setminus A\) onto \(\Lambda '\setminus A'\), splitting before or after relabelling agrees:
Substituting (390) into the partial trace and reindexing the complementary sum by the bijection \(\upsilon '\mapsto \upsilon '\circ e_B\),
which is (387). Thus \(\rho _{A'}^{\Omega '}=P\rho _A^\Omega P^{-1}\) for the permutation matrix \(P\) of \(\sigma '\mapsto \sigma '\circ e_A\). Conjugation preserves the eigenvalues, so (388) follows.
27.38.1 Finite improvement of the box exponent
For \(e_0\in \mathbb R\), set \(e_*=2\cdot 10^{-6}\); the parameters used in the proof of [ 247 , Proposition 9.5 ] are
These parameters remain fixed as the entropy exponent decreases.
With the parameters of (394) and (395), the condition \(0{\lt}e_0{\lt}1\) gives \(0{\lt}g_0{\lt}1/2\), \(0{\lt}\ell {\lt}1\), and \(\kappa {\gt}0\). If \(e_0{\lt}1\) and \(e\le e_0\), the density-error exponents satisfy
For \(e_0{\lt}1\), write \(p=\ell +2\kappa \) for the energy-prefactor exponent. Then
For \(e_0{\gt}0\) and \(e{\gt}e_*\), one has \(\kappa {\lt}(1-\ell )e\) and \(g_0-\ell -\nu /4{\lt}1\).
If \(0{\lt}e_0{\lt}1\), then \(g_0=(1-e_0)/2\) lies in \((0,1/2)\). Hence \(\ell =g_0/100000\) lies in \((0,1)\), and \(\kappa =\min \{ \ell ,e_*/4\} {\gt}0\). Substituting (394) and (395) gives (396). Since \(g_0{\gt}0\) when \(e_0{\lt}1\), the identities imply (397) for \(e\le e_0\). The bound \(\kappa \le \ell \) gives (398). For the comparison bounds, substitute the same formulas and use \(\kappa \le e_*/4\), \(e{\gt}e_*\), and \(e_0{\gt}0\).
Fix \(0{\lt}e_0{\lt}1\), use the fixed parameters of Definition 27.38.1.1, and fix \(C,C_\delta \ge 0\) and \(\gamma ,\gamma _\delta \in \mathbb R\). For an integer \(n\ge 2\) and a real exponent \(e\le e_0\), put
All quotients have the convention \(x/0=0\). There is an integer \(N\ge 2\), depending only on the fixed constants and \(e_0\), such that every integer \(n\ge N\) and every real \(e\le e_0\) satisfy \(A_n(e)\le n^{-\nu /2}\). This is the numerical principal term in the energy estimate of [ 247 , Section 9, lines 731–755 ] .
Apply the numerical scale estimates with the fixed exponents and auxiliary entropy exponent \(\mu \). For some fixed \(C_1\ge 1\) and all sufficiently large \(n\), cancellation of the positive \(D_n^2\) gives
Increase the threshold so that
These bounds follow because \(\nu {\gt}0\) and every fixed power of \(\log n\) grows more slowly than every positive power of \(n\). The parameter inequalities \(g_0-\ell \ge 45\nu \) and \(e-\mu \le -10\nu \) hold uniformly for \(e\le e_0\). Hence
Since \(\nu {\lt}100\), the sum in parentheses in \(A_n(e)\) is at most \(3n^{-\nu }\). The ceiling bound \(D_n\le 2n^\kappa \) therefore yields
where the last inequality uses \(\ell +2\kappa \le 3\nu /100\). Every enlargement of the threshold depends only on the fixed constants and \(e_0\).
Under the hypotheses and notation of Theorem 27.38.1.3, put
There is an integer \(N\ge 2\), depending only on the fixed constants and \(e_0\), such that every integer \(n\ge N\) and every real \(e\le e_0\) satisfy
The threshold in the principal energy estimate suffices.
Since \(0{\lt}g_0{\lt}1/2\) and \(\nu =g_0/1000\), we have \(\nu /2{\lt}1000\). For \(n\ge 2\), it follows that \(n^{-1000}\le n^{-\nu /2}\). As \(C\ge 0\), the principal energy estimate gives
The fixed contribution \(Cn^{-1000}\) is included in \(E_n(e)\). The remaining error in the scanner energy estimate tends to zero with the replica index at each fixed system size.
Fix the scanner exponents described in Section 28.8, and put
There is an integer \(N\ge 1\), depending only on these fixed exponents, such that \(K_n\ge 1\) and \(a_n{\gt}0\) for every integer \(n\ge N\). Define \(h(\tau )=-\tau \log |\tau |-(1-\tau )\log |1-\tau |\) for \(\tau \in \mathbb R\), with the convention \(0\log 0=0\). On \([0,1]\), this is the binary entropy. For every such \(n\) and every \(\tau \in \mathbb R\),
The coefficient is absolute, and the threshold precedes \(\tau \) and every physical instance. This bounds the binary-entropy term in the proof of [ 247 , Proposition 9.5 ] for the actual rounded band count.
Since \(\mu {\lt}1-\ell \), the numerical scale estimates of Lemma 28.8.1 give \(K_n\ge 1\) for all sufficiently large \(n\). They also give \(L_n\le n\), and integer division gives \(K_n\le L_n\). The extension satisfies \(h(\tau )\le \log 2\) for every real \(\tau \), so
Fix \(e_*{\lt}e_0{\lt}1\), where \(e_*=2\cdot 10^{-6}\), and use the parameters of (394) and (395). Set \(\eta =1-\ell \) and
There is an integer \(N\ge 1\), depending only on \(e_0\), such that \(K_n\ge 1\) and \(a_n{\gt}0\) for every integer \(n\ge N\) and, simultaneously,
for every \(e_*{\lt}e\le e_0\) and every real \(\tau \), where \(h(\tau )=-\tau \log |\tau |-(1-\tau )\log |1-\tau |\) with the convention \(0\log 0=0\). The threshold is chosen before \(e\) and \(\tau \). This is a numerical estimate for the additive terms following the comparison in the proof of [ 247 , Proposition 9.5 ] .
Apply the binary-entropy estimate with the fixed parameters chosen from \(e_0\). It gives one \(N\ge 1\) such that \(K_n\ge 1\), \(a_n{\gt}0\), and \(h(\tau )/a_n\le (\log 2/4)n\) for every \(n\ge N\) and every real \(\tau \). Fix such an \(n\) and an exponent \(e_*{\lt}e\le e_0\), and set \(P=n^{1+\eta e}\). Since \(\eta {\gt}0\), \(e{\gt}0\) and \(n\ge 1\), we have \(1\le n\le P\). The floor bound \(L_n\le n^\eta \) yields \(nL_n^e\le P\). Since \(\kappa {\gt}0\) and \(n\ge 1\), we have \(n^\kappa \ge 1\). The parameter inequality \(\kappa {\lt}\eta e\) gives
and hence \(nD_n\le 2P\). Also \(n^{3/5}\le P\), and the nonnegative coefficient \(\log 2/4\) gives \(h(\tau )/a_n\le (\log 2/4)P\). Adding these four estimates and \(1\le P\) proves the result, with coefficient \(1+2+1+\log 2/4+1=5+\log 2/4\).
Fix \(0{\lt}e_0{\lt}1\), set \(e_*=2\cdot 10^{-6}\), and take \(\ell \) from (395). Let \(F_i:\mathbb N\to \mathbb R\) be an arbitrary family, and suppose there are constants \(V\ge 0\) and \(C_0{\gt}0\) such that, for every \(i\) and \(r\ge 1\),
Assume the following implication for every \(e_*{\lt}e\le e_0\): if there is a constant \(C_e{\gt}0\) such that \(F_i(r)\le C_er^{1+e}\) for every \(i\) and \(r\ge 1\), then there are a constant \(C'_e{\gt}0\) and an integer \(N_e\) such that
for every \(i\) and \(r\ge N_e\). Then there is one constant \(C{\gt}0\) such that \(F_i(r)\le Cr^{1+e_*}\) for every \(i\) and every \(r\ge 1\).
Write \(a=1-\ell \), so \(0{\lt}a{\lt}1\). At each improvement, enlarge the constant using the quadratic bound in (419) to include \(1\le r{\lt}N_e\). The enlarged constant is independent of \(i\). Starting at \(e_0\), apply (420) while the current exponent exceeds \(e_*\). Since \(a^ne_0\to 0\), only finitely many steps are needed. Once \(a^ne_0\le e_*\), the inequality \(r^{1+a^ne_0}\le r^{1+e_*}\) for \(r\ge 1\) gives the result.
27.39 Rounded scales in a safe rectangle
Fix \(C_2{\gt}0\), \(C_r\ge 0\), and \(0{\lt}\ell {\lt}1\). There is an integer \(S\ge 1\) such that, for every integer \(s\ge S\), setting \(n=\lceil C_2s\rceil \), \(L=\lfloor n^{1-\ell }\rfloor \), and \(r_0=\lceil C_r\log ^2n\rceil \) gives \(1\le L\le s\) and, for every integer \(D_0\ge 1\),
The threshold depends only on \(C_2,C_r,\ell \). These are the rounded clearance estimates used in [ 247 , proof of Proposition 9.5, lines 707–729 ] .
Since \(\ell {\gt}0\), \(n^{-\ell }\to 0\), and \((\log n)^2/n\to 0\). Consequently
For \(s\ge 1\), rounding gives \(C_2s\le n\le (C_2+1)s\). Choose the threshold so that the ratio is less than \(1/(C_2+1)\). The floor and ceiling estimates then give \(4L+20r_0{\lt}s\). Also \(n\ge 1\) and \(1-\ell {\gt}0\) imply \(L\ge 1\). Since \(D_0\ge 1\), these inequalities imply both stated budgets.
For the rounded scales of Theorem 27.39.1, one threshold precedes every finite physical domain, cut, integer \(D_0\ge 1\) and safe parent rectangle \(Q\) of size \(s\). For all parent sizes beyond that threshold, whenever \(j\le L\) and \(2^{K_{\rm cap}}\le L\), every square selected by the capped dyadic partition of \(Q^{+j}\setminus Q\) is safe, and every parent site has sup-norm distance strictly greater than \(2L+10r_0\) from every crossing-edge endpoint. The dyadic cap exponent \(K_{\rm cap}\) is independent of the scanner band count.
Beyond the threshold, the first rounded budget reads
so Theorem 27.124.4 makes every selected square safe. Native parent safety is strict, so the second rounded budget gives the asserted strict endpoint clearance.
27.39.1 Rounded collar clearance
The scale choice in [ 247 , Section 9, lines 32–41 ] and its use in the proof of Proposition 9.5, lines 693–729, require a collar whose depth is small compared with the rectangle size. The following numerical estimates supply a sufficient clearance budget.
Let \(C_2{\gt}0\) and \(0{\lt}\eta {\lt}1\) be real numbers. There is an integer \(N\in \mathbb N\), depending only on \(C_2\) and \(\eta \), such that for every \(s\in \mathbb N\) with \(s\ge N\), on setting \(n_s=\lceil C_2s\rceil \) and \(L_s=\lfloor n_s^\eta \rfloor \), one has
Since \(1-\eta {\gt}0\), choose \(N\ge 1\) such that \(s^{1-\eta }\ge 2(C_2+1)^\eta \) for every integer \(s\ge N\). The ceiling bound gives \(n_s{\lt}C_2s+1\le (C_2+1)s\), and hence
Let \(D_0,L,s\in \mathbb N\) satisfy \(D_0\ge 1\) and \(2L\le s\). Then
Multiplication by \(L\ge 0\) gives \(L\le D_0L\). Therefore
Fix real numbers \(C_2{\gt}0\) and \(0{\lt}e_0{\lt}1\), and let \(\ell =(1-e_0)/200000\), as in the fixed bootstrap parameters. There is an integer \(N\ge 1\), depending only on \(C_2\) and \(e_0\), such that for every \(s\in \mathbb N\) with \(s\ge N\), on setting \(n_s=\lceil C_2s\rceil \) and \(L_s=\lfloor n_s^{1-\ell }\rfloor \), one has \(L_s\ge 1\) and \(2L_s\le s\), and for every \(D_0\in \mathbb N\) with \(D_0\ge 1\),
The same threshold is chosen before the safety parameter \(D_0\).
The fixed parameter bounds give \(0{\lt}\ell {\lt}1\), so \(0{\lt}1-\ell {\lt}1\). Apply Theorem 27.39.1.1 with \(\eta =1-\ell \) to choose \(N\) depending only on \(C_2\) and \(e_0\). Enlarge \(N\) to at least one. Since \(C_2s{\gt}0\), the integer \(n_s\) is at least one, so \(L_s\ge 1\). For every \(s\ge N\), one also has \(2L_s\le s\). For each \(D_0\ge 1\), Theorem 27.39.1.2 now gives the asserted budget without changing \(N\).
27.40 Minimum norm of regularized regional filters
Let \(V\) be a finite ordered set of sites, with finite-dimensional coordinate Hilbert spaces \(\mathcal H_v\), and set \(\mathcal H_X=\bigotimes _{v\in X}\mathcal H_v\). The empty tensor product is \(\mathbb {C}\). For \(X\subseteq V\), use the regional extension \(\widehat K_X\) of Definition 26.1.2.3. Fix regions \(X_0,\ldots ,X_{m-1}\) and a unit vector \(\Omega \in \mathcal H_V\). The following argument proves attainment and the elementary norm bounds in the regularized variational problem of the September 24, 2026 polynomial-PEPS manuscript.
Let \(b{\gt}0\) and \(a_0,\ldots ,a_{m-1}\in \mathbb {R}\), and put \(A=\sum _j a_j\). Define
The densities are independent for different indices, including indices with the same region. The powers act on the full regional space. For \(m=0\), the product is the identity and \(F=1\).
In the special case of two occurrences \(X_0=X_1=X\), write \(K_j=(x_j+bI_X)^{-a_j/2}\). The upper line below carries \(\mathcal H_X\); the lower line carries the complementary tensor factor \(\mathcal H_{V\setminus X}\), which may be one-dimensional. This picture illustrates repeated support only; the general family of regions remains arbitrary.
![\begin{tenkz}[rows={wire,wire}, cols=5, bonds=none]
\tn[at=(1,1), name=filterone, skin=box,
ports={180:physical, 0:physical}]{K_1}
\tn[at=(1,3), name=filterzero, skin=box,
ports={180:physical, 0:physical}]{K_0}
\tn[at=(2,1), name=identityone, skin=box,
ports={180:physical, 0:physical}]{I}
\tn[at=(2,3), name=identityzero, skin=box,
ports={180:physical, 0:physical}]{I}
\tn[at=(1,5), name=reference, skin=box, wires=2,
ports={180@1:physical, 180@2:physical}]{\Omega}
\tnwire[name=insideoutput]{open w}{filterone.180}
\tnwire{filterone.0}{filterzero.180}
\tnwire{filterzero.0}{reference.180@1}
\tnwire[name=outsideoutput]{open w}{identityone.180}
\tnwire{identityone.0}{identityzero.180}
\tnwire{identityzero.0}{reference.180@2}
\tnmark[form=label, label pos=n]{on insideoutput 0.5}{$i$}
\tnmark[form=label, label pos=s]{on outsideoutput 0.5}{$\alpha$}
\end{tenkz}](tenkz_svg/tenkz-40f3ffea4415c103.svg)
Reading from \(\Omega \) on the right, \(K_0\otimes I\) acts first and \(K_1\otimes I\) acts second. Thus the displayed output is \(\Psi (x)=(K_1\otimes I)(K_0\otimes I)\Omega \).
For a qubit example, take \(X=V\), \(b=1\), \(a_0=a_1=2\), \(x_0=|0\rangle \! \langle 0|\) and \(x_1=|+\rangle \! \langle +|\), where \(|+\rangle =(|0\rangle +|1\rangle )/\sqrt2\). Then
The product order matters: \((K_1K_0)_{01}=-1/4\), whereas \((K_0K_1)_{01}=-1/8\). The complementary space is \(\mathbb {C}\) in this example, so the lower identities are scalar identities.
The set \(\mathcal D\) is compact and nonempty. Each \(L_j\) and the objective \(F\) are continuous on \(\mathcal D\).
In finite dimension, the positive semidefinite trace-one matrices form a compact set. Their finite product is compact. Since \(\| \Omega \| =1\), some global configuration has a nonzero coefficient in \(\Omega \). Restricting that configuration to each region gives \(d_j=\dim \mathcal H_{X_j}{\gt}0\). The densities \(I_{X_j}/d_j\) prove nonemptiness. For \(b{\gt}0\), the shifted matrices are positive definite, and their fixed real powers vary continuously. Identity extension, finite ordered multiplication, evaluation on \(\Omega \), and the norm preserve continuity.
Suppose \(a_j\ge 0\) for every \(j\). For every \(x\in \mathcal D\), every \(j\), and every \(\xi \in \mathcal H_V\),
Consequently, every \(x\in \mathcal D\) satisfies
A positive semidefinite trace-one matrix satisfies \(0\le x_j\le I\). The eigenvalues of \(x_j+bI\) therefore lie in \([b,1+b]\). The negative power has operator norm at most \(b^{-a_j/2}\), while its inverse has norm at most \((1+b)^{a_j/2}\). Apply the identity-extension inequality to the factor and its inverse to obtain the upper and lower bounds. Apply these bounds successively in the order \(L_0,\ldots ,L_{m-1}\), and use \(\| \Omega \| =1\). This argument never exchanges two factors.
Define the minimum value and normalized output by
Under the nonnegative-weight hypothesis, the preceding theorem makes this normalization well defined for every feasible tuple.
There exists \(x_*\in \mathcal D\) such that \(F(x_*)\le F(x)\) for all \(x\in \mathcal D\). Every feasible minimizer has objective value \(N\). If all \(a_j\ge 0\), then
At every feasible minimizer \(x_*\), \(\widehat\Psi (x_*)=N^{-1}M(x_*)\Omega \).
A continuous real-valued function on a nonempty compact set attains its minimum. The value at any minimizing point is the least element of \(F(\mathcal D)\), hence its infimum. Apply the bounds for every feasible tuple at a minimizer. Positivity permits division by \(F(x)\); at a minimizer this divisor equals \(N\).
Assume all \(a_j\ge 0\) and set \(b=e^{-R}\). For every real \(R\) and every feasible tuple \(x\),
For \(R\ge 0\), the minimum satisfies
Substitute \(b=e^{-R}{\gt}0\). If \(R\ge 0\), then \(1+e^{-R}\le 2\), so the nonpositive exponent \(-A/2\) gives \((1+e^{-R})^{-A/2}\ge 2^{-A/2}\).
The argument allows empty or repeated regions, zero weights, and singular densities. Neither nestedness of the regions nor commutation of their filters is required. It gives the same minimum value for every minimizer, without asserting uniqueness or smooth dependence of a minimizer on \(R\).
27.41 Coordinate variation and final-state marginals
Retain the regions, identity extensions, and ordered product of Definition 27.40.1. Write \(K_j=(x_j+bI_{X_j})^{-a_j/2}\), so \(L_j=\widehat{K_j}_{X_j}\). The coordinate variation below is the unstripped first variation in the proof of the regularized patch estimate. Its marginals are those of the same final normalized vector defined in the variational problem.
Fix an index \(j\) and a unitary \(U\) on \(\mathcal H_{X_j}\). Define \(x^{j,U}\) by replacing \(x_j\) with \(Ux_jU^*\) and leaving every other coordinate unchanged. Put
Empty products are identities. The map \(\mathcal I_j\) is complex linear, and \(\mathcal I_j(K_j)=\Psi (x)\). For \(B^*=-B\) and real \(t\), the exponential \(U(t)=e^{tB}\) is unitary. Equal regions at different indices still have independent coordinates.
For \(x\in \mathcal D\), \(b{\gt}0\), arbitrary real weights, and a unitary \(U\) on \(\mathcal H_{X_j}\), one has \(x^{j,U}\in \mathcal D\) and
Unitary conjugation preserves positivity and trace. It fixes the identity, so \(Ux_jU^*+bI=U(x_j+bI)U^*\). Unitary covariance of the real power gives (431). Every factor except the one at index \(j\) is unchanged, giving (432). The argument also applies to singular \(x_j\) because \(x_j+bI\) is positive definite on the full regional space.
Let \(x\in \mathcal D\), \(b{\gt}0\), and \(B^*=-B\) on \(\mathcal H_{X_j}\). For arbitrary real weights, the filtered output satisfies
If \(F(x)\le F(y)\) for every \(y\in \mathcal D\), then
Set \(D_j=\widehat{BK_j-K_jB}_{X_j}\) and \(\Delta _j=P_j^+D_jP_j^-\Omega \). Every line in the following diagram carries the full space \(\mathcal H_V\).
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Starting at \(\Omega \) on the right, \(P_j^-\) acts first, \(D_j\) second, and \(P_j^+\) last. The two outer products need not be supported on \(X_j\).
Differentiate \(e^{tB}K_je^{-tB}\) at zero to obtain \(BK_j-K_jB\). Apply the continuous linear map \(\mathcal I_j\) to obtain (433). The entire real curve is feasible. Thus \(t\mapsto \| \Psi (x^{j,e^{tB}})\| ^2\) has a local minimum at zero, and its derivative there is \(2\operatorname {Re}\langle \Psi (x),\mathcal I_j(BK_j-K_jB)\rangle \). Fermat’s theorem gives (434).
Identify configurations indexed by the full region \(V\) with global configurations by forgetting the redundant membership condition. This bijection induces a unitary coordinate identification \(J\). For the same final vector \(\phi =\widehat\Psi (x)\) and every region \(X\subseteq V\), define
At a feasible minimizer with nonnegative weights, \(\phi =N^{-1}M(x)\Omega \). No intermediate filtered vector enters (435).
For every global vector \(\xi \), region \(X\), and arbitrary complex operator \(K\) on \(\mathcal H_X\),
In particular, \(\langle \phi ,\widehat K_X\phi \rangle =\operatorname{tr}(\rho _XK)\). Each \(\rho _X\) is positive semidefinite. If \(a_j\ge 0\), \(b{\gt}0\), \(\| \Omega \| =1\), and \(x\in \mathcal D\), then \(\operatorname{tr}\rho _X=1\). These assertions include the empty region.
In regional coordinates, \(\widehat K_X=K\otimes I\) and the reduced matrix is the partial trace of \(|J\xi \rangle \! \langle J\xi |\) over the second factor. The partial-trace identity gives \(\operatorname{tr}(\operatorname {Tr}_{V\setminus X}(P)K)=\operatorname{tr}(P(K\otimes I))\). For \(P=|J\xi \rangle \! \langle J\xi |\) this is (436); no real part or Hermiticity assumption is needed. Partial trace preserves positivity and trace. Finally \(J\) preserves the norm and \(\| \phi \| =1\), so \(\operatorname{tr}\rho _X=1\).
For the sitewise basis relabellings of Theorem 26.1.2.4, the marginal \(\rho _X\) of the same final vector \(\phi =\widehat\Psi (x)\) satisfies
This identity holds for every tuple \(x\) and every choice of weights and regulator.
27.42 Sparse coordinate strips
Let \(F\subset \mathbb Z^2\) be finite and let \(m\ge 1\) be an integer. For residues \(a,b\in \{ 0,\ldots ,m-1\} \), define
For every finite \(F\subset \mathbb Z^2\) and integer \(m\ge 1\), there are residues \(a,b\in \{ 0,\ldots ,m-1\} \) such that
This is the finite averaging step in the sparse-strip construction of [ 247 , Section 11 ] .
For any map \(f:F\to \{ 0,\ldots ,m-1\} \), its \(m\) fibres partition \(F\). One fibre therefore has at most \(\lfloor |F|/m\rfloor \) elements. Apply this assertion to the two coordinate residue maps and choose their residues independently. Each selected strip contains at most \(\lfloor |F|/m\rfloor \) cells, so their union satisfies \(|B_{a,b}(F)|\le 2\lfloor |F|/m\rfloor \). Multiplying by \(m\) and using \(m\lfloor |F|/m\rfloor \le |F|\) gives (439). The argument includes \(F=\varnothing \) and \(m=1\).
27.43 Finite cell counts
For a finite induced domain \(\Lambda \) and a cut \(A\subseteq \Lambda \), the boundary endpoints are the integer endpoints of the crossing edges,
For a finite set \(P\subset \mathbb Z^2\) of parent indices, its dyadic child indices are
These are the four children of each parent in a translated dyadic mesh.
For every finite \(T\subset \mathbb Z^2\), integer \(r\ge 0\), finite parent set \(P\), and finite induced domain with cut \(A\), one has
These cardinality estimates are ingredients in the layer count of [ 247 , Section 11 ] ; no connectedness assumption is needed.
Each point of \(T\) contributes an integer square with \((2r+1)^2\) sites; counting their union gives the first bound. Each crossing edge contributes at most two endpoints, giving the second. In each coordinate the remainder modulo two identifies the child label, and division by two then identifies the parent index. Thus distinct parent and child-label pairs give distinct indices, proving the last equality. All three assertions include empty sets.
27.44 Translated dyadic neighborhoods
Fix an origin \(o\in \mathbb R^2\). For every integer \(k\ge 0\) and index \(z\in \mathbb Z^2\), define the half-open cell and its index map by
The map \(p\) gives the index of a cell’s parent. The arbitrary origin includes the translated origin used in [ 247 , Section 11 ] .
Every point lies in the unique cell indexed by \(c_k(x)\), and passing to parents agrees with increasing the scale:
For every integer \(C_0\ge 0\), two integer indices at sup distance at most \(C_0\) have parents at sup distance at most \(C_0\). Consequently \(p\) maps the integer \(C_0\)-neighborhood of a finite index set into the \(C_0\)-neighborhood of its parent image. An index is a child of a member of \(P\) precisely when its parent belongs to \(P\).
The defining inequalities of the floor function give the cell membership criterion. The identity \(\lfloor t/2\rfloor =\lfloor \lfloor t\rfloor /2\rfloor \) gives the parent formula. Integer division by two preserves the stated coordinate bounds, including negative indices. Finally the remainder modulo two is either zero or one, identifying the unique child in each coordinate.
Let \(Z\subset \mathbb Z^2\) be finite and let \(C_0\ge 0\) be an integer. For each integer \(k\ge 0\), set
These are actual subsets of the plane and finite sets of indices; no nesting or distance bound is assumed in their definitions. The picture below shows one endpoint with \(C_0=1\): the layer \(D_k\) is the ring of scale-\(k\) cells that lie in \(N_{k+1}\) but not in \(N_k\).
![\begin{tenkz}[size=s, rows={ket,ket,ket,ket,ket,ket,ket,ket}, cols=8, bonds=none]
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For every origin, finite \(Z\), integer \(C_0\ge 0\), and integer \(k\ge 0\),
The statements include the empty set of endpoints and radius zero.
Apply the parent identity to every point of \(Z\). The cell membership criterion identifies membership in \(N_k\) with membership of its index in \(P_k\). The parent preserves the integer radius, proving nesting. A point belongs to \(N_{k+1}\) precisely when its scale-\(k\) index is a child of an index in \(P_{k+1}\). Removing \(N_k\) therefore removes precisely the indices in \(P_k\), which gives the last equality.
For every origin, finite \(Z\), integer \(C_0\ge 0\), and integer \(k\ge 0\),
When \(Z\) is the set of endpoints of the crossing edges of a cut in a finite induced domain, with \(b=|\partial _\Lambda A|\), this gives
The constant depends only on the chosen radius, independently of the origin, scale, domain and cut. These are the layer-cell estimates in [ 247 , Section 11 ] .
The occupied-index image has at most \(|Z|\) elements. Its integer \(C_0\)-neighborhood has at most \((2C_0+1)^2|Z|\) elements, and each parent has exactly four children. Taking a set difference can only decrease the count. Each crossing edge contributes at most two endpoints, giving the second bound.
For every origin, nonempty finite endpoint set \(Z\), and integer \(C_0\ge 2\), every point belongs to some \(N_k\), with \(k\ge 0\). Consequently,
This is the exhaustion assertion in [ 247 , Section 11 ] .
Choose \(z\in Z\). For any \(x\in \mathbb R^2\), take \(2^k\) larger than the sum of the four absolute values of the coordinates of \(x-o\) and \(z-o\). Each translated coordinate divided by \(2^k\) then lies in \((-1,1)\), so its floor is either \(-1\) or \(0\). The indices \(c_k(x)\) and \(c_k(z)\) therefore differ by at most one in each coordinate. Since \(c_k(z)\) is occupied and \(C_0\ge 2\), the index \(c_k(x)\) belongs to \(P_k\). Thus \(x\in N_k\), as required.
For every origin, nonnegative scale and cell index, the closure of a half-open dyadic cell is the corresponding closed rectangle. For every finite endpoint set and nonnegative integer radius, the actual layer satisfies
In particular, the closure of every dyadic cell is convex. These formulas give the closed regions used in the distance estimates of [ 247 , Section 11 ] .
The side length \(2^k\) is positive, so the closure of each half-open coordinate interval is its closed interval. Closure commutes with finite products. The exact layer union is finite, and closure commutes with finite unions, which gives the second formula. Each closed coordinate interval is convex, and a product of convex sets is convex.
27.45 The half-open layer partition
The nested neighborhoods \(N_k\) determine the layers \(D_k=N_{k+1}\setminus N_k\). Their half-open convention gives a unique layer assignment before taking closures of birth regions; see [ 247 , Section 11 ] .
For arbitrary origin, finite endpoint set and nonnegative integer width, suppose that \(\ell \le k\) and that \(z\) is an actual fine-layer index at exponent \(\ell \). Then its half-open cell belongs to \(D_k\).
The exact union of the fine cells in \(D_k\) contains the cell indexed by \(z\).
For arbitrary origin, finite endpoint set and nonnegative integer width, \(D_k\cap D_h=\varnothing \) whenever \(k\ne h\).
If \(k{\lt}h\), then \(D_k\subseteq N_{k+1}\subseteq N_h\), whereas \(D_h\) is disjoint from \(N_h\). The other order follows by symmetry.
Under the same hypotheses, fix a lower index \(k_0\). For every \(k\ge k_0\), the dummy neighborhood \(N_{k_0}\) is disjoint from \(D_k\).
Nesting gives \(N_{k_0}\subseteq N_k\), and \(D_k=N_{k+1}\setminus N_k\).
Suppose that \(Z\) is nonempty and \(C_0\ge 2\). For every lower index \(k_0\) and every \(x\notin N_{k_0}\), there is a unique \(k\ge k_0\) such that \(x\in D_k\).
Exhaustion gives a least \(K\) for which \(x\in N_K\). Nesting and \(x\notin N_{k_0}\) imply \(K{\gt}k_0\). Minimality gives \(x\notin N_{K-1}\), so \(x\in D_{K-1}\). Pairwise disjointness gives uniqueness.
For nonempty \(Z\), integer \(C_0\ge 2\) and every lower index \(k_0\),
A point either belongs to \(N_{k_0}\) or lies in its unique later layer.
27.46 Distances between dyadic layers
Retain the cells, neighborhoods and layers of Definition 27.44.3. The origin \(o\in \mathbb R^2\) is arbitrary, \(Z\subset \mathbb Z^2\) is finite, and \(C_0\ge 0\) is an integer. All scale indices are nonnegative integers. Distances in the plane are taken in the sup metric, so \(d_\infty (x,y)=\max \{ |x_1-y_1|,|x_2-y_2|\} \).
Let \(u,v\in \mathbb Z^2\), \(x\in \overline{C_{k,u}}\), and \(y\in \overline{C_{k,v}}\). If \(|u_i-v_i|\le C_0\) for both coordinates \(i\in \{ 1,2\} \), then
If \(|u_i-v_i|{\gt}C_0\) in at least one coordinate, then
Each coordinate of a closed cell lies in its closed interval of length \(2^k\). Indices differing by at most \(C_0\) therefore give coordinate distances at most \((C_0+1)2^k\). If two integer indices differ by more than \(C_0\), they differ by at least \(C_0+1\); the corresponding closed intervals are separated by at least \(C_0 2^k\). Taking the maximum of the two coordinate distances proves both bounds.
For every \(x\in \overline{D_k}\) and every \(z\in Z\),
For every \(x\in \overline{D_k}\) there is a point \(z_*\in Z\) such that
In particular, the existence of such an \(x\) implies \(Z\ne \varnothing \). Writing \(d_\infty (x,Z)=\inf _{z\in Z}d_\infty (x,z)\), one obtains
This is the layer-distance estimate in [ 247 , Section 11 ] . Empty layers are allowed.
The closure of the layer is a finite union of closed scale-\(k\) cells. A layer index lies outside \(P_k\), so it differs from every occupied index in at least one coordinate by more than \(C_0\). Applying the closed-cell lower bound to the cell containing \(z\) proves the first inequality.
The inclusion \(D_k\subseteq N_{k+1}\) persists after taking closures. A closed cell in \(N_{k+1}\) has an index within \(C_0\) of an occupied index. Choose a point \(z_*\) of \(Z\) in the latter cell and apply the closed-cell upper bound at scale \(k+1\). Its side length is \(2^{k+1}\), giving the second inequality. The infimum is bounded below by the first inequality and above by the distance to \(z_*\).
For every pair of scales \(k,h\), every \(x\in \overline{D_k}\), and every \(y\in \overline{D_h}\),
If \(h\ge k+2\), then
This is the nonadjacent-layer estimate in [ 247 , Section 11 ] . The last lower bound is positive under the source’s choice \(C_0\ge 2\). Both inequalities hold for every \(C_0\ge 0\), including when the stated lower bound is nonpositive.
Choose \(z\in Z\) with \(d_\infty (x,z)\le 2(C_0+1)2^k\). The endpoint lower bound gives \(d_\infty (y,z)\ge C_0 2^h\). The triangle inequality proves the first assertion. If \(h\ge k+2\), then \(4\cdot 2^k\le 2^h\), and hence
This proves the second assertion.
27.47 The dummy neighborhood meets only the initial layer
The dummy tile is the initial neighborhood \(N_{k_0}\). Its distance bound distinguishes it from the later layers; see [ 247 , Section 11 ] . The conclusions below concern closures, with distances measured in the sup metric.
For arbitrary origin, finite endpoint set \(Z\), nonnegative integer width \(C_0\) and index \(k\), every \(x\in \overline{N_k}\) admits an endpoint \(z\in Z\) with
The point lies in a closed scale-\(k\) cell whose index is within \(C_0\) of an occupied index in each coordinate. Choose an endpoint in that occupied cell and apply the closed-cell distance bound.
Under the same hypotheses, let \(k\ge k_0+1\), \(x\in \overline{N_{k_0}}\), and \(y\in \overline{D_k}\). Then
In particular, this is a positive separation when \(C_0\ge 2\).
Choose \(z\in Z\) with \(d_\infty (x,z)\le (C_0+1)2^{k_0}\). The layer bound gives \(C_0 2^k\le d_\infty (y,z)\), while \(2^k\ge 2\cdot 2^{k_0}\). The triangle inequality now yields
Suppose that \(C_0\ge 2\) and \(k\ge k_0\). If \(\overline{N_{k_0}}\cap \overline{D_k}\) is nonempty, then \(k=k_0\).
If \(k\ne k_0\), then \(k\ge k_0+1\). A shared point has distance zero from itself, contradicting the positive separation.
27.48 Fine cells and sparse belts
The origin \(o\in \mathbb R^2\) remains arbitrary. All scale indices and refinement depths are nonnegative integers. Let \(Z\subset \mathbb Z^2\) be finite, and let \(C_0\ge 0\) be an integer. Retain the layer \(D_k\) and its index set \(I_k\) from Definition 27.44.3.
For every refinement depth \(d\) and fine index \(u\in \mathbb Z^2\), define its coarse ancestor by
For every finite set \(P\subset \mathbb Z^2\), define its refinement by
For every finite \(P\), depth \(d\), and index \(u\in \mathbb Z^2\),
For every pair of scales \(\ell \le k\) and every \(x\in \mathbb R^2\),
Euclidean division by \(2^d\) gives a unique integer quotient and a remainder in \(\{ 0,\ldots ,2^d-1\} \) in each coordinate. This proves the membership criterion and the cardinality formula, including negative indices. The identity \(\lfloor t/n\rfloor =\lfloor \lfloor t\rfloor /n\rfloor \) for a positive integer \(n\) gives the point-index identity.
For every fine scale \(\ell \le k\), set
For every \(\ell \le k\), the actual layer and its fine-cell count satisfy
If \(Z\) is the set of endpoints of the crossing edges of a cut in a finite induced domain and \(b\) is the number of those edges, then
These formulas include empty layers and equal scales. The half-open fine cells are disjoint and nonempty, so \(F_{k,\ell }\) is exactly the set of fine indices whose cells meet \(D_k\). These are the predetermined cells used in the belt count of [ 247 , Section 11 ] .
A point’s fine index belongs to \(R_{k-\ell }(I_k)\) precisely when its coarse index belongs to \(I_k\). The cell-membership criterion therefore proves the exact union. Multiply the coarse-cell count by \(4^{k-\ell }\) and apply the endpoint and crossing-edge bounds.
With the fixed exponents \(\delta _0\) and \(\zeta =1-\delta _0/2\), define, for every scale \(k\),
Both floor arguments are nonnegative, so the displayed floors are the ordinary integer floors. The fine cells have side \(t_k\), while \(s_k\) is the belt pitch, as in [ 247 , Section 11 ] .
For every scale \(k\),
The inequalities \(\zeta \le 1\) and \(\delta _0\ge 0\) give the scale order. Ordered exponents give divisibility of the corresponding powers of two. Each floor is greater than its argument minus one. Adding the two floor bounds and using \(\zeta =1-\delta _0/2\) proves the last inequality.
Let \(F\subset \mathbb Z^2\) be finite and let \(m\ge 1\) be an integer. For every pair of residues \(a,b\in \{ 0,\ldots ,m-1\} \), define
For a fine layer with pitch exponent \(p\ge k\ge \ell \), the modulus is \(m=2^{p-\ell }\), the ratio of pitch to fine-cell side.
Let \(Z\) be the crossing-edge endpoint set of a finite induced-domain cut, with crossing-edge count \(b\). For every fine scale \(\ell \le k\) and every pitch exponent \(p\ge 0\), put \(F=F_{k,\ell }\) and \(m=2^{\max (p-\ell ,0)}\). There are residues \(a,b'\in \{ 0,\ldots ,m-1\} \) such that
When \(p\ge \ell \), put \(r=2^k\), \(t=2^\ell \), and \(s=2^p\); then \(m=s/t\) and
Apply coordinate-residue averaging to the fixed set \(F\). The selected belt cells satisfy \(m|B|\le 2|F|\). The actual fine-cell count gives the first inequality. Divide by the positive modulus and use \(4^{k-\ell }/2^{p-\ell }=r^2/(st)\) for the second.
For every scale \(k\),
The ratio equals \(2^{2k-p_k-\ell _k}\). The exponent bound gives \(2^{2k-p_k-\ell _k}\le 2^{2-\delta _0 k/2}=4\, 2^{-\delta _0 k/2}\). The factor four accounts for the two floor-rounding losses.
Let \(Z\) be the crossing-edge endpoint set of a finite induced-domain cut, with crossing-edge count \(b\). For every scale \(k\), put \(F=F_{k,\ell _k}\) and \(m=2^{p_k-\ell _k}=s_k/t_k\). There are residues \(a,b'\in \{ 0,\ldots ,m-1\} \) such that
This is the sparse-belt estimate in [ 247 , Section 11 ] . The constant is fixed after the radius \(C_0\) and is independent of the domain, cut, origin and scale. The residues are chosen separately in each layer. Empty endpoint sets and layers are included.
The scale order permits the fine-cell shift theorem with \(\ell =\ell _k\) and \(p=p_k\). Its coefficient \(16(2C_0+1)^2\) multiplied by the scale-ratio factor four gives \(64(2C_0+1)^2\).
27.49 Unique fine cells outside the dummy region
The half-open fine cells refine the actual layer partition. Retain the common origin, finite endpoint set \(Z\), integer width \(C_0\) and fine exponents \(\ell _k\); see [ 247 , Section 11 ] .
Suppose that \(Z\) is nonempty and \(C_0\ge 2\). For every lower index \(k_0\) and every \(x\notin N_{k_0}\), there is a unique pair \((k,z)\) such that \(k\ge k_0\), \(z\in F_{k,\ell _k}\) and \(x\in C_{\ell _k,z}\).
The point belongs to a unique layer \(D_k\) with \(k\ge k_0\). Since \(\ell _k\le k\), the exact union of its fine cells supplies \(z\in F_{k,\ell _k}\) with \(x\in C_{\ell _k,z}\). Any other admissible fine cell belongs to its actual layer, whose index must therefore equal \(k\). The unique cell-index criterion at exponent \(\ell _k\) then forces its index to equal \(z\).
For nonempty \(Z\), integer \(C_0\ge 2\) and every lower index \(k_0\),
Every point either belongs to \(N_{k_0}\) or belongs to the fine cell of its unique admissible pair \((k,z)\).
27.49.1 Capped contained dyadic squares
For \(k\in \mathbb {N}\) and \(z\in \mathbb {Z}^2\), let \(C_{k,z}\) be the integer sites of \(\lbrack 2^kz_1,2^k(z_1+1))\times \lbrack 2^kz_2,2^k(z_2+1))\).
Membership is given by coordinatewise integer division: \(x\in C_{k,z}\) if and only if \(\lfloor x/2^k\rfloor =z\). Equivalently, \(C_{k,z}=\mathbb {Z}^2\cap 2^k\bigl(z+[0,1)^2\bigr)\) consists of the integer points of the half-open cell of Definition 27.44.1 with origin \(o=0\); this is why the same symbol is used for both. Thus \(|C_{k,z}|=4^k\) and \(C_{0,z}=\{ z\} \). Two intersecting squares are nested in scale order. In particular, \(C_{k,z}\subseteq C_{k+1,\lfloor z/2\rfloor }\).
The floor description of membership in the real cell at origin zero gives the identification with its integer points. Refine a single index to unit scale. Integer ancestry agrees with the floor index at origin zero. Composing ancestry maps shows that all sites of a fine square have the same coarse index, which proves nesting.
For a finite set \(S\subseteq \mathbb {Z}^2\) and \(K\in \mathbb {N}\), let \(\mathcal P_K(S)\) consist of the pairs \((k,z)\) with \(k\le K\), \(C_{k,z}\subseteq S\), and either \(k=K\) or \(C_{k+1,\lfloor z/2\rfloor }\not\subseteq S\). This is the selection in the covering argument of [ 247 , Lemma 9.4 ] .
The squares selected by \(\mathcal P_K(S)\) partition \(S\). In particular,
The empty set has an empty partition, and \(\mathcal P_0(S)=\{ (0,x):x\in S\} \).
For each \(x\in S\), the unit square \(C_{0,x}=\{ x\} \) is contained in \(S\). Choose
If \(k{\lt}K\), maximality shows that the parent \(C_{k+1,\lfloor x/2^{k+1}\rfloor }\) is not contained in \(S\), so \((k,\lfloor x/2^k\rfloor )\in \mathcal P_K(S)\). Conversely, every selected square is contained in \(S\). Let \((k,u),(l,v)\in \mathcal P_K(S)\) with \(k{\lt}l\) and \(C_{k,u}\cap C_{l,v}\neq \varnothing \). Nesting gives
and \(k{\lt}l\le K\), contradicting the selection of \((k,u)\). At equal scales the common site forces equal indices. Counting the disjoint union gives the sum identity; restricting its nonnegative summands to scale \(K\) gives the cap bound.
Let \(M_k(S)\) be the indices \(z\) for which \(C_{k,z}\) meets both \(S\) and its complement.
Let \(a,b,u\in \mathbb Z\) with \(a\le b\) and \(u{\gt}0\). Then
The interval contains \(\lfloor b/u\rfloor -\lfloor a/u\rfloor +1\) integers. Apply \(u\lfloor b/u\rfloor \le b\) and \(a{\lt}u(\lfloor a/u\rfloor +1)\).
Let \(Q=[x_0,x_1]\times [y_0,y_1]\cap \mathbb Z^2\) be a nonempty integer rectangle, and put \(u=2^k\) for \(k\in \mathbb N\). If \(M_k(Q)\) is the set of indices of dyadic squares meeting both \(Q\) and its complement, then
This is the boundary-line count used in [ 247 , proof of Proposition 9.5, lines 717–729 ] .
Set \(X=[\lfloor x_0/u\rfloor ,\lfloor x_1/u\rfloor ]\cap \mathbb Z\) and \(Y=[\lfloor y_0/u\rfloor ,\lfloor y_1/u\rfloor ]\cap \mathbb Z\). An inside point places a mixed square’s index in \(X\times Y\). An outside point in the same square crosses one of the four sides of \(Q\). Monotonicity of integer division then forces the corresponding index coordinate to equal an endpoint of \(X\) or \(Y\). Consequently
The quotient-interval estimate gives \(u\, \# M_k(Q)\le 2(x_1-x_0)+2(y_1-y_0)+8u\le 4\operatorname {size}(Q)+8u\).
If \((k,z)\in \mathcal P_K(S)\) and \(k{\lt}K\), then \(\lfloor z/2\rfloor \in M_{k+1}(S)\). Consequently,
A selected square is nonempty and contained in \(S\), so its parent has an inside site. Below the cap, selection supplies an outside site in the parent. All selected indices at scale \(k\) therefore belong to the one-step refinement of \(M_{k+1}(S)\), whose cardinality is \(4|M_{k+1}(S)|\).
Let \(P\) be one of the real polygons of a template and put \(S=P\cap \mathbb Z^2\). For \(y\in \mathbb Z\), write \(R_y(S)=\{ x\in \mathbb Z:(x,y)\in S\} \). Every nonempty real horizontal section of \(P\) is a closed interval \([a,b]\), and
This integer interval may be empty. If it is nonempty, its endpoints are its attained minimum and maximum. For every real \(c\) and integers \(m,y\), rounding commutes with the integer shift:
A finite convex hull is compact and convex. Its intersection with a horizontal line is compact, and convexity fills the interval between its attained extrema. Exact lattice sampling then rounds the endpoints. Translation by an integer commutes with both floor and ceiling. This is the initial row argument in the proof of Lemma 9.4 of [ 247 ] .
If \(C_{\rm tpl}\ge 1\), a template satisfies \(|T|\le 9ns_0\).
Each nonempty sample lies in an integer square of radius \(s_0\) about any one of its sites, so its cardinality is at most \((2s_0+1)^2\). Empty samples contribute zero. Sum over the pieces and use \((2s_0+1)^2\le 9s_0(s_0+1)\) and \(N_0(s_0+1)\le n\).
For a finite \(S\subset \mathbb Z^2\), let \(S_r=S+([-r,r]^2\cap \mathbb Z^2)\). Membership in \(S_r\) means that both coordinate differences from some point of \(S\) are at most \(r\) in absolute value. Dilation is monotone in both \(S\) and \(r\), satisfies \((S_r)_t=S_{r+t}\), and
For a sampled polygon, each nonempty row in this union contributes the interval from its minimum minus \(r\) to its maximum plus \(r\). If \(S=\bigcup _{i\in I}S_i\) is a finite union, then, for \(r,t\ge 0\),
The pieces may be empty, overlapping, or disconnected.
The coordinate inequalities give the row formula. A point newly present in the dilation of a union has a witness in one constituent, and cannot belong to the old dilation of that constituent. Counting the union gives the cardinality inequality without requiring disjointness.
Each polygon is the intersection of the supporting strips for the four linear forms \(x,y,x+y,x-y\). Equivalently, the polygon is the intersection of all closed supporting slabs with allowed normal directions.
Every supporting slab contains the convex hull. Conversely, the three oriented side functionals of a triangle give nonnegative barycentric coordinates after division by its nonzero signed determinant. Both signs of the determinant are allowed. For a rectangle, the orthogonal side coordinates lie in \([0,1]\), giving four nonnegative convex weights. Every allowed normal is a scalar multiple of one of the four listed forms. Their extrema over the finite vertex set are attained.
For each sampled polygon there are eight integers \(l_x,u_x,l_y,u_y,l_+,u_+,l_-,u_-\) such that, on the vertical strip,
Outside \([l_y,u_y]\) the row is empty; the displayed interval may also be empty. Nonempty rows form an integer interval, and the attained lower and upper endpoints each change by at most one between consecutive nonempty rows. Restricting to a vertical window also preserves consecutive occupied columns. A sampled point can be moved to an occupied adjacent row with a horizontal change of at most one.
Round each real lower strip bound upward and each upper bound downward. Integer row shifts commute with rounding. The lower endpoint is a maximum of three affine functions with slopes \(0,-1,1\), and the upper endpoint is their minimum counterpart. Pairwise lower–upper comparisons give an interval of nonempty rows. Each affine function changes by at most one in one row step, as do its maximum or minimum.
Let \(S\) be one sampled piece. Its dilated horizontal rows are intervals. Increasing the radius from \(r\) to \(r+1\) adds at most four sites to each old nonempty row. Every radius-\(r\) row has at most \(2s_0+2r+1\) sites. If \(S\) is nonempty, there are integers \(a\le b\) with \(b-a\le 2s_0\) such that its radius-\(r\) occupied rows are exactly \([a-r,b+r]\cap \mathbb Z\). For \(1\le j\le s_0\), including empty samples,
Intersect the original sample with a vertical radius window. Its occupied columns are consecutive, so horizontal radius extension gives an interval. Compare adjacent windows by clamping an original row index to the smaller window. An occupied adjacent row supplies a sampled point with horizontal displacement at most one. The additional horizontal radius adds one more, so each old endpoint extends by at most two. The piece diameter bounds every row width. Count the old occupied rows, each contributing at most four sites, and the two new extreme rows. Their total is at most \(12s_0+12j-2\le 24(s_0+1)\).
If \(C_{\rm tpl}\ge 24\) and \(1\le j\le s_0\), the template satisfies
Apply union-layer subadditivity to the pieces and then the scale bound:
Overlap, empty pieces, and disconnected unions require no extra condition.
These geometric estimates do not establish the entropy or physical area-law conclusions.
27.49.2 Ambient boundaries of template dilations
Write \(\mathcal N(p)\) for the four nearest neighbors of \(p\in \mathbb Z^2\). For a finite set \(S\), its ambient boundary \(\partial S\) consists of unordered nearest-neighbor edges with one endpoint in \(S\) and the other outside \(S\). These are ambient lattice edges; they are not boundary edges of an intersection with a physical cut.
Regard \(S\) as a subset of its unit sup-norm dilation \(S_1\). Take its unordered crossing edges in the induced nearest-neighbor graph on \(S_1\), and map the endpoints back to \(\mathbb Z^2\).
The boundary consists exactly of edges \(\{ p,q\} \) with \(p\in S\), \(q\notin S\), and \(q\in \mathcal N(p)\). The inside–outside orientation is unique, and mapping back from the finite enclosure preserves the number of unordered edges. Each point has at most four neighbors; a nearest-neighbor step from \(S_r\) lies in \(S_{r+1}\). The empty set has empty boundary.
Every neighbor of a point of \(S\) lies in \(S_1\), so the enclosure loses no crossing edge. Its inclusion into the lattice is injective, as is the induced map on unordered pairs. The endpoint in \(S\) determines the orientation uniquely. The four neighbors change exactly one coordinate by one, proving the radius assertion.
The neighbor relation is symmetric. If a finite set \(R\) meets every edge of \(\partial S\), then \(|\partial S|\le 4|R|\). In particular,
Cover the crossing edges by the four incident edges at each point of \(R\). For the first bound use the outside endpoints. For the second, an inside endpoint in \(S_{j-1}\) would put its outside neighbor in \(S_j\), a contradiction. No connectedness or nonemptiness assumption is needed.
For finite lattice sets \(S,U\),
An edge leaving \(S\setminus U\) either leaves \(S\) or enters \(U\). Count the union of these two sets of unordered edges.
If \(C_{\rm tpl}\ge 24\) and \(0\le j\le s_0\), then
For positive \(j\), count inside endpoints in \(T_j\setminus T_{j-1}\) and apply the depth-layer bound. At \(j=0\), use outside endpoints in \(T_1\setminus T\); the template assumption \(s_0\ge 1\) permits this layer. Thus the largest radius \(s_0\) never needs a bound at radius \(s_0+1\). Apply the set-difference estimate for the shell.
27.50 Entropy cost of a partial depth row
Let \(\Omega \) be a unit vector on a finite lattice domain with local dimension \(q\). For disjoint regions \(R,B\), one has
Subadditivity gives
Complements are taken in the physical domain. Since \(R\cap B=\varnothing \),
Apply subadditivity to this disjoint union and use purity, \(S_\Omega (X)=S_\Omega (X^c)\), to obtain
Let \(T\) be a template with \(C_{\rm tpl}\ge 24\), let \(1\le j\le s_0\), and let \(Y\subseteq T_j\setminus T_{j-1}\). For a unit vector \(\Omega \), local dimension \(q\ge 1\), and any physical region \(A\), one has
This is the partial-row entropy step in the proof of [ 247 , Lemma 9.4 ] .
Set \(R=A\cap (T_{j-1}\setminus T)\) and \(B=A\cap Y\). These regions are disjoint and \(|B|\le |Y|\le n\). The regional increment bound and \(S_\Omega (B)\le |B|\log q\) (Theorem 27.110.1) give the claim since \(\log q\ge 0\).
27.50.1 Finite weighted dyadic summation
Let \(K\ge 0\) be an integer, \(e{\gt}0\), \(B\ge 0\), \(A\in \mathbb R\), and \(a_k\in \mathbb R\). Suppose \(2^K a_K\le A\) and \(2^k a_k\le B\) for \(0\le k{\lt}K\). Then
This is the finite summation step in [ 247 , Lemma 9.4 ] ; template counting and the entropy application are separate statements.
Factor each summand as \((2^k a_k)(2^k)^e\). The cap contributes at most \(A(2^K)^e\). Since \(2^e{\gt}1\), the lower scales contribute at most
The same formula includes \(K=0\), when the lower sum is empty.
27.51 Safe rectangles in template coverings
Let \(T\) be separated from the cut, let \(D_0\ge 1\) be an integer, and let \(0\le j\le s_0\). Every integer rectangle \(Q\subseteq T_j\) of size at most \(s_0\) is safe. In particular, if \(2^K\le s_0\), every selected square of the capped partitions of \(T\) and \(T_j\setminus T\) is a safe rectangle. The standing choice \(D_0{\gt}2R+10\) implies \(D_0\ge 1\). This is the clearance step of [ 247 , Lemma 9.4 ] .
For \(x\in Q\subseteq T_j\), choose \(p\in T\) with \(\| p-x\| _\infty \le j\). For every cut endpoint \(z\),
Since \(j\le s_0\le D_0s_0\), this gives \(D_0\operatorname {size}(Q)\le D_0s_0{\lt}\| x-z\| _\infty \). Selected cells are contained in the partitioned set, and their sizes satisfy \(2^k\le 2^K\le s_0\).
Let \(T\) be a template with \(C_{\rm tpl}\ge 24\), let \(e{\gt}0\), and choose \(K\ge 0\) with \(2^K\le s_0{\lt}2^{K+1}\). For the actual capped dyadic partition \(\mathcal P\) of \(T\),
This is the core-covering moment estimate in [ 247 , Lemma 9.4 ] . It requires neither separation from a physical cut nor an entropy hypothesis.
The side budgets are \(18n\) at scale \(K\) and \(4n\) below it. Apply Theorem 27.123.2 with \(L=s_0\), \(A=18n\) and \(B=4n\), and factor out \(n\).
27.52 Weighted coverings of template shells
Let \(T\) be a template with \(C_{\rm tpl}\ge 24\) and \(0\le j\le L\le s_0\). Then \(|T_j\setminus T|\le nj\). If \(L{\lt}2^{K+1}\), the number \(a_K\) of cap-scale squares in the capped dyadic partition of \(T_j\setminus T\) satisfies \(2^K a_K\le 2n\).
Sum the depth-layer estimates to obtain \(|T_j\setminus T|\le nj\). Disjointness gives \(4^K a_K\le nj\le nL{\lt}2n2^K\); divide by \(2^K{\gt}0\).
Let \(T\) be a template with \(C_{\rm tpl}\ge 24\), \(0\le j\le L\le s_0\), \(2^K\le L{\lt}2^{K+1}\), and \(e{\gt}0\). For the maximal contained dyadic squares \(Q\) of \(T_j\setminus T\) with side capped at \(2^K\), one has
This is the weighted covering step of [ 247 , Lemma 9.4 ] ; safety and the entropy estimate remain separate.
Write \(a_k\) for the number of selected squares of side \(2^k\). The mixed-square bound gives \(2^k a_k\le 14n\) for \(k{\lt}K\). The cap bound gives \(2^K a_K\le 2n\). Apply Theorem 27.123.2 with \(A=2n\), \(B=14n\) and the cap bounds \(2^K\le L{\lt}2^{K+1}\), and factor out \(n\).
27.53 Entropy from a uniform safe-box estimate
Fix a finite physical domain, a cut \(A\), a normalized vector \(\Omega \), local dimension \(q\ge 1\), and a native safety parameter \(D_0\ge 1\). Assume explicitly that for every \(\varepsilon \in (0,1)\) there is \(C_\varepsilon \ge 0\) such that every safe native rectangle satisfies
Fix \(e\in (0,1)\). The same constant \(C_e\), independent of the template, works for all separated templates with \(C_{\rm tpl}\ge 24\). For every \(1\le L\le s_0\) and \(0\le j\le L\), one has
If \(1\le j\le L\) and \(Y\subseteq T_j\setminus T_{j-1}\) is arbitrary, the entropy of \(A\cap ((T_{j-1}\setminus T)\cup Y)\) is at most the displayed shell bound plus \(n\log q\). The fixed-exponent statements require only the safe-box estimate at that exponent and \(e{\gt}0\). The quantified statement retains the source range \(0{\lt}e{\lt}1\) and the uniform choice of \(C_e\).
Every selected square is safe by separation and the scale bounds. Apply the safe-box estimate to its physical rectangle region, sum by subadditivity, and use the actual weighted core and shell estimates. For a prefix, first use the shell estimate at depth \(j-1\), then add the proved partial-row cost \(n\log q\). The uniform constant is chosen from the safe-box hypothesis before any template or scale is selected. The quantified theorem chooses the caps as \(\lfloor \log _2 s_0\rfloor \) and \(\lfloor \log _2 L\rfloor \); the integer logarithm inequalities give the required comparable powers of two. This composes the conditional entropy step of [ 247 , Lemma 9.4 ] ; it does not derive the safe-box hypothesis from an initial existential exponent.
27.53.1 Physical boundaries of separated templates
Let \(\Lambda \subset \mathbb Z^2\) be any finite induced domain, \(A\subseteq \Lambda \), and \(Z\) the endpoints of its unordered crossing edges. No connectedness assumption is imposed on \(\Lambda \) or \(A\).
For any finite ambient set \(U\) disjoint from \(Z\), inclusion into the lattice sends \(\partial _\Lambda (A\cap U)\) injectively into \(\partial U\). In particular, \(|\partial _\Lambda (A\cap U)|\le |\partial U|\).
Orient a physical crossing edge from \(p\in A\cap U\) to \(q\notin A\cap U\). If \(q\notin A\), then \(p\in Z\), contrary to \(U\cap Z=\varnothing \). Thus \(q\in A\) and \(q\notin U\), and the edge crosses \(U\). Inclusion preserves unordered pairs injectively.
Suppose \(D_0\ge 1\), \(s_0\ge 1\), and \(\operatorname {dist}_\infty (T,Z){\gt}4D_0s_0\). For \(0\le j\le s_0\), the ambient dilation \(T_j\) is disjoint from \(Z\).
A point \(z\in T_j\cap Z\) would be within distance \(j\le s_0\) of a point of \(T\), contradicting \(4D_0s_0{\lt}\operatorname {dist}_\infty (T,Z)\).
Let \(T\) be a template separated from the cut, with \(C_{\rm tpl}\ge 24\) and \(D_0\ge 1\). For every \(0\le j\le s_0\),
These are the physical edge estimates in [ 247 , Lemma 9.4 ] .
The preceding separation estimate excludes cut endpoints from \(T_j\) and its shell. By Theorem 27.53.1.1 and Lemma 27.49.2.5,
The core is the case \(j=0\). The manuscript fixes \(D_0{\gt}2R+10\), which implies \(D_0\ge 1\) for \(R\ge 0\).
Suppose \(\operatorname {dist}_\infty (T,Z){\gt}4D_0s_0\). For every \(j\ge 0\), \(p\in T_j\), and \(z\in Z\),
If \(D_0\ge 1\), \(j\le s_0\), and \(u\le s_0\), this distance exceeds \(D_0u\). In particular, if \(C_{k,c}\subseteq T_j\) and \(2^k\le s_0\), every site of \(C_{k,c}\) has distance greater than \(D_0 2^k\) from every endpoint of a physical crossing edge. The square is exactly the product of integer intervals
each of which has \(2^k\) sites. This is the strict-clearance step in [ 247 , Lemma 9.4 ] .
Choose \(q\in T\) with \(\| p-q\| _\infty \le j\). The triangle inequality gives \(\| p-z\| _\infty \ge \| q-z\| _\infty -j{\gt}4D_0s_0-j\). Since \(s_0{\gt}0\), \(D_0\ge 1\), and \(j\le s_0\), we have \(4D_0s_0-j{\gt}D_0s_0\ge D_0u\). Both endpoints of every crossing edge belong to \(Z\). The integer interval description follows directly from the half-open dyadic square inequalities.
Let \(\Lambda \subset \mathbb Z^2\) be finite, \(A\subseteq \Lambda \), and \(T\subset \mathbb Z^2\) finite. For integers \(L,R\ge 0\), write \(T_L=T+[-L,L]^2_{\mathbb Z}\). Suppose \(\| t-z\| _\infty {\gt}L+R\) for every \(t\in T\) and every endpoint \(z\) of a physical crossing edge of \(A\). Then
The statement includes empty targets and disconnected domains. This is the cut-exclusion step in the proof of [ 247 , Theorem 1.1 ] ; separation from the earlier members of a family is a further geometric condition.
Let \(x\) be joined to \(y\in A\cap T_L\) by a path of length at most \(R\). Then \(T\) is nonempty. The depth \(d(v)=\min _{t\in T}\| v-t\| _\infty \) satisfies \(d(y)\le L\) and changes by at most one along each graph edge. Every endpoint of a prefix of the path therefore has depth at most \(L+R\). If the path leaves \(A\), its first crossing gives a physical cut endpoint of depth at most \(L+R\), contrary to the hypothesis. Thus \(x\in A\); the depth bound at \(x\) gives \(x\in T_{L+R}\).
27.53.2 Mixed-square counts for template shells
Put \(u=2^k\). The integer sites of \(C_{k,z}\) satisfy \(uz_i\le x_i{\lt}uz_i+u\) in each coordinate. For arbitrary finite lattice sets and finite unions,
In particular, the corresponding mixed-square counts are subadditive.
An inside site of a union belongs to one constituent, whereas an outside site lies outside every constituent. For a difference, the inside site lies in \(S\) and outside \(U\). An outside site either lies outside \(S\) or belongs to \(U\). The same two sites therefore witness mixedness for one of the defining sets.
For an actual sampled template piece \(S_a=P_a\cap \mathbb {Z}^2\), let \((x,y)\in (S_a)_j\). If row \(z\) is occupied and \(|z-y|\le d\), there is \((w,z)\in (S_a)_j\) with \(|w-x|\le d\). The same assertion holds for the undilated sample.
The four pairs of integer supporting inequalities describe the sampled rows. Their slopes belong to \(\{ -1,0,1\} \), so clamping \(x\) into the occupied target row changes it by at most \(d\). For the dilated piece, first clamp the original source row into the target window of radius \(j\). Consecutiveness of occupied source rows keeps this row occupied. Apply the preceding displacement estimate, then clamp the horizontal coordinate into the resulting integer interval of radius \(j\).
For every actual sampled template piece \(S_a\) of diameter at most \(s_0\), every integer \(j\ge 0\) and \(u=2^k\),
Empty samples and thin polygons are included. This gives the estimate in the proof of [ 247 , Lemma 9.4 ] .
In an interior block of \(u\) rows, every row is occupied. Moving an inside site and the two endpoints of a reference row by at most \(u\) shows that a mixed square has column index within two of an endpoint’s column index. Thus at most ten columns occur in each interior block. For the first and last occupied blocks, count all columns meeting the diameter enclosure. If \(N_x,N_y\) count the enclosure’s column and row blocks, respectively, then
Combining these inequalities gives the stated bound. The empty sample has no mixed squares.
Let \(T\) be an actual template with \(N_0\) pieces. For every \(j\ge 0\),
If \(C_{\rm tpl}\ge 24\) and \(0\le j\le s_0\), then
No disjointness or connectedness of the pieces is required.
Integer dilation commutes with the actual finite union of sampled pieces. Sum the piece estimates and apply \(24N_0(s_0+1)\le n\). The difference bound is the sum of the core and dilated-template bounds.
Suppose \(C_{\rm tpl}\ge 24\), \(0\le j\le s_0\), \(k{\lt}K\) and \(u=2^k\le s_0\). The number \(N_k\) of selected squares of side \(u\) in the capped partition of \(T_j\setminus T\) satisfies \(uN_k\le 14n\).
Each selected square has a mixed parent, and each parent has four children. Hence \(uN_k\le 2(2u)|M_{k+1}(T_j\setminus T)|\). The mixed-square estimate and \(24N_0u\le n\) give \(uN_k\le 2(3n+96N_0u)\le 14n\).
Let \(T\) be a template with \(C_{\rm tpl}\ge 24\), and suppose \(2^K\le s_0{\lt}2^{K+1}\). Write \(N_k\) for the number of selected squares of side \(2^k\) in the capped partition of \(T\). Then
This proves the geometric core-covering counts in the proof of Lemma 9.4. No separation from a cut is needed for these counts.
The cap squares are disjoint, so \(4^K N_K\le |T|\le 9ns_0\le 18n2^K\). Below the cap, every selected square has a mixed parent and each parent has four children. Since \(2^{k+1}\le 2^K\le s_0\),
where \(24N_0s_0\le n\) follows from the template scale inequality.
27.54 Uniform separation of the dyadic scales
Retain the fixed exponents and scales of Definition 27.48.5. In particular, \(\delta _0=2\cdot 10^{-7}\), \(\zeta =1-\delta _0/2\), \(t_k=2^{\ell _k}\), \(r_k=2^k\), and \(s_k=2^{p_k}\). All indices below are nonnegative integers. The following statements give the uniform scale separation used in [ 247 , Section 11 ] .
For every pair of indices \(d,k\) satisfying \(10^7d\le k\),
The threshold gives \(d\le k\) and \(\zeta k=k-10^{-7}k\le k-d\). Taking the floor proves the first inequality. Likewise \((1+\delta _0)k\ge k+2d\ge k+d\), so its floor is at least the integer \(k+d\).
For every pair of indices \(d,k\) satisfying \(10^7d\le k\),
Apply the increasing function \(n\mapsto 2^n\) to the two index inequalities and use \(2^{a+b}=2^a2^b\).
For every real number \(M\), there is a nonnegative integer \(K\) such that, for every nonnegative integer \(k\ge K\),
The threshold is chosen from \(M\) and the fixed exponents alone, independently of the origin, domain, cut, and neighborhood radius.
Choose a nonnegative integer \(d\) with \(M{\lt}2^d\), using the unboundedness of powers of two, and set \(K=10^7d\). For every \(k\ge K\), the preceding theorem gives \(2^dt_k\le r_k\) and \(2^dr_k\le s_k\). The cell sides are positive, so replacing \(2^d\) by \(M\) proves both inequalities.
27.55 Polynomial factors in the dyadic budget
Retain the fixed exponent \(\delta _0=2\cdot 10^{-7}\). For every real number \(p\) and nonnegative integer \(k\), set
The power \((k+1)^p\) is the ordinary real power of a positive number. The following estimates give the numerical series bound used in [ 247 , Section 11 ] . To bound the number of repairs by this series, one must also bound the descendants of each initial mark and relate the initial marks to the sparse belts.
For every real number \(p\), there is a nonnegative integer \(K\) such that, for every nonnegative integer \(k\ge K\),
The threshold depends only on \(p\) and the fixed exponent.
Set \(a=\delta _0\log 2/4{\gt}0\). Exponential decay dominates every fixed real power, so \((k+1)^p\exp (-ak)\) tends to zero as \(k\) tends to infinity. Choose \(K\) so that this expression is at most one for every \(k\ge K\). Multiplying by the positive number \(2^{-\delta _0 k/4}\) gives the asserted inequality.
For every real number \(p\), the nonnegative series
converges.
Put \(q=2^{-\delta _0/4}\). Positivity of \(\delta _0\) gives \(0{\lt}q{\lt}1\). The absorption estimate bounds \(w_p(k)\) by \(q^k\) for all sufficiently large \(k\). The geometric series converges, and adding finitely many initial terms preserves convergence.
For every real number \(p\), there is a real number \(B{\gt}0\) such that, for every finite set \(F\) of nonnegative integers,
The bound is chosen from \(p\) and the fixed exponent alone, independently of the origin, domain, cut, and neighborhood radius. It therefore applies to every finite selection of scales above any chosen lower scale.
Let \(S\) be the sum of the convergent nonnegative series and take \(B=S+1\). Nonnegativity gives \(S\ge 0\) and bounds every finite subsum by \(S\), proving both \(B{\gt}0\) and the stated estimate.
27.56 Primary birth regions and adjacent fine meshes
Retain the half-open cells \(C_{k,z}\), the layer \(D_k\), and its finite index set \(I_k\) from Definition 27.44.3. The origin \(o\in \mathbb R^2\) is arbitrary, the endpoint set \(Z\subset \mathbb Z^2\) is finite, and the neighborhood radius \(C_0\) is a nonnegative integer. All distances and diameters use the sup metric on \(\mathbb R^2\).
The primary regions in [ 247 , Section 11 ] are obtained by intersecting a pitch interior with an actual layer and taking its closure. All fragments in one pitch interior receive the same primary identifier, even when their union is disconnected.
Fix nonnegative integer indices \(k,\ell ,p\), put \(r=2^k\), \(t=2^\ell \), and \(s=2^p\), and choose integer shifts \(a,b\). For every \(j\in \mathbb Z^2\), set
Here \(U_j\) is the open pitch interior, \(F_{j,z}\) is a closed cell fragment, and \(R_j\) is the corresponding primary birth region. These definitions also allow empty interiors and empty fragments. In the geometric construction, \(\ell =\ell _k\), \(p=p_k\), and \(a,b\) represent the selected residue classes modulo \(s/t\). In the picture below, with \(s=4t\) and \(r=2t\), the open pitch interiors \(U_j\) and \(U_{j'}\), for \(j'=j+(1,0)\), share one vertical interval and are separated by the vertical strip of width \(t\) between them, and each scale-\(r\) cell of the layer contributes one closed fragment to the primary region of each pitch interior it meets.
![\begin{tenkz}[size=s, rows={ket,ket,ket,ket}, cols=8, bonds=none]
\tndeclare{species}{belt}{hue=source:orange}
\tndeclare{species}{layer}{hue=source:blue}
\tnmark[form=enclosure, species=belt, tint]{(1,1) .. (4,8) - (1,2) .. (3,4) - (1,6) .. (3,8)}{}
\tnmark[form=enclosure, species=layer, tint]{(1,3) .. (2,4)}{}
\tnmark[form=enclosure, species=layer, tint]{(3,3) .. (3,4)}{}
\tnmark[form=enclosure, species=layer, tint]{(1,6) .. (2,6)}{}
\tnmark[form=enclosure, species=complement]{(3,3) .. (4,4)}{}
\tnmark[form=enclosure, species=complement]{(1,5) .. (2,6)}{}
\tnmark[form=enclosure, species=selected, label pos=n]{(1,2) .. (3,4)}{$U_j$}
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For every choice of the preceding data,
The exact cell union gives \(D_k=\bigcup _{z\in I_k}C_{k,z}\). Intersect this finite union with \(U_j\) and commute closure with the finite union. Empty intersections have empty closures, so retaining precisely the indices in \(J_j\) leaves the union unchanged.
For \(i\in \{ 1,2\} \), write \(\alpha _{i,z}=o_i+rz_i\) and \(\beta _{i,z}=o_i+r(z_i+1)\). If \(C_{k,z}\cap U_j\ne \varnothing \), then
Both coordinate intervals have strictly positive length. Every fragment, including an empty one, satisfies the following containment and diameter bounds. The distance bound holds for arbitrary points \(x,y\in F_{j,z}\).
A point of the nonempty intersection lies below both upper endpoints in each coordinate. It lies at or above the cell’s lower endpoint and strictly above the pitch interior’s lower endpoint. Hence the maximum lower endpoint is strictly below the minimum upper endpoint. The coordinate intersection contains the open interval between these endpoints and is contained in their closed interval. Taking closures therefore gives that closed interval. Closure commutes with products, proving the rectangle formula.
Containment in the closed cell follows from monotonicity of closure. Apply the closed-cell distance bound with identical cell indices and index radius zero. This gives the pointwise distance bound, and hence the diameter bound.
For every primary index \(j\), the following containment and diameter bounds hold. The distance bound holds for arbitrary points \(x,y\in R_j\).
For distinct indices \(i,j\), every \(x\in R_i\) and \(y\in R_j\) satisfy
Monotonicity of closure gives \(R_j\subseteq \overline{U_j}\). This closure is contained in the product of the closed intervals \([v_{j,q}+t,v_{j,q}+s]\), for \(q=1,2\). Two points in this product have coordinate differences at most \(s\), proving the distance and diameter bounds.
Distinct integer indices differ in at least one coordinate. Suppose \(i_q{\lt}j_q\). The upper endpoint for index \(i_q\) is at most \(o_q+t(a,b)_q+sj_q\), while the lower endpoint for index \(j_q\) is \(o_q+t(a,b)_q+sj_q+t\). Thus the two coordinate intervals are separated by at least \(t\). Taking the maximum coordinate distance proves the separation claim; the other ordering is symmetric.
If \(k\le p\), then every primary index satisfies
This bound is independent of the endpoint set, neighborhood radius, origin, and selected shifts.
Put \(N=2^{p-k}\), so \(s=Nr\), and let \(q_i=\lfloor (v_{j,i}-o_i)/r\rfloor \). If a cell indexed by \(z\) meets \(U_j\), a point in that intersection lies above \(v_{j,i}\) and below \(v_{j,i}+s\). The half-open cell inequalities and the defining floor inequalities consequently give, for each coordinate \(i\in \{ 1,2\} \),
Thus \(J_j\) lies in an integer rectangle with \(N+1\) indices in each coordinate. Its cardinality is at most \((N+1)^2\), which is at most \((N+2)^2\).
The fixed fine exponents \(\ell _k=\lfloor \zeta k\rfloor \) are nondecreasing. For every nonnegative integer \(k\),
Since \(0\le \zeta \le 1\), multiplication by \(\zeta \) and taking floors preserve the order of the layer indices. Also \(\zeta (k+1)\le \zeta k+1\), and adding one commutes with taking the floor. Therefore \(\ell _k\le \ell _{k+1}\le \ell _k+1\). Exponentiating the two possible integer values gives the side-length assertion.
27.57 The number of primary identifiers in a layer
Retain the origin, layer and pitch interiors from Definition 27.56.1. Write \(I_k\) for the layer-cell index set, \(r=2^k\), \(t=2^\ell \), \(s=2^p\), and \(o'=o+t(a,b)\). The count below is the primary count in [ 247 , Section 11 ] .
For every \(z\in I_k\), let \(L_z\) and \(H_z\) be the lower and upper corners of the closed cell of side \(r\). For each coordinate \(i\), define
Let \(B_z\) be the integer rectangle \([q_{z,1}^-,q_{z,1}^+]\times [q_{z,2}^-,q_{z,2}^+]\). The finite set of primary pitch indices is
For every choice of the nonnegative scale indices and every \(j\in \mathbb Z^2\),
Only the reverse implication needs a proof. A point \(x\) in the intersection belongs to a cell \(C_{k,z}\) with \(z\in I_k\). Each coordinate lies between those of \(L_z\) and \(H_z\), so monotonicity of the floor gives \(q_{z,i}^-\le \lfloor (x_i-o'_i)/s\rfloor \le q_{z,i}^+\). The open pitch inequalities imply \(j_i{\lt}(x_i-o'_i)/s{\lt}j_i+1\); hence the middle integer is \(j_i\). Thus \(j\in B_z\), and the defining filter retains it.
If \(k\le p\), then
The scale ordering gives \(r\le s\). In each coordinate, \(H_{z,i}=L_{z,i}+r\) and therefore
The upper bound follows by comparing \((H_{z,i}-o'_i)/s\) with \((L_{z,i}-o'_i)/s+1\) and using \(\lfloor u+1\rfloor =\lfloor u\rfloor +1\). Thus \(|B_z|\le 4\). A finite union has cardinality at most the sum of the cardinalities of its members, and the nonempty-intersection filter can only decrease the cardinality. This proves the bound.
Take \(Z\) to be the endpoint set of the crossing edges of a cut \(A\) in a finite domain \(\Lambda \). If \(k\le p\), then
The bound is uniform in the domain, cut, origin and selected shifts.
Combine \(|\mathcal P_k|\le 4|I_k|\) with \(|I_k|\le 8(2C_0+1)^2|\partial _\Lambda A|\).
27.58 The marked points of belt cells
The belt construction marks each cell at its center, corners and side midpoints. We first count these actual points for a finite family of cells at one scale, as in [ 247 , Section 11 ] . Coincident points contribute only once to the union.
For a cell \(C_{\ell ,z}=o+t(z+[0,1)^2)\), where \(t=2^\ell \), put
These are its four corners, four side midpoints and center.
For any finite cell-index set \(F\subset \mathbb Z^2\), define
Every point of \(M_{\ell ,z}\) belongs to \(\overline{C_{\ell ,z}}\).
Each coordinate of \(e/2\) lies in \([0,1]\), so both coordinates of the mark lie in the corresponding closed cell intervals.
For every origin, scale and cell index, \(|M_{\ell ,z}|=9\).
The side \(2^\ell \) is positive. Thus distinct elements of \(\{ 0,1,2\} ^2\) give distinct points, and there are nine such elements.
For every finite cell-index set \(F\), including the empty set,
The cardinality of a finite union is at most the sum of the cardinalities of its members. Each member has nine points.
Let \(F_k(a,b)\) be the actual belt-cell indices in the refined layer for the selected coordinate residues \(a,b\). There is a pair of residues such that
The estimate is uniform in the finite domain, cut, origin and scale.
Choose the pair of residues supplied by the sparse-belt estimate, then multiply its cell count by nine. The coefficient is \(9\cdot 64=576\).
27.59 Separation on the translated quarter mesh
The numerical estimates used in the initial-star argument come from a common translated lattice. We record them separately from the construction of the active interfaces; see [ 247 , Section 11 ] .
Let \(C_0\ge 2\) and \(k\ge 50{,}000{,}000\), and put \(t_k=2^{\ell _k}\). If \(x\in \overline{D_k}\), \(y\in \overline{D_h}\), and the layer indices differ by at least two, then
The threshold is uniform in the finite endpoint set and origin.
The scale-ratio estimate with exponent gap five gives \(32t_k\le 2^k\). The nonadjacent-layer bound gives \(\| x-y\| _\infty \ge (C_0-1)2^{\max (k,h)}/2\ge 2^k/2\ge 16t_k\).
Under the same conditions on \(C_0\) and \(k\), if \(x\in \overline{D_k}\), \(y\in \overline{D_h}\) and \(\| x-y\| _\infty {\lt}10t_k\), then
A difference of at least two would give distance at least \(16t_k\), contradicting the strict upper bound since \(t_k{\gt}0\).
For \(o\in \mathbb R^2\) and \(q\in \mathbb R\), define
Let \(j,\ell \) be nonnegative integers with \(\ell \le j+1\). For every finite cell-index set \(F\),
In particular, cells of sides \(t/2\), \(t\) and \(2t\), where \(t=2^\ell \) and \(\ell \ge 1\), have all their marks on the same mesh \(\mathcal M(o,t/4)\).
A mark of a cell of side \(2^j\) is \(o+2^j(z+e/2)\), where \(e\in \{ 0,1,2\} ^2\). Since \(\ell \le j+1\), the vector \(2^{j+1-\ell }(2z+e)\) is integral, and the mark equals \(o+(2^\ell /4)2^{j+1-\ell }(2z+e)\).
If \(q{\gt}0\) and \(x,y\in \mathcal M(o,q)\) are distinct, then
At least one coordinate of the integer vectors indexing \(x\) and \(y\) differs. Its difference has absolute value at least one, so the corresponding coordinate difference of \(x\) and \(y\) has absolute value at least \(q\).
Suppose \(q{\gt}0\) and \(u,v,x\in \mathcal M(o,q)\). Let \(L\) be the affine span of \(u,v\). Suppose either \(u=v\), or \(L\) has slope \(0\), \(\infty \), \(1\) or \(-1\). If \(x\notin L\), then
Thus on a quarter mesh the clearance is at least \(t/8\).
If \(u=v\), use the point-separation bound. Otherwise, the supporting line has a constant coordinate, coordinate sum or coordinate difference. Its value at a missing mesh point differs from the line’s value by a nonzero integer multiple of \(q\). A coordinate difference is bounded by the sup distance; a difference of sums or differences is bounded by twice that distance. The result follows.
Let \(q{\gt}0\) and \(a,b,p\in \mathcal M(o,q)\). Suppose that the segment \([a,b]\) is horizontal, vertical or diagonal, with coincident endpoints also permitted. If \(p\notin [a,b]\), then every \(x\in [a,b]\) satisfies \(\| p-x\| _\infty \ge q/2\). This estimate is used in the isolation arguments for the initial stars and recursive repair in [ 247 , Section 11 ] .
If \(a=b\), then \(x=a\ne p\), so mesh-point separation gives the stronger bound \(q\). Otherwise let \(L\) be the supporting line of \([a,b]\). If \(p\notin L\), apply the line-clearance estimate. If \(p\in L\setminus [a,b]\), one endpoint \(e\in \{ a,b\} \) lies between \(p\) and \(x\). Since \(p\ne e\), mesh-point separation and distance additivity give
In both cases the asserted half-spacing bound follows.
27.59.1 Clipping an incident segment
The local repair construction in [ 247 , Section 11 ] intersects the segments incident to a mark with a small closed square. After normalization, the following calculation determines their endpoints. It applies to any real normed space, and hence to the maximum norm used for the planar geometry.
Let \(E\) be a real normed space and let \(a,b\in E\) satisfy \(\| b-a\| =1\). If \(0\le \eta \le 1\), then
Write a point of \([a,b]\) as \((1-t)a+tb\), where \(0\le t\le 1\). Its distance from \(a\) is \(t\| b-a\| =t\). It therefore belongs to the closed ball precisely when \(0\le t\le \eta \). These points form the segment on the right-hand side.
27.60 The finite family of initial marks
The sparse shifts may be chosen in every distance layer. Their cell counts have a summable geometric bound, so only finitely many layers contribute belt cells. We can therefore form the finite union of their actual marked points and assign to each point its smallest incident cell side; see [ 247 , Section 11 ] .
For every finite domain \(\Lambda \), cut \(A\), origin \(o\) and nonnegative integer \(C_0\), there are coordinate residues \(a_k,b_k\) at every scale. Write \(F_k\) for their actual belt-cell index set. These choices satisfy
and the set \(\{ k:F_k\ne \varnothing \} \) is finite.
Choose a sparse pair of residues at each scale. The integer counts \(|F_k|\) are dominated by a fixed multiple of a summable geometric sequence. Their real-valued series is therefore summable, so its terms tend to zero. Eventually \(|F_k|{\lt}1\), which forces \(F_k\) to be empty. Only finitely many indices remain.
There is a positive constant \(B\) such that, for every finite domain, cut, origin, nonnegative integer \(C_0\) and lower index \(k_0\), one can choose the residues from the preceding theorem and a finite set \(M\) of actual marks with
The chosen residues retain their per-layer cell estimate and finite support. The constant \(B\) precedes all the geometric data; the support cutoff may depend on those data.
Let \(B_0{\gt}0\) be the uniform finite-sum budget for the zero polynomial exponent, and take \(B=576B_0\). Form the union of the actual marks over the finite support restricted to \(k\ge k_0\). Its membership is exactly the stated condition. A finite union has at most the sum of the individual cardinalities, and each cell contributes at most nine marks. Thus its count is at most
where \(H\) is the finite retained support.
Fix an origin \(o\) and a finite endpoint set \(Z\), and form all layers from these data with \(C_0\ge 2\). Let \(k,h\ge 50{,}000{,}000\). If \(x\) and \(y\) are distinct marks of actual fine-layer cells in layers \(k\) and \(h\), respectively, then
Cell-boundary marks are included, and the estimate applies to every fine-layer cell, hence to every selected belt cell.
Suppose that the distance is less than \(t_k/4\). Each mark is in the closure of its actual layer. Layer locality forces \(k\le h+1\). Monotonicity and the adjacent-scale estimate give \(\ell _k\le \ell _h+1\), so both marks lie on the translated mesh of spacing \(t_k/4\). Distinct mesh points cannot be closer than this spacing. Reversing \(k\) and \(h\) proves both lower bounds and hence their maximum.
There is \(B{\gt}0\) such that, for every finite domain, cut, origin, integer \(C_0\ge 2\) and lower index \(k_0\ge 50{,}000{,}000\), there are sparse residues with the per-layer estimates and finite support, the exact finite union \(M\) of their actual marks above \(k_0\), and a positive side \(s(v)\) for each \(v\in M\), satisfying
For each \(v\), some selected belt cell in a layer \(k\ge k_0\) has \(v\) as an actual mark and side \(s(v)=t_k\). Every selected belt cell in a layer \(h\ge k_0\) that has \(v\) as one of its actual marks has side at least \(s(v)\).
Choose the counted finite union and residues from the preceding construction. For each point of \(M\), take the least layer index admitting an incident selected cell. The fine-cell exponent is nondecreasing, so this gives a positive smallest incident side. Apply fine-layer mark separation to the two witnessing cells. When \(M\) is empty the assignment and pairwise condition are empty.
27.61 The fan of a belt cell
The local fan is constructed before its triangles are colored. Each side may remain whole or be divided at its midpoint, according to the corners of the opposing cells. The following construction permits every such choice of subdivisions; see [ 247 , Section 11 ] .
Let \(Q\) be a closed dyadic square of side \(t=2^\ell \) and center \(c\). On each of its four sides choose either the whole side or its two half sides. The resulting segments are called elementary segments. For each elementary segment \(e=[a_e,b_e]\), let
These are nondegenerate closed triangles. Their perimeter and radial edges have slopes \(0,\infty ,1,-1\). The picture below shows one admissible choice: the upper and right sides are halved, the lower and left sides are kept whole, and the fan has six triangles.
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Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\). Let \(Q=o+2^\ell (z+[0,1)^2)\) be the half-open dyadic cell, with center \(c=o+2^\ell (z+(1/2,1/2))\) and half-side \(r=2^\ell /2\). Then
The coordinate description of \(\overline Q\) gives \(x\in \overline Q\) if and only if \(|x_j-c_j|\le r\) for \(j=1,2\). These inequalities are equivalent to \(\| x-c\| _\infty \le r\).
The center and both endpoints of every elementary segment belong to the actual nine-point mark set \(M_{\ell ,z}\) of the same cell.
The center has relative coordinates \((1/2,1/2)\). Every segment endpoint is a corner or side midpoint, so both relative coordinates are among \(0,1/2,1\).
Every such fan has between four and eight triangles.
Each of the four sides contributes either one or two elementary segments.
The closed fan triangles cover exactly the closed dyadic square:
Every point of \(Q\) lies on the segment from \(c\) to some point of the perimeter. Every perimeter point belongs to an elementary segment, so it is in the corresponding fan triangle. Conversely, every vertex lies in \(Q\), and convexity keeps the triangle in \(Q\).
For any two elementary segments \(e,e'\) of the same fan,
Thus a shared perimeter endpoint gives a shared radial segment, while disjoint elementary segments give intersection \(\{ c\} \).
A point different from \(c\) determines its unique perimeter endpoint on the outward ray from \(c\). It belongs to both triangles exactly when that endpoint lies in both elementary segments. The center belongs to every triangle.
27.61.1 The primary containing a non-belt fine cell
For nonnegative fine and pitch exponents \(\ell \le p\), integer shift representatives \(a,b\) and fine-cell index \(z\), put \(m=2^{p-\ell }\) and define
These are the coordinate indices of the shifted pitch square.
Fix an origin \(o\), finite endpoint set \(Z\), layer index \(k\) and nonnegative integer width \(C_0\). Suppose that \(\ell \le k\) and \(\ell \le p\). Choose residues \(0\le a,b{\lt}m=2^{p-\ell }\), and let \(z\) index an actual fine cell in this layer that is omitted by both selected belt residues. Its closed square satisfies
where \(I_{a,b}(j)\) is the corresponding open pitch interior.
Since neither residue is selected, each shifted coordinate remainder lies between \(1\) and \(m-1\). The open fine-cell interior consequently lies strictly inside the indicated pitch interior and belongs to the actual layer. Its closure is the whole closed fine square. Taking closures gives the inclusion, including when a fine-cell side touches a pitch-interior boundary.
Under the preceding hypotheses, there is exactly one retained primary pitch index whose birth region contains the whole closed fine square. This index is \(j(z)\).
The open fine square is nonempty and lies in the actual layer–pitch intersection, so the pitch index is retained. If two different primary birth regions contained this square, an interior point would belong to both. Their positive same-layer separation rules this out.
For any origin, natural exponent \(\ell \), integer cell index and optional midpoint subdivisions, let \(c\) be the cell center and \(e=[a_e,b_e]\) an actual elementary base. Every \(x\in e\) satisfies
This is the actual base-radius identity in the fan construction of [ 247 , Section 11 ] .
In the affine parametrization of a side, one relative coordinate is \(\pm 2^\ell /2\) and the other is \((2^\ell /2)u\) with \(|u|\le 1\). The whole side and either midpoint half all satisfy this bound. The maximum of the two absolute coordinates is therefore \(2^\ell /2\).
27.62 Interiors of actual fan triangles
Fix an arbitrary origin \(o\in \mathbb R^2\), natural dyadic exponent \(\ell \) and integer cell index \(z\in \mathbb Z^2\). Choose any optional midpoint subdivisions of the four sides. Write \(c\) for the cell center and \(P_e=\operatorname {conv}\{ c,a_e,b_e\} \) for each actual fan triangle; see [ 247 , Section 11 ] .
Every actual fan triangle satisfies
The defining determinant \(\det (a_e-c,b_e-c)\) is nonzero. Thus the two edge vectors are linearly independent and span the plane, and the three vertices affinely span \(\mathbb R^2\). Their convex hull consequently has nonempty interior.
The convex hull of these finitely many vertices is closed. A convex set with nonempty interior has the same closure as its interior, so \(\overline{\operatorname {int}(P_e)}=\overline{P_e}=P_e\).
27.63 Allowed lines and the eight open fan triangles
Fix an arbitrary origin, natural dyadic exponent and integer cell index. Split every side of the actual square at its midpoint, giving eight triangles \(P_e\) with common center \(c\). These are the elementary open sectors used in the local construction of [ 247 , Section 11 ] . Put \(\pi _1(x)=x_1\), \(\pi _2(x)=x_2\) and \(\Phi =\{ \pi _1,\pi _2,\pi _1+\pi _2,\pi _1-\pi _2\} \). The level lines \(\phi (x)=\phi (c)\) are exactly the four allowed lines through \(c\).
Every one of the eight actual triangles satisfies
Equivalently, the vector \(x-c\) has none of the slopes \(0,\infty ,1,-1\) whenever \(x\in \operatorname {int}(P_e)\).
Let \(\phi \) be any of the four surjective linear functionals. The whole-side coordinates and midpoint-half descriptions show that the two perimeter vertices \(a_e,b_e\) lie in one weak half-plane through \(c\): either both functional values are at least \(\phi (c)\), or both are at most \(\phi (c)\).
In the first case, convexity places \(P_e\) in \(\phi ^{-1}([\phi (c),\infty ))\). Since a surjective continuous linear functional is open, the interior of this half-plane is \(\phi ^{-1}((\phi (c),\infty ))\). Hence \(\phi (x){\gt}\phi (c)\) for every \(x\in \operatorname {int}(P_e)\). Applying the same argument to \(-\phi \) handles the other weak side. Thus none of the four level equations can hold in the interior.
27.64 Restriction of concentric dyadic fans
Use the maximum norm on \(\mathbb R^2\). Fix a common center \(c\in \mathbb R^2\) and a natural exponent \(\ell \), and put \(r=2^\ell /2\). The smaller square has origin \(o_{\rm S}=c-r(1,1)\), exponent \(\ell \) and cell index \((0,0)\); the larger square has origin \(o_{\rm L}=c-2r(1,1)\), exponent \(\ell +1\) and the same index. Choose identical optional midpoint subdivisions on corresponding sides, and denote the corresponding triangles by \(P_e^{\rm S}\) and \(P_e^{\rm L}\). This restriction supplies the margin in the local construction of [ 247 , Section 11 ] .
For every pair of corresponding elementary bases \(e\),
The subdivisions may be arbitrary; they need only agree between the two fans.
The homothety \(H(y)=c+(y-c)/2\) fixes \(c\) and sends the two large perimeter vertices to their corresponding small vertices. Affine maps preserve convex hulls, so \(H(P_e^{\rm L})=P_e^{\rm S}\).
A point of the larger triangle has the form \(x=c+\lambda (y-c)\), where \(0\le \lambda \le 1\) and \(y\) lies on its elementary base. The base-radius identity gives \(\| y-c\| _\infty =2r\), hence \(\| x-c\| _\infty =2r\lambda \). If \(x\in \overline B_\infty (c,r)\), then \(\lambda \le 1/2\), and \(x=c+2\lambda (H(y)-c)\) belongs to the smaller triangle. Conversely, halving a radial coefficient in \([0,1]\) keeps the resulting point in the larger triangle and within distance \(r\) of \(c\). This proves both inclusions.
27.65 Actual initial regions in eight sectors
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), an integer width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at every layer, and retain the resulting actual initial identifiers \(\mathcal I\). Write \(X_i^0\) for their closed birth regions and \(U_i=\operatorname {int}(X_i^0)\) for their open interiors. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be any of the nine marks of its defining cell. Put
The late-layer bound implies \(\ell _k\ge 5\). Use the maximum norm on \(\mathbb R^2\). Center two fans at \(v\), with origins \(o_{\rm S}=v-r(1,1)\) and \(o_{\rm L}=v-2r(1,1)\), exponents \(\ell \) and \(\ell +1\), and cell index \((0,0)\). Divide every side at its midpoint. Let \(\mathcal J\) index the eight corresponding elementary bases, and write \(P_s^{\rm S}\) and \(P_s^{\rm L}\) for the corresponding closed triangles. Their half-sides are \(r=t_k/64\) and \(2r=t_k/32\). This local construction is part of [ 247 , Section 11 ] .
Let \(c\in \mathbb R^2\) and \(m\in \mathbb N\), and put \(\rho =2^m/2\). Let \(Q=c-\rho (1,1)+2^m[0,1)^2\) be the zero-index half-open dyadic cell at origin \(c-\rho (1,1)\). In the maximum metric,
This is the centered square used in the local fan construction of [ 247 , Section 11 ] .
The closure of the zero-index dyadic cell is the product of the closed coordinate intervals whose lower endpoints are \(c_1-\rho \) and \(c_2-\rho \). Each interval has length \(2^m=2\rho \), so its upper endpoint is \(c_j+\rho \). Their product is precisely the closed maximum-metric ball with center \(c\) and radius \(\rho \).
Let \(k\ge 50{,}000{,}000\), write \(\ell _k\) for its fine-cell exponent, and put
Then
Thus the closed half-radius square has half-side \(t_k/128\). This is the numerical scale relation used in the initial-star geometry of [ 247 , Section 11 ] .
The explicit lower bound on \(k\) gives \(\ell _k\ge 5\). Hence \(2^{\ell _k}=32\, 2^{\ell _k-5}\). Dividing by \(128\) and substituting the definition of \(r\) proves (628).
There is a unique function \(\sigma :\mathcal J\to \mathcal I\) satisfying both assertions below. Every working triangle satisfies
For every initial identifier \(i\in \mathcal I\),
The union is empty when no triangle is assigned to \(i\).
The mark belongs to the closed reference fine cell and hence to the closed actual layer. The layer’s endpoint-distance assertion supplies an element of \(Z\), so the actual closed birth regions cover the plane. The late-layer bound and monotonicity of the fine exponent give \(\ell _k\ge 5\), and exponent arithmetic gives \(2r=t_k/32\).
Put \(S_s=\operatorname {int}(P_s^{\rm L})\) and \(W=B_\infty (v,2r)\). The coordinate interval description of the centered square and the fan cover give \(\bigcup _sP_s^{\rm L}=\overline B_\infty (v,2r)\). Taking interiors of the inclusion of each triangle into this square shows \(S_s\subseteq W\). Each \(S_s\) is nonempty by triangle regularity, and is preconnected by convexity.
No \(S_s\) meets the frontier of any \(X_i^0\). Indeed, a frontier point \(x\in S_s\) would satisfy \(\| x-v\| _\infty {\lt}t_k/32\), so the nearby frontier theorem would give an allowed direction for \(x-v\). The all-midpoint triangle assertion excludes that direction. Choose a point of \(S_s\) and an actual birth region containing it. Frontier exclusion puts this point in the region’s interior. Since \(X_i^0=\overline{U_i}\), the same exclusion gives \(\overline{U_i}\cap S_s\subseteq U_i\). Preconnectedness then propagates membership to all of \(S_s\). Its nonemptiness and the pairwise disjointness of the \(U_i\) make its identifier unique. Choosing these unique identifiers defines \(\sigma \).
Let \(D=\bigcup _sS_s\). Triangle regularity and the finiteness of \(\mathcal J\) give \(\overline D=\bigcup _sP_s^{\rm L}=\overline B_\infty (v,2r)\). For an arbitrary identifier \(i\), set \(E_i=\bigcup _{\sigma (s)=i}P_s^{\rm L}\). This is a closed finite union, including when it is empty. The assignment and disjointness give \(U_i\cap D\subseteq E_i\). Since \(U_i\) and \(W\) are open and \(W\subseteq \overline D\),
Conversely, taking closures of the assigned interiors gives \(E_i\subseteq X_i^0\). The smaller closed square lies inside \(W\) because \(r{\lt}2r\), and hence \(X_i^0\cap \overline B_\infty (v,r)=E_i\cap \overline B_\infty (v,r)\). Distribute this intersection through the finite union and apply the concentric restriction to each assigned triangle. This yields the asserted union of smaller triangles for every identifier. Finally, any other assignment has the same identifier on each nonempty working sector, and thus equals \(\sigma \).
The initial-sector assignment is the unique function \(\sigma :\mathcal J\to \mathcal I\) satisfying the open working-triangle containments and smaller closed-square decomposition of Theorem 27.65.3.
27.66 Colors of adjacent initial sectors
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), an integer width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at every layer. Let \(\mathcal I\) be the resulting actual initial identifiers, let \(X_i^0\) be their closed birth regions, and write \(\chi (i)\in \{ 0,1\} \) for their initial colors. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be one of the nine marks of its defining cell. Put \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Divide every side of the square \(\overline B_\infty (v,r)=v+[-r,r]^2\) at its midpoint, and join its center to the eight elementary bases. Let \(\mathcal J\) be the set of these eight bases, and denote their closed triangles by \(P_s^{\rm S}\). Write \(b_s^-\) and \(b_s^+\) for the initial and final endpoints of base \(s\) in the perimeter orientation. Let \(\sigma :\mathcal J\to \mathcal I\) be the unique actual assignment of Theorem 27.65.3, obtained from the working fan of half-side \(2r=t_k/32\).
This assertion concerns adjacent members of the eight fixed sectors in [ 247 , Section 11 ] .
Let \(s,t\in \mathcal J\) satisfy \(b_s^+=b_t^-\) or \(b_t^+=b_s^-\). Then
The exact closed restriction of the assignment gives \(P_s^{\rm S}\subseteq X_{\sigma (s)}^0\) for each \(s\): the triangle is one of the terms in the union describing \(X_{\sigma (s)}^0\cap \overline B_\infty (v,r)\).
Suppose first that \(b_s^+=b_t^-\), and put \(e=b_s^+\). The base-radius identity gives \(\| e-v\| _\infty =r{\gt}0\), so \(e\ne v\). Each triangle is the convex hull of its center and two base endpoints. Consequently,
When \(b_t^+=b_s^-\), interchange \(s\) and \(t\) to obtain the same nondegenerate shared radial segment. If \(\sigma (s)\ne \sigma (t)\), the initial-interface theorem gives \(\chi (\sigma (s))\ne \chi (\sigma (t))\). Conversely, equality of the identifiers immediately gives equality of their colors.
27.67 Actual initial identifiers on sector runs
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), an integer width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at every layer. Let \(\mathcal I\) be the actual initial identifiers, let \(X_i^0\) be their closed birth regions, put \(U_i=\operatorname {int}(X_i^0)\), and write \(\chi (i)\in \{ 0,1\} \) for their initial colors. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be any of the nine marks of its defining cell. Put \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Use the all-midpoint fans centered at \(v\), with origins \(o_{\rm S}=v-r(1,1)\) and \(o_{\rm L}=v-2r(1,1)\), exponents \(\ell \) and \(\ell +1\), and cell index \((0,0)\). Let \(\mathcal J\) index their eight corresponding elementary bases, and write \(P_s^{\rm S}\) and \(P_s^{\rm L}\) for the closed triangles. Let \(\sigma :\mathcal J\to \mathcal I\) be the unique actual assignment of Theorem 27.65.3. For the smaller fan, write \(b_s^-\) and \(b_s^+\) for the initial and final endpoints of the oriented base \(s\). Two bases are adjacent when \(b_s^+=b_t^-\) or \(b_t^+=b_s^-\).
Give base \(s\) the color \(\chi (\sigma (s))\). Form the existing fan graph \(G\), joining distinct adjacent bases exactly when their colors agree. Let \(\mathcal R\) be its connected components, and write \([s]\) for the component of \(s\). For \(R\in \mathcal R\), its closed run region is \(Y_R=\bigcup _{s\in R}P_s^{\rm S}\). These local runs occur in [ 247 , Section 11 ] .
There is a unique function \(\lambda :\mathcal R\to \mathcal I\) satisfying \(\lambda ([s])=\sigma (s)\) for every \(s\in \mathcal J\) and the following identity for every \(i\in \mathcal I\):
Distinct components carrying one identifier remain distinct terms of the union. If no sector has identifier \(i\), the union is empty.
An edge of \(G\) joins adjacent sectors with equal colors. The adjacent-sector color theorem therefore gives equality of their assigned identifiers. Equality is preserved along paths, so \(\sigma \) is constant on every connected component. It thus determines \(\lambda ([s])=\sigma (s)\). Every component contains a sector, which also makes this function unique.
The actual closed restriction is \(X_i^0\cap \overline B_\infty (v,r) =\bigcup _{\sigma (s)=i}P_s^{\rm S}\). Every sector belongs to its component, and all sectors in a component labeled \(i\) have assigned identifier \(i\). Grouping the terms by their connected components gives the asserted identity. The grouping retains every separated component carrying the same identifier. When there are no such components, both unions are empty.
Suppose that \(\sigma (s)=\sigma (t)\) for every adjacent pair \(s,t\in \mathcal J\). Then there is a unique identifier \(i\in \mathcal I\) such that \(\overline B_\infty (v,r)\subseteq U_i\).
The constant-colored fan graph is connected, including across the last and first bases. The assumption makes it a subgraph of \(G\): neighboring equal identifiers have equal initial colors. Hence \(G\) has one connected component, and descent shows that \(\sigma \) is constant at some actual identifier \(i_0\).
For each working triangle, \(\operatorname {int}(P_s^{\rm L})\subseteq U_{i_0}\). Taking closures and using triangle and birth-region regularity gives \(P_s^{\rm L}\subseteq X_{i_0}^0\). The actual fan cover and the centered-cell identity therefore yield
Taking interiors gives \(B_\infty (v,2r)\subseteq U_{i_0}\). Since \(r{\gt}0\), the smaller closed square lies in this larger open square, and thus \(\overline B_\infty (v,r)\subseteq U_{i_0}\). Finally \(v\) belongs to the smaller square. Pairwise disjointness of the actual open regions prevents another identifier from containing that square, proving uniqueness.
27.68 Active radial interfaces near an initial mark
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), an integer width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at every layer. Let \(\mathcal I\) be the actual initial identifiers, let \(X_i^0\) be their closed birth regions, and put \(U_i=\operatorname {int}(X_i^0)\). Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be one of the nine marks of its defining cell. Put \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Join \(v\) to the eight elementary bases of the all-midpoint fan in \(\overline B_\infty (v,r)\). Let \(\mathcal J\) index those bases, write \(P_s\) for the corresponding closed triangle, and write \(b_s^-\) and \(b_s^+\) for its initial and final base endpoints in the perimeter orientation. Let \(\sigma :\mathcal J\to \mathcal I\) be the unique actual assignment of Theorem 27.65.3. Thus, for every \(i\in \mathcal I\),
An oriented neighboring pair \((s,t)\) is active when \(b_s^+=b_t^-\) and \(\sigma (s)\ne \sigma (t)\). Let \(\mathcal A\) be the set of such pairs. This describes the radial interfaces in the local construction of [ 247 , Section 11 ] .
For every \(x\in B_\infty (v,r)\),
In the picture below each triangle carries its identifier \(\sigma (s)\); the solid radial segments are the active ones, and they are exactly the frontier points of the birth regions inside the open square.
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Put \(K=\sigma (\mathcal J)\) and, for each \(i\in K\), put \(E_i=\bigcup _{\sigma (s)=i}P_s\). This is a finite closed family covering \(\overline B_\infty (v,r)\). By (640), \(X_i^0\) and \(E_i\) agree on a neighborhood of every point of \(B_\infty (v,r)\). Their interior membership therefore agrees there. An identifier outside \(K\) has no birth-region point in this neighborhood.
Suppose that \(x\in \partial X_i^0\) and \(x\ne v\). The finite-cover frontier theorem gives a different identifier \(j\in K\) with \(x\in E_i\cap E_j\); its outer-frontier alternative is excluded by \(x\in B_\infty (v,r)\). Choose \(s,t\in \mathcal J\) with \(\sigma (s)=i\), \(\sigma (t)=j\) and \(x\in P_s\cap P_t\). Both triangles also contain \(v\), so their intersection is nontrivial. The actual triangle-contact theorem gives, after interchanging \(s\) and \(t\) if necessary, \(b_s^+=b_t^-\) and \(P_s\cap P_t=[v,b_s^+]\). Since \(i\ne j\), this pair is active.
If \(x=v\) and no pair is active, every adjacent pair has the same assigned identifier. The no-active theorem then places the smaller closed square in one actual open region \(U_j\). A point of \(U_j\) is on no birth frontier: it is interior to \(X_j^0\), and disjointness of the open regions, together with \(X_i^0=\overline{U_i}\), excludes membership in \(X_i^0\) for \(i\ne j\). Thus a frontier point at \(v\) forces an active pair, whose radial segment contains \(v\).
Conversely, let \((s,t)\in \mathcal A\) and \(x\in [v,b_s^+]\). The exact fan-intersection formula includes this segment in \(P_s\cap P_t\): the common base endpoint belongs to both bases, and its radial join with \(v\) belongs to both triangles. By (640), \(x\in X_{\sigma (s)}^0\cap X_{\sigma (t)}^0\). Since the two identifiers are different, \(U_{\sigma (s)}\) is disjoint from \(X_{\sigma (t)}^0=\overline{U_{\sigma (t)}}\). Hence \(x\notin U_{\sigma (s)}\) and \(x\in \partial X_{\sigma (s)}^0\), as required.
For every \(x\in \overline B_\infty (v,r/2)\),
The closed half-side is \(r/2=t_k/128\). If the reference cell realizes the smallest incident side \(S_v=t_k\), it is \(S_v/128\).
The fan radius is positive. Consequently, \(\overline B_\infty (v,r/2)\subseteq B_\infty (v,r)\), and the open-square equivalence applies to every point of this closed square.
Retain the origin, endpoint set, width, initial layer, arbitrary valid pitch residues, actual cell and mark fixed above. Let \(\tau \) be the perimeter successor, and put
Then
Each ray includes \(v\). If \(\mathcal A_*\) is empty, the right-hand union is empty and there is no birth frontier in this closed square. Identifiers may occur in several separated sectors; the change set records successive sectors without identifying such occurrences.
Endpoint adjacency is exactly the successor relation, so the active pairs are \((s,\tau (s))\) with \(s\in \mathcal A_*\). The closed radial description therefore identifies the frontier points with the segments \([v,b_s^+]\) at these changes. Since \(\overline B_\infty (v,r/2)\subseteq \overline B_\infty (v,r)\), the radial restriction theorem replaces each segment by \(R_s\) throughout the smaller closed square. This includes the center, and proves 644.
27.69 Opposing corners divide a side only at its midpoint
The subdivisions induced by actual opposing cells follow from the common dyadic origin and neighboring-layer locality; see [ 247 , Section 11 ] .
For origin \(o\), natural exponent \(\ell \), integer cell index \(z\) and \(\varepsilon \in \{ 0,1\} ^2\), the corresponding corner is \(o+2^\ell (z+\varepsilon )\).
Fix a common origin \(o\), finite endpoint set \(Z\) and integer width \(C_0\ge 2\). Let \(z\) and \(w\) index actual fine cells of layers \(k\) and \(h\), respectively, with \(k\ge 50\, 000\, 000\). Write \([a,b]\) for any whole side of the cell indexed by \(z\). If a corner \(x\) of the cell indexed by \(w\) belongs to \([a,b]\), then \(x=a\), \(x=(a+b)/2\), or \(x=b\).
The shared point belongs to both closed layers, so \(h\le k+1\) and \(k\le h+1\). If \(\ell _j\) is the fine exponent of layer \(j\), it follows that \(\ell _k\le \ell _h+1\). Put \(t=2^{\ell _k}\). The integer coordinates of the opposing corner place it on \(o+(t/2)\mathbb Z^2\). Along the reference side its varying coordinate, measured from \(a\) in units of \(t/2\), is therefore an integer between \(0\) and \(2\). These three integers give the two endpoints and the midpoint.
27.70 Ordered endpoints of a whole cell side
The four sides of a dyadic square are traversed counterclockwise, starting with the right side. The resulting endpoint identities hold at every natural scale and every integer cell index; see [ 247 , Section 11 ] .
Fix an origin \(o\in \mathbb R^2\), natural exponent \(\ell \) and integer cell index \(z\in \mathbb Z^2\). Put \(t=2^\ell \) and \(q=o+tz\). If \((a_s,b_s)\) is the ordered pair of endpoints of the unsplit side \(s\in \{ 0,1,2,3\} \), then
In particular, for every \(s\) there exist \(\varepsilon _s,\eta _s\in \{ 0,1\} ^2\) such that
Thus both endpoints are actual corners of the same dyadic cell.
Substitute the four side indices into the affine parametrization of the fan boundary. The resulting coordinate pairs are those displayed above. The corner pairs \((\varepsilon _s,\eta _s)\), in the same order, are \(((1,0),(1,1))\), \(((1,1),(0,1))\), \(((0,1),(0,0))\) and \(((0,0),(1,0))\).
27.71 Subdivisions determined by actual corners
The opposing corners determine the subdivisions used in the cell fans; see [ 247 , Section 11 ] . Retain the common origin, endpoint set and dyadic layers, and fix a lower index \(k_0\). All actual fine cells at indices at least \(k_0\) participate in this construction.
Fix an arbitrary origin \(o\in \mathbb R^2\), a natural exponent \(\ell \), and a signed integer cell index \(z\in \mathbb Z^2\). Choose any optional midpoint subdivisions of the cell sides. Write \(A_s,B_s\) for the ordered whole-side endpoints, and put \(L_s(u)=(1-u)A_s+uB_s\). Let \(\varepsilon _s\in \{ 0,1\} \) record whether side \(s\) is divided. Index its elementary segments by \(h=0\) when \(\varepsilon _s=0\), and by \(h\in \{ 0,1\} \) when \(\varepsilon _s=1\). Their initial and final endpoints satisfy
Parametrize the whole side by a signed coordinate \(u\in [-1,1]\). Its affine parameter is \((u+1)/2\). For an undivided side the initial and final signed coordinates are \(-1\) and \(1\), giving parameters \(0\) and \(1\). For half \(h\) of a divided side they are \(h-1\) and \(h\), giving parameters \(h/2\) and \((h+1)/2\). The defining side coordinates give the asserted ordered endpoint identities.
Fix any dyadic cell and optional midpoint subdivision. Let \([a,b]\) be a whole side and \([u,v]\) one of its elementary segments, ordered counterclockwise. One of the following descriptions applies: the side is undivided and \((u,v)=(a,b)\); the side is divided and \((u,v)=(a,(a+b)/2)\); or the side is divided and \((u,v)=((a+b)/2,b)\). In every case \([u,v]\subseteq [a,b]\).
In the affine parametrization of the whole side by \([0,1]\), the elementary endpoints have parameters \(0,1\) for an undivided side, \(0,1/2\) for the first half, and \(1/2,1\) for the second half. Each resulting interval belongs to \([0,1]\).
For the fine cell indexed by \(z\) in layer \(k\), write \([a_s,b_s]\) for its whole side \(s\). Set \(\varepsilon _s=1\) if the midpoint \((a_s+b_s)/2\) is a corner of some fine cell in a layer \(h\ge k_0\), and \(\varepsilon _s=0\) otherwise. The elementary segments are the whole side if \(\varepsilon _s=0\), and its two midpoint halves if \(\varepsilon _s=1\).
Side \(s\) of the fine cell indexed by \(z\) in layer \(k\) is divided if and only if there are a layer \(h\ge k_0\), an index \(w\in F_{h,\ell _h}\) and a corner \(\varepsilon \in \{ 0,1\} ^2\) such that the corner \(\varepsilon \) of the fine cell \(w\) equals \((a_s+b_s)/2\).
This is the defining condition of \(\varepsilon _s\).
Suppose that \(C_0\ge 2\), \(k\ge 50\, 000\, 000\), and \(z\) indexes an actual fine cell of layer \(k\). Let \(w\) index an actual fine cell of any layer \(h\ge k_0\). If a corner of the cell indexed by \(w\) belongs to an elementary segment of the reference cell under the actual midpoint subdivision, then it is an endpoint of that segment.
Every elementary segment lies on its whole side. The whole-side assertion gives its start, midpoint or end as the only possible opposing corners. For an undivided side, a corner at the midpoint would be a witness requiring subdivision. For either midpoint half, the opposite endpoint of the whole side lies outside that half. The remaining two possibilities are exactly its endpoints.
The source takes the lower index \(k_0\) sufficiently large without fixing a value [ 247 , Section 11 ] . The bound \(k\ge 50\, 000\, 000\) is one admissible explicit choice of that threshold, so the statement above implies the qualitative assertion for all sufficiently large layers.
27.72 Contacts between actual fine cells
Fix an origin \(o\), finite endpoint set \(Z\) and nonnegative integer width \(C_0\). Retain the fine exponents \(\ell _k\) and index sets \(F_{k,\ell _k}\), and write \(Q_{k,z}=C_{\ell _k,z}\) for an actual fine cell. The common dyadic origin restricts positive-length contacts to whole sides and midpoint halves; see [ 247 , Section 11 ] .
For every exponent \(\ell \ge 0\) and cell index \(z\), number the coordinate sides of \(\overline{C_{\ell ,z}}\) counterclockwise, starting with the right side. If \(a_{\ell ,z,s}\) and \(b_{\ell ,z,s}\) are the corresponding unsplit fan endpoints, put \(S_{\ell ,z,s}=[a_{\ell ,z,s},b_{\ell ,z,s}]\). Side indices are read modulo four. Writing \(a=a_{\ell ,z,s}\) and \(b=b_{\ell ,z,s}\), its midpoint halves are \(H_{\ell ,z,s,0}=[a,(a+b)/2]\) and \(H_{\ell ,z,s,1}=[(a+b)/2,b]\).
For arbitrary \(C_0\ge 0\) and layer indices \(k,h\), suppose that \(z\in F_{k,\ell _k}\), \(w\in F_{h,\ell _h}\) and \((k,z)\ne (h,w)\). Then \(Q_{k,z}\cap Q_{h,w}=\varnothing \).
If \(k=h\), the unique cell-index criterion implies that any common point would give \(z=w\). If \(k\ne h\), the two fine cells belong to the disjoint half-open layers \(D_k\) and \(D_h\), since \(\ell _k\le k\) and \(\ell _h\le h\).
Suppose that \(C_0\ge 2\), \(k\ge 50\, 000\, 000\), \(z\in F_{k,\ell _k}\), \(w\in F_{h,\ell _h}\) and \((k,z)\ne (h,w)\). If \(I=\overline{Q_{k,z}}\cap \overline{Q_{h,w}}\) contains two distinct points, there are opposite side indices \(s,t\), with \(t=s+2\) modulo four, for which one of the following alternatives holds:
\(\ell _k=\ell _h\) and \(I=S_{\ell _k,z,s}=S_{\ell _h,w,t}\).
\(\ell _h=\ell _k+1\) and \(I=S_{\ell _k,z,s}=H_{\ell _h,w,t,n}\) for some \(n\in \{ 0,1\} \).
\(\ell _k=\ell _h+1\) and \(I=S_{\ell _h,w,t}=H_{\ell _k,z,s,n}\) for some \(n\in \{ 0,1\} \).
The opposing index \(h\) is unrestricted.
A common point belongs to both closed layers. The nearby-layer estimate gives \(h\le k+1\) and \(k\le h+1\), so the fine exponents differ by at most one. Order the cells by their fine exponents. In units of the smaller side, translate the larger cell’s lower corner to the origin. The two half-open cells are then \(\lbrack a,a+1)\times \lbrack b,b+1)\) and \(\lbrack 0,m)^2\), where \(a,b\) are integers and \(m\in \{ 1,2\} \).
Closed contact gives \(-1\le a,b\le m\). Disjointness excludes \(0\le a,b{\lt}m\), and two distinct common points exclude the case where both coordinates lie in \(\{ -1,m\} \). Thus one coordinate equals \(-1\) or \(m\), while the other is an integer in \(\lbrack 0,m)\). The common set is therefore a whole smaller side on the opposite coordinate side of the larger square. For \(m=1\) it is the whole larger side; for \(m=2\) it is one of its midpoint halves. Swapping the two cells gives the third alternative.
27.73 Dummy-neighborhood corners on fine-cell sides
Retain the common origin, finite endpoint set and integer width \(C_0\), and fix a lower layer \(k_0\). Let \(P_{k_0}\) be the indices of the coarse cells contributing to \(N_{k_0}\). Their corners need not be boundary vertices of the union. The following assertions concern each contributing cell separately; see [ 247 , Section 11 ] .
Suppose that \(C_0\ge 2\), \(k\ge k_0\), \(z\in F_{k,\ell _k}\) and \(w\in P_{k_0}\). For \(\varepsilon \in \{ 0,1\} ^2\), put \(x=o+2^{k_0}(w+\varepsilon )\). If \(x\in S_{\ell _k,z,s}\), then \(x=a_{\ell _k,z,s}\) or \(x=b_{\ell _k,z,s}\).
The corner belongs to \(\overline{N_{k_0}}\). The whole side lies in the convex square \(\overline{Q_{k,z}}\subseteq \overline{D_k}\), so dummy-contact localization gives \(k=k_0\). Put \(q=2^{\ell _{k_0}}\). Since \(\ell _{k_0}\le k_0\), the corner and both side endpoints belong to \(o+q\mathbb Z^2\). If \(x\) differed from both endpoints \(a,b\), mesh separation would give
The equality follows from \(x\in [a,b]\), and the last inequality is the cell’s sup-diameter bound. This contradicts \(q{\gt}0\).
Suppose that \(C_0\ge 2\), \(k\ge k_0\), \(z\in F_{k,\ell _k}\) and \(w\in P_{k_0}\). Choose any optional midpoint subdivision of \(\overline{Q_{k,z}}\) and one of its elementary segments \([u,v]\). For \(\varepsilon \in \{ 0,1\} ^2\), if the corner \(x=o+2^{k_0}(w+\varepsilon )\) of the contributing dummy cell belongs to \([u,v]\), then \(x=u\) or \(x=v\).
The elementary segment belongs to its whole side \([a,b]\), so \(x=a\) or \(x=b\). For an undivided side this is the desired conclusion. On \([a,(a+b)/2]\), the endpoint \(b\) cannot occur unless \(a=b\); on \([(a+b)/2,b]\), the endpoint \(a\) cannot occur unless \(a=b\). If the endpoints coincide, the conclusion holds directly. Thus every possible case gives one of the elementary endpoints.
27.74 Matching elementary sides under the actual subdivision
When two actual fine cells have a common side segment, their midpoint choices must agree on that segment. The following statement applies to each elementary segment under the subdivision determined by all actual corners at or above the fixed lower index; see [ 247 , Section 11 ] .
Fix a common origin \(o\in \mathbb R^2\), finite endpoint set \(Z\), integer width \(C_0\ge 2\) and natural indices \(k_0,k,h\) with \(k\ge 50\, 000\, 000\) and \(k_0\le k,h\). Let \(z\in F_{k,\ell _k}\) and \(w\in F_{h,\ell _h}\) be actual fine-cell indices with \((k,z)\ne (h,w)\), and write \(Q_{k,z}=Q_{\ell _k}(z)\) and \(Q_{h,w}=Q_{\ell _h}(w)\). Let \([a_i,b_i]\) be any elementary segment of the reference cell under its actual midpoint subdivision relative to \(k_0\). Suppose that \([a_i,b_i]\cap \overline{Q_{h,w}}\) contains two distinct points. Then there is an elementary segment \([a_j,b_j]\) of the other cell under its actual midpoint subdivision relative to \(k_0\) such that
where \(s(i)\) and \(s(j)\) are their counterclockwise side indices.
The closed-cell contact is either a whole side of each cell or a whole side of the smaller cell equal to a midpoint half of a side of the larger cell. Contact containing two distinct points identifies the facing side of the reference cell: two different sides of the same square meet in at most one point.
In the whole-side case, the two whole sides have the same midpoint. Their subdivision conditions therefore have exactly the same corner witnesses. If they are undivided, the whole sides match. If they are divided, the first half on one side matches the second half on the opposite side, and conversely.
In the unequal-size case, a corner of the smaller cell is the midpoint of the larger side. The smaller cell lies at or above \(k_0\), so this corner forces the larger side to be divided. The smaller side is undivided. Indeed, a corner at its midpoint would lie strictly inside the corresponding elementary half of the larger side, contrary to the elementary-corner endpoint assertion. That assertion is applicable to the larger cell: its layer is at least that of the smaller cell, so the required late-layer bound follows from the reference bound whenever the reference cell is the smaller one.
Hence the whole smaller side matches the appropriate half of the larger side. When the reference cell is larger, contact containing two distinct points selects that half, since the two halves meet only at their midpoint. Finally, counterclockwise traversal of opposite facing sides gives the reversed endpoint equalities.
Theorem 27.75.1 supplies an opposing closed region for every elementary segment, and Theorem 27.76.2 proves that its actual fine-cell or dummy identifier is unique. Consistent region labels and the isolated-star construction remain further steps in the two-family argument.
27.75 An opposing closed region for every elementary side
The actual plane cover supplies an opposing rectangle at the midpoint of each elementary segment. Corner exclusion extends this contact to containment of the whole segment; see [ 247 , Section 11 ] .
Fix a common origin \(o\in \mathbb R^2\), finite endpoint set \(Z\), integer width \(C_0\ge 2\) and natural indices \(k_0\le k\) with \(k\ge 50\, 000\, 000\). Let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(e=[a_i,b_i]\) be any elementary segment of \(Q_{k,z}=Q_{\ell _k}(z)\) under its actual midpoint subdivision relative to \(k_0\). Then either
or there exist \(h\ge k_0\) and \(w\in F_{h,\ell _h}\) such that
The reference fine cell is nonempty, so its actual layer membership implies \(Z\ne \varnothing \). Put \(x=(a_i+b_i)/2\) and take the points in the ball of radius \(10\, 2^{\ell _k}\) about \(x\) that lie outside the closed reference cell. Their closure contains \(x\), since \(x\) belongs to a side of that square.
Every such point lies either in \(N_{k_0}\) or in an actual fine cell of a layer \(h\ge k_0\). Neighboring-layer locality restricts these latter indices to \(k-1\le h\le k+1\). There are finitely many fine cells at these indices, and the reference cell is excluded by the choice of points. Taking the closure of this finite cover places \(x\) either in the closed dummy neighborhood or in one distinct closed fine cell. In the dummy case, the finite union defining the neighborhood places \(x\) in one contributing closed coarse cell.
In either case the interiors of the selected rectangle and the reference square are disjoint. Since the midpoint of \(e\) belongs to the selected rectangle, its normal coordinate is a boundary coordinate of that rectangle. If the rectangle failed to contain either end of \(e\), a rectangle corner would lie strictly inside \(e\). The corresponding dummy-corner or actual fine-corner endpoint assertion excludes this. Thus the selected closed rectangle contains the whole segment. A contributing dummy rectangle belongs to \(\overline{N_{k_0}}\), yielding the first alternative; a distinct actual fine cell yields the second.
This assertion provides existence and whole-segment containment. The uniqueness assertion below identifies the opposing actual fine cell or dummy neighborhood. Consistent primary and fan-run labels and the isolated-star construction remain further obligations.
27.76 Uniqueness of the opposing region
The preceding existence assertion can be sharpened to uniqueness of the opposing fine cell or dummy neighborhood. This identifies the region containing an elementary side, before assigning primary or fan-run labels; see [ 247 , Section 11 ] .
Fix an origin \(o\in \mathbb R^2\), a natural exponent \(\ell \), a signed cell index \(z\in \mathbb Z^2\), and any optional midpoint subdivision of the sides of \(Q_\ell (z)\). Put \(t=2^\ell \), \(p=o+tz\) and \(q=p+(t,t)\). For the ordered endpoints \(a,b\) of every elementary segment, one has \(a\ne b\) and either
or
No layer-membership or late-scale assumption is required.
Each whole side has length \(t{\gt}0\) and a constant coordinate equal to a boundary coordinate of the square. An elementary segment is either this whole side or one of its two midpoint halves. Its endpoints retain that constant coordinate and differ in the other coordinate by \(t\) or \(t/2\).
Fix a common origin \(o\in \mathbb R^2\), finite endpoint set \(Z\), integer width \(C_0\ge 2\) and natural indices \(k_0\le k\) with \(k\ge 50\, 000\, 000\). Let \(z\in F_{k,\ell _k}\) index an actual fine cell. For any elementary segment \(e=[a_i,b_i]\) of this cell under its actual midpoint subdivision relative to \(k_0\), consider the identifiers
Associate to them the closed regions
There is exactly one \(P\in \mathcal I\) such that \(e\subseteq R_P\). The dummy neighborhood is a single region even when several coarse cells contribute to it. No further lower bound on the opposing layer index is imposed beyond \(h\ge k_0\).
Existence follows from the preceding opposing-region theorem. To prove uniqueness, put \(x=(a_i+b_i)/2\). The segment is nondegenerate and lies on one side of the reference square, so \(x\) is interior to that side in its tangential coordinate.
Suppose first that two distinct actual fine cells contain the whole segment in their closures. Their interiors and the interior of the reference cell are pairwise disjoint. Neither opposing square can have a corner at \(x\), since an opposing corner on an elementary segment must be an endpoint. Consequently the tangential coordinate of \(x\) lies strictly between the two tangential bounds of each opposing square.
Write \(x=(\xi ,\eta )\) in tangential and normal coordinates, and write the closures of the reference square and the two opposing squares as \([u_j,v_j]\times [u'_j,v'_j]\) for \(j=0,1,2\). Then
Since the tangential open intervals overlap, pairwise disjoint interiors force the normal open intervals \((u'_j,v'_j)\) to be pairwise disjoint. Each of them has positive length and contains \((\eta -\epsilon ,\eta )\) or \((\eta ,\eta +\epsilon )\) for some \(\epsilon {\gt}0\). Two of the three therefore meet, a contradiction. Interchanging the coordinates handles a horizontal elementary segment.
If both the dummy neighborhood and a distinct actual fine cell contain the segment, the finite union defining the dummy neighborhood supplies a contributing closed coarse cell containing \(x\). Its interior is disjoint from both fine-cell interiors because their layer indices are at least \(k_0\). A dummy corner on the elementary segment is again an endpoint, so \(x\) is not such a corner. The same three-rectangle argument gives a contradiction. Thus the existing opposing identifier is unique.
These assertions identify the opposing actual fine cell or dummy neighborhood. Their composition with primary identifiers and fan-run labels, the isolated-star construction and the complete two-family partition remain further obligations.
27.77 Contact and reciprocal selection of opposing regions
The unique opposing region determines a selection for each actual elementary side. Contact with an actual fine cell characterizes this selection. When the lower index is sufficiently large, the corresponding elementary segments select each other; see [ 247 , Section 11 ] .
Fix a common origin \(o\in \mathbb R^2\), finite endpoint set \(Z\), integer width \(C_0\ge 2\), and natural indices \(k_0\le k\) with \(k\ge 50\, 000\, 000\). For an actual fine-cell index \(z\in F_{k,\ell _k}\) and a slot \(i\) of its actual midpoint subdivision, write \(e_i=[a_i,b_i]\). Denote by \(P_{k,z}(i)\) the unique opposing identifier whose closed region contains \(e_i\): the dummy symbol \(\star \) represents \(\overline{N_{k_0}}\), while an indexed cell \((h,w)\) represents \(\overline{Q_{\ell _h}(w)}\) with \(h\ge k_0\), \(w\in F_{h,\ell _h}\) and \((h,w)\ne (k,z)\). The dummy neighborhood is one identifier even if several coarse cells contribute to it. No late bound on \(k_0\) is required.
Retain the assumptions of the selected-opponent definition. Let \(w\in F_{h,\ell _h}\) be an actual candidate fine-cell index at a layer \(h\ge k_0\). Then
The candidate layer \(h\) has no additional late bound.
If the candidate is selected, its indexed cell differs from the reference and its closure contains the whole segment. The two elementary endpoints are distinct, so both belong to the contact.
Conversely, distinct indexed cells with two-point contact have matching actual elementary segments with reversed endpoints. The candidate segment lies in the closed candidate square: its endpoints belong to that square and the square is convex. Thus the whole reference segment lies in the same closed square. Uniqueness of the opposing identifier makes the candidate the selected opponent.
Fix a common origin and finite endpoint set, integer width \(C_0\ge 2\), and natural indices \(k,h\ge k_0\ge 50\, 000\, 000\). Let \(z\in F_{k,\ell _k}\) and \(w\in F_{h,\ell _h}\) be actual fine-cell indices. For every actual elementary slot \(i\) of the reference cell, the equality \(P_{k,z}(i)=(h,w)\) holds if and only if \((k,z)\ne (h,w)\) and there exists an actual elementary slot \(j\) of the candidate cell such that
Here \(s(i)\) and \(s(j)\) are the counterclockwise side indices, and both subdivisions use the same lower index \(k_0\). No matching or return selection is supplied as a premise.
The lower bound on \(k_0\) gives the required late bounds on both \(k\) and \(h\). If the candidate is selected, the contact characterization gives distinctness and two-point contact. Actual matching supplies the slot \(j\), opposite side index and reversed endpoints. These endpoints belong to the closed reference square, which is convex; they are distinct. Applying the contact characterization with the cells exchanged shows that \(j\) selects the reference cell in return.
Conversely, the reversed endpoints make the two elementary segments equal. The candidate segment is nondegenerate and lies in its closed square. Hence the reference segment has two-point contact with that square. Distinctness and the contact characterization give \(P_{k,z}(i)=(h,w)\).
These assertions establish reciprocal selection of actual fine-cell opponents. They do not assign primary identifiers or colors. Consistent global labels, the isolated-star construction and the complete two-family partition remain further obligations.
27.78 Opposite labels on actual belt-cell sides
Use the selected opposing region to assign labels to each belt-cell fan, as in [ 247 , Section 11 ] . The residue choices at every layer are arbitrary; the label assignment requires no sparsity estimate.
Fix a common origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\), integer width \(C_0\ge 2\) and natural lower index \(k_0\ge 50\, 000\, 000\). For every layer \(h\), choose residues \(0\le a_h,b_h{\lt}m_h\), where \(m_h=2^{p_h-\ell _h}\), \(\ell _h\) is the fine exponent and \(p_h\) is the pitch exponent. Let \(\mathcal B_h\) be the corresponding actual belt-cell index set. For \(k\ge k_0\) and \(z\in \mathcal B_k\), denote the elementary segments of the actual fan by \(e_i=[a_i,b_i]\) and their selected opposing identifiers by \(P_{k,z}(i)\). Order the triples \((k,z_1,z_2)\) lexicographically. Define \(c_{k,z}(i)\in \mathbb Z/2\mathbb Z\) by
The indexed cells are distinct whenever the opponent is a fine cell, so the last two alternatives are exhaustive.
Retain the data of the label definition. If an actual elementary segment \(e_i\) lies in \(\overline{N_{k_0}}\), then \(c_{k,z}(i)=(k_0-1)+1\ne k_0-1\) in \(\mathbb Z/2\mathbb Z\). Thus its incident triangle has the label opposite the dummy tile.
The dummy identifier satisfies the whole-segment containment condition. Uniqueness therefore gives \(P_{k,z}(i)=\star \). The asserted label follows from its definition; adding one changes each residue modulo two.
Retain the data of the label definition. Let \(h\ge k_0\) and let \(w\in F_{h,\ell _h}\setminus \mathcal B_h\) be an actual non-belt index. If \(e_i\cap \overline{Q_{\ell _h}(w)}\) contains two distinct points, put \(J=j(w)\), the shifted pitch index at residues \(a_h,b_h\). Then \(J\) is retained in the actual primary index set \(\mathcal P_h\), the whole segment \(e_i\) lies in the primary birth region \(\overline{D_h\cap I_{a_h,b_h}(J)}\), and \(c_{k,z}(i)=h+1\ne h\) in \(\mathbb Z/2\mathbb Z\). The primary identifier is \((h,J)\), including when its region is disconnected.
Belt and non-belt membership imply \((k,z)\ne (h,w)\). The contact characterization gives \(P_{k,z}(i)=(h,w)\), and the selected opponent contains the whole segment in its closed square. Closed non-belt-cell containment consequently gives \(e_i\subseteq \overline{Q_{\ell _h}(w)}\subseteq \overline{D_h\cap I_{a_h,b_h}(J)}\). Since \(e_i\) contains its first endpoint, this last closure is nonempty. The layer–pitch intersection itself is therefore nonempty, and the exact retained-index characterization gives \(J\in \mathcal P_h\). The non-belt clause in the label definition gives \(h+1\), which differs from \(h\).
Retain the common data and residue choices. Let \((k,z)\ne (h,w)\) be actual belt-cell pairs at \(k,h\ge k_0\), and let \(e_i,e_j\) be arbitrary elementary segments in their respective actual subdivisions. If \(e_i\cap e_j\) contains two distinct points, then \(c_{k,z}(i)\ne c_{h,w}(j)\). No matching slot or reversed endpoints are assumed.
Each elementary segment lies in its incident triangle: the segment’s convex hull is contained in the convex hull of the three triangle vertices. The fan cover then places it in its own closed cell. Thus the assumed intersection gives two-point contact of each segment with the other closed cell. The contact characterization yields \(P_{k,z}(i)=(h,w)\) and \(P_{h,w}(j)=(k,z)\). The distinct lexicographic keys have opposite strict orders, so the two belt clauses assign zero and one in opposite order.
The local labels can be used in the equal-label runs of each fan. Global primary–run interfaces, point contacts, boundary sectors and the isolated-star construction remain further parts of the two-family construction.
27.79 Merging equal-colored triangles in a cell
Fix a fan in one dyadic square \(Q\) and assign each triangle one of the two colors \(0,1\). The merging below applies to every such assignment; see [ 247 , Section 11 ] .
Write \([a_e,b_e]\) for the elementary perimeter segment of triangle \(P_e\), oriented counterclockwise, and let \(\lambda (e)\in \{ 0,1\} \) be its color. Two distinct triangles are adjacent when \(b_e=a_{e'}\) or \(b_{e'}=a_e\). Retain an adjacency precisely when the two colors agree. A run is a connected component of this retained graph. Its region is
Distinct components remain distinct identifiers, including when their colors agree.
Any two triangles in the same run have the same color.
Along every retained edge the colors agree. Following a finite path in a component gives the assertion.
The run regions cover exactly the closed square:
Every elementary segment belongs to one component. Thus the union of all run regions is the union of all fan triangles, which is \(Q\).
Suppose that two adjacent fan triangles belong to distinct runs. Their colors are different, and hence are opposite members of \(\{ 0,1\} \).
If their colors agreed, their adjacency would be retained and they would lie in the same component.
If every triangle has the same color, all triangles have the same run identifier, and every run region is the whole closed square.
The one or two elementary segments on each side are connected in their perimeter order. The last segment of each side is adjacent to the first segment of the next side, including the fourth-to-first transition. Under constant coloring all these adjacencies are retained, so the graph is connected. Its single component contains every triangle, whose union is \(Q\).
27.80 Geometric contacts within one cell fan
The merging of triangles in [ 247 , Section 11 ] uses their geometric contacts. The following local assertions hold for every origin, dyadic exponent, signed cell index and optional side-midpoint subdivision. They require no assignment of labels outside the cell.
Fix a fan with center \(c\) in a translated dyadic square. Write \(e_i=[a_i,b_i]\) for its elementary perimeter segments, oriented counterclockwise, and \(P_i=\operatorname {conv}\{ c,a_i,b_i\} \) for its triangles. If \(i\ne j\) and \(P_i\cap P_j\) contains two distinct points, then one of the following alternatives holds:
Thus geometric contact determines consecutive perimeter segments and identifies the complete common radial edge.
If the two perimeter segments were disjoint, the fan intersection identity would give \(P_i\cap P_j=\{ c\} \). Their intersection is therefore nonempty. Two distinct elementary segments on the same side are its first and second halves, whose intersection is the midpoint. Elementary segments on distinct sides can meet only at a common corner of consecutive sides; any segment containing that corner has it as the corresponding endpoint. In either case their intersection is the singleton of a last-to-first endpoint. Substituting into the fan intersection identity gives the asserted radial segment.
Give the triangles of this fan arbitrary labels in \(\{ 0,1\} \). Let \(R\ne T\) be two actual run identifiers, with regions \(T_R,T_T\). If \(T_R\cap T_T\) contains two distinct points, then every triangle in \(R\) has a different label from every triangle in \(T\).
Choose a common point \(x\ne c\). It belongs to triangles \(P_i,P_j\) in the two runs. Their slots are distinct, and both triangles also contain \(c\), so their intersection contains two distinct points. The preceding theorem gives endpoint adjacency. Distinct runs then force the two labels to differ. Label constancy on each run gives the assertion for every pair of constituent triangles.
These assertions concern a single fan, whose triangles share one center. Across different cells, a shared nondegenerate segment must be distinguished from isolated common corners. The classification of global interfaces is a further part of the two-family construction.
27.81 Cyclic order of the eight midpoint fan triangles
Let \(Q\) be a translated dyadic square with arbitrary origin, natural exponent \(\ell \) and signed integer cell index. Put \(t=2^\ell \), and write \(c\) for its center. Orient its four sides counterclockwise, with indices \(s\in \{ 0,1,2,3\} \) and ordered endpoints \(A_s,B_s\). When every side is divided at its midpoint, index the elementary bases by \(\mathcal J=\{ 0,1,2,3\} \times \{ 0,1\} \), where the second index records the first or second half of the side. Write \(a_i,b_i\) for the ordered endpoints of base \(i\) and \(P_i=\operatorname {conv}\{ c,a_i,b_i\} \). This is the perimeter order in the fan construction of [ 247 , Section 11 ] .
Number the eight bases by \(e(s,h)=2s+h\) modulo \(8\). Let \(\tau \) be the permutation obtained by increasing this number by one. Equivalently,
where the side index is reduced modulo \(4\).
For every \(i\in \mathcal J\), the final endpoint of base \(i\) is the initial endpoint of its successor:
The ordered whole-side coordinates give \(B_s=A_{s+1}\), with the side index reduced modulo \(4\). The indexed affine parameters give
The first pair meets the second pair at the midpoint of side \(s\). The second pair meets the first pair on side \(s+1\) at \(B_s=A_{s+1}\). These are exactly the two cases in the successor definition, including the transition from side \(3\) to side \(0\).
More generally, choose any optional midpoint subdivision of the four sides of \(Q\), and let \(\mathcal E\) index its elementary bases. The map \(i\mapsto b_i\) from \(\mathcal E\) to the square perimeter is injective.
Suppose that \(i\ne j\) and \(b_i=b_j=p\). The center and \(p\) belong to both closed triangles. The base-radius identity gives \(\| p-c\| _\infty =t/2{\gt}0\), so these are two distinct intersection points. The contact classification therefore gives either \(b_i=a_j\) or \(b_j=a_i\). In the first case \(a_j=b_j\); in the second case \(a_i=b_i\). Both contradict the nondegeneracy of the corresponding elementary segment. Thus \(i=j\).
For \(i,j\in \mathcal J\) in the all-midpoint fan,
Apply the successor endpoint identity at \(\tau ^{-1}(j)\) to obtain \(a_j=b_{\tau ^{-1}(j)}\). If \(b_i=a_j\), injectivity of final endpoints gives \(i=\tau ^{-1}(j)\), and hence \(j=\tau (i)\). The reverse implication is the successor endpoint identity.
27.82 Directed rays of an elementary fan
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\) be arbitrary, and let \(Q=o+2^\ell (z+[0,1]^2)\) be the corresponding closed dyadic square. Write \(c\) for its center. On each side choose either the whole side or its two midpoint halves, and let \(\mathcal E\) index the resulting ordered elementary bases \([a_i,b_i]\). By Theorem 27.61.1.4, every final endpoint satisfies \(\| b_i-c\| _\infty =2^\ell /2{\gt}0\).
For nonzero vectors \(u,w\in \mathbb {R}^2\), write \(u\sim w\) if \(w=\lambda u\) for some \(\lambda {\gt}0\). Let \(\mathcal R=(\mathbb {R}^2\setminus \{ 0\} )/{\sim }\) be the set of directed rays, and write \([u]\) for the class of \(u\). Thus opposite nonzero vectors represent distinct directed rays. These are the directions from a cell center in the fan construction of [ 247 , Section 11 ] .
For every \(i,j\in \mathcal E\),
The two vectors have the same positive norm \(r=2^\ell /2\). If \(b_j-c=\lambda (b_i-c)\) with \(\lambda {\gt}0\), homogeneity of the norm gives \(r=\lambda r\), hence \(\lambda =1\). Thus \(b_i=b_j\). Injectivity of final endpoints gives \(i=j\). Conversely, equal indices give equal vectors, so the positive multiple \(\lambda =1\) proves that they have the same direction.
Assign to each elementary base its final directed ray from the center:
The positive common radius makes every vector in this definition nonzero. Only positive multiples are identified: \([u]\ne [-u]\) for every \(u\ne 0\).
The map \(\rho :\mathcal E\to \mathcal R\) is injective.
Equality of the two ray classes means that their nonzero representatives have the same direction. The endpoint-direction equivalence then gives equality of their indices.
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\) be arbitrary. Let \(c\) be the center of the closed dyadic square \(Q=o+2^\ell (z+[0,1]^2)\), and choose independently on each side either the whole side or its two midpoint halves. For an elementary base \([a_s,b_s]\), put \(r=2^\ell /2\) and \(R_s=\{ c+\lambda (b_s-c):\lambda \ge 0\} \). Then
Here the full ray \(R_s\) includes its center \(c\), and the assertion also holds when \(\ell =0\). This is the local restriction of a radial direction used in the initial-star construction of [ 247 , Section 11 ] .
Every point of \([c,b_s]\) belongs to \(R_s\). Conversely, let \(x\in R_s\) satisfy \(\| x-c\| _\infty \le r\). The points lie on the same directed line from \(c\), so either \(x\in [c,b_s]\) or \(b_s\in [c,x]\). In the second case, distance addition along the segment gives
The radius bound forces \(\| x-b_s\| _\infty =0\), hence \(x=b_s\). This proves the reverse inclusion. Distance addition along a segment is valid in the maximum norm and requires no strict convexity.
27.83 Parity of changes of initial sector identifiers
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), a natural width \(C\ge 2\), and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at each layer. Let \(\mathcal I\) be the resulting actual initial identifiers, write \(X_i^0\) for their closed birth regions, and write \(\chi (i)\in \{ 0,1\} \) for their initial colors. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be one of the nine marks of its defining cell. Put \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Use the all-midpoint fan of the square \(\overline B_\infty (v,r)=v+[-r,r]^2\) and its eight-element index set \(\mathcal J\). Let \(\tau \) be the counterclockwise successor permutation of Definition 27.81.1. Let \(\sigma :\mathcal J\to \mathcal I\) be the unique actual assignment of Theorem 27.65.3, obtained from the working fan of half-side \(2r=t_k/32\).
The number of changes of actual initial identifier around these eight fan members is even:
Put \(c_s=\chi (\sigma (s))\). The successor endpoint identity makes \(s\) and \(\tau (s)\) adjacent, so the actual adjacent-color assertion gives
Let \(A=\{ s\in \mathcal J:c_s=0\} \) and \(B=\tau ^{-1}(A)\). The binary values of \(c_s\) show that the change set is \((A\setminus B)\cup (B\setminus A)\), a disjoint union. Since \(\tau \) is a permutation, \(|A|=|B|\), and therefore \(|A\setminus B|=|B\setminus A|\). With \(d=|A\setminus B|\), the cardinality of the change set is \(d+d\), which is even.
27.84 Cardinality of the distinct active directed rays
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), a natural width \(C\ge 2\), and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues at every layer. Let \(\mathcal I\) be the resulting actual initial identifiers, and write \(X_i^0\) for their closed birth regions. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be one of the nine marks of its defining cell. Put \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Use the all-midpoint fan of \(v+[-r,r]^2\), with its eight-element index set \(\mathcal J\) and counterclockwise successor \(\tau \). Let \(\sigma :\mathcal J\to \mathcal I\) be the unique actual assignment of Theorem 27.65.3. Write \(P_s\) for the final endpoint of elementary base \(s\). The endpoint vector \(P_s-v\) is nonzero, and \([P_s-v]\) denotes its directed ray in the notation of Definition 27.82.2. Define
The set of distinct directed rays at changes of the actual initial identifier has even cardinality at most eight:
Injectivity of the fan’s directed rays gives \(|\mathcal D|=|A|\). The parity theorem gives \(|A|\in 2\mathbb N\). Since \(A\subseteq \mathcal J\) and \(|\mathcal J|=8\), one also has \(|\mathcal D|\le 8\).
For every \(x\in B_\infty (v,r)\), one has
The actual radial-frontier characterization gives two neighboring fan members \(s,t\) whose common endpoint is \(P_s\), whose assigned identifiers differ, and whose shared radial segment contains \(x\). Endpoint adjacency is equivalent to \(t=\tau (s)\). Thus these two members give precisely \(s\in A\) and \(x\in [v,P_s]\). Conversely, \(s\in A\) and \(x\in [v,P_s]\) give the neighboring pair \(s,\tau (s)\) in the radial-frontier characterization.
27.85 Radial incidence and convexity in an elementary fan
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\) be arbitrary, and let \(Q=o+2^\ell (z+[0,1]^2)\) be the corresponding closed dyadic square. Write \(c\) for its center. On each side choose either the whole side or its two midpoint halves. Orient the perimeter counterclockwise, and orient each chosen base accordingly. Let \(\mathcal E\) index the resulting ordered bases \([a_i,b_i]\), and put
These are the triangles and radial segments of the elementary fan in [ 247 , Section 11 ] . All endpoints satisfy \(\| b_i-c\| _\infty =2^\ell /2{\gt}0\), and the directed rays determined by \(b_i-c\), for \(i\in \mathcal E\), are pairwise distinct.
For \(i,t\in \mathcal E\) and \(x\in R_t\) with \(x\ne c\),
The segment \(R_t\) is contained in \(P_t\) by convexity. If \(t=i\), this proves the reverse implication. If \(b_t=a_i\), then \(R_t=[c,a_i]\subseteq P_i\), which gives the other reverse case.
Conversely, suppose \(x\in P_i\) and \(t\ne i\). The distinct points \(c,x\) belong to \(P_t\cap P_i\). The classification of contacts between distinct fan triangles therefore gives either \(b_t=a_i\), as required, or
In the second case \(x\in R_t\cap R_i\) and \(x\ne c\). Thus there are \(u,v{\gt}0\) with
The two endpoint vectors consequently determine the same directed ray. Distinctness of these directed rays gives \(t=i\), contradicting the assumption. This proves (698).
For every \(i\in \mathcal E\), the ordered triple \((c,a_i,b_i)\) is an affine basis of \(\mathbb {R}^2\).
The triangle \(P_i\) has nonempty interior. Its affine span is therefore the whole plane. The vector space parallel to that affine span has dimension \(2\). Since there are three vertices, they are affinely independent and form an affine basis.
For every \(i,t\in \mathcal E\), the set \(P_i\setminus R_t\) is convex.
Every \(R_t\) contains \(c\). Applying (698) away from \(c\) therefore gives the exact alternatives
The three vertices of \(P_i\) form an affine basis by the preceding theorem. Let \(\lambda _0,\lambda _1,\lambda _2\) be the barycentric coordinates for \(c,a_i,b_i\), respectively. On \(P_i\) these coordinates are nonnegative, sum to \(1\), and reconstruct each point as \(x=\lambda _0(x)c+\lambda _1(x)a_i+\lambda _2(x)b_i\). It follows that
Each set in (708) is convex, being the intersection of a triangle with a strict affine half-plane. Hence (705) shows that \(P_i\setminus R_t\) is convex for every \(t\in \mathcal E\).
For every \(i\in \mathcal E\) and every subset \(A\subseteq \mathcal E\), the set \(P_i\setminus \bigcup _{t\in A}R_t\) is convex. The set \(A\) may be empty.
By the preceding theorem, \(P_i\setminus R_t\) is convex for every \(t\in \mathcal E\). Moreover,
The intersection of convex sets is convex. When \(A=\varnothing \), the bounded intersection is the whole plane, so the last expression is \(P_i\); no exclusion of the center is made in this case.
27.86 Connected pieces of the midpoint fan
A vector \((u_1,u_2)\in \mathbb {R}^2\) has an allowed direction if \(u_1=0\), \(u_2=0\), \(u_1=u_2\) or \(u_1=-u_2\). The following observation will be used for the radial segments of the elementary fans in [ 247 , Section 11 ] .
Let \(p,q\in \mathbb {R}^2\) and suppose that \(q-p\) has an allowed direction. If \(v,x\in \operatorname {aff}\{ p,q\} \), then \(x-v\) has an allowed direction. The points \(p,q\) may coincide.
Write \(v=(1-r)p+rq\) and \(x=(1-s)p+sq\) with \(r,s\in \mathbb {R}\). Then \(x-v=(s-r)(q-p)\). Each of the four equations defining an allowed direction is preserved by real scalar multiplication.
In an elementary fan, the vectors \(b_i-a_i\) and \(c-b_i\) have allowed directions. Each side may be left whole or subdivided at its midpoint.
The fan triangle with vertices \(c,a_i,b_i\) is an allowed polygonal piece. Its edge directions include \(b_i-a_i\) and \(c-b_i\), so both have one of the four prescribed slopes.
For the next result, subdivide every side of the dyadic square at its midpoint. There are then eight triangles \(P_i\) and eight radial segments \(R_i=[c,b_i]\), with the notation of the preceding section. The plane carries the maximum norm.
Let \(A\) be any subset of the eight radial indices and let \(r{\gt}0\). For each triangle index \(i\), the set
is nonempty and connected. The set \(A\) may be empty.
The triangle \(P_i\) is the closure of its interior and contains \(c\). Hence \(B_\infty (c,r)\) contains a point \(x\) of the interior of \(P_i\). The vector \(x-c\) has none of the four allowed directions. Each radial has an allowed direction, and so does every difference of points in its affine span. Thus \(x\) lies on no radial, proving nonemptiness even when all eight radials are deleted.
The triangle with the selected radials deleted is convex, as is the ball. Their intersection is therefore a nonempty convex subset of a real normed space. It is path connected and hence connected. If \(A\) is empty, no radial is deleted and the center is retained.
The connected elementary pieces of Theorem 27.86.3 arise from fan triangles whose base and radial directions are prescribed by Theorem 27.86.2. The following assertion concerns the radial contacts described in Theorem 27.80.1.
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\) be arbitrary, and let \(Q=o+2^\ell (z+[0,1]^2)\) be the corresponding closed dyadic square. Write \(c\) for its center and \(r=2^\ell /2{\gt}0\) for its half-side. Subdivide each side at its midpoint and write \(P_i=\operatorname {conv}\{ c,a_i,b_i\} \) for the eight fan triangles, indexed in perimeter order by \(\mathcal J\). Let \(\tau \) be the cyclic successor and assign arbitrary colors \(f_i\in \{ 0,1\} \). Define
Here \(B_\infty (c,r)\) is the open maximum-norm ball. The sets \(S_i\) are the retained fan pieces. The change set \(A\) may be empty. This is the local radial deletion used in the initial-star construction of [ 247 , Section 11 ] .
Let \(S_i\) be the retained pieces of Definition 27.86.4. If \(i,j\) are consecutive perimeter positions and \(f_i=f_j\), then \(S_i\cap S_j\) is nonempty. Here either \(j=\tau (i)\) or \(i=\tau (j)\) is permitted. The change set \(A\) may be empty.
27.87 Connected fan pieces in a smaller closed square
The following local assertions concern the eight-triangle fan used in the initial-star construction of [ 247 , Section 11 ] . The origin, dyadic exponent, cell index and binary coloring are arbitrary. The exponent zero and the empty change set are included; distinct runs carrying the same color remain distinct.
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\), \(z\in \mathbb {Z}^2\) and the colors \(f_i\in \{ 0,1\} \) be arbitrary. Use the center \(c\), half-side \(r=2^\ell /2{\gt}0\), triangles \(P_i\), change set \(A\), deleted radials \(L\) and retained pieces \(S_i\) of Definition 27.86.4. Define
The closed maximum-norm ball is the closed square of radius \(r/2\). The equality follows from \(\overline B_\infty (c,r/2)\subset B_\infty (c,r)\). When \(A=\varnothing \), the center belongs to every \(K_i\).
If \(i,j\) are consecutive perimeter positions with \(f_i=f_j\), then \(K_i\cap K_j\) is nonempty. Either \(j=\tau (i)\) or \(i=\tau (j)\) is permitted, and the change set may be empty.
Suppose first that \(j=\tau (i)\), so \(b_i=a_j\), and set \(m=(c+b_i)/2\). The base-radius identity gives \(\| m-c\| _\infty =r/2{\gt}0\), so \(m\ne c\) and \(m\in \overline B_\infty (c,r/2)\subset B_\infty (c,r)\). The noncentral radial-incidence characterization places \(m\) in both \(P_i\) and \(P_j\).
If \(m\in [c,b_t]\), then \(b_t-c\) and \(b_i-c\) have the same directed ray. The distinctness of the endpoint directions gives \(t=i\). But \(f_i=f_j=f_{\tau (i)}\), so \(i\notin A\). Thus \(m\notin L\) and \(m\in K_i\cap K_j\). For \(i=\tau (j)\), interchange the indices and use symmetry of intersection.
Every \(K_i\) is nonempty and connected.
Apply Theorem 27.86.3 at radius \(r/2\) to obtain a point of \((P_i\setminus L)\cap B_\infty (c,r/2)\), which belongs to \(K_i\). The deleted triangle is convex by Theorem 27.85.4. Both centered balls are convex, so their intersection with the deleted triangle is a nonempty convex set and hence connected.
Retain an adjacency between consecutive triangles precisely when their colors agree. For any connected component \(R\) of this graph, put \(T_R=\bigcup _{i\in R}P_i\). Then \((T_R\setminus L)\cap \overline B_\infty (c,r/2)\) is nonempty and connected. Components carrying the same color remain distinct.
Each \(K_i\) is nonempty and connected by Theorem 27.87.3, and Theorem 27.87.2 gives a nonempty intersection along every retained edge. The component \(R\) is nonempty, and any two of its vertices are joined by a finite path within \(R\). Successive unions along these paths show that \(\bigcup _{i\in R}K_i\) is nonempty and connected. Distributing deletion and intersection over the union gives
Hence the set on the left of (715) is nonempty and connected. If \(A=\varnothing \), no radial is removed and the center is retained.
Retain the arbitrary origin, natural exponent, cell index and binary coloring of Definition 27.87.1. With its center \(c\), half-side \(r\), triangles \(P_i\) and deleted radial union \(L\), put
Write \(\mathcal C(\Omega _{1/2})\) for the connected components in the relative topology, and \(C_{\Omega _{1/2}}(x)\) for the component containing \(x\in \Omega _{1/2}\). There is a unique bijection \(\Phi :\mathcal C(\Omega _{1/2})\longrightarrow \mathcal R\) such that, for every \(x\in \Omega _{1/2}\) and \(R\in \mathcal R\),
The assertion includes exponent zero and an empty change set. In the latter case the center is retained. Distinct runs carrying the same color correspond to distinct connected components. This is a local ingredient of the initial-star construction in [ 247 , Section 11 ] .
For each run \(R\), set \(V_R=T_R\cap \Omega _{1/2}\). Theorem 27.87.4 shows that \(V_R\) is nonempty and connected. These sets are pairwise disjoint by Theorem 27.89.1. The smaller closed square lies in \(\overline B_\infty (c,r)\), which is the closed cell by Theorem 27.61.2. The run cover therefore gives \(\Omega _{1/2}=\bigcup _{R\in \mathcal R}V_R\).
Each triangle is a finite convex hull, hence compact and closed, as in Theorem 27.49.1.9. Every \(T_R\) is a finite union of such triangles, so \(V_R\) is relatively closed in \(\Omega _{1/2}\). The disjoint cover gives
There are finitely many runs, so this complement is relatively closed. Thus every \(V_R\) is relatively open as well.
A connected component meeting \(V_R\) contains the connected set \(V_R\) and cannot meet its relatively open and closed complement. It is therefore exactly \(V_R\). These sets give every component exactly once. For a retained point, membership in \(V_R\) is precisely membership in \(T_R\), proving (717). Every component contains a point, so the same characterization proves uniqueness of the bijection.
27.88 Connected runs after radial deletion
The connected pieces of Theorem 27.86.3 meet across every retained adjacency by Theorem 27.86.5. The following gluing assertion applies to the runs of Definition 27.79.1 in the fan construction of [ 247 , Section 11 ] .
Consider the arbitrary fan, coloring and retained pieces \(S_i\) of Definition 27.86.4. Retain the adjacency between consecutive triangles precisely when their colors agree. For each connected component \(R\) of this graph, put \(T_R=\bigcup _{i\in R}P_i\). Then \((T_R\setminus L)\cap B_\infty (c,r)\) is nonempty and connected. The change set \(A\) may be empty. Components with the same color remain distinct runs.
Each retained piece \(S_i\) is nonempty and connected by Theorem 27.86.3. A retained edge joins consecutive triangles with equal colors, so Theorem 27.86.5 gives \(S_i\cap S_j\ne \varnothing \) along every such edge.
The component \(R\) is nonempty, and any two of its vertices are joined by a finite path of retained edges within \(R\). Gluing the connected sets along these paths shows that \(\bigcup _{i\in R}S_i\) is connected. It is nonempty because \(R\) contains a vertex and every \(S_i\) is nonempty. Distributing deletion and intersection over the union gives
This proves the assertion. If \(A\) is empty, no radial is removed and the center is retained.
27.89 Disjoint runs after radial deletion
Let \(o\in \mathbb {R}^2\), \(\ell \in \mathbb {N}\) and \(z\in \mathbb {Z}^2\) be arbitrary, and let \(Q=o+2^\ell (z+[0,1]^2)\) be the corresponding closed dyadic square with center \(c\). Subdivide every side at its midpoint, obtaining eight fan triangles \(P_i=\operatorname {conv}\{ c,a_i,b_i\} \) indexed in perimeter order by \(\mathcal J\). Let \(\tau \) be the cyclic successor and assign arbitrary colors \(f_i\in \{ 0,1\} \). Define the change set and its radial union by
Retain the adjacency between consecutive triangles precisely when their colors agree. For each connected component \(R\) of this graph, put \(T_R=\bigcup _{i\in R}P_i\). The sets \(T_R\setminus K\) are pairwise disjoint: if \(R\ne T\), then
The change set \(A\) may be empty. Distinct components of the retained graph remain distinct runs even when their colors agree.
Suppose that \(R\ne T\) and that \(x\in (T_R\setminus K)\cap (T_T\setminus K)\). Choose \(i\in R\) and \(j\in T\) with \(x\in P_i\cap P_j\). Since the components differ, \(i\ne j\).
First suppose that \(x=c\). Every radial contains \(c\), so \(c\notin K\) implies \(f_t=f_{\tau (t)}\) for every \(t\in \mathcal J\). The endpoint-successor equivalence then makes the colors equal across every perimeter adjacency. With constant color zero, the full perimeter graph is connected by Theorem 27.79.5. Each of its edges is also retained under the coloring \(f\). Thus \(i\) and \(j\) are connected in the retained graph, contradicting their membership in distinct components.
Now suppose that \(x\ne c\). By Theorem 27.61.6, both triangles contain \(c\). Thus \(P_i\cap P_j\) contains two distinct points. The contact classification gives either \(b_i=a_j\) and \(P_i\cap P_j=[c,b_i]\), or the reverse orientation \(b_j=a_i\) and \(P_i\cap P_j=[c,b_j]\). In the first case, \(j=\tau (i)\). Since the adjacent triangles belong to distinct runs, Theorem 27.79.4 gives \(f_i\ne f_j\). Hence \(i\in A\) and \(x\in [c,b_i]\subset K\), a contradiction. In the reverse orientation, the same argument gives \(j\in A\) and \(x\in [c,b_j]\subset K\).
When \(A\) is empty, the center is retained and every perimeter adjacency is retained. There is therefore a single run, as the first part of the argument shows. The assertion is an auxiliary to the initial-star construction in [ 247 , Section 11 ] and holds before any ball restriction.
27.90 Connected components after radial deletion
The retained fan pieces of Definition 27.86.4 combine along each run to form a connected component of the cut square. The following correspondence is an auxiliary to the initial-star construction of [ 247 , Section 11 ] .
Consider the arbitrary dyadic square \(Q\), all-midpoint fan, coloring and radial union \(L\) of Definition 27.86.4. Retain an adjacency between consecutive triangles precisely when their colors agree, and write \(\mathcal R\) for the connected components of this finite graph. Put \(\Omega =B_\infty (c,r)\setminus L\). For each \(R\in \mathcal R\), set
Write \(\mathcal C(\Omega )\) for the connected components of \(\Omega \) in its relative topology, and \(C_\Omega (x)\) for the component containing \(x\in \Omega \). There is a unique bijection \(\Phi :\mathcal C(\Omega )\longrightarrow \mathcal R\) such that, for every \(x\in \Omega \) and \(R\in \mathcal R\),
Equivalently, every connected component of \(\Omega \) is \(U_R\) for exactly one run \(R\), and every \(U_R\) is a connected component. The assertion includes \(\ell =0\) and an empty change set \(A\). Distinct runs with the same color correspond to distinct components.
Theorem 27.61.2 gives \(Q=\overline B_\infty (c,r)\), so \(\Omega \subset Q\). By Theorem 27.79.3, the run regions cover \(Q\). Hence the sets \(U_R\) cover \(\Omega \). Theorem 27.89.1 makes this cover pairwise disjoint. Since \(U_R=(T_R\setminus L)\cap B_\infty (c,r)\), Theorem 27.88.1 shows that every \(U_R\) is nonempty and connected. Passing from the ambient plane to the relative topology of \(\Omega \) preserves this connectedness.
Each triangle \(P_i\) is the convex hull of a finite set of vertices, so it is compact and closed. Since each run has finitely many triangles, \(T_R\) is closed, and \(U_R\) is relatively closed in \(\Omega \). There are finitely many runs. The disjoint cover gives \(\Omega \setminus U_R=\bigcup _{T\in \mathcal R,\ T\ne R}U_T\). This finite union is relatively closed, so \(U_R\) is also relatively open.
A connected component meeting the nonempty connected set \(U_R\) contains it: their union is connected, and the component is maximal. It cannot meet the complement of the relatively open and closed set \(U_R\), since that would separate the component. Consequently that component is exactly \(U_R\). The sets \(U_R\) thus give all the connected components of \(\Omega \), each exactly once, and determine the required bijection. For \(x\in \Omega \), membership in \(U_R\) is precisely membership in \(T_R\), which gives (726). Every component contains a point, so this characterization also proves uniqueness of the bijection.
If \(A\) is empty, then \(L\) is empty and every perimeter adjacency is retained. There is a single run, and \(\Omega =B_\infty (c,r)\) has a single connected component. The same argument therefore includes this case.
27.91 Connected components near an initial mark
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), a natural width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). Choose arbitrary valid pitch residues \(a_h,b_h\) modulo \(2^{p_h-\ell _h}\) at every layer \(h\). Let \(\mathcal I\) be the resulting actual initial identifiers, let \(X_i^0\) be their closed birth regions, put \(U_i=\operatorname {int}(X_i^0)\), and write \(\chi (i)\in \{ 0,1\} \) for their initial colors. Let \(k\ge k_0\), let \(z\in F_{k,\ell _k}\) be an actual fine-cell index, and let \(v\) be any of the nine marks of its defining cell. Set \(\ell =\ell _k-5\) and \(r=2^\ell /2=t_k/64\).
Use the eight triangles obtained by subdividing every side of the square centered at \(v\) with half-side \(r\) at its midpoint. Their origin is \(o_{\rm S}=v-r(1,1)\), their exponent is \(\ell \), and their cell index is \((0,0)\). Write \(\mathcal J\) for the perimeter positions, \(\tau \) for their cyclic successor, \(P_s^{\rm S}\) for the closed triangles and \(b_s^+\) for their final base endpoints. Let \(\sigma :\mathcal J\to \mathcal I\) be the canonical assignment of Definition 27.65.4, obtained from the concentric working fan of half-side \(2r\). Color position \(s\) by \(\chi (\sigma (s))\). Retain an adjacency between distinct consecutive positions precisely when these colors agree. Write \(\mathcal R\) for the runs of this graph and \([s]\) for the run containing \(s\). Their closed regions are \(Y_R=\bigcup _{s\in R}P_s^{\rm S}\). Let \(\lambda :\mathcal R\to \mathcal I\) be the unique assignment of Theorem 27.67.1; thus \(\lambda ([s])=\sigma (s)\). Define
These are the local initial regions used in [ 247 , Section 11 ] .
Write \(\mathcal C(\Omega )\) for the connected components of \(\Omega \) in its relative topology, and \(C_\Omega (x)\) for the component containing \(x\in \Omega \). There is a unique bijection \(\Phi :\mathcal C(\Omega )\longrightarrow \mathcal R\) such that, for every \(x\in \Omega \) and \(R\in \mathcal R\),
The change set \(A\) may be empty; the center is then retained. Separated runs with the same initial identifier correspond to distinct connected components.
The smaller fan has center \(v\), since its origin is \(v-r(1,1)\) and its half-side is \(r\). The successor endpoint theorem makes \(s\) and \(\tau (s)\) adjacent. The actual adjacent-sector color theorem gives \(\chi (\sigma (s))=\chi (\sigma (\tau (s)))\) if and only if \(\sigma (s)=\sigma (\tau (s))\). Thus the color-change indices are exactly \(A\), and their radial union is precisely \(L\). Apply Theorem 27.90.1 to the smaller fan and its derived coloring. Its cut square is the set \(\Omega \) in (729), its graph runs are \(\mathcal R\), and its closed run regions are \(Y_R\). The theorem therefore gives the unique bijection with (730).
Retain all the arbitrary choices and hypotheses fixed at the beginning of this section, including the pitch residues at every layer and the canonical assignment \(\sigma \). With the same runs \(\mathcal R\), closed run regions \(Y_R\) and actual identifier-change radial union \(L\), put
Write \(\mathcal C(\Omega _{1/2})\) for its connected components in the relative topology, and \(C_{\Omega _{1/2}}(x)\) for the component containing \(x\in \Omega _{1/2}\). There is a unique bijection \(\Phi :\mathcal C(\Omega _{1/2})\longrightarrow \mathcal R\) such that, for every \(x\in \Omega _{1/2}\) and \(R\in \mathcal R\),
The change set \(A\) may be empty; the center is then retained. Separated runs carrying the same initial identifier correspond to distinct connected components. This is a local assertion in the initial-star construction of [ 247 , Section 11 ] .
The smaller fan has origin \(v-r(1,1)\), exponent \(\ell \) and cell index \((0,0)\), so its center is \(v\). Color its triangle \(s\) by \(\chi (\sigma (s))\). The successor endpoint theorem makes \(s\) and \(\tau (s)\) adjacent. The adjacent-sector color theorem therefore gives
Consequently its color-change indices are exactly \(A\), and the radial union removed from the fan is exactly \(L\).
Apply Theorem 27.87.5 to this smaller fan with the derived coloring. Its retained closed square is (731), its graph runs are \(\mathcal R\), and its closed run regions are \(Y_R\). The resulting unique bijection has precisely (732).
For every run \(R\in \mathcal R\) and every point \(x\in \Omega \cap Y_R\), one has \(x\in U_{\lambda (R)}\). This includes an empty change set and does not identify separated runs assigned to the same initial identifier.
The canonical run decomposition gives
The term \(T=R\) contains \(x\), so \(x\in X_{\lambda (R)}^0\). Suppose that \(x\) also belongs to its frontier. Since \(x\in B_\infty (v,r)\), Theorem 27.84.2 gives an index \(s\in A\) with \(x\in [v,b_s^+]\). This places \(x\) in \(L\), contrary to \(x\in \Omega \). Hence \(x\in X_{\lambda (R)}^0\setminus \partial X_{\lambda (R)}^0 =\operatorname {int}(X_{\lambda (R)}^0)=U_{\lambda (R)}\).
Retain all the choices and hypotheses fixed in this section. For every connected component \(Q\) of \(\Omega _{1/2}=\overline B_\infty (v,r/2)\setminus L\), there is a unique initial identifier \(i\in \mathcal I\) such that \(Q\subseteq U_i\). Distinct components may carry the same identifier, and the change set may be empty. This is a local auxiliary to the initial-star assertion of [ 247 , Section 11 ] .
The component correspondence of the closed half-square identifies each component \(Q\) with a run \(R\) so that the points of \(Q\) are exactly the retained points of the closed run region \(Y_R\). Since \(\overline B_\infty (v,r/2)\subseteq B_\infty (v,r)\), every point of \(Q\) is a retained point of \(B_\infty (v,r)\setminus L\) lying in \(Y_R\), hence belongs to the open region \(U_{\lambda (R)}\) assigned to its run. This exhibits the identifier \(i=\lambda (R)\) with \(Q\subseteq U_i\). For uniqueness, pick any point of \(Q\); it lies in \(U_{\lambda (R)}\), and were it also contained in some \(U_{i'}\) with \(i'\neq \lambda (R)\) the two open regions would meet, contradicting their pairwise disjointness. Hence the identifier is unique.
27.92 The closed fine-cell cover of a primary region
The primary regions in [ 247 , Section 11 ] have an exact decomposition into whole closed non-belt fine cells. One primary identifier is retained even when its region is disconnected.
Fix an arbitrary common origin, finite endpoint set and nonnegative integer neighborhood width. Let \(k,\ell ,p\) be natural exponents with \(\ell \le k\) and \(\ell \le p\), and put \(m=2^{p-\ell }\). Choose canonical residues \(0\le a,b{\lt}m\). Denote the actual fine-layer index set by \(F_{k,\ell }\) and its belt subset by \(\mathcal B\). For each signed pitch index \(J\in \mathbb Z^2\), the primary birth region satisfies
The equality includes empty layers, non-retained pitch indices, disconnected regions and the case \(p=\ell \).
Each closed non-belt fine cell with shifted pitch index \(J\) lies in the primary birth region, giving one inclusion. For the other, choose the actual half-open fine cell containing a point of \(D_k\cap U_J\). In either coordinate, let \(z\) be its integer index and \(J\) the pitch index, and put \(r=z-a-mJ\) for the corresponding residue shift \(a\). The strict pitch inequalities and the half-open cell inequalities imply \(1\le r{\lt}m\). Euclidean division therefore gives \(\lfloor (z-a)/m\rfloor =J\), and the residue is different from the belt residue. Doing this in both coordinates places the chosen cell in the indicated finite union. That union is closed, so it also contains \(\overline{D_k\cap U_J}\). If \(p=\ell \), the inequalities \(1\le r{\lt}m\) have no solution and both sides are empty.
The equality supplies actual fine-cell witnesses for later interface arguments. A disconnected primary may have several isolated contacts with another region; two arbitrary common points do not by themselves specify a nondegenerate interface segment.
27.93 Fan contacts along cell boundaries
The labels assigned across cell sides in [ 247 , Section 11 ] are labels of elementary perimeter segments. The following equalities identify precisely where the incident triangles can meet those boundaries. Empty and single-point intersections are included.
Fix an arbitrary translated dyadic square, a natural exponent, a signed cell index and an optional side-midpoint subdivision. For any actual triangle \(P_i\) of the resulting fan, write \(e_i=[a_i,b_i]\) for its elementary base. If \(E\) is any whole side of its own square, then
Let \(Q\) be the half-open square and \(c\) its center. The elementary base lies in \(\overline Q\), and \(P_i\) is the convex join of \(\{ c\} \) and \(e_i\). Since \(Q\) is convex and \(c\in \operatorname {int}Q\), every point of \(P_i\) outside \(e_i\) lies in \(\operatorname {int}Q\). A whole side has one coordinate fixed at a lower or upper endpoint of the square, and so is disjoint from its interior. This gives \(P_i\cap E\subseteq e_i\cap E\); base containment gives the reverse inclusion.
Fix an arbitrary common origin, finite endpoint set and nonnegative integer neighborhood width. Let \(z\in F_{k,\ell _k}\) and \(w\in F_{h,\ell _h}\) be actual fine-cell indices with \((k,z)\ne (h,w)\). Choose an arbitrary optional midpoint mask for the cell \(Q_{\ell _k}(z)\), and let \(P_i\) and \(e_i\) be any resulting triangle and its elementary base. Then
No lower scale bound or positive-length contact assumption is needed.
The two actual half-open fine cells are disjoint. The open interior of the reference cell is therefore disjoint from the other cell and from its closure. As in the preceding proof, every point of \(P_i\) outside its base lies in that interior. Such a point cannot belong to the other closed cell, proving one inclusion. The other follows from \(e_i\subseteq P_i\).
These exact identities preserve the locations of contacts when passing from triangles to their bases. Interfaces between run regions in different cells will be considered through shared nondegenerate segments, rather than through two arbitrary common points.
27.94 Interfaces of belt runs with primary regions and other belt runs
The local perimeter labels in [ 247 , Section 11 ] determine the labels along positive-length interfaces of the resulting regions. A shared nondegenerate segment is retained as the geometric hypothesis, since two isolated common points of disconnected regions need not lie on a common interface.
Fix a common origin, finite endpoint set, width \(C\ge 2\) and initial layer \(k_0\ge 50{,}000{,}000\), with arbitrary residue choices in every layer. Let \(R\) be an actual run of the labelled fan in an actual belt cell of layer \(k\ge k_0\). For any \(h\ge k_0\) and signed pitch index \(J\), suppose a closed segment \([u,v]\), with \(u\ne v\), lies in both the run region and the primary birth region of index \(J\) at layer \(h\). Then \(J\) is retained, and every triangle of \(R\) has label \(h+1\ne h\) modulo two. Retention is a conclusion; the primary region need not be connected.
The primary is a finite union of closed actual non-belt fine cells. The run is a finite union of its fan triangles. Their constituent intersections cover the infinite segment \([u,v]\), so at least one contains two distinct points. The belt cell and the selected non-belt cell have distinct indexed identifiers. The triangle/base contact equality transfers this contact to the elementary base. The local non-belt opponent rule then identifies its pitch index as a retained primary and assigns the opposite primary label. Constancy on the run gives that same label for every constituent triangle.
Under the same common data and scale bounds, let \(R\) and \(T\) be actual runs in actual belt cells of layers \(k,h\ge k_0\), whose indexed cell identifiers are distinct. If a nondegenerate closed segment belongs to both run regions, then every triangle of \(R\) has a label different from every triangle of \(T\).
The segment is covered by finitely many pairs of constituent triangles. Infinitude again yields a pair with two distinct common points. Each triangle lies in its own closed cell. Applying the triangle/base equality in both directions places these two points on both elementary bases. The local belt-contact rule gives opposite labels for that pair. Constancy on each run extends the conclusion to all of their triangles.
Dummy-region interfaces, global region identifiers, isolated contact stars and the subsequent repairs remain separate steps of the two-family argument.
27.95 Interfaces with the initial dummy region
The initial dummy region has label \(k_0-1\) modulo two in [ 247 , Section 11 ] . The following results pass from an actual contact to the elementary perimeter assignment and then to the label of an entire belt run. Write \(N_0=N_{k_0}\) for the half-open initial neighborhood and \(\ell _k\) for the fine exponent at layer \(k\).
Fix an arbitrary common origin, finite endpoint set and width \(C\ge 2\). Let \(k\ge k_0\) and \(z\in F_{k,\ell _k}\) be an actual fine-cell index. Choose any optional midpoint subdivision of its sides, and let \(e_i=[a_i,b_i]\) be an elementary base of the resulting fan. If \(x\in (a_i,b_i)\cap \overline{N_0}\), then \(e_i\subseteq \overline{N_0}\). No lower bound on \(k_0\) is required.
Choose a coarse cell in the finite dummy union whose closure contains \(x\). Its interior is disjoint from the interior of the reference fine cell. After interchanging coordinates if necessary, the elementary side is vertical. Since \(x\) lies strictly between its endpoints, disjointness forces its normal coordinate to be a boundary coordinate of the coarse cell. If a tangential boundary coordinate lay strictly between the elementary endpoints, the corresponding coarse-cell corner would lie in the open elementary side. The dummy-corner theorem excludes this. Thus the coarse-cell closure contains the entire elementary segment, and is itself contained in \(\overline{N_0}\).
For arbitrary common origin, finite endpoint set and natural width \(C\), let \(k\ge k_0\) and \(z\in F_{k,\ell _k}\) be an actual fine-cell index. Choose an arbitrary optional midpoint mask. Every resulting fan triangle \(P_i\), with elementary base \(e_i\), satisfies
Empty and single-point contacts are included.
The open interior of the reference fine cell is disjoint from the dummy neighborhood and therefore from its closure. A common point belongs to the closed reference square, so it lies on its frontier. The product-interval frontier formula places it on one of the four whole sides. The own-side contact equality then places it on \(e_i\). This proves one inclusion; the other follows from \(e_i\subseteq P_i\).
Fix a common origin, finite endpoint set, width \(C\ge 2\) and initial layer \(k_0\ge 50{,}000{,}000\), with arbitrary residue choices in every layer. Let \(R\) be an actual run in an actual belt cell of layer \(k\ge k_0\). If a closed segment with distinct endpoints lies in both the run region and \(\overline{N_0}\), every constituent triangle of \(R\) has label \(k_0\ne k_0-1\) modulo two.
The shared segment is infinite, whereas the collection of elementary endpoints in the fan is finite. Choose a shared point outside that collection and a constituent triangle containing it. The preceding contact equality puts the point on the triangle’s base, away from both endpoints, hence in the open base segment. Open-contact continuation puts the whole base in the dummy closure. The local dummy assignment gives the opposite dummy parity, and constancy on the run transfers this label to every constituent triangle.
The whole-run conclusion retains a shared nondegenerate segment. Two isolated common points of disconnected unions do not supply this premise. Global identifiers, isolated stars and recursive repairs remain further obligations.
27.96 The initial family of regions
Fix an arbitrary origin \(o\in \mathbb R^2\), a finite endpoint set \(Z\subset \mathbb Z^2\), an integer width \(C\ge 2\) and an initial layer \(k_0\ge 50{,}000{,}000\). At each layer \(k\ge k_0\), choose arbitrary residues \(a_k,b_k\) modulo \(2^{p_k-\ell _k}\). Retain the primary indices, belt cells, midpoint subdivisions and triangle colors of [ 247 , Section 11 ] .
Let \(G_k\) be the retained primary pitch indices and \(B_k\) the belt-cell indices at layer \(k\). For \(z\in B_k\), let \(\mathcal R_{k,z}\) be the runs of its colored fan. The identifier family is the disjoint union
The symbol \(*\) identifies the dummy region. A primary identifier retains its layer and pitch index, including when its region has disconnected pieces. A run identifier retains its layer and cell. Distinct runs remain distinct even when their colors agree.
Write \(N_0=N_{k_0}\) for the initial half-open neighborhood, \(R_{k,J}\) for a primary birth region and \(T_{k,z,R}\) for a belt-run region. For \(i\in \mathcal I\), define
The initial open region with identifier \(i\) is \(\operatorname {int}(X_i^0)\).
Assign \(\lambda (i)\in \mathbb Z/2\mathbb Z\) by
Run-color constancy makes the last value well defined.
If \(e\) is any constituent triangle of a belt run \(R\) in layer \(k\) and cell \(z\), then \(\lambda (k,z,R)\) equals the color of \(e\).
The run is the component containing \(e\). By definition, \(\lambda (k,z,R)\) is the color that this component carries, which is the color of \(e\).
If \(Z\ne \varnothing \), then the closed birth regions cover the plane:
Fix \(x\in \mathbb R^2\). If \(x\in N_0\), then \(x\in \overline{N_0}=X_*^0\). Otherwise fine-cell exhaustion gives a fine cell at some layer \(k\ge k_0\) containing \(x\). Its closure also contains \(x\). If the cell is in the belt, its closed square is covered by its run regions, so \(x\) lies in one \(T_{k,z,R}\). If the cell is outside the belt, the unique retained primary containing its closure also contains \(x\). Each case places \(x\) in a birth region with an identifier.
27.97 The translated origin and integer lattice points
The geometric construction in [ 247 , Section 11 ] uses a translated origin whose coordinates, sum and difference are nonintegral.
Set \(o=(\sqrt2,\sqrt3)\in \mathbb R^2\).
None of \(o_1\), \(o_2\), \(o_1+o_2\) and \(o_1-o_2\) is an integer.
Squaring positive rational bounds gives \(4/3{\lt}\sqrt2{\lt}2\) and \(5/3{\lt}\sqrt3{\lt}2\); also \(\sqrt2{\lt}\sqrt3\). Thus \(o_1,o_2\in (1,2)\), \(o_1+o_2\in (3,4)\) and \(o_1-o_2\in (-1,0)\). No integer lies strictly between consecutive integers.
For every \(m\in \mathbb Z\), none of the four lines
contains a point of \(\mathbb Z^2\).
If \(x\in \mathbb Z^2\) satisfies one of these equations, subtracting \(m\) expresses the corresponding coordinate, sum or difference of \(o\) as an integer. Each possibility contradicts nonintegrality.
27.98 Disjoint interiors of the initial regions
The initial regions in [ 247 , Section 11 ] may share boundary segments or isolated points. Their open interiors are disjoint.
Fix any translated dyadic square, optional midpoint mask and assignment of two colors to its fan triangles. If \(R\ne T\) are two runs, then \(\operatorname {int}(T_R)\cap \operatorname {int}(T_T)=\varnothing \).
A real segment in the plane is nowhere dense: it is closed, and the affine span of its two endpoints is proper, so its interior is empty. Distinct runs have disjoint sets of constituent triangles. For every triangle \(P_e\) in \(R\) and \(P_f\) in \(T\), their intersection is either an actual radial segment or the common center. In either case it is nowhere dense. Since both runs have finitely many triangles,
is nowhere dense. Hence its interior, which equals \(\operatorname {int}(T_R)\cap \operatorname {int}(T_T)\), is empty.
Fix an arbitrary origin, finite endpoint set, integer width \(C\ge 2\) and initial layer \(k_0\ge 50{,}000{,}000\), with arbitrary valid residue choices at every layer. For any distinct actual initial identifiers \(i,j\), \(\operatorname {int}(X_i^0)\cap \operatorname {int}(X_j^0)=\varnothing \). No nonemptiness condition on the endpoint set is required.
A half-open dyadic square \(Q\) is convex with nonempty interior, so \(\overline{\operatorname {int}(Q)}=\overline Q\). Suppose that \(Q\) is disjoint from each half-open cell \(T_i\) in an opposing family. Since \(\operatorname {int}(Q)\) is open, it is also disjoint from each closure \(\overline{T_i}\), and thus from their union. The open set \(U=\operatorname {int}\bigl(\bigcup _i\overline{T_i}\bigr)\) is therefore disjoint from \(\operatorname {int}(Q)\), hence from its closure:
Union over the first family’s cells to obtain disjoint interiors of the two unions of closures.
Each primary is exactly a finite union of closed nonbelt fine cells, while every run is contained in its own closed cell. Different primary identifiers cannot share an indexed nonbelt cell: a cell has one layer and one primary pitch index. A belt cell cannot be such a nonbelt cell. Distinct indexed actual fine cells are disjoint before closure, so the preceding closure argument treats all different-cell cases. Later cells are likewise disjoint from the half-open dummy neighborhood; the same two open-set closure steps treat dummy contacts. The only remaining case is two distinct runs in one cell, for which the first theorem applies.
27.99 Initial boundaries and lattice membership
Use the prescribed origin \(o=(\sqrt2,\sqrt3)\) and the actual initial birth regions \(X_i^0\) of [ 247 , Section 11 ] . Fix a finite endpoint set, integer width \(C\ge 2\) and initial layer \(k_0\ge 50{,}000{,}000\), with arbitrary valid residue choices.
For every actual initial identifier \(i\), \(\partial X_i^0\cap \mathbb Z^2=\varnothing \).
First allow an arbitrary origin. Every dyadic square boundary lies on a line \(x_1=o_1+m\) or \(x_2=o_2+m\) with \(m\in \mathbb Z\), since its side length is an integer power of two. The boundary of a finite union is contained in the union of its members’ boundaries, and \(\partial \overline E\subseteq \partial E\). The dummy region and the exact finite closed-cell decomposition of each primary therefore have boundaries on these coordinate lines.
For a fan triangle \(P_e\) in a square \(Q\), its boundary satisfies
Indeed, the finite closed fan covers \(\overline Q\). A boundary point of \(P_e\) away from \(\partial Q\) and all other triangles would have an open neighborhood in \(\operatorname {int}(Q)\) contained in \(P_e\), a contradiction. Each intersection with another triangle is a radial segment or the common center. At every actual layer \(k\ge k_0\), the fine exponent \(\ell _k\) is at least one. The center and elementary endpoints are marks, and the mark-mesh theorem with spacing one places them in \(o+\mathbb Z^2\). Each radial direction has an allowed slope. Thus its segment lies on an integer translate of one of the four lines with linear forms \(x_1\), \(x_2\), \(x_1+x_2\) and \(x_1-x_2\).
A run has finitely many triangles, so its boundary is contained in their boundary union. This gives the four supporting-line forms for every initial birth boundary; no finiteness of the ambient identifier family is used. At \(o=(\sqrt2,\sqrt3)\), every such integer-translated line avoids \(\mathbb Z^2\), proving the assertion.
Every actual initial identifier \(i\) satisfies \(\partial \operatorname {int}(X_i^0)\cap \mathbb Z^2=\varnothing \).
The inclusion \(\partial \operatorname {int}(X_i^0)\subseteq \partial X_i^0\) transfers the birth-boundary avoidance to the open region.
For every identifier \(i\) and every \(v\in \mathbb Z^2\), \(v\in X_i^0\) if and only if \(v\in \operatorname {int}(X_i^0)\).
A point in \(X_i^0\) outside its boundary is an interior point. Birth-boundary avoidance gives this implication at every lattice point. The converse is the inclusion \(\operatorname {int}(X_i^0)\subseteq X_i^0\).
27.100 The initial partition of the integer lattice
Fix the origin \(o=(\sqrt2,\sqrt3)\), a nonempty finite endpoint set, integer width \(C\ge 2\) and initial layer \(k_0\ge 50{,}000{,}000\), with arbitrary valid residue choices at each layer. Every \(v\in \mathbb Z^2\) belongs to exactly one actual initial open region \(\operatorname {int}(X_i^0)\) of [ 247 , Section 11 ] .
Closed coverage gives an identifier \(i\) with \(v\in X_i^0\). At a lattice point, birth membership agrees with open membership, so \(v\in \operatorname {int}(X_i^0)\). If another identifier \(j\) has this property, pairwise disjointness of the open regions forces \(j=i\).
27.101 Colors across initial interfaces
Retain the actual initial identifiers, closed birth regions \(X_i^0\) and colors \(\lambda (i)\) of [ 247 , Section 11 ] . The origin, finite endpoint set and valid residue choices are arbitrary, with \(C\ge 2\) and \(k_0\ge 50{,}000{,}000\).
Let \(i\ne j\) be actual initial identifiers. If \(u,v\in \mathbb R^2\) are distinct and their entire closed segment satisfies
then \(\lambda (i)\ne \lambda (j)\). No nonemptiness condition on the endpoint set is required.
For an interface involving a belt run and a primary or the dummy, the corresponding run-interface theorem gives opposite colors. The color of a run agrees with every constituent triangle. The same argument handles runs in different indexed cells. If two distinct runs lie in one cell, the shared segment supplies two distinct common points; the fan-run contact theorem gives opposite colors.
Let primaries \(i=(k,J)\) and \(j=(h,K)\) with \(k\le h\) share a point \(x\). They lie in neighboring closed layers, so \(h\in \{ k,k+1\} \). If \(h=k\), then \(J\ne K\), and positive separation gives some \(t{\gt}0\) with
which is impossible. Hence \(h=k+1\), and
A primary \(i=(k,J)\) contacting the dummy lies in layer \(k=k_0\), so
Two dummy identifiers cannot be distinct. These cases exhaust the identifier family.
The hypothesis retains the whole segment. Two isolated common corners of possibly disconnected regions do not constitute such an interface.
27.102 Regularity of initial birth regions
Retain the actual initial family of [ 247 , Section 11 ] , with arbitrary origin, finite endpoint set and valid residues, \(C\ge 2\) and \(k_0\ge 50{,}000{,}000\).
For every actual initial identifier \(i\),
The endpoint set may be empty. Empty regions and disconnected primary regions are included.
Each fan triangle is the closed convex hull of vertices \(a,b,c\) with
This determinant is that of the edge vectors \(b-a\) and \(c-a\), so they are linearly independent and span the plane, and the three vertices affinely span the plane. Thus the triangle has nonempty interior. A convex set with nonempty interior has the same closure as its interior; closedness therefore gives the asserted identity for each triangle.
One unsplit fan triangle is contained in each closed dyadic square. Its nonempty interior supplies nonempty interior of the square. The same convexity argument proves regularity of the square.
A finite union of regular closed sets is regular closed. Indeed, each member’s interior lies in the union’s interior, so taking closures shows that the union lies in the closure of that interior. The reverse inclusion follows from closedness of the finite union. This argument also includes the empty union.
The dummy is a finite union of closed coarse squares, each primary has its exact finite cover by closed nonbelt fine cells, and each run is a finite union of its actual fan triangles. The finite-union argument proves the identity in all three identifier cases.
27.103 Primitive frontiers of actual fans
The local mesh argument of [ 247 , Section 11 ] needs the actual endpoints of boundary segments. Retain any origin, dyadic exponent, integer cell index and optional midpoint subdivisions. Write \(Q\) for the half-open square, \(c\) for its center, \(\mathcal E\) for its cyclic fan slots, \(P_e\) for the corresponding closed triangle and \(v_e\) for its final base endpoint.
Let \(Q\subseteq \mathbb R^2\), let \(\mathcal E\) be a finite index set, and let \(P_e\) be closed sets satisfying \(\bigcup _{e\in \mathcal E}P_e=\overline Q\). For every \(e\in \mathcal E\),
This finite-cover assertion is used in the local geometric argument of [ 247 , Section 11 ] .
Let \(x\in \partial P_e\) with \(x\notin \partial Q\). Closedness gives \(x\in P_e\subseteq \overline Q\), hence \(x\in \operatorname {int}(Q)\). If \(x\) belonged to no other member, the complement of their finite closed union would be an open neighborhood of \(x\). Its intersection with \(\operatorname {int}(Q)\) would be contained in \(P_e\) by the cover identity. This would put \(x\) in \(\operatorname {int}(P_e)\), contradicting \(x\in \partial P_e\).
For every actual fan slot \(e\in \mathcal E\),
No layer, contact or positive-exponent condition is required.
Apply Theorem 27.103.1 to the actual closed fan triangles, whose union is \(\overline Q\). A shared point is either the center or, together with the center, gives two distinct common points. In the latter case the triangle-contact classification places it on an actual radial segment. The center itself belongs to every such segment. This proves the inclusion.
The inclusion retains primitive edges, including seams that may disappear when neighboring triangles are merged. Active boundaries are determined by distinct region identifiers in the subsequent local analysis.
27.104 Layers and meshes near a cell mark
Fix an arbitrary origin \(o\), a finite endpoint set \(Z\subset \mathbb Z^2\) and width \(C\ge 2\). Let \(\ell _k\) be the existing fine exponent and \(M_{k,z}\) the nine actual marks of the fine cell indexed by \(z\) at layer \(k\). The estimates below supply the numerical geometry in [ 247 , Section 11 ] .
Let \(k\ge 50{,}000{,}000\) and let \(z,w\) be actual fine-cell indices in layers \(k,h\), respectively. Suppose \(v\in M_{k,z}\) and the closed cell indexed by \(w\) meets the closed square of sup radius \(10t\) about \(v\), where \(t=2^{\ell _k}\). Then
Only the reference layer carries the stated late bound.
The reference mark and a contact point belong to their respective closed actual layers, and their distance is at most \(10t\). Nonadjacent layers are separated by at least \(16t\), so their indices differ by at most one. Monotonicity of the fine exponent and its successor increment give the three alternatives. Both exponents are therefore at least \(\ell _k-1\). The existing mesh inclusion, applied to each singleton cell family, puts both nine-point sets in the same quarter mesh.
Suppose \(k\ge 50{,}000{,}000\) and \(k_0+1\le k\). If \(x\in \overline{N_{k_0}}\) and \(y\in \overline{D_k}\), then
No nonemptiness condition on \(Z\) is supplied.
The actual dummy membership supplies an endpoint \(u\in Z\) with \(\| x-u\| _\infty \le (C+1)2^{k_0}\), while \(\| y-u\| _\infty \ge C2^k\). The triangle inequality and \(2\, 2^{k_0}\le 2^k\) give
The established scale comparison at the stated threshold gives \(32\, 2^{\ell _k}\le 2^k\), proving the result.
27.105 Whole sides of a dyadic square
The primitive boundary analysis in [ 247 , Section 11 ] uses whole square sides. Let \(Q\) be an actual translated half-open dyadic square, with arbitrary origin, exponent and integer index, and let \(S_0,S_1,S_2,S_3\) be its four closed whole sides.
For every \(x\in \partial Q\), there is an index \(s\in \{ 0,1,2,3\} \) such that \(x\in S_s\).
Write \(Q=\lbrack A,A')\times \lbrack B,B')\), where \(A{\lt}A'\) and \(B{\lt}B'\). The product frontier identity gives
These four closed segments are exactly the four whole sides, by the established endpoint coordinates.
27.106 Initial frontiers near a mark
Fix an arbitrary origin, finite endpoint set and valid residue choices for the initial construction, with \(C\ge 2\) and \(k_0\ge 50{,}000{,}000\). The estimate below is the boundary consequence of the local mesh argument in [ 247 , Section 11 ] .
Let \(k\ge k_0\), let \(z\) be an actual fine-cell index in layer \(k\), and let \(v\) be one of its nine marks. Put \(t=2^{\ell _k}\). For any actual initial identifier \(i\) and any \(x\in \partial X_i^0\) satisfying \(\| x-v\| _\infty {\lt}t/32\), the vector \(x-v\) has one of the four allowed slopes. Thus \(x\) lies on one of the four allowed lines through \(v\).
The frontier of a finite union is contained in the union of the constituent frontiers. For a primary, its exact finite closed-cell cover therefore supplies a square frontier; for a fan run, its finite triangle support supplies an outer square side or an actual radial segment. The endpoints of each primitive segment are actual cell marks, and its slope is allowed by the triangle construction.
Contact with a primitive segment supplies a point of its actual closed fine cell within \(t/32\) of \(v\). The nearby-cell estimate puts both segment endpoints and \(v\) on the common quarter mesh. An allowed supporting line missing \(v\) has distance at least \(t/8\) from \(v\), a contradiction. Both points therefore lie on the same allowed line, giving the asserted direction.
A dummy frontier point belongs to the closed dummy neighborhood. If \(k{\gt}k_0\), the sixteen-fine-side separation contradicts its stated distance from \(v\). Hence \(k=k_0\). A constituent coarse-square frontier supplies a whole side, whose endpoints lie on the same quarter mesh by the established mesh inclusion and \(\ell _k\le k\). The same line-clearance argument applies.
This estimate concerns primitive boundary directions. Assigning actual identifiers to the local sectors and identifying the active rays are subsequent assertions.
27.107 Removal squares and repair children
The removal step in [ 247 , Section 11 ] takes place before replacement regions are created. Every old region loses the same union of closed squares, including squares whose interiors will be left unassigned at the stopping scale.
Retain the fixed exponent \(\rho _0=(1+\beta )/2\). For a positive parent scale \(S\) and a mark \(v\in \mathbb R^2\), define
The floor and the exponent are integers. Equivalently, \(Q(v,S)\) is the closed ball of radius \(d(S)\) for the maximum metric.
The exponent satisfies \(0{\lt}\rho _0{\lt}1\). For every \(S{\gt}0\),
and \(d(S){\lt}S\) whenever \(S{\gt}1\). For every \(a_0{\gt}0\) there is \(S_0\) such that every \(S\ge S_0\) satisfies \(d(S)\le a_0S/1000\). Hence a stopping scale \(S_*\) with the margins of [ 247 , Section 11 ] exists and may be taken as large as required.
Put \(x=\rho _0\log _2 S\), so that \(S^{\rho _0}=2^x\) and \(d(S)=2^{\lfloor x\rfloor }\). The inequalities \(\lfloor x\rfloor \le x{\lt}\lfloor x\rfloor +1\) give 774. The exponent gaps give \(\rho _0{\lt}1\), so \(S^{\rho _0}{\lt}S\) for \(S{\gt}1\). Finally \(S^{\rho _0}=S^{\rho _0-1}S\) and \(S^{\rho _0-1}\to 0\) as \(S\to \infty \).
Let \((v_j,S_j)_{j\in J}\) be a finite family of marks, each with a location \(v_j\in \mathbb R^2\) and a positive assigned scale \(S_j\); distinct indices may share a location. Let \((R_i)_{i\in I}\) be the current actual regions of the old identifiers. Set
This formula applies to every \(i\in I\). The labels and remembered birth regions retain their values at the removal step.
Let \((v_i,S_i)_{i\in I}\) be the current parent family, with \(v_i\in \mathbb R^2\) and \(S_i{\gt}0\), and retain the stopping threshold \(S_*\). Set \(I_+=\{ i\in I:d(S_i)\ge S_*\} \) and \(P=\{ -1,0,1\} ^2\). The next children are indexed by \(I_+\times P\). For each \((i,p)\in I_+\times P\), their locations and assigned scales are
These offsets give the center, corners and side midpoints of \(Q(v_i,S_i)\). Each child retains its parent and offset as indices; coincident locations do not identify these indices. A parent with \(d(S_i){\lt}S_*\) contributes no children, and its square is terminal. This is the marking rule of [ 247 , Section 11 ] .
27.107.1 Depth and descendant count of a repair branch
Every child of a parent of scale \(S\) receives the scale \(d(S)\). The marks of generation \(j\) descended from one initial mark of scale \(S\) therefore all have the scale \(d^{j}(S)\), the \(j\)-fold iterate of \(d\), and each generation has at most \(9^j\) marks. The following estimates concern these iterated scales only; they do not construct the replacements.
For every \(S{\gt}0\) and \(j\ge 0\),
For every \(S_*{\gt}1\) there is a constant \(C{\gt}0\) such that, for every \(S{\gt}0\) and every \(j\ge 0\) with \(d^{j}(S)\ge S_*\),
The half-side \(d(S)\) is a power of two and \(d(S)\le S^{\rho _0}\), so \(\log d(S)\le \rho _0\log S\). Induction on \(j\) gives (777). If \(d^{j}(S)\ge S_*\), then \(0{\lt}\log S_*\le \rho _0^{j}\log S\), so \(\log S{\gt}0\). Taking logarithms,
Since \(\log \log S\le \log \log (e^{e}+S)\) and the latter is at least one, the constant \(C=(\lvert \log \log S_*\rvert +1)/\log (1/\rho _0)\) suffices.
For every \(S_*{\gt}1\) there is a constant \(C{\gt}0\) such that, for every \(S{\gt}0\) and every \(m\ge 0\) with \(d^{m}(S)\ge S_*\),
For every \(M{\gt}0\) there is moreover a constant \(C{\gt}0\), depending only on \(S_*\) and \(M\), such that the same sum is at most \(C(k+1)^{C}\) whenever \(S\le M2^{k}\) for a nonnegative integer \(k\).
The sum is at most \(9^{m+1}\). The depth bound gives \(m+1\le C_1+1+C_1\log \ell \) with \(\ell =\log (e^{e}+S)\), so \(9^{m+1}\le 9^{C_1+1}\ell ^{p}\) with \(p=C_1\log 9\). The hypothesis forces \(S{\gt}1\), hence \(\log (S+2)\ge 1\), and \(\ell \le e+\log (S+2)\le 4\log (S+2)\). Enlarging the exponent from \(p\) to \(C\ge p\) only increases a power of a number at least one. For the second assertion, \(S+2\le (M+2)2^{k}\) gives \(\log (S+2)\le (\log (M+2)+1)(k+1)\).
Let \(c_1{\gt}0\) and \(C_B\ge 0\). Every sufficiently large real number \(u\) satisfies
For \(u\ge 1\), \(\lceil C_Bu\rceil \le (C_B+1)u\), so the right side is at most \(2(C_B+1)^{\beta }u^{\beta }\). The exponent gaps give \(\beta {\lt}\rho _0\), and \(u^{\beta -\rho _0}\) tends to zero.
27.108 Entropy cancellation for ordered labelled regions
Let \(\Lambda \subset \mathbb Z^2\) be finite, with local dimension \(q\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Suppose \(A=D\mathbin {\dot\cup }\bigcup _iX_i\) is an ordered two-family partition, with family labels \(f_i\in \{ 0,1\} \). Put \(B=\Lambda \setminus A\) and \(P_i=\bigcup _{j{\lt}i,\, f_j=f_i}X_j\). If \(I_\Omega (X_i:B\cup P_i)\le \varepsilon _i\) for every index, then
This is [ 247 , Lemma 11.1 ] , with no Hamiltonian or geometric hypothesis. Empty pieces, families, and residual sets are allowed. The same bound holds for a unit vector in any finite tensor product of finite-dimensional spaces, with site-dependent local dimensions; this is the generality of the source statement, and the lattice statement above is its specialization to sites in \(\mathbb Z^2\) with one local dimension \(q\).
Restrict the common index order to each family label. Its preceding union is exactly \(P_i\). The finite-domain partial trace agrees with the generic finite-product partial trace after identifying the complementary configuration bases. The generic two-family entropy cancellation therefore applies. The two disjoint family index sets exhaust the common index set, so their error sums add to \(\sum _i\varepsilon _i\).
27.109 Conditional two-family entropy estimate
Let \(\Omega \) be a normalized pure state on a finite domain, and let \(G,B,C\) be pairwise-disjoint regions. Suppose an ordered two-family partition of \(G\) has remainder \(D\), pieces \(X_i\) and family labels \(f_i\in \{ 0,1\} \). Put \(P_i=\bigcup _{j{\lt}i,\, f_j=f_i}X_j\). Assume \(I_\Omega (X_i:C\cup B\cup P_i)\le \varepsilon _i\) when \(f_i=0\) and \(I_\Omega (X_i:(G\cup B)^c\cup P_i)\le \varepsilon _i\) when \(f_i=1\). Then
This is the entropy estimate used before the geometric counting argument in [ 248 , Lemma 3.2 ] .
Filter the common order by the two labels. In each family, its preceding union is exactly \(P_i\). Since \(C\) is disjoint from \(G\cup B\), \(C\cup (G\cup B\cup C)^c=(G\cup B)^c\), so the second-family hypothesis has the required complementary-exterior form. Apply the generic pure-state conditional two-family bound and combine the two disjoint index sums. The finite-domain regional entropies agree with the corresponding generic finite-product reduced-state entropies.
27.110 Regional entropy dimension bounds
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Every region \(A\subseteq \Lambda \) satisfies \(S_\Omega (A)\le |A|\log q\). This is the regional dimension estimate used in the proof of [ 247 , Theorem 1.1 ] .
The regional density matrix has trace one and is positive semidefinite. Its entropy is at most the logarithm of its rank. Its rank is at most the dimension \(q^{|A|}\) of the regional Hilbert space, so \(S_\Omega (A)\le \log (q^{|A|})=|A|\log q\).
Under the hypotheses of Theorem 27.108.1, suppose also that \(q\ge 1\), \(|D|\le c_D|\partial _\Lambda A|\), and \(\sum _i\varepsilon _i\le c_E|\partial _\Lambda A|\). Then
Two-family entropy cancellation gives \(S_\Omega (A)\le S_\Omega (D)+\frac12\sum _i\varepsilon _i\). Apply the regional dimension bound to \(D\). Since \(\log q\ge 0\), substituting the two assumed bounds proves the claimed inequality.
27.111 Regional entropy continuity
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega ,\Phi \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be unit vectors. Put \(\delta =\frac12\bigl\| |\Omega \rangle \langle \Omega | -|\Phi \rangle \langle \Phi |\bigr\| _1\). Every region \(A\subseteq \Lambda \) satisfies
If \(q=0\), the unit-vector hypotheses force \(\Lambda \) and \(A\) to be empty; in this case \(|A|\log q\) is read as zero. This is the regional form of [ 247 , Lemma 2.1 ] used in its scanner entropy estimate.
Partial trace contracts trace distance, so the reduced states have trace distance at most \(\delta \). Entropy continuity on their common space of dimension \(q^{|A|}\) bounds the entropy difference by \(4\sqrt\delta \log (e q^{|A|})\). Each reduced state has trace one, so this dimension is nonzero. The logarithm is \(1+|A|\log q\).
27.112 Entropy and dimension of a selected Schmidt spectrum
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Fix \(X\subseteq \Lambda \) and write \(S_X=S_{\Omega }(X)\). Let \((p_i)\) be the eigenvalues of its reduced state on \(X\), and let \(E\) be a set of spectral indices with mass \(z=\sum _{i\in E}p_i{\gt}0\). Suppose that \(w\in \mathbb R\) and that every \(i\in E\) satisfies \(e^{-S_X-w}\le p_i\le e^{-S_X+w}\). In the corresponding Schmidt decomposition, write \(v_i\) for the complementary Schmidt vectors, and let \(r:E\to \{ 0,\ldots ,|E|-1\} \) be the fixed enumeration of the selected indices. Define
Then
These are the selected-spectrum entropy estimates used in [ 247 , Section 8 ] .
Every selected eigenvalue is positive. The normalized restriction \(\rho _E=z^{-1}\sum _{i\in E}p_i|i\rangle \! \langle i|\) therefore has rank \(|E|\). The exponential window bounds both \(|\log \operatorname {rank}\rho _E-S_X|\) and \(|S(\rho _E)-S_X|\) by \(w+|\log z|\). The auxiliary marginal of \(\psi \) is diagonal with entries \(p_i/z\) in the fixed enumeration, so \(S'=S(\rho _E)\). Finally, normalization of \(\Omega \) gives \(z\le 1\), whence \(|\log z|=-\log z\).
27.113 Boundary entropy from birth-target information
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\ge 1\) be an integer, and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Suppose
is a finite ordered two-family partition, with labels \(f_i\in \{ 0,1\} \). For each piece, put
Let \(Y_i\subseteq \Lambda \) be birth targets satisfying
The birth targets need not be pairwise disjoint. Let \(n_i\in \mathbb N\), \(c_D,c_S\in \mathbb R\) and \(c_I\ge 0\). Suppose that, for every \(i\),
Suppose also that
Here \(I_\Omega (U:W)=S_\Omega (U)+S_\Omega (W)-S_\Omega (U\cup W)\) is ordinary mutual information of the actual regional reductions. At a zero scale the weight is defined by \(0^{-100}=0\); only positive scales occur in the manuscript application. Then
This is the conditional implication in the proof of [ 247 , Theorem 1.1, lines 838–852 of Section 11 ] . For that application \(Y_i=A\cap T_i\). The construction of the partition and birth targets, the stated estimates, and the uniform choice of their constants remain separate requirements for the full area law.
Since \(Y_i\subseteq A\) and \(Y_i\cap P_i=\varnothing \), the birth target is disjoint from the entire subsystem \(J_i\). Discarding the part \(Y_i\setminus X_i\) of its first system therefore gives
Put \(\varepsilon _i=c_I n_i^{-100}\). Because \(c_I\ge 0\),
Apply Theorem 27.110.2 with error coefficient \(c_Ic_S\) and residual coefficient \(c_D\). That estimate combines the two-family cancellation of Theorem 27.108.1 with the regional dimension bound of Theorem 27.110.1, and gives the claimed coefficient.
27.114 Collar and radius scales
For the fixed amplification parameters, let \(\gamma =(1-\varepsilon )/\alpha \). This is the exponent in the radius bound of [ 247 , Proposition 10.2 ] .
The fixed parameters satisfy \(0{\lt}\gamma \) and \(1-\varepsilon {\lt}\gamma {\lt}\beta \). For every real constant \(C\), \([L^{1-\varepsilon }+2CL^\gamma ]/L^\beta \to 0\) as the positive integer \(L\) tends to infinity.
The inequalities follow by substituting the fixed parameters. Each power in the numerator has exponent strictly smaller than \(\beta \).
For every \(C\in \mathbb R\) and \(\eta {\gt}0\), there is \(N\ge 2\) such that every integer \(L\ge N\) and every nonnegative integer radius \(r\le CL^\gamma \) satisfy \(\lfloor L^{1-\varepsilon }\rfloor +2r{\lt}\eta L^\beta \). Consequently, for every sequence of nonnegative integer radii satisfying \(r(L)\le CL^\gamma \) eventually, \(\lfloor L^{1-\varepsilon }\rfloor +2r(L)=o(L^\beta )\). The threshold depends only on \(C\) and \(\eta \).
Apply the preceding limit and use \(\lfloor L^{1-\varepsilon }\rfloor \le L^{1-\varepsilon }\). This proves the scale comparison in the final paragraph of the proof of [ 247 , Proposition 10.2 ] ; the geometric clearance and amplified mutual-information estimate require separate arguments.
27.114.1 The clock time and a good realization
The following entries formalize the choice of the clock time and the truncation radius in the proof of [ 247 , Proposition 10.2 ] , and the selection of one clock realization on which the excited component, the omitted events and the radius truncation are all small. The three averaged estimates that precede this selection are inputs.
For \(g{\gt}0\), real numbers \(\mathcal I,w\) and integers \(K,n\ge 1\), put \(t_*=(6/g)(\mathcal I+w+K\log n)\). For real \(C'\) and \(\alpha \), put \(r=\lceil (C'(t_*+K\log n))^{1/\alpha }\rceil \). This is the choice of time and radius of [ 247 , Proposition 10.2 ] , with the constant \(C=6/g\), which gives \(gt_*\ge \mathcal I+3w+6K\log n\) when \(\mathcal I,w\ge 0\).
Let \(X,Y,Z\) be real random variables on a probability space, with \(Y,Z\ge 0\) almost surely and \(X^2,Y,Z\) integrable. Suppose that \(\mathbb EX^2\le a\), \(\mathbb EY\le b\) and \(\mathbb EZ\le c\), and that \(a/\varepsilon ^2+b/\varepsilon +c/\varepsilon {\lt}1\) for some \(\varepsilon {\gt}0\). Then some realization satisfies \(X,Y,Z\le \varepsilon \).
By Markov’s inequality the events \(X^2\ge \varepsilon ^2\), \(Y\ge \varepsilon \) and \(Z\ge \varepsilon \) have probabilities at most \(a/\varepsilon ^2\), \(b/\varepsilon \) and \(c/\varepsilon \). Their union has probability less than one, so its complement is nonempty.
Let \(c,\alpha {\gt}0\). There is \(B\ge 1\), depending only on \(c\) and \(\alpha \), such that \((1+r)^2e^{-cr^\alpha }\le B\) for every \(r\ge 0\).
For \(r\le 1\) the left side is at most \(4\). For \(r\ge 1\) choose an integer \(m\ge 2/\alpha \); comparing with the \(m\)-th term of the exponential series gives \(e^{cr^\alpha }\ge c^mr^{\alpha m}/m!\ge c^mr^2/m!\), so the left side is at most \(4m!/c^m\).
Let \(g,c,\alpha {\gt}0\), let \(v,C,C_0\ge 0\) and let \(K\ge 1\) be an integer. There are \(C'{\gt}0\) and \(N\ge 2\) with the following property. For every integer \(n\ge N\), all \(\mathcal I,w\ge 0\) and every integer \(0\le s\le C_0n^2\), the time \(t_*\) and radius \(r\) of Definition 27.114.1.1 and \(M_0=\exp ((\mathcal I+3w)/2)\) satisfy
The first bound follows from \(gt_*=6(\mathcal I+w+K\log n)\). Since \(\log M_0\le gt_*/4\) and \(r^\alpha \ge C'(t_*+K\log n)\), the choice \(cC'=g/2+2v+12\) leaves at most a factor \(C\, C_0\, n^2\, t_*e^{-t_*}\, n^{-6K}\) in the second bound. In the third, half of the stretched exponential absorbs \((1+r)^2\) into a constant \(B\ge 1\) depending only on \(c\) and \(\alpha \), by comparing a power of \(r^\alpha \) with the exponential series. Finally \(t_*e^{-t_*}\le 1\), and \(n\ge CC_0B\) gives the remaining factor \(n^{-3K}\). This is the explicit control of the scale choices in [ 247 , Proposition 10.2 ] ; no bound on \(\mathcal I\) is needed for it.
Let \(g,c,\alpha ,v,C,C_0,K\) be as in Theorem 27.114.1.4. There are \(C'{\gt}0\) and \(N\ge 2\) such that, for all \(n\ge N\), all \(\mathcal I,w\ge 0\) and every integer \(0\le s\le C_0n^2\), the following holds with \(t_*,r,M_0\) as there. Let \(X,Y,Z\) be real random variables on one probability space, with \(Y,Z\ge 0\) almost surely, such that
Then one realization satisfies \(X,Y,Z\le n^{-K}\). In the proof of [ 247 , Proposition 10.2 ] , with \(s=|S_0|\), the three variables are \(\| D(I-P_\Omega )\zeta \| \), \(\| (D-D_{\rm ret})\zeta \| \) and \(M_0\| D_{\rm ret}-L'\| \), and the conclusion is the choice of a good realization there. This entry is that step only, with the three averaged estimates as hypotheses; it is not the proposition.
By Theorem 27.114.1.4 the three averages are at most \(n^{-6K}\), \(n^{-3K}\) and \(n^{-3K}\). With \(\varepsilon =n^{-K}\le 1/2\), the failure probabilities sum to at most \(\varepsilon ^4+2\varepsilon ^2{\lt}1\), and Theorem 27.114.1.2 applies.
27.115 The final entropy comparison of the amplification
The proof of [ 247 , Proposition 10.2 ] ends with a unitary \(U'\) that acts on neither the target system nor a distant region \(J\) and that moves the state close to a vector whose joint reduction on these two systems is a product. The results of this section turn that unitary estimate into the bound \(C n^{-M}\) on mutual information. They form the last step of the proof and do not by themselves establish the proposition: the construction of the unitary is a separate argument.
Throughout, \(V\) is a finite set of sites with finite local spaces \(\mathcal H_v\), and \(\mathcal H=\bigotimes _{v\in V}\mathcal H_v\). For a region \(R\subseteq V\) put \(\mathcal H_R=\bigotimes _{v\in R}\mathcal H_v\), and for a vector \(\psi \in \mathcal H\) let \(\rho _{\psi ,R}\) be the reduced state of \(|\psi \rangle \langle \psi |\) on \(R\). For disjoint regions \(X,J\) write \(I_\psi (X:J)=S(\rho _{\psi ,X})+S(\rho _{\psi ,J})-S(\rho _{\psi ,X\cup J})\). Ancillary systems are included by enlarging \(V\).
For unit vectors \(\psi ,\varphi \) of a finite-dimensional space,
For unit vectors of \(\mathcal H\) the reduced states on every region \(R\subseteq V\) are no farther apart in trace distance than the two pure states.
The trace distance of the two pure states is \(\sqrt{1-|\langle \psi ,\varphi \rangle |^2}\). With \(t=|\langle \psi ,\varphi \rangle |\le 1\) we have \(1-t^2\le 2-2t\le 2-2\Re \langle \psi ,\varphi \rangle =\| \psi -\varphi \| ^2\). Partial trace contracts trace distance.
Let \(R\subseteq V\) and let \(U\) be a unitary on \(\mathcal H_{V\setminus R}\). For every \(\varphi \in \mathcal H\), the vectors \(\varphi \) and \((I_R\otimes U)\varphi \) have the same reduced state on \(R\). In particular, for disjoint regions \(X,J\) with \(R=X\cup J\), they have the same mutual information \(I(X:J)\).
The reduced state of \((I_R\otimes U)\varphi \) is the partial trace over \(V\setminus R\) of \((I\otimes U)|\varphi \rangle \langle \varphi |(I\otimes U)^\dagger \). Cyclicity of the partial trace in the traced factor and \(U^\dagger U=I\) remove \(U\). The mutual information of \(X\) and \(J\) is a function of the reduced state on \(X\cup J\).
Let \(\psi ,\eta \in \mathcal H\) be unit vectors and let \(X,J\subseteq V\) be disjoint with \(I_\eta (X:J)=0\). Then
Only the dimension of \(\mathcal H_X\) enters.
Apply [ 247 , Lemma 2.1 ] to \(S(X)\) and \(S(X|J)\) for the joint reduced states on \(X\cup J\), and use \(I(X:J)=S(X)-S(X|J)\). The two moduli sum to \(8\sqrt\delta \log (e\dim \mathcal H_X)\), where \(\delta \) is the trace distance of the joint reductions. By Lemma 27.115.1, \(\delta \le \| \psi -\eta \| \).
Let \(X,J\subseteq V\) be disjoint, let \(U\) be a unitary on \(\mathcal H_{V\setminus (X\cup J)}\), and let \(\varphi ,\eta \in \mathcal H\) be unit vectors with \(I_\eta (X:J)=0\). Put \(\delta =\| (I_{X\cup J}\otimes U)\varphi -\eta \| \). Then
This is the comparison in the final part of the proof of [ 247 , Proposition 10.2 ] .
In the setting of Theorem 27.115.4, let \(n\ge 1\) be an integer and suppose that \(\delta \le C_1n^{-K/2}\) and \(\log \dim \mathcal H_X\le C_2n^2\). Then
If \(M,K\) are integers with \(K\ge 4(M+2)\), then every \(X'\subseteq X\) satisfies \(I_\varphi (X':J)\le 8\sqrt{C_1}\, (1+C_2)\, n^{-M}\). The constant depends only on \(C_1\) and \(C_2\). The choice \(K\ge 4(M+3)\) in the proof of [ 247 , Proposition 10.2 ] satisfies this condition.
The square root of \(C_1n^{-K/2}\) is \(\sqrt{C_1}\, n^{-K/4}\), and \(\log (e\dim \mathcal H_X)\le 1+C_2n^2\le (1+C_2)n^2\) since \(n\ge 1\). Multiply the two bounds. Since \(\log \dim \mathcal H_X\ge 0\), the constant \(C_2\) is nonnegative, and \(2-K/4\le -M\) gives \(n^{2-K/4}\le n^{-M}\). Discarding part of \(X\) does not increase mutual information.
27.116 Cuts with no crossing edge
This section proves Lemma 2.2 of [ 247 ] and the degenerate cases recorded before it in Section 2 of that preprint.
Let \(H=H_A\otimes I+I\otimes H_B\) with \(H_A,H_B\) Hermitian. If a unit vector \(\omega \) satisfies \(H\omega =E_0\omega \) and \(H-E_0I\ge \Delta (I-|\omega \rangle \langle \omega |)\) with \(\Delta {\gt}0\), then \(\omega \) is a product vector, and its reduced state on the first factor has zero entropy.
Let \(a,b\) be the least eigenvalues of \(H_A,H_B\) with unit eigenvectors \(u,v\). Then \(H\ge a+b\), so \(E_0=\langle \omega ,H\omega \rangle \ge a+b\), while \(H(u\otimes v)=(H_Au)\otimes v+u\otimes (H_Bv)=(a+b)(u\otimes v)\). The gap inequality on \(u\otimes v\) gives \(\Delta (1-|\langle \omega ,u\otimes v\rangle |^2)\le a+b-E_0\le 0\), hence \(|\langle \omega ,u\otimes v\rangle |=1\) and \(\omega \) is a multiple of \(u\otimes v\). The reduced state is then a rank-one projection.
Every support of finite induced-graph diameter lies in one connected component of the domain graph. If \(\partial _\Lambda A=\varnothing \), every such support lies in \(A\) or in \(\Lambda \setminus A\). The set of sites of any union of connected components has no crossing edge.
A walk inside the domain cannot pass from \(A\) to its complement without traversing an edge of \(\partial _\Lambda A\), and any two sites of a support are joined by such a walk. Adjacent sites lie in the same component.
Under the hypotheses of Theorem 27.37.7, if every admissible support \(X\) satisfies \(X\subseteq A\) or \(X\cap A=\varnothing \), then \(S_\Omega (A)=0\).
Every term of the Hamiltonian acts inside \(A\) or inside its complement, so in the coordinates of the cut \(H=H_A\otimes I+I\otimes H_{\Lambda \setminus A}\); replacing both factors by their Hermitian parts keeps this form. The ground vector is then a product vector across the cut and its reduced state on \(A\) has zero entropy.
For range \(R=0\) every regional entropy of the gapped ground vector vanishes. For \(q=1\) or an empty domain every regional entropy of every unit vector vanishes. Every union of connected components, and in particular the empty region and the full domain, has zero regional entropy for the gapped ground vector.
For \(R=0\) every support is a single site and so is never split by a cut, so Theorem 27.116.3 applies. For \(q=1\) or \(\Lambda =\varnothing \) the regional Hilbert space is one-dimensional, and a unit-trace state on it has zero entropy. A union of connected components has no crossing edge by Theorem 27.116.2, so Theorem 27.116.4 applies.
27.116.1 Vertex-boundary formulations
For a finite induced domain \(\Lambda \subset \mathbb Z^2\) and \(A\subseteq \Lambda \), define
These are the conventions of [ 247 , Corollary 1.2 ] .
Both \(\partial _{\rm in}A\) and \(Z_A\) are empty when \(A=\varnothing \) or \(A=\Lambda \).
In both cases \(\partial _\Lambda A=\varnothing \), since no edge joins a site of \(A\) to a site of \(\Lambda \setminus A\).
Every site of a finite induced square-lattice domain has degree at most four.
The neighbors of \((a,b)\) form a subset of \(\{ (a+1,b),(a-1,b),(a,b+1),(a,b-1)\} \).
For every cut in a finite induced lattice domain, \(|\partial _\Lambda A|\le 4|\partial _{\rm in}A|\) and \(|\partial _\Lambda A|\le 2|Z_A|\). No connectedness assumption is required.
Let \(G_A\) have exactly the crossing edges, with degrees \(d_A(x)\). It is bipartite between the inner boundaries of \(A\) and its complement, and its nonisolated sites are exactly \(Z_A\). Counting degrees on one side and on both sides gives
Empty boundaries satisfy the same identities.
Assume that, for fixed \(q,R,J,\Delta \), a nonnegative constant \(C\) gives \(S_\Omega (A)\le C|\partial _\Lambda A|\) uniformly for the finite-domain Hamiltonians and ground vectors of Section 27.37. Then the same constant gives \(S_\Omega (A)\le 4C|\partial _{\rm in}A|\) and \(S_\Omega (A)\le 2C|Z_A|\) uniformly for those instances. The two implications also hold for a single entropy inequality.
Multiply the two boundary comparisons by \(C\ge 0\) and apply the assumed edge-boundary entropy inequality. The constant is chosen before the domain, Hamiltonian, ground vector, and cut.
27.117 Marginal surprisal tails at a lattice cut
The marginal moment and tail bounds of [ 247 , Lemma 3.1 ] hold for any finite tensor product with designated supports. Here they are instantiated for the finite-range Hamiltonians of Definition 27.37.2.
For a cut \(A\subseteq \Lambda \) put \(\mathcal B_A=1+\sum _X\log ^2(\mathrm e\, q^{|X|})\), the sum running over the admissible supports \(X\) that meet both \(A\) and its complement.
An operator in the algebra generated by products acting as the identity outside \(S\) has vanishing matrix elements between configurations that differ outside \(S\), and its other matrix elements do not depend on the common values outside \(S\).
For a product \(X=\bigotimes _im_i\) of one-site operators with \(m_i=I\) for \(i\notin S\),
which vanishes when \(\sigma \) and \(\tau \) differ outside \(S\) and otherwise depends only on \(\sigma |_S,\tau |_S\). Both properties are preserved by sums and scalar multiples.
Let \(\Omega \) be a unit vector satisfying (343) and (344) with \(\Delta {\gt}0\) for a Hamiltonian whose terms have norm at most \(J\ge 0\), and let \(A\subseteq \Lambda \). Let \(K\) take the value \(-\log p_j\) with probability \(p_j\) for the eigenvalues \(p_j\) of \(\rho _A\), and put \(\vartheta =J/\Delta \). Then \(\log \mathbb E\, \mathrm e^{uK}\le uS_\Omega (A)+512\mathrm e\vartheta \mathcal B_Au^2\) for \(|u|\le 1/(32\sqrt{(1+\vartheta )\mathcal B_A})\), and for every real \(w\)
Writing \(\Omega \) in the coordinates of \(\mathcal H_A\otimes \mathcal H_{A^c}\),
Both sides have entries \(\sum _{z}\Omega (x\cup z)\overline{\Omega (y\cup z)}\) over configurations \(z\) of \(A^c\).
Under the hypotheses and notation of Theorem 27.117.3, let \(n\ge 0\) be an integer and \(M\) real, and put \(w=32\sqrt{(1+\vartheta )\mathcal B_A}\, (M\log (n+2)+\mathrm e/2+\log 2)\). The positive eigenvalues \(p_j\) of \(\rho _A\) with \(|-\log p_j-S_\Omega (A)|\le w\) carry total mass at least \(1-(n+2)^{-M}\). This is the concentration statement following Lemma 3.1 of [ 247 ] .
At this width the second argument of the minimum in (787) equals \((n+2)^{-M}\), so the tail probability is at most \((n+2)^{-M}\); for \(M{\lt}0\) this bound exceeds one and the conclusion is trivial. The indices of positive weight outside the typical set are exactly those of the tail event.
Put \(v_R=1+2R(R+1)\) and \(\mu _R=2^{v_R-1}\). At most \(\mu _R\) admissible supports contain a given site, and for every \(A\subseteq \Lambda \) at most \(|\partial _\Lambda A|\, v_R\mu _R\) admissible supports meet both \(A\) and its complement.
Let \(X\) meet \(A\) in \(v\) and its complement in \(w\). A walk of length at most \(R\) from \(v\) to \(w\) in the induced graph crosses an edge \(\{ x,y\} \in \partial _\Lambda A\) with \(x\in A\), and its initial segment joins \(v\) to \(x\) in at most \(R\) steps. Thus \(X\) contains a site of the ball of radius \(R\) about the endpoint in \(A\) of a boundary edge. This ball has at most \(v_R\) sites by Theorem 36.4.2, and each of them lies in at most \(\mu _R\) admissible supports by Theorem 36.4.3.
Let \(q\ge 1\) and put \(C=v_R\mu _R\log ^2(\mathrm e\, q^{v_R})\). Then \(C\ge 1\), and for every \(A\subseteq \Lambda \),
This is the bound on the tail parameter stated after Lemma 3.1 of [ 247 ] ; the constant depends only on \(q\) and \(R\).
An admissible support has at most \(v_R\) sites, so each summand of \(\mathcal B_A\) is at most \(\log ^2(\mathrm e\, q^{v_R})\), and by Theorem 27.117.6 there are at most \(|\partial _\Lambda A|\, v_R\mu _R\) summands. Hence \(\mathcal B_A\le 1+C|\partial _\Lambda A|\le C(1+|\partial _\Lambda A|)\), the last step because \(C\ge 1\).
Under the hypotheses and notation of Theorem 27.117.3, let \(q\ge 1\), let \(n\ge 0\) be an integer and \(M\ge 0\), and put
with \(C\) as in Theorem 27.117.7. The positive eigenvalues \(p_j\) of \(\rho _A\) with
carry total mass at least \(1-(n+2)^{-M}\). This is the concentration statement following Lemma 3.1 of [ 247 ] , which is stated there for \(n\ge 1\) and \(M{\gt}0\).
Since \(\log (n+2)\ge \log 2\),
and (789) gives \(\sqrt{(1+\vartheta )\mathcal B_A}\le \sqrt{(1+\vartheta )C(1+|\partial _\Lambda A|)}\). Both factors being nonnegative, the width of Theorem 27.117.5 is at most \(C_M\sqrt{1+|\partial _\Lambda A|}\log (n+2)\), and the mass of the typical set does not decrease when the width grows.
27.118 Conditional movement and skew estimates on lattice regions
The conditional movement estimate and the conditional skew estimate of [ 247 , Lemmas 5.1 and 5.3 ] are stated for a unit vector on four finite-dimensional systems. Here the four systems are four regions partitioning a finite domain \(\Lambda \), and every operator acts on its region as in Definition 27.37.2. The vector is an arbitrary unit vector; neither estimate uses a Hamiltonian or a gap. For a positive semidefinite \(\rho \) write \(\hat\rho =\rho +\Pi _{\ker \rho }\), and let \(\rho ^{[s]}\) be the real power of \(\rho \) on its support, extended by zero on its kernel.
Let \(P,x,Y,F\) partition \(\Lambda \) and let \(\theta \) be a unit vector. Then
where \(S_\theta (x|R)=S_\theta (R\cup x)-S_\theta (R)\).
Complementary regions of a pure vector have equal entropy, so \(S_\theta (x\cup Y)=S_\theta (F\cup P)\) and \(S_\theta (Y)=S_\theta (x\cup F\cup P)\), which gives the first identity; the second follows by exchanging \(P\) and \(Y\). Nonnegativity is strong subadditivity.
There are universal constants \(C,c{\gt}0\) such that the following holds. Let \(P,x,Y,F\) partition a finite domain with local dimension \(q\), let \(\theta \) be a unit vector, let \(\sigma \ge 0\) on \(P\cup x\) and \(\tau \ge 0\) on \(x\cup Y\) have trace at most one, put \(\ell =\log (\mathrm e\, q^{|x|})\), and let \(0{\lt}a\ell \le c\). Then
This is [ 247 , Lemma 5.1 ] with \(U=Y\).
The estimate holds for four regions partitioning any finite set of sites with arbitrary local dimensions: ordering the configurations as \((x\times Y)\times (P\times F)\) turns the regional states into the marginals of \(\theta \), operators on a region into tensor products with the identity, and the exponent into \(\eta _\theta \), so this regional form is [ 247 , Lemma 5.1 ] . Specialize it to the sites of \(\Lambda \) with local dimension \(q\); the regional states, entropies and placements of the finite-domain model are those of the general regional form.
Let \(P_0,P_1,x,U,F\) partition a finite domain with local dimension \(q\), put \(P=P_0\cup P_1\) and \(Y=x\cup U\), let \(\theta \) be a unit vector and let \(0\le h\le 1\) act on \(P_0\cup x\). Put
\(\ell _h=\log (\mathrm e\, q^{|P_0|+|x|})\) and \(\eta _\theta =S_\theta (x|P)+S_\theta (x|U)\). If \(0{\lt}a\ell _h\le 1/8\), then
This is [ 247 , Lemma 5.3 ] .
The estimate holds for five regions partitioning any finite set of sites with arbitrary local dimensions: in the coordinates \((P_0\times P_1)\times ((x\times U)\times F)\) the function \(f\) is the function of [ 247 , Lemma 5.3 ] for the reordered vector, whose conditional mutual information \(I(x:F|U)\) is \(S_\theta (x|P)+S_\theta (x|U)\). Specialize this regional form to the sites of \(\Lambda \) with local dimension \(q\), identifying the regional data of the finite-domain model as in Theorem 27.118.1.
27.119 Conditional entropy in a buffered rectangle
This section formalizes [ 247 , Lemma 3.2 ] on the finite domains of Definition 27.37.2. An integer rectangle is a product of two finite nonempty integer intervals; its size is the larger number of sites along an axis, and \(Q^{+d}=Q+([-d,d]^2\cap \mathbb Z^2)\).
Let \(A\subseteq \Lambda \) and let \(Z\) be the set of endpoints of the edges crossing \(A\). For \(D_0{\gt}2R+10\) a rectangle \(Q\) is safe if its sup-norm distance to \(Z\) exceeds \(D_0\operatorname {size}(Q)\), the distance to the empty set being infinite. The physical region of an ambient rectangle \(Q\) is \(A\cap Q\).
Let \(Q\) be safe and let \(Q_0^{+d'}\subseteq Q\). An admissible support with a site of \(A\) in \(Q\) lies in \(A\). If \(d+R\le d'\), no admissible support is split by both \(A\cap Q_0^{+d}\) and \(A\cap Q_0^{+d'}\). If \(R+1\le d'\), at most \(8(R+1)\mu _R(\operatorname {size}(Q_0)+d')\) admissible supports are split by \(A\cap Q_0^{+d'}\), where \(\mu _R=2^{2R(R+1)}\) bounds the number of admissible supports containing a site.
The picture shows the nested rectangles for range \(R=1\), \(d=1\) and \(d'=2\), with two admissible supports that are each split by exactly one contour.
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If a support had a site \(x\in A\cap Q\) and a site outside \(A\), a walk of length at most \(R\) between them would cross an edge of the cut at sup-norm distance at most \(R{\lt}D_0\operatorname {size}(Q)\) from \(Q\). The ambient diameter of a support is at most its graph diameter \(R\), so a support split by \(A\cap Q_0^{+d}\) lies in \(Q_0^{+(d+R)}\) and is not split by \(A\cap Q_0^{+d'}\). A support split by \(A\cap Q_0^{+d'}\) has a site in the boundary layer \(Q_0^{+d'}\setminus Q_0^{+(d'-R-1)}\), which has at most \(4(R+1)(\operatorname {size}(Q_0)+2d')\) lattice points.
For \(q\ge 1\), \(R\), \(J\ge 0\) and \(\Delta {\gt}0\) there are an integer \(C_{\mathrm{pad}}\ge 1\) and a constant \(C_{\mathrm{buf}}\) with the following property. Let \(\Omega \) be a unit vector satisfying (343) and (344), let \(Q\) be safe, let \(r\ge 1\), and let \(Q_0\) be a rectangle of size at most \(r\) with \(Q_0^{+C_{\mathrm{pad}}r}\subseteq Q\). For \(X=A\cap Q_0\) and \(T=(A\cap Q_0^{+C_{\mathrm{pad}}r})\setminus X\),
This is [ 247 , Lemma 3.2 ] .
Put \(D=R+1\), \(m=\lfloor (C_{\mathrm{pad}}-1)r/D\rfloor \) and \(X_j=A\cap Q_0^{+d_j}\) with \(d_j=r+jD\), \(1\le j\le m\). By Theorem 27.119.2 every admissible support splits at most one \(X_j\), and the number of split supports and the parameter \(\mathcal B_j\) of [ 247 , Lemma 3.1 ] are at most a constant times \(r+d_j\). The harmonic weights \(a_j=2/(Z_r(r+d_j))\) with \(Z_r=\sum _j1/(r+d_j)\) have total mass two and quadratic cost \(4/Z_r\), and \(Z_r\ge D^{-1}\log ((C_{\mathrm{pad}}+1)/(2+D))\). Choosing \(C_{\mathrm{pad}}\) makes \(a_j\le 1/2\), the energy of the optimized filters at most \(\Delta /2\), and every \(a_j/(1-a_j)\) admissible for [ 247 , Lemma 3.1 ] , uniformly in \(r\). The comparison of the optimized filter norms with Schmidt pinning then gives \(2S(X\mid T)\le S(X)+C+m\log 2\), and \(m\le C_{\mathrm{pad}}r\).
27.120 Entropy cost of a partial rectangle layer
Let \(\Lambda \subseteq \mathbb Z^2\) be finite, let \(q\in \mathbb N\) with \(q\ge 1\), and let \(\Omega \) be a unit vector on the domain with local dimension \(q\). Let \(A\subseteq \Lambda \) be arbitrary, let \(Q\) be a nonempty integer rectangle, and put \(s=\operatorname {size}(Q)\). Let \(j\in \mathbb N\) satisfy \(1\le j\le s\), and let \(Y\subseteq Q^{+j}\setminus Q^{+(j-1)}\) be arbitrary. Define \(R=A\cap (Q^{+(j-1)}\setminus Q)\) and \(R_Y=A\cap ((Q^{+(j-1)}\setminus Q)\cup Y)\). Then
This rectangle estimate follows from the partial-row argument of [ 247 , Lemma 9.4, lines 666–667 ] and is used in the rectangle covering argument in the proof of Proposition 9.5, lines 717–729. It includes an empty \(Y\), thin rectangles and missing lattice sites in the domain.
Put \(B=A\cap Y\). Then \(R\) and \(B\) are disjoint and \(R_Y=R\cup B\). The boundary-layer count recorded in the proof of Theorem 27.119.2 gives
The first inequality uses the injective inclusion of physical sites into the ambient lattice. By the regional increment and dimension bounds,
where \(\log q\ge 0\) because \(q\ge 1\).
27.121 The initial safe-box estimate
For fixed \(q,R,J,\Delta \) and \(D_0{\gt}2R+10\), let \(F_{\mathrm{box}}(r)\) be the supremum of \(S_\Omega (A\cap Q)\) over all finite-range Hamiltonians on finite domains with gapped ground vectors, all cuts \(A\), and all safe rectangles \(Q\) of size at most \(r\) (Definition 27.119.1).
The function \(F_{\mathrm{box}}\) defined before [ 247 , Proposition 3.3 ] .
For \(q\ge 1\) the function \(F_{\mathrm{box}}\) is nonnegative, nondecreasing in \(r\), and satisfies \(F_{\mathrm{box}}(r)\le r^2\log q\).
A region \(A\cap Q\) has at most \(\operatorname {size}(Q)^2\) sites, so its entropy is at most \(r^2\log q\); entropies of unit vectors are nonnegative; and enlarging \(r\) enlarges the collection in the supremum.
For fixed \(q,R,J,\Delta ,r\) with \(q\ge 1\), write \(F_{\mathrm{box},D_0}(r)\) for the safe-box entropy at safety parameter \(D_0\). If \(D_0,D_1\in \mathbb N\) and \(D_0\le D_1\), then \(F_{\mathrm{box},D_1}(r)\le F_{\mathrm{box},D_0}(r)\).
Every rectangle safe at threshold \(D_1\) is safe at threshold \(D_0\), so the supremum is taken over a smaller collection.
Partition each axis of a rectangle \(Q\) of size at most \(Mr\) into chunks of \(r\) sites and one final shorter chunk. The products of chunks are at most \(M^2\) pairwise disjoint rectangles of size at most \(r\) covering \(Q\). If a point of one child lies in the \(Cr\)-padding of another, their chunk positions differ by at most \(C\) along each axis. A child at least \(C\) positions from the first chunk and \(C+2\) from the last along both axes has its \(Cr\)-padding inside \(Q\).
Coordinates in a chunk at position \(a\) lie in \([ar,ar+r-1]\) relative to the corner of \(Q\), and the padding adds \(Cr\) on both sides.
Let \(\Omega \) be a unit vector and write \(S=S_\Omega \). Let \(X_i\) be pairwise disjoint regions carrying colors in \(\{ 0,\dots ,N-1\} \), and suppose each \(i\) in a set \(G\) has a buffer \(T_i\) contained in the union of the regions of the other colors, with \(S(X_i\mid T_i)\le S(X_i)/2+\beta \). Then
For each of the \(N\) cyclic rotations of the colors, order the regions by rotated color and apply the chain rule. Every term is at most \(S(X_i)\). For \(i\in G\), in the rotation putting the color of \(i\) last, all of \(T_i\) precedes \(X_i\), so the term is at most \(S(X_i\mid T_i)\). Average over the rotations.
With \(C_{\mathrm{pad}},C_{\mathrm{buf}}\) from Theorem 27.119.3, \(P=2C_{\mathrm{pad}}+1\) and \(p=1/(2P^2)\), for every \(M\) and \(r\ge 1\),
Color the children of a safe parent by their chunk positions modulo \(P\). The padding of a child meets only children within \(C_{\mathrm{pad}}\) positions, which have other colors. Children are safe and of size at most \(r\), and interior children satisfy the hypotheses of Theorem 27.119.3. At most \(2PM\) children are not interior. Apply Theorem 27.121.5 with \(N=P^2\).
For fixed \(q\ge 1\), \(R\), \(J\ge 0\) and \(\Delta {\gt}0\), there are constants \(C\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), such that for every integer \(D_0{\gt}2R+10\) and every integer \(r\ge 1\), \(F_{\mathrm{box},D_0}(r)\le Cr^{1+e_0}\). This is [ 247 , Proposition 3.3 ] .
First take the safety parameter \(2R+11\). Choose \(M=8(C_{\mathrm{pad}}+1)\), so \(2PM\le M^2/2\) and the coefficient is at most \(\lambda =(1-p/2)M^2\), with \(M{\lt}\lambda {\lt}M^2\). Iterating the recurrence at powers of \(M\) and using monotonicity gives \(F_{\mathrm{box},2R+11}(r)\le Cr^{\log _M\lambda }\), with \(1{\lt}\log _M\lambda {\lt}2\). Theorem 27.121.3 gives the same bound at every integer safety parameter \(D_0{\gt}2R+10\).
Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(A\subseteq \Lambda \), and let \(Z\) be the endpoints of its crossing edges. Let \(D_0,j\in \mathbb N\), and let \(Q,Q'\) be integer rectangles, each a product of two finite nonempty integer intervals. Suppose that every point of \(Q\) has sup-norm distance greater than \(D_0\operatorname {size}(Q)\) from every point of \(Z\), and that
Then every point of \(Q'\) has sup-norm distance greater than \(D_0\operatorname {size}(Q')\) from every point of \(Z\). Here the size counts sites along the longer axis and \(Q^{+j}=Q+([-j,j]^2\cap \mathbb Z^2)\). This is the clearance step in [ 247 , proof of Proposition 9.5, lines 723–729 ] .
Suppose that \(p\in Q'\) and \(z\in Z\) satisfy \(\| p-z\| _\infty \le D_0\operatorname {size}(Q')\). The containment and radius inequality give
Thus some \(v\in Q\) satisfies \(\| v-z\| _\infty \le D_0\operatorname {size}(Q)\), contrary to the assumed safety of \(Q\).
Fix a local dimension \(q\ge 1\), an interaction range \(R\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), such that the following holds simultaneously for every integer \(D_0{\gt}2R+10\). Let \(\Lambda \) be any finite induced square-lattice domain, let \(H\) be a Hamiltonian on \(\Lambda \) as in Definition 27.37.2, and let \(\Omega \) be its normalized ground vector at energy \(E_0\) with the full-system projector gap \(\Delta \). For every cut \(A\subseteq \Lambda \) and every safe integer rectangle \(Q\),
Here \(Q\) is a product of two finite nonempty integer intervals, \(\operatorname {size}(Q)\) is its larger number of sites along an axis, and safety means that every point of \(Q\) has sup-norm distance greater than \(D_0\operatorname {size}(Q)\) from every endpoint of a cut edge. The physical region \(A\cap Q\) includes only sites of \(\Lambda \). Empty cuts, empty physical intersections, disconnected domains, and domains with holes are permitted. This is the pointwise consequence of the safe-box supremum and [ 247 , Proposition 3.3 ] at its initial exponent.
The initial safe-box estimate supplies \(C\) and \(e_0\in (0,1)\) uniformly over all safety parameters \(D_0{\gt}2R+10\). At the admissible threshold \(D_0=2R+11\) and scale \(r=1\), its nonnegative supremum satisfies
Thus the same constant is nonnegative. Every integer rectangle has \(\operatorname {size}(Q)\ge 1\). By the definition of the safe-box supremum and the initial estimate,
which proves (806). The constant and exponent were chosen before the safety parameter, domain, Hamiltonian, energy, ground vector, cut, and rectangle.
27.122 Nested square patches
Fix a finite induced lattice domain \(\Lambda \subset \mathbb Z^2\) and a centre \(c\in \mathbb R^2\). The square patches in the proof of [ 248 , Proposition 4.1 ] are samples of closed ambient squares. Intersecting with \(\Lambda \) permits clipping at the physical boundary and does not add edges to the induced graph.
For \(r\in \mathbb R\), define
Thus \(x\in K(c,r)\) exactly when \(x\in \Lambda \) and \(|x_1-c_1|\le r\), \(|x_2-c_2|\le r\). Sites on the square boundary are included. Equivalently, each integer coordinate lies between \(\lceil c_a-r\rceil \) and \(\lfloor c_a+r\rfloor \), for \(a=1,2\). The sample is empty for \(r{\lt}0\) and increases with \(r\).
For a lattice point \(x\), set \(\rho (x)=\max \{ |x_1-c_1|,|x_2-c_2|\} \). Then \(K(c,r)=\{ x\in \Lambda :\rho (x)\le r\} \).
Adjacent sites \(x,y\) satisfy \(|x_a-y_a|\le 1\) for \(a=1,2\) and \(|\rho (x)-\rho (y)|\le 1\). In particular, if \(x\in K(c,r)\), then \(y\in K(c,r+1)\). For any cut \(A\), an unordered pair belongs to \(\partial _\Lambda A\) precisely when its sites are adjacent and exactly one belongs to \(A\). Consequently, for adjacent \(x,y\),
The interval is closed at the smaller radial distance and open at the larger radial distance.
One coordinate of adjacent sites differs by one and the other is equal, so \(|x_a-y_a|\le 1\). For \(a=1,2\) the triangle inequality gives
hence \(\rho (y)\le \rho (x)+1\); interchanging \(x\) and \(y\) gives \(|\rho (x)-\rho (y)|\le 1\). An unordered pair crosses \(A\) exactly when
For \(A=K(c,r)\), membership of a site \(z\) is \(\rho (z)\le r\), and the disjunction becomes \((\rho (x)\le r{\lt}\rho (y))\vee (\rho (y)\le r{\lt}\rho (x))\), which is (810).
If \(r\ge 0\), then \(|K(c,r)|\le (2r+1)^2\). If also \(r\le 4\), then \(|K(c,r)|\le 81\).
Each coordinate interval has at most \(2r+1\) integer points. The sample is a subset of their Cartesian product. When \(r\le 4\), its cardinality is at most \(9^2=81\).
For every \(r\ge 0\), the number of unordered induced edges crossing the sampled square satisfies \(|\partial _\Lambda K(c,r)|\le 8r+4\).
A horizontal crossing edge lies on an integer row meeting \([c_2-r,c_2+r]\). On each such row there are at most two crossing edges, one at each end of the integer interval inside the square. An interval of length \(2r\) contains at most \(2r+1\) integers. The same argument applies to vertical edges. Hence the total is at most \(2(2r+1)+2(2r+1)=8r+4\). Removing sites outside \(\Lambda \) cannot create an induced crossing edge.
For \(u\ge 0\), put \(m=\lfloor u/2\rfloor +1\). For each integer \(0\le j{\lt}m\), define
The family contains \(m\ge 1\) patches and \(m{\gt}u/2\). Every selected radius satisfies \(u\le r_j\le 2u\) and every selected patch satisfies \(|\partial _\Lambda K_j|\le 16u+4\). For indices \(i{\lt}j\), one has \(r_i+2\le r_j\). For \(i\le j\), one has \(K_i\subseteq K_j\); clipping can make these inclusions equalities.
The floor inequalities give \(u/2{\lt}m\) and \(j\le \lfloor u/2\rfloor \le u/2\). Thus \(u\le u+2j\le 2u\), and the crossing-edge estimate applies. Increasing an integer index by at least one increases the radius by at least two; inclusion follows from the closed coordinate inequalities.
Suppose \(e\in \partial _\Lambda K(c,r)\). If \(s+1\le r\), then the two sites of \(e\) are disjoint from \(K(c,s)\). If \(r+1\le t\), then both sites belong to \(K(c,t)\).
Write \(e=\{ x,y\} \) with \(x\in K(c,r)\) and \(y\notin K(c,r)\). If either site belonged to \(K(c,s)\), the nearest-neighbour enlargement would put \(y\) in \(K(c,s+1)\subseteq K(c,r)\), a contradiction. Conversely, \(x\in K(c,r)\) implies \(y\in K(c,r+1)\), so both sites belong to every \(K(c,t)\) with \(r+1\le t\).
Let \(e=\{ x,y\} \) be an induced nearest-neighbour edge. If \(e\in \partial _\Lambda K_j\), then, for every \(i{\lt}j\), \(\{ x,y\} \cap K_i=\varnothing \), and, for every \(j{\lt}k\), \(\{ x,y\} \subseteq K_k\). Consequently, an edge crosses at most one patch in the family, and the crossing-edge sets of distinct patches are disjoint.
Earlier and later radii differ from \(r_j\) by at least two, so the separated-radius support statements apply. If an edge crossed two patches, order their distinct indices. Crossing the earlier patch places both sites in the later patch, contradicting crossing of the later one.
The following finite example uses \(\Lambda =\{ 0,\ldots ,10\} \times \{ 0,1,2\} \), \(c=(5/4,3/2)\), and \(u=4\). Its three clipped samples are \(K_0=\{ 0,\ldots ,5\} \times \{ 0,1,2\} \), \(K_1=\{ 0,\ldots ,7\} \times \{ 0,1,2\} \), and \(K_2=\{ 0,\ldots ,9\} \times \{ 0,1,2\} \). The marked edge \(e=\{ (7,1),(8,1)\} \) crosses \(K_1\); it is disjoint from \(K_0\) and contained in \(K_2\). The rounded outlines enclose the clipped sets of physical sites; they are not the boundaries of the ambient real squares.
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Let \(u\ge 4\), \(\kappa \in \mathbb R\), and assign each selected patch the same weight \(t_j=\kappa /m\). Then
Since \(m{\gt}u/2\) and \(u\ge 4\), one has \(16u+4\le 36m\). The constant weights satisfy \(\sum _{j=0}^{m-1}t_j^2=\kappa ^2/m\), which gives (812). Bounding each boundary size by \(16u+4\) gives
27.122.1 Dyadic counts for rectangle shells
The rectangle shell covering in [ 247 , proof of Proposition 9.5, lines 717–729 ] uses the maximal contained squares of Definition 27.49.1.3. The following estimates give the total side lengths at and below the cap, uniformly in the aspect ratio of the rectangle.
Let \(Q\) be a nonempty integer rectangle and let \(j\in \mathbb N\). Then
Write \(Q=[x_0,x_1]\times [y_0,y_1]\cap \mathbb Z^2\). Its side lengths are \(x_1-x_0+1\) and \(y_1-y_0+1\). Dilation adds \(2j\) to each, so their maximum also increases by \(2j\).
Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,L,K\in \mathbb N\) satisfy \(j\le L\le s\) and \(L{\lt}2^{K+1}\). Put \(S_j=Q^{+j}\setminus Q\) and \(N_K=\# \{ (r,z)\in \mathcal P_K(S_j):r=K\} \). Then
The estimate includes \(j=0\) and \(K=0\).
The boundary-layer count in the proof of Theorem 27.119.2 gives
Set \(v=2^K\). The exact capped partition and \(L{\lt}2v\) imply
Cancel \(v{\gt}0\).
Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,K,k\in \mathbb N\) satisfy \(j\le s\), \(k{\lt}K\) and \(2^{k+1}\le s\). Put \(S_j=Q^{+j}\setminus Q\) and \(N_k=\# \{ (r,z)\in \mathcal P_K(S_j):r=k\} \). Then
Set \(u=2^{k+1}\). The mixed-square count for a difference, the rectangle count and Theorem 27.122.1.1 give
Each selected square below the cap has a mixed parent, and a parent has four children. Hence
27.123 Weighted dyadic covering of rectangle shells
Let \(S\subseteq \mathbb Z^2\) be finite, let \(K\in \mathbb N\), and let \(f:\mathbb N\to \mathbb R\) be arbitrary. Then
This is the finite summation by scale used in the covering argument of [ 247 , Lemma 9.4, lines 656–664 ] . The identity includes \(S=\varnothing \) and \(K=0\) and imposes no sign condition on \(f\).
Membership in \(\mathcal P_K(S)\) implies \(r\le K\). Group its finitely many terms according to \(r\). On each group with \(r=k\), the summand is the constant \(f(k)\), so
Summing over \(k=0,\ldots ,K\) gives the identity.
Let \(S\subseteq \mathbb Z^2\) be finite and let \(K,L\in \mathbb N\) satisfy \(2^K\le L\). Let \(e,A,B\in \mathbb R\) with \(e{\gt}0\) and \(B\ge 0\). Write \(N_k=\# \{ (r,z)\in \mathcal P_K(S):r=k\} \). Suppose \(2^K N_K\le A\) and \(2^k N_k\le B\) for every \(0\le k{\lt}K\). Then
This is the finite moment estimate used in the covering argument of [ 247 , Lemma 9.4, lines 656–664 ] . It includes \(S=\varnothing \) and \(K=0\).
Group the selected squares by scale and apply the finite weighted dyadic-sum estimate:
The cap bound gives \(A\ge 0\), while \(2^e-1{\gt}0\). Hence the coefficient is nonnegative, and the last inequality follows from \(e{\gt}0\) and \(2^K\le L\).
Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,L,K\in \mathbb {N}\) satisfy \(j\le L\le s\) and \(2^K\le L{\lt}2^{K+1}\), and let \(e{\gt}0\). Set \(S_j=Q^{+j}\setminus Q\), with both rectangles viewed as sets of lattice sites. Then the capped maximal-contained dyadic partition satisfies
This is the geometric weighted covering estimate used in [ 247 , proof of Proposition 9.5, lines 717–729 ] .
Let \(N_k\) denote the number of selected squares of scale \(k\). The rectangle-shell counting estimates give the following bounds. The second holds for every \(k\) with \(0\le k{\lt}K\).
For the second estimate, the required bounds are \(j\le L\le s\) and \(2^{k+1}\le 2^K\le L\le s\).
Apply Theorem 27.123.2 with \(A=24s\), \(B=64s\) and the cap bounds \(2^K\le L{\lt}2^{K+1}\), and factor out \(s\). If \(j=0\), the shell and its partition are empty. If \(K=0\), the lower-scale sum is empty and the cap inequalities force \(L=1\). The argument also applies to thin rectangles and negative coordinates.
27.124 Entropy of safe rectangle shells
For \(k\in \mathbb {N}\) and \(z\in \mathbb {Z}^2\), let
This is the integer rectangle underlying the dyadic square in the covering argument of [ 247 , Lemma 9.4, lines 641–649 ] .
For every \(k\in \mathbb {N}\) and \(z\in \mathbb {Z}^2\), the lattice points of \(R_{k,z}\) are exactly \(C_{k,z}\), and \(\operatorname {size}(R_{k,z})=2^k\). For a finite physical domain \(\Lambda \) and a region \(A\subseteq \Lambda \), its physical region is \(A\cap C_{k,z}\).
The half-open coordinate interval from \(2^kz_i\) to \(2^k(z_i+1)\) contains exactly the integers from \(2^kz_i\) to \(2^kz_i+2^k-1\). Each coordinate interval contains \(2^k\) sites. Intersecting with \(A\) gives the equality of physical regions.
Fix a finite lattice domain \(\Lambda \), a region \(A\subseteq \Lambda \), a finite lattice set \(S\), and \(K\in \mathbb {N}\). The physical regions \(A\cap R_{k,z}\) for \((k,z)\in \mathcal P_K(S)\) are pairwise disjoint, and their union is \(A\cap S\). For a unit physical vector \(\Omega \) on \(\Lambda \) with local dimension \(q\in \mathbb {N}\), one has
This is the physical summation step of [ 247 , Lemma 9.4, lines 651–664 ] .
The selected ambient squares partition \(S\). Intersecting them with \(A\) preserves pairwise disjointness and gives union \(A\cap S\). Apply finite disjoint-union subadditivity, starting with the empty region, whose entropy is zero. This also includes \(S=\varnothing \).
Fix a finite lattice domain \(\Lambda \), a physical region \(A\), and \(D_0\in \mathbb {N}\). Let \(Q\) be a safe integer rectangle and put \(s=\operatorname {size}(Q)\). Suppose \(j,L,K\in \mathbb {N}\) satisfy \(j\le L\), \(2^K\le L\), and
Then each selected square of \(\mathcal P_K(Q^{+j}\setminus Q)\), interpreted as its integer rectangle, is safe. This is a sufficient clearance condition for the covering argument in [ 247 , proof of Proposition 9.5, lines 723–729 ] .
A selected square \(R_{k,z}\) is contained in \(Q^{+j}\) and has \(k\le K\) and \(\operatorname {size}(R_{k,z})=2^k\le L\). Hence
Safety inside a dilated parent rectangle gives the result.
Fix a finite lattice domain \(\Lambda \) with local dimension \(q\), a unit physical vector \(\Omega \), a region \(A\), and \(D_0\in \mathbb {N}\). Let \(S\) be a finite set of lattice sites, let \(K\in \mathbb {N}\), and let \(e,C\in \mathbb {R}\). Suppose that each square selected by \(\mathcal P_K(S)\) is safe, and that every safe integer rectangle \(R\) satisfies
Then
This is the conditional summation step in [ 247 , proof of Lemma 9.4, lines 651–664 ] .
The physical regions \(A\cap R_{k,z}\) are pairwise disjoint and have union \(A\cap S\). Each \(R_{k,z}\) is an integer rectangle of size \(2^k\). Subadditivity and the assumed estimate give
Fix a finite lattice domain \(\Lambda \) with local dimension \(q\), a unit physical vector \(\Omega \), a region \(A\), and \(D_0\in \mathbb {N}\). Let \(e{\gt}0\) and \(C\ge 0\), and assume that every safe integer rectangle \(R\) satisfies
Let \(Q\) be safe and put \(s=\operatorname {size}(Q)\). For \(j,L,K\in \mathbb {N}\) with \(j\le L\le s\), \(2^K\le L{\lt}2^{K+1}\), and \(D_0L+L\le D_0s\), one has
This is the conditional shell entropy estimate used in [ 247 , proof of Proposition 9.5, lines 717–729 ] .
The actual selected squares give disjoint physical rectangle regions whose union is \(A\cap (Q^{+j}\setminus Q)\). Each is safe. Subadditivity and the assumed safe-box estimate give
The last step multiplies the weighted covering estimate by \(C\ge 0\).
27.125 Initial-exponent entropy of safe rectangle shells
Fix a local dimension \(q\in \mathbb N\) with \(q\ge 1\), an interaction range \(R\in \mathbb N\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), with the following property simultaneously for every integer safety parameter \(D_0{\gt}2R+10\). Let \(\Lambda \) be any finite induced square-lattice domain, let \(H\) be an admissible Hamiltonian on \(\Lambda \) with range \(R\) and term norm bound \(J\), and let \(\Omega \) be its normalized ground vector at energy \(E_0\) with full-system projector gap \(\Delta \). Let \(A\subseteq \Lambda \) be any cut, let \(Q\) be any safe integer rectangle, and put \(s=\operatorname {size}(Q)\). For all \(j,L\in \mathbb N\) satisfying \(j\le L\le s\) and \(D_0L+L\le D_0s\), one has
The constant and exponent are chosen before the safety parameter, domain, Hamiltonian, energy, original ground vector, cut, parent rectangle and all shell scales. This is the initial-exponent consequence of [ 247 , Proposition 3.3, lines 590–604 ] for the rectangle-shell entropy input in the proof of Proposition 9.5, lines 717–729.
Choose \(C\ge 0\) and \(e_0\in (0,1)\) from Theorem 27.121.9. For every safety parameter and physical instance in the statement, that same ground vector satisfies, for every safe integer rectangle \(P\),
If \(L=0\), then \(j=0\), the shell is empty, and both sides of the desired inequality vanish. If \(L{\gt}0\), choose \(K=\lfloor \log _2L\rfloor \), so that \(2^K\le L{\lt}2^{K+1}\). Apply Theorem 27.124.6 with this dyadic cap, the pointwise estimate, the normalization of \(\Omega \), and the stated clearance conditions. It gives the asserted bound with the same \(C\) and \(e_0\).
27.125.1 Initial entropy estimate for rounded collars
The rounded collar scale in [ 247 , Section 9, lines 32–41 ] allows the initial rectangle estimate to be expressed as a strict reduction of its exponent on the shell. The following statements provide this initial input to the proof of Proposition 9.5, lines 693–729.
Let \(C_2,\eta ,e\ge 0\) be real numbers, and let \(s\in \mathbb N\) with \(s\ge 1\). Put \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). Then
The ceiling estimate gives \(\lceil C_2s\rceil {\lt}C_2s+1\le (C_2+1)s\). Since both exponents are nonnegative, monotonicity of real powers gives
Fix a local dimension \(q\ge 1\), an interaction range \(R\in \mathbb N\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), with the following property. For each real \(C_2{\gt}0\) there is an integer threshold \(N\ge 1\), chosen before all safety parameters and physical instances, such that the following holds.
Let \(D_0{\gt}2R+10\) be an integer, let \(\Lambda \) be any finite induced square-lattice domain, and let \(H\) be an admissible Hamiltonian on \(\Lambda \) with range \(R\) and term norm bound \(J\). Let \(\Omega \) be its normalized ground vector at energy \(E_0\) with full-system projector gap \(\Delta \). Let \(A\subseteq \Lambda \) be any cut and \(Q\) any safe integer rectangle of size \(s\ge N\). Put \(\ell =(1-e_0)/200000\), \(\eta =1-\ell \), and \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). For every integer \(0\le j\le L_s\), one has
The exponent \(1+\eta e_0\) is strictly smaller than \(1+e_0\). This estimates the shell; the improvement of the safe-box exponent still requires the remaining physical comparison argument.
Choose \(C\) and \(e_0\) from Theorem 27.125.1. The fixed parameter bounds give \(0{\lt}\ell {\lt}1\), and hence \(0{\lt}\eta {\lt}1\). For each \(C_2{\gt}0\), Theorem 27.39.1.3 supplies a threshold \(N\ge 1\) for which \(2L_s\le s\) and \(D_0L_s+L_s\le D_0s\) for every allowed safety parameter. In particular, \(L_s\le s\). The initial shell estimate and the homogeneous bound now give
The same \(C\), \(e_0\), and \(N\) apply to every physical instance in the statement. Finally, \(\ell e_0{\gt}0\) gives \(1+\eta e_0=1+e_0-\ell e_0{\lt}1+e_0\).
27.125.2 Uniform coefficients for rounded shell estimates
The fixed collar exponent in [ 247 , Section 9, lines 693–703 ] remains unchanged while the current entropy exponent decreases. The following auxiliary estimates make the shell coefficient uniform on any interval with a positive lower endpoint. The pointwise entropy input is stated explicitly; these estimates do not establish the physical improvement in Proposition 9.5.
Let \(b,e,e_0,\eta ,C_2\in \mathbb R\) satisfy \(0{\lt}b\le e\le e_0\), \(\eta \ge 0\), and \(C_2\ge 0\). Then
The denominator \(2^b-1\) is positive and is at most \(2^e-1\). Therefore the first factor decreases when \(b\) is replaced by \(e\). Since \(C_2+1\ge 1\) and \(\eta e\le \eta e_0\), the second factor is bounded by \((C_2+1)^{\eta e_0}\). Multiplying these two bounds, whose factors are nonnegative, gives the result.
Fix \(C_2{\gt}0\) and \(e_0\in (0,1)\), and set \(\ell =(1-e_0)/200000\) and \(\eta =1-\ell \). There is an integer \(N\ge 1\), depending only on \(C_2\) and \(e_0\), with the following property. Let \(b{\gt}0\), let \(\Lambda \) be any finite induced square-lattice domain, let \(q\in \mathbb N\), and let \(\Omega \) be a unit vector on its configurations. Let \(D_0\ge 1\) be an integer, let \(A\subseteq \Lambda \) be any cut, and let \(Q\) be any safe integer rectangle with \(s=\operatorname {size}(Q)\ge N\). Suppose that \(b\le e\le e_0\) and that \(C\in \mathbb R\) satisfies, for every safe integer rectangle \(P\),
Put \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). For every integer \(0\le j\le L_s\), one has
The threshold precedes the lower endpoint, current exponent, coefficient and all geometric and physical data. Both entropy estimates concern the same vector. No ground-state assumption is needed for this conditional statement.
The fixed parameter bounds give \(0{\lt}\eta {\lt}1\). Theorem 27.39.1.3 chooses a threshold \(N\ge 1\) for which \(L_s\ge 1\), \(2L_s\le s\) and \(D_0L_s+L_s\le D_0s\) for every allowed safety parameter. In particular, \(L_s\le s\). Choose \(K=\lfloor \log _2L_s\rfloor \).
Applying the pointwise estimate to the parent rectangle and using entropy nonnegativity gives \(0\le Cs^{1+e}\). Since \(s{\gt}0\), this implies \(C\ge 0\). The conditional rectangle-shell estimate and the homogeneous rounded-power bound now yield
The last step uses the uniform coefficient theorem. The choice of threshold depends only on the rounded geometric scale.
27.125.3 Replica limits before large-scale entropy absorption
The proof of Proposition 9.5 in [ 247 , Section 9, lines 755–769 ] takes the replica limit at each fixed physical scale before choosing the large-scale threshold. The following auxiliary scalar statements express that order. The two norm comparisons and the common entropy are explicit inputs; their physical derivation remains separate.
Let \(S,C,\tau ,E,B_u,B_l\in \mathbb R\) satisfy \(S\ge 0\), \(C\ge 0\), \(\tau {\gt}0\) and
Let \(F,d,r_u,r_l,r_e\) be real sequences. Suppose that \(r_u(k),r_l(k),r_e(k)\) tend to zero as \(k\to \infty \), and that for every sufficiently large \(k\),
Then
The entropy \(S\) and all fixed coefficients are independent of \(k\). The two budgets include any substitutions of the typical entropy and auxiliary dimension made before these scalar comparisons.
Since \(C,S\ge 0\) and \(\tau {\gt}0\), the defect bound gives
Combining the two comparisons therefore yields
The right-hand side tends to \(B_u+B_l\), whereas the left-hand side is constant. Passing to the limit gives the same inequality without the three remainders. Finally, \(2-C(\tau +E/\tau )\ge 1\) and \(S\ge 0\) imply the conclusion.
Fix \(e_0\in (0,1)\) and \(C,C_E\in \mathbb R\), and let \(\nu =(1-e_0)/2000\). There is an integer \(N\ge 1\), depending only on these fixed coefficients, with the following property. For every integer \(n\ge N\), put
Then
This is the numerical threshold used after the fixed-scale replica limit. It does not require a replica convergence rate uniform in \(n\).
The fixed parameter bounds give \(\nu {\gt}0\). At positive integer scales, the quotient of powers gives
Both \(\tau _n\) and the last expression tend to zero. Choose one threshold for the bounds \(\tau _n{\lt}1/2\) and \(C(1+C_E)n^{-\nu /4}{\lt}1\), and enlarge it to at least one. Positivity of \(n\) gives \(\tau _n{\gt}0\) and the asserted inequalities.
Fix a unit state \(\Omega _t\) on a finite subset of the square lattice, a region \(X\), and a spectral set \(E\) for its reduced density on \(X\). Write \(S=S_{\Omega _t}(X)\). Suppose that the selected mass \(z\) is positive and that every selected eigenvalue \(p_i\) satisfies
Let \(d_*=|E|\), let \(\psi \) be the normalized selected Schmidt vector, with the selected Schmidt indices enumerated in its auxiliary register, and let \(S'\) be the entropy of that register. Put \(\eta =w-\log z\). Fix \(C\ge 0\), \(0{\lt}\tau {\lt}1/2\), \(a{\gt}0\), \(E_{\rm def}\ge 0\) and real budgets \(B_{\rm sh},B_{\rm exc}\) such that
Let \(F,d,r_u,r_l,r_e\) be real sequences. Suppose that \(r_u(k),r_l(k),r_e(k)\) tend to zero and that, for all sufficiently large \(k\),
Then
This is a conditional consequence of the rough comparisons, before the shell and mismatch budgets are estimated in [ 247 , Proposition 9.5 ] . It retains the same physical state and selected spectrum; the physical derivation of the comparisons is a separate result.
The selected auxiliary density has nonnegative entropy, so \(\max \{ 0,2S'\} =2S'\). Apply the fixed-scale replica absorption theorem to \(S'\), with coefficient \(2C\) and budgets
This gives
Only now use the selected-spectrum bounds \(|S'-S|\le \eta \) and \(|\log d_*-S|\le \eta \). They imply \(\eta \ge 0\) and \(S\le R+(5+4C\tau )\eta \). The smallness condition and \(E_{\rm def}\ge 0\) give \(2C\tau \le 1\), hence the displayed bound.
27.126 Geometry of the dyadic distribution protocol
This section treats the geometric part of [ 248 , Section 7 ] , September 24, 2026, 06-geometry.tex: the bands along the edges of a repainted dyadic square, the angular separation of [ 248 , Lemma 7.2 ] (lines 254–319), the point-treatment estimates (lines 357–415), the absence of changed true vertices in the bulk operations and the tags (lines 375–398 and 610–631), and parts of the geometry of the small-patch covers, of the direct repaintings and of the hole resizings (lines 341–351 and 447–533), and the counts of the padded dyadic hierarchy behind items 1–3 of [ 248 , Proposition 7.1 ] (lines 12–53 and 633–652). The protocol of Proposition 7.1 itself, whose changes are implemented by the information estimate and the contraction gates of the earlier sections, is not treated here.
The clearance estimates are first proved for every guide with the properties that the proof of Lemma 7.2 uses: the normal word of the operation on the band \(\{ -8{\lt}x{\lt}2\} \) and no change off it, the label \(A\) on the open square, and the label \(C\) on the exterior band \(\{ -8{\lt}x{\lt}0\} \). The schedule of guides is then constructed, these properties are derived from the disjointness of the bands, and the estimates are moved to every square of the hierarchy. This gives Theorem 27.126.13, which is Lemma 7.2 for the labellings that the schedule specifies; Lemma 7.2 for the guides sampled under the source’s tie convention is Theorem 27.126.20. A guide of the source is a partition of the plane into open chambers, with labels on boundaries sampled after one small generic displacement; here a guide is a labelling of the whole plane, and two labellings that agree off finitely many interface curves represent the same guide. The source’s sampled guides are identified with readings of these labellings in Lemma 27.126.19. The earlier scope restriction and this convention are recorded in [ 62 ] . Distances are sup distances in the plane, and labels are elements of an arbitrary set; the labels \(A\), \(B\), \(C\) below are permitted to coincide.
Let \(S=[0,n]^2\). For an edge of \(S\) let \(s\) be the coordinate parallel to the edge, measured from one endpoint, and \(d\) the normal coordinate, positive into \(S\). Put
The band \(\{ \alpha {\lt}x{\lt}\beta \} \) is the set of points with \(0{\lt}s{\lt}n\) and \(\alpha w(s){\lt}d{\lt}\beta w(s)\); its closed version has \(0\le s\le n\) and \(\alpha w(s)\le d\le \beta w(s)\). The lens of the edge is the band \(Y=\{ -7{\lt}x{\lt}-7/2\} \), and the exterior neighboring square is \(\{ 0{\lt}s{\lt}n,\ -n{\lt}d{\lt}0\} \). For a point \(y\), \(d_{\mathcal V}(y)\) is its distance from the two endpoints of the edge, or from the four corners of \(S\).
The bands \(\{ -8{\lt}x{\lt}2\} \) along distinct edges of \(S\) are disjoint, and the band \(\{ -8{\lt}x{\lt}0\} \) lies in the exterior neighboring square of its edge. Two points of a closed band with \(-8\le \alpha \le \beta \le 8\) are at distance at most \(n\). Consequently the closure of the changed set of each elementary edge birth or death, and the closure of the central region of Theorem 27.126.6, have diameter at most \(n\).
Write \(w\le s/1000\) and \(w\le (n-s)/1000\). For opposite edges the bands lie within \(n/250\) of their edges. For adjacent edges a common point at distances \(r_1,r_2{\gt}0\) from the two edge lines would satisfy \(r_1{\lt}0.002\, r_2\) and \(r_2{\lt}0.002\, r_1\). On \(\{ -8{\lt}x{\lt}0\} \) one has \(0{\gt}d{\gt}-8w\ge -n/250\). The diameter bound follows from \(|d|\le 8w\le n/250\) in each closed band. The changed set of an elementary operation lies in the closed band of its changed interval, and the central region lies in \(S\).
A normal word lists labels in increasing \(x\) separated by interfaces; at an interface curve the label on its right is taken, whereas the source samples after one generic displacement; the two conventions differ only on the interface curves. An edge operation starts from the main word \(C\mid A\mid B\) with interfaces \(0,1\). On an auxiliary sheet, uniform at \(C\), the births of \(B\) in \((-6,-1)\), of \(A\) in \((-5,-2)\) and of \(C\) in \((-4,-3)\) give
After the lens exchange the main word is \(C\mid B\mid A\mid C\mid A\mid B\) at \(-6,-5,-4,0,1\). The \(C\) band \((-4,0)\) then dies into \(A\), the \(A\) band \((-5,1)\) dies into \(B\), \(C\) is born in \((-3,0)\), and the \(B\) band \((-6,-3)\) dies into \(C\), leaving \(C\mid B\) at \(0\). Each of these seven births and deaths changes its word only on a closed interval, and its surrounding word (the word before a birth, after a death) carries one label on a strictly larger open interval inside \((-8,2)\), with margins at least \(1/2\). The seven normal words are drawn below at one fixed \(0{\lt}s{\lt}n\) over \(-8\le x\le 2\): row (1) is the starting main word, row (2) the auxiliary word (878) with the lens \(Y\), row (3) the main word after the exchange, and rows (4)–(7) the main words after the death of \(C\) on \((-4,0)\), the death of \(A\) on \((-5,1)\), the birth of \(C\) on \((-3,0)\) and the death of \(B\) on \((-6,-3)\).
![\begin{tenkz}[rows={wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire,wire}, cols=20, bonds=none, size=s, metrics=compact]
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\tndeclare{species}{lens}{hue=source:red}
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\tnmark[form=label]{0.5 w of bandTick0}{$-8$}
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\tnmark[form=label]{0.5 e of bandTick2}{$-6$}
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\tnmark[form=label]{0.5 e of bandTick4}{$-4$}
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\tn[at=(1,20), name=bandTick10, skin=none]{}
\tnmark[form=label]{0.5 e of bandTick10}{$2$}
% main, start: C \mid A \mid B at 0,1
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\tn[at=(3,2), skin=none]{}
\tn[at=(3,3), skin=none]{}
\tn[at=(3,4), skin=none]{}
\tn[at=(3,5), skin=none]{}
\tn[at=(3,6), skin=none]{}
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\tn[at=(3,8), skin=none]{}
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\tn[at=(3,12), skin=none]{}
\tn[at=(3,13), skin=none]{}
\tn[at=(3,14), skin=none]{}
\tn[at=(3,15), skin=none]{}
\tn[at=(3,16), skin=none]{}
\tn[at=(3,17), skin=none]{}
\tn[at=(3,18), skin=none]{}
\tn[at=(3,19), skin=none]{}
\tn[at=(3,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(3,1) .. (3,16)}{$C$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(3,17) .. (3,18)}{$A$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(3,19) .. (3,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (3,1)}{(1)}
% auxiliary: C \mid B \mid A \mid C \mid A \mid B \mid C at -6,-5,-4,-3,-2,-1
\tn[at=(5,1), skin=none]{}
\tn[at=(5,2), skin=none]{}
\tn[at=(5,3), skin=none]{}
\tn[at=(5,4), skin=none]{}
\tn[at=(5,5), skin=none]{}
\tn[at=(5,6), skin=none]{}
\tn[at=(5,7), skin=none]{}
\tn[at=(5,8), skin=none]{}
\tn[at=(5,9), skin=none]{}
\tn[at=(5,10), skin=none]{}
\tn[at=(5,11), skin=none]{}
\tn[at=(5,12), skin=none]{}
\tn[at=(5,13), skin=none]{}
\tn[at=(5,14), skin=none]{}
\tn[at=(5,15), skin=none]{}
\tn[at=(5,16), skin=none]{}
\tn[at=(5,17), skin=none]{}
\tn[at=(5,18), skin=none]{}
\tn[at=(5,19), skin=none]{}
\tn[at=(5,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(5,1) .. (5,4)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(5,5) .. (5,6)}{$B$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(5,7) .. (5,8)}{$A$}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(5,9) .. (5,10)}{$C$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(5,11) .. (5,12)}{$A$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(5,13) .. (5,14)}{$B$}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(5,15) .. (5,20)}{$C$}
\tnmark[form=label, label pos=w]{0.5 w of (5,1)}{(2)}
% main, exchanged: C \mid B \mid A \mid C \mid A \mid B at -6,-5,-4,0,1
\tn[at=(8,1), skin=none]{}
\tn[at=(8,2), skin=none]{}
\tn[at=(8,3), skin=none]{}
\tn[at=(8,4), skin=none]{}
\tn[at=(8,5), skin=none]{}
\tn[at=(8,6), skin=none]{}
\tn[at=(8,7), skin=none]{}
\tn[at=(8,8), skin=none]{}
\tn[at=(8,9), skin=none]{}
\tn[at=(8,10), skin=none]{}
\tn[at=(8,11), skin=none]{}
\tn[at=(8,12), skin=none]{}
\tn[at=(8,13), skin=none]{}
\tn[at=(8,14), skin=none]{}
\tn[at=(8,15), skin=none]{}
\tn[at=(8,16), skin=none]{}
\tn[at=(8,17), skin=none]{}
\tn[at=(8,18), skin=none]{}
\tn[at=(8,19), skin=none]{}
\tn[at=(8,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(8,1) .. (8,4)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(8,5) .. (8,6)}{$B$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(8,7) .. (8,8)}{$A$}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(8,9) .. (8,16)}{$C$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(8,17) .. (8,18)}{$A$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(8,19) .. (8,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (8,1)}{(3)}
% death of $C$: C \mid B \mid A \mid B at -6,-5,1
\tn[at=(10,1), skin=none]{}
\tn[at=(10,2), skin=none]{}
\tn[at=(10,3), skin=none]{}
\tn[at=(10,4), skin=none]{}
\tn[at=(10,5), skin=none]{}
\tn[at=(10,6), skin=none]{}
\tn[at=(10,7), skin=none]{}
\tn[at=(10,8), skin=none]{}
\tn[at=(10,9), skin=none]{}
\tn[at=(10,10), skin=none]{}
\tn[at=(10,11), skin=none]{}
\tn[at=(10,12), skin=none]{}
\tn[at=(10,13), skin=none]{}
\tn[at=(10,14), skin=none]{}
\tn[at=(10,15), skin=none]{}
\tn[at=(10,16), skin=none]{}
\tn[at=(10,17), skin=none]{}
\tn[at=(10,18), skin=none]{}
\tn[at=(10,19), skin=none]{}
\tn[at=(10,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(10,1) .. (10,4)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(10,5) .. (10,6)}{$B$}
\tnmark[form=enclosure, species=bandA, tint, label pos=n]{(10,7) .. (10,18)}{$A$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(10,19) .. (10,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (10,1)}{(4)}
% death of $A$: C \mid B at -6
\tn[at=(12,1), skin=none]{}
\tn[at=(12,2), skin=none]{}
\tn[at=(12,3), skin=none]{}
\tn[at=(12,4), skin=none]{}
\tn[at=(12,5), skin=none]{}
\tn[at=(12,6), skin=none]{}
\tn[at=(12,7), skin=none]{}
\tn[at=(12,8), skin=none]{}
\tn[at=(12,9), skin=none]{}
\tn[at=(12,10), skin=none]{}
\tn[at=(12,11), skin=none]{}
\tn[at=(12,12), skin=none]{}
\tn[at=(12,13), skin=none]{}
\tn[at=(12,14), skin=none]{}
\tn[at=(12,15), skin=none]{}
\tn[at=(12,16), skin=none]{}
\tn[at=(12,17), skin=none]{}
\tn[at=(12,18), skin=none]{}
\tn[at=(12,19), skin=none]{}
\tn[at=(12,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(12,1) .. (12,4)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(12,5) .. (12,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (12,1)}{(5)}
% birth of $C$: C \mid B \mid C \mid B at -6,-3,0
\tn[at=(14,1), skin=none]{}
\tn[at=(14,2), skin=none]{}
\tn[at=(14,3), skin=none]{}
\tn[at=(14,4), skin=none]{}
\tn[at=(14,5), skin=none]{}
\tn[at=(14,6), skin=none]{}
\tn[at=(14,7), skin=none]{}
\tn[at=(14,8), skin=none]{}
\tn[at=(14,9), skin=none]{}
\tn[at=(14,10), skin=none]{}
\tn[at=(14,11), skin=none]{}
\tn[at=(14,12), skin=none]{}
\tn[at=(14,13), skin=none]{}
\tn[at=(14,14), skin=none]{}
\tn[at=(14,15), skin=none]{}
\tn[at=(14,16), skin=none]{}
\tn[at=(14,17), skin=none]{}
\tn[at=(14,18), skin=none]{}
\tn[at=(14,19), skin=none]{}
\tn[at=(14,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(14,1) .. (14,4)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(14,5) .. (14,10)}{$B$}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(14,11) .. (14,16)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(14,17) .. (14,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (14,1)}{(6)}
% death of $B$: C \mid B at 0
\tn[at=(16,1), skin=none]{}
\tn[at=(16,2), skin=none]{}
\tn[at=(16,3), skin=none]{}
\tn[at=(16,4), skin=none]{}
\tn[at=(16,5), skin=none]{}
\tn[at=(16,6), skin=none]{}
\tn[at=(16,7), skin=none]{}
\tn[at=(16,8), skin=none]{}
\tn[at=(16,9), skin=none]{}
\tn[at=(16,10), skin=none]{}
\tn[at=(16,11), skin=none]{}
\tn[at=(16,12), skin=none]{}
\tn[at=(16,13), skin=none]{}
\tn[at=(16,14), skin=none]{}
\tn[at=(16,15), skin=none]{}
\tn[at=(16,16), skin=none]{}
\tn[at=(16,17), skin=none]{}
\tn[at=(16,18), skin=none]{}
\tn[at=(16,19), skin=none]{}
\tn[at=(16,20), skin=none]{}
\tnmark[form=enclosure, species=bandC, tint, label pos=n]{(16,1) .. (16,16)}{$C$}
\tnmark[form=enclosure, species=bandB, tint, label pos=n]{(16,17) .. (16,20)}{$B$}
\tnmark[form=label, label pos=w]{0.5 w of (16,1)}{(7)}
\tnmark[form=enclosure, species=lens, label pos=s]{(5,3) .. (5,9)}{$Y$}
\end{tenkz}](tenkz_svg/tenkz-877952425adbb54f.svg)
Replacing the starting main word by the auxiliary word (878) on \(-7{\lt}x{\lt}-7/2\) gives the main word \(C\mid B\mid A\mid C\mid A\mid B\) at \(-6,-5,-4,0,1\), at every value of \(x\).
Both thresholds \(-7\) and \(-7/2\) lie in \(C\) intervals of both words; compare the two words on each interval.
Let \(a_0=1/5000\), \(\alpha \le \beta \), \(\alpha '+1/2\le \alpha \), \(\beta +1/2\le \beta '\) and \(-8\le \alpha '\), \(\beta '\le 8\). If \(y\) lies in the closed band \(\{ \alpha \le x\le \beta \} \) and \(|y-z|{\lt}a_0\min (n,d_{\mathcal V}(y))\), then \(z\) lies in the open band \(\{ \alpha '{\lt}x{\lt}\beta '\} \).
Put \(m=\min (s,n-s)\) at \(y\), so \(w=m/1000\) and \(|d|\le m\); hence \(d_{\mathcal V}(y)\le m\). If \(|y-z|{\lt}m/5000\), the parallel and normal coordinates move by less than \(m/5000\), and \(w\) is \(10^{-3}\)-Lipschitz, so \(|\beta '(w(s_z)-w(s_y))|\le 8m/(5\cdot 10^{6})\). Then \(d_z{\lt}\beta w(s_y)+m/5000\le \beta ' w(s_z)-m/2000+m/5000+8m/(5\cdot 10^6)\), and symmetrically at \(\alpha '\).
Let \(R\) be the set of points of \((0,n)^2\) with \(x{\gt}1\) for every edge, and let \(f\) be a labelling of the plane equal to \(A\) on \((0,n)^2\). For every \(y\) in the closure of \(R\) and every \(z\) with \(f(z)\ne A\),
where \(\mathcal V\) is the set of corners of \(S\) and \(a_0=1/5000\). This is the clearance estimate for the central birth in the proof of [ 248 , Lemma 7.2 ] , for any surrounding guide equal to \(A\) on the interior of \(S\). The guide after the central birth is sketched below, with the bands \(\{ 0{\lt}x{\lt}1\} \) drawn far wider than they are and their narrowing towards the corners drawn as four strings; the dots are the corners in \(\mathcal V\).
![\begin{tenkz}[lattice={9x9}, bonds=none, size=s] % chktex 29
\tndeclare{species}{bandA}{hue=source:orange}
\tndeclare{species}{bandB}{hue=source:cyan}
\tndeclare{species}{contact}{hue=source:red}
\tn[at=(1,1), skin=none]{}
\tn[at=(1,2), skin=none]{}
\tn[at=(1,3), skin=none]{}
\tn[at=(1,4), skin=none]{}
\tn[at=(1,5), skin=none]{}
\tn[at=(1,6), skin=none]{}
\tn[at=(1,7), skin=none]{}
\tn[at=(1,8), skin=none]{}
\tn[at=(1,9), skin=none]{}
\tn[at=(2,1), skin=none]{}
\tn[at=(2,2), name=centralCornerNW, skin=dot, species=contact]{}
\tn[at=(2,3), skin=none]{}
\tn[at=(2,4), skin=none]{}
\tn[at=(2,5), skin=none]{}
\tn[at=(2,6), skin=none]{}
\tn[at=(2,7), skin=none]{}
\tn[at=(2,8), name=centralCornerNE, skin=dot, species=contact]{}
\tn[at=(2,9), skin=none]{}
\tn[at=(3,1), skin=none]{}
\tn[at=(3,2), skin=none]{}
\tn[at=(3,3), name=centralTipNW, skin=none]{}
\tn[at=(3,4), skin=none]{}
\tn[at=(3,5), skin=none]{}
\tn[at=(3,6), skin=none]{}
\tn[at=(3,7), name=centralTipNE, skin=none]{}
\tn[at=(3,8), skin=none]{}
\tn[at=(3,9), skin=none]{}
\tn[at=(4,1), skin=none]{}
\tn[at=(4,2), skin=none]{}
\tn[at=(4,3), skin=none]{}
\tn[at=(4,4), skin=none]{}
\tn[at=(4,5), skin=none]{}
\tn[at=(4,6), skin=none]{}
\tn[at=(4,7), skin=none]{}
\tn[at=(4,8), skin=none]{}
\tn[at=(4,9), skin=none]{}
\tn[at=(5,1), skin=none]{}
\tn[at=(5,2), skin=none]{}
\tn[at=(5,3), skin=none]{}
\tn[at=(5,4), skin=none]{}
\tn[at=(5,5), skin=none]{}
\tn[at=(5,6), skin=none]{}
\tn[at=(5,7), skin=none]{}
\tn[at=(5,8), skin=none]{}
\tn[at=(5,9), skin=none]{}
\tn[at=(6,1), skin=none]{}
\tn[at=(6,2), skin=none]{}
\tn[at=(6,3), skin=none]{}
\tn[at=(6,4), skin=none]{}
\tn[at=(6,5), skin=none]{}
\tn[at=(6,6), skin=none]{}
\tn[at=(6,7), skin=none]{}
\tn[at=(6,8), skin=none]{}
\tn[at=(6,9), skin=none]{}
\tn[at=(7,1), skin=none]{}
\tn[at=(7,2), skin=none]{}
\tn[at=(7,3), name=centralTipSW, skin=none]{}
\tn[at=(7,4), skin=none]{}
\tn[at=(7,5), skin=none]{}
\tn[at=(7,6), skin=none]{}
\tn[at=(7,7), name=centralTipSE, skin=none]{}
\tn[at=(7,8), skin=none]{}
\tn[at=(7,9), skin=none]{}
\tn[at=(8,1), skin=none]{}
\tn[at=(8,2), name=centralCornerSW, skin=dot, species=contact]{}
\tn[at=(8,3), skin=none]{}
\tn[at=(8,4), skin=none]{}
\tn[at=(8,5), skin=none]{}
\tn[at=(8,6), skin=none]{}
\tn[at=(8,7), skin=none]{}
\tn[at=(8,8), name=centralCornerSE, skin=dot, species=contact]{}
\tn[at=(8,9), skin=none]{}
\tn[at=(9,1), skin=none]{}
\tn[at=(9,2), skin=none]{}
\tn[at=(9,3), skin=none]{}
\tn[at=(9,4), skin=none]{}
\tn[at=(9,5), skin=none]{}
\tn[at=(9,6), skin=none]{}
\tn[at=(9,7), skin=none]{}
\tn[at=(9,8), skin=none]{}
\tn[at=(9,9), skin=none]{}
\tnmark[form=enclosure, species=bandA, tint]{(2,3) .. (2,7)}{$A$}
\tnmark[form=enclosure, species=bandA, tint]{(8,3) .. (8,7)}{$A$}
\tnmark[form=enclosure, species=bandA, tint]{(3,2) .. (7,2)}{$A$}
\tnmark[form=enclosure, species=bandA, tint]{(3,8) .. (7,8)}{$A$}
\tnmark[form=enclosure, species=bandB, tint]{(3,3) .. (7,7)}{$B$}
\tnwire[kind=string, species=bandB]{centralTipNW}{centralCornerNW}
\tnwire[kind=string, species=bandB]{centralTipNE}{centralCornerNE}
\tnwire[kind=string, species=bandB]{centralTipSW}{centralCornerSW}
\tnwire[kind=string, species=bandB]{centralTipSE}{centralCornerSE}
\tnmark[form=label]{(1,5)}{$C_e$}
\tnmark[form=label]{(9,5)}{$C_e$}
\tnmark[form=label]{(5,1)}{$C_e$}
\tnmark[form=label]{(5,9)}{$C_e$}
\end{tenkz}](tenkz_svg/tenkz-884ec5190d5dcc6e.svg)
With \(u=\min (y_1,n-y_1)\) and \(v=\min (y_2,n-y_2)\), the four conditions \(d\ge w\) give \(v\ge u/1000\) and \(u\ge v/1000\). The nearest corner is at distance \(\max (u,v)\le 1000\min (u,v)\), so a point within \(a_0\min (n,d_{\mathcal V}(y))\) of \(y\) is within \(\min (u,v)/5\) of \(y\) and lies in \((0,n)^2\).
Consider one of the seven elementary births and deaths of an edge operation, and labellings \(f_b\), \(f_a\) and \(f_s\) of the plane whose normal words on \(\{ -8{\lt}x{\lt}2\} \) are its words before, after and surrounding, with \(f_b=f_a\) off that band. Let \(P_\circ \) be its surrounding label. For every \(y\) in the closure of \(\{ f_b\ne f_a\} \) and every \(z\) with \(f_s(z)\ne P_\circ \), \(|y-z|\ge a_0\min \{ n,d_{\mathcal V}(y)\} \), with \(\mathcal V\) the two endpoints of the edge. These are the clearance estimates for the edge births and deaths in the proof of [ 248 , Lemma 7.2 ] , for any guides with the stated words.
The set \(\{ f_b\ne f_a\} \) lies in the closed band of the changed interval, which is closed. If \(|y-z|{\lt}a_0\min \{ n,d_{\mathcal V}(y)\} \), the conical band estimate puts \(z\) in the open band of the surrounding sector, where \(f_s\) equals \(P_\circ \).
Let the main labelling be \(C\) on the exterior band \(\{ -8{\lt}x{\lt}0\} \), and let the auxiliary labelling have the word (878) on \(\{ -8{\lt}x{\lt}2\} \) and be \(C\) off it. Every point \(y\) in the closure of the positions that are not \(C\) on both labellings satisfies \(|y-z|\ge a_0\min \{ n,d_{\mathcal V}(y)\} \) for every \(z\) on the boundary of the lens \(Y\). This is the clearance estimate for the lens exchange in the proof of [ 248 , Lemma 7.2 ] , for any guides with the stated properties.
A boundary point of \(Y\) is an endpoint of the edge, where the bound is immediate, or satisfies \(d=-7w\) or \(d=-\tfrac 72w\) with \(0{\lt}s{\lt}n\). If \(y\) lies outside the exterior band \(\{ -8{\lt}x{\lt}0\} \), then \(d_{\mathcal V}(y)\le d_{\mathcal V}(z)+|y-z|\) and \(d_{\mathcal V}(z)\le m_z=\min (s_z,n-s_z)\); a distance below \(a_0d_{\mathcal V}(y)\) is then below \(m_z/2000\), which keeps \(y\) in that band because \(-7m_z/1000\le d_z\le -\tfrac 72m_z/1000\) and \(w\) is \(10^{-3}\)-Lipschitz. If \(y\) lies in one of the closed auxiliary bands \(\{ -6\le x\le -4\} \) or \(\{ -3\le x\le -1\} \), apply the conical band estimate with the open bands \(\{ -\tfrac {13}2{\lt}x{\lt}-\tfrac 72\} \) or \(\{ -\tfrac 72{\lt}x{\lt}-\tfrac 12\} \), neither of which meets the two boundary curves.
A repainting of \(S=[0,n]^2\), \(n{\gt}0\), starts from a guide \(g\) equal to \(A\) on the open square and to one label \(C_e\) on the open exterior neighboring square across each edge \(e\). The guide after the central birth is \(B\) on the central region and \(g\) elsewhere. For a list \(E\) of edges, the completed guide replaces, for each \(e\in E\), the band \(\{ -8{\lt}x{\lt}2\} \) of \(e\) by the word \(C_e\mid B\) at \(0\). During the construction along \(e\) after the edges of \(E\), the main guide with normal word \(W\) is the completed guide with the band of \(e\) replaced by \(W\), and the auxiliary guide with normal word \(W\) is the uniform guide \(C_e\) with the band of \(e\) replaced by \(W\). The three auxiliary births and the four main births and deaths are those of Definition 27.126.3. Guides are labellings of the plane; at the start of the construction along \(e\) the schedule passes from the completed guide to the main guide with the starting word, which agrees with it off the curves \(x=0\) and \(x=1\) of the band of \(e\).
Let \(e\notin E\). On the band of \(e\), off the curves \(x=0\) and \(x=1\), the completed guide carries the word \(C_e\mid A\mid B\) at \(0,1\); so it agrees with the main guide with that word off those curves. Exchanging the starting main guide and the auxiliary guide with word (878) on the lens \(Y\) gives the main guide with word \(C_e\mid B\mid A\mid C_e\mid A\mid B\) at \(-6,-5,-4,0,1\). The auxiliary births start from the uniform word \(C_e\), the main operations start from that main word, consecutive operations on one sheet match, and the last main death leaves the completed guide for the list \(E\) with \(e\) added. Once all four edges are completed, the open square carries \(B\), and every point outside the closed square keeps its starting label.
On the band of \(e\), the bands of the completed edges are absent by the band disjointness. Where \(x{\lt}0\) the point lies in the exterior neighboring square; where \(0{\lt}x{\lt}1\) it lies in the open square and outside the central region; where \(1{\lt}x{\lt}2\) it lies in the central region, since for every other edge it lies in the open square and outside that edge’s band, so its normal ratio there is at least \(2\). The exchange is Lemma 27.126.4 on the band, and the identity off it. A point of the open square outside the central region lies in the band \(\{ 0{\lt}x\le 1\} \) of some edge, where the final word is \(B\); a point outside the closed square lies in no band or in the part \(x{\lt}0\) of a band, where the final word is \(C_e\).
For \(n{\gt}0\) and labels \(\ell (Q)\) of the blocks \(Q\in \mathbb Z^2\), the block guide has label \(\ell (\lfloor p_1/n\rfloor ,\lfloor p_2/n\rfloor )\) at \(p\). The repainting of the block \(S=(r,s)\) to the label \(B\) starts from the block guide moved by \(-(nr,ns)\), which is \(\ell (S)\) on the open square and \(\ell \) of the neighboring block across each edge; all its guides are then moved back by \((nr,ns)\). During the pass to level \(n\), with a set of already repainted blocks, a repainted block of the root square carries its new label, another block of the root square the label of its parent \(2n\)-block, and a block outside the root square the placeholder. The picture shows one stage of the pass to level \(n\) for a root square of four blocks per side, before the repainting of the shaded block \(S\) to its new label; \(\ast \) is the placeholder.
![\begin{tenkz}[lattice={6x6}, bonds=none, size=s] % chktex 29
\tndeclare{species}{repainted}{hue=source:cyan}
\tndeclare{species}{current}{hue=source:orange}
\tn[at=(1,1), skin=none]{}
\tnmark[form=label]{(1,1)}{$\ast$}
\tn[at=(1,2), skin=none]{}
\tnmark[form=label]{(1,2)}{$\ast$}
\tn[at=(1,3), skin=none]{}
\tnmark[form=label]{(1,3)}{$\ast$}
\tn[at=(1,4), skin=none]{}
\tnmark[form=label]{(1,4)}{$\ast$}
\tn[at=(1,5), skin=none]{}
\tnmark[form=label]{(1,5)}{$\ast$}
\tn[at=(1,6), skin=none]{}
\tnmark[form=label]{(1,6)}{$\ast$}
\tn[at=(2,1), skin=none]{}
\tnmark[form=label]{(2,1)}{$\ast$}
\tn[at=(2,2), skin=box, species=repainted]{B_1}
\tn[at=(2,3), skin=box, species=repainted]{B_2}
\tn[at=(2,4), name=dyadicCurrent, skin=box, species=current]{A_2}
\tn[at=(2,5), skin=box]{A_2}
\tn[at=(2,6), skin=none]{}
\tnmark[form=label]{(2,6)}{$\ast$}
\tn[at=(3,1), skin=none]{}
\tnmark[form=label]{(3,1)}{$\ast$}
\tn[at=(3,2), skin=box, species=repainted]{B_3}
\tn[at=(3,3), skin=box, species=repainted]{B_4}
\tn[at=(3,4), skin=box]{A_2}
\tn[at=(3,5), skin=box]{A_2}
\tn[at=(3,6), skin=none]{}
\tnmark[form=label]{(3,6)}{$\ast$}
\tn[at=(4,1), skin=none]{}
\tnmark[form=label]{(4,1)}{$\ast$}
\tn[at=(4,2), skin=box]{A_3}
\tn[at=(4,3), skin=box]{A_3}
\tn[at=(4,4), skin=box]{A_4}
\tn[at=(4,5), skin=box]{A_4}
\tn[at=(4,6), skin=none]{}
\tnmark[form=label]{(4,6)}{$\ast$}
\tn[at=(5,1), skin=none]{}
\tnmark[form=label]{(5,1)}{$\ast$}
\tn[at=(5,2), skin=box]{A_3}
\tn[at=(5,3), skin=box]{A_3}
\tn[at=(5,4), skin=box]{A_4}
\tn[at=(5,5), skin=box]{A_4}
\tn[at=(5,6), skin=none]{}
\tnmark[form=label]{(5,6)}{$\ast$}
\tn[at=(6,1), skin=none]{}
\tnmark[form=label]{(6,1)}{$\ast$}
\tn[at=(6,2), skin=none]{}
\tnmark[form=label]{(6,2)}{$\ast$}
\tn[at=(6,3), skin=none]{}
\tnmark[form=label]{(6,3)}{$\ast$}
\tn[at=(6,4), skin=none]{}
\tnmark[form=label]{(6,4)}{$\ast$}
\tn[at=(6,5), skin=none]{}
\tnmark[form=label]{(6,5)}{$\ast$}
\tn[at=(6,6), skin=none]{}
\tnmark[form=label]{(6,6)}{$\ast$}
\tnmark[form=enclosure]{(2,2) .. (3,3)}{}
\tnmark[form=enclosure]{(2,4) .. (3,5)}{}
\tnmark[form=enclosure]{(4,2) .. (5,3)}{}
\tnmark[form=enclosure]{(4,4) .. (5,5)}{}
\end{tenkz}](tenkz_svg/tenkz-4c1a12aa026619c0.svg)
Moving all guides by a vector \(o\) moves the changed regions and the obstructing sets by \(o\), and the mark distances become the distances from the moved marks; sup distances are translation invariant. A lower bound on the distance from a point to every point of a set holds for its closure.
Translation is a homeomorphism, so it commutes with closures and boundaries, and \(|(y-o)-(z-o)|=|y-z|\). The set of points at distance at least \(c\) from \(y\) is closed.
Let \(a_0=1/5000\), \(n{\gt}0\), and consider the repainting of a block \(S\) of side \(n\) from a block guide, with any block labels and any final label, through the guides that the schedule specifies at each operation. Every elementary birth or death of its schedule, the central birth and the seven births and deaths along each edge after any completed edges, has changed region \(R\) of diameter at most \(n\), and its surrounding guide \(f\) satisfies, at every point \(y\in \overline R\),
where \(\mathcal V\) is the set of corners of \(S\) for the central birth and the two endpoints of the edge otherwise. For each lens exchange, every point \(y\) in the closure of the positions that are not common \(C_e\) on its two guides before the exchange satisfies \(\operatorname {dist}(y,\partial Y)\ge a_0\min \{ n,d_{\mathcal V}(y)\} \). This is [ 248 , Lemma 7.2 ] for the labellings that the schedule specifies; for the guides sampled under the source’s tie convention it is Theorem 27.126.20. The positions inside \(Y\) that are not common \(C_e\) before the exchange, which contain the positions the exchange changes, also form a set whose closure has diameter at most \(n\); the source uses this bound in its proof of the exchange step, lines 421–424.
The guides of an edge construction are obtained from one guide of their sheet by replacing the band of the edge by a word, so they have the properties assumed in Theorems 27.126.6, 27.126.7 and 27.126.8: such a guide carries its word on the band, two such guides of one sheet agree off the band, and the starting main word is \(C_e\) on \(\{ -8{\lt}x{\lt}0\} \); the starting guide of the central birth is \(A\) on the open square. Lemma 27.126.10 identifies these guides with the guides reached by the schedule. The diameter bound is Lemma 27.126.2. The exchange changes the main guide from the starting guide to the guide with the main word, and the auxiliary guide to the starting main guide on \(Y\), so it changes only points of \(Y\) where the two sheets differ; these are not common \(C_e\). The closure of \(Y\) lies in the closed band \(\{ -7\le x\le -7/2\} \), of diameter at most \(n\) by Lemma 27.126.2. Translate by Lemma 27.126.12.
Once the four edges of a block \(S\) of the root square are completed, the main guide agrees, off the boundary of \(S\), with the block guide in which \(S\) also carries its new label. The block guide of side \(2n\) at the end of the pass to level \(2n\) equals the block guide of side \(n\) at the start of the pass to level \(n\). The block labels of a level are the placeholder, old labels or new labels; at the start of the level no new label occurs and at its end no old label. Every normal word of an edge construction, the starting word and the words before, after and surrounding each of its births and deaths, takes only the values \(A\), \(B\), \(C_e\). Hence every value of every main or auxiliary guide of a repainting is the placeholder, an old label or a new label, and outside the closed square and its exterior bands \(\{ -8{\lt}x{\lt}0\} \) the main guides keep their starting values. The labels \(A\), \(B\), \(C_e\) of the repainting of \(S\) are owned by \(S\), its parent, a neighboring block or its parent, or the placeholder; a neighboring block differs from \(S\) by at most one in each index. These are the guide part of the invariant on active labels in [ 248 , Section 7 ] , lines 597–618; its part on tags is Lemma 27.126.28.
Inside the open square the final guide is \(B\) by Lemma 27.126.10; outside the closed square it is the starting block guide, and there the block index differs from \(S\). For the level transition, \(\lfloor x/(2n)\rfloor =\lfloor \lfloor x/n\rfloor /2\rfloor \) and a block lies in the root square of \(2M\) blocks per side if and only if its parent lies in that of \(M\) blocks. The words use only \(A\), \(B\), \(C_e\), which are values of the starting guide or the final label.
Let the side \(n\) be an integer. No cell center \((i+\tfrac 12,j+\tfrac 12)\) lies on the boundary of a block, or on the curves \(x=0\) and \(x=1\) of the band of an edge of a block. Hence, at every cell center, the guide reached at the start of the construction along an edge equals the main guide with the starting word, and the guide left by a repainting equals the block guide of the next repainting. The two replacements of guides in the schedule therefore leave the labels at the lattice sites unchanged, as in the sampling convention of [ 248 , Section 7 ] , lines 62–80.
Say that a labelling \(f'\) of the plane reads a labelling \(f\) after the displacement \(v\) if \(f'(p)=f(p+\varepsilon v)\) for every point \(p\) and all sufficiently small \(\varepsilon {\gt}0\). If \(f_1'\) and \(f_2'\) read \(f_1\) and \(f_2\) after the same \(v\), then the set where \(f_1'\ne f_2'\) lies in the closure of the set where \(f_1\ne f_2\); the set where \(f'\ne c\) lies in the closure of the set where \(f\ne c\); and the positions, inside an open set or anywhere, that are not common \(C\) on \(f_1'\) and \(f_2'\) lie in the closure of those on \(f_1\) and \(f_2\). Hence the estimates of Theorem 27.126.13 hold, with the same constant \(a_0=1/5000\), for any labellings that read the guides of the schedule after one displacement \(v\), whatever \(v\) is.
If \(f_1=f_2\) near \(p\), then \(f_1(p+\varepsilon v)=f_2(p+\varepsilon v)\) for all small \(\varepsilon {\gt}0\), so \(f_1'(p)=f_2'(p)\). Off the closure of the set where \(f_1\ne f_2\) the two labellings agree near every point. A constant labelling reads itself. All estimates of Theorem 27.126.13 are stated for closures, which can only shrink.
Let \(v\) be any vector. The block guide, every guide of the repainting of a block (after the central birth, after completed edges, and along an edge with a normal word on its band, on the main or on the auxiliary sheet), and each of these guides homogenized on the open squares of radius \(t\) about finitely many grid corners, have exactly one reading after \(v\): at every point \(p\) the label at \(p+\varepsilon v\) is the same for all small \(\varepsilon {\gt}0\).
Along a ray \(\varepsilon \mapsto p+\varepsilon v\) the coordinates, the parallel and normal coordinates of an edge, and the band width \(w(s)=10^{-3}\min (s,n-s)\) agree for small \(\varepsilon {\gt}0\) with affine functions \(a+b\varepsilon \). Such a function has, for small \(\varepsilon {\gt}0\), a constant sign: that of \(a\) if \(a\ne 0\), and that of \(b\) otherwise. Hence every strict inequality between such functions, and every bound \(x{\lt}t\) on the normal ratio \(x=d/w(s)\), is eventually decided; the floor \(\lfloor a+b\varepsilon \rfloor \) is eventually \(\lfloor a\rfloor \) if \(b\ge 0\) and \(\lceil a\rceil -1\) if \(b{\lt}0\). The bands, the central region, the sup squares and the blocks are finite Boolean combinations of such conditions, and a guide is a finite choice among them. Two readings agree, since both are the eventual label.
The source specifies each guide by its open polygonal chambers and reads it at a point after one small generic displacement into an incident chamber ( [ 248 , Section 7 ] , lines 69–80). The guides of the schedule take, on an interface curve \(x=t\), the label for \(x\ge t\), while a fixed displacement \(v\) whose two components are not small relative to each other crosses to \(x{\gt}t\) along two edges of a square and to \(x{\lt}t\) along the other two; the two conventions differ on the interface curves of those two edges. Lemma 27.126.17 shows that the guides of the schedule have readings after every displacement, and Lemma 27.126.16 gives the clearance estimates of Lemma 7.2 for every such reading, on all four edges. The next three results identify the source’s sampled guides with these readings and state Lemma 7.2 for them.
The open chamber of a label \(l\) of a labelling \(f\) of the plane is the interior of \(f^{-1}(l)\). A labelling \(g\) is sampled from the open chambers of \(f\) after the displacement \(v\) if, at every point \(p\), the point \(p+\varepsilon v\) lies in the open chamber of the label \(g(p)\) for all sufficiently small \(\varepsilon {\gt}0\); such a \(g\) reads \(f\) after \(v\), and two of them agree. The source’s open polygonal chambers are modelled in this way as the interiors of the label regions of the labellings that the schedule specifies: these agree with the source’s descriptions on every open chamber, merging adjacent chambers of one label does not change the sampled label, and a label confined to boundaries is never sampled. Let \(T=\{ -8,-6,-5,-4,-3,-2,-1,0,1,2\} \), the values of the normal ratio at which a guide of the schedule, or the region of a band that a word replaces, can have a boundary, and let \(\mathcal N\) be the set of the vectors \((\pm t,1000)\) and \((1000,\pm t)\) with \(t\in T\). A displacement \(v\) is generic if \(N\cdot v\ne 0\) for every \(N\in \mathcal N\): it is parallel to none of the lines with normals in \(\mathcal N\), so it avoids finitely many lines through the origin. The vector \((1,1)\) is generic, and so is every nonzero multiple of a generic vector.
The block guide, every guide of the repainting of a block (after the central birth, after completed edges, and along an edge with a word of the construction on its band, on the main or on the auxiliary sheet), and each of these homogenized on the open squares of radius \(t\) about finitely many grid corners, is constant near every point off a set that lies near every point in finitely many lines with normals in \(\mathcal N\). So its open chambers are the open sets on which it is constant, off a locally finite family of segments. Consequently, for every generic displacement \(v\), each of these guides has exactly one guide sampled from its open chambers after \(v\), and a labelling is sampled from its open chambers after \(v\) if and only if it reads it after \(v\).
In the edge coordinates of a square of side \(n\), the interface curve \(x=t\) of a band is \(1000\, d=t\min (s,n-s)\), a union of two segments on lines with normals \((\pm t,1000)\) or \((1000,\pm t)\), and \(w(s)=0\) only on the lines \(s=0\) and \(s=n\). The edges, the sides of the treated squares and the grid lines are lines with \(t=0\), and near every point only two grid lines in each direction can occur. A strict inequality between continuous functions is locally constant where the two sides differ; the bound \(d/w(s){\lt}t\) is locally constant where \(w(s)\ne 0\) and \(d\ne t\, w(s)\); the integer part of a continuous function is locally constant where the function is not an integer. The guides are finite Boolean combinations and choices among such conditions, and the thresholds of the bands and of the words of the construction lie in \(T\). A line not parallel to \(v\) meets the ray \(p+\varepsilon v\) in at most one point, so for generic \(v\) and small \(\varepsilon {\gt}0\) the point \(p+\varepsilon v\) avoids the walls, where the guide is locally constant; with the eventual label of Lemma 27.126.17 it lies in the open chamber of that label.
Let \(a_0=1/5000\), \(n{\gt}0\), and let \(v\) be a generic displacement. The guides of the repainting of a block \(S\) of side \(n\), read as in the source by sampling their open chambers after \(v\), exist, and the estimates of Theorem 27.126.13 hold for them: every elementary birth or death has changed region of diameter at most \(n\) and satisfies 880 for its sampled surrounding guide, and every point \(y\) in the closure of the positions that are not common \(C_e\) on the two sampled guides before a lens exchange satisfies \(\operatorname {dist}(y,\partial Y)\ge a_0\min \{ n,d_{\mathcal V}(y)\} \). This is [ 248 , Lemma 7.2 ] under the source’s tie convention, lines 69–80, with the source’s hypotheses.
Let \(0{\lt}t\le n\) and \(h_n=\epsilon _0\min (n,t)=\epsilon _0t\). Homogenize every involved guide of an elementary operation to its surrounding label (to \(C_e\) for an exchange) on the open sup squares of radius \(t\) about its marks. Every point \(y\) of the closure of the remaining changed region satisfies \(|y-z|\ge a_0\max \{ t,\min (n,d_{\mathcal V}(y))\} \) for every position \(z\) of another label in the homogenized surrounding guide, and every point \(y\) of the closure of the remaining positions that are not common \(C_e\) satisfies the same bound for every point \(z\) of \(\partial Y\). This is the logarithmic floor of [ 248 , Section 7 ] , lines 357–367, with \(a_1=a_0\).
Homogenization only removes changed positions, only inside the treated squares, and only removes other-label positions. A point of the closure of the remaining changed region lies outside the open treated squares, so \(d_{\mathcal V}(y)\ge t\) and \(\min (n,d_{\mathcal V}(y))\ge t\). Apply Theorem 27.126.13.
A true vertex, a point in the closures of the regions of three distinct labels, lies in the closure of the positions of labels other than any given label. If \(2\epsilon _0\le a_0/2\), every point within \(2\epsilon _0t\) of a true vertex of the homogenized surrounding guide lies at distance at least \(\tfrac {a_0}2\max \{ t,\min (n,d_{\mathcal V}(y))\} \) from every point \(y\) of the closure of the remaining changed region, for the central birth and for every birth or death along an edge. This is the enlargement of [ 248 , Section 7 ] , lines 368–373.
Of three distinct labels at least one differs from the given label. With \(Q=\max \{ t,\min (n,d_{\mathcal V}(y))\} \ge t\), the floor gives \(|y-c|\ge a_0Q\) at the vertex \(c\), hence \(|y-x|\ge a_0Q-2\epsilon _0t\ge a_0Q-\tfrac {a_0}2t\ge \tfrac {a_0}2Q\).
Let \(g\) and \(h\) be \(1\)-Lipschitz functions, \(a,t{\gt}0\) and \(2\epsilon _0(a+1)\le a/2\). If a center \(c\) satisfies \(g(c)\ge aQ_c\) with \(Q_x=\max \{ t,\min (n,h(x))\} \), then every \(x\) with \(|x-c|\le 2\epsilon _0t\) satisfies \(g(x)\ge \tfrac a2Q_x\). With \(g\) the distance from a point of the lens boundary and \(h\) the mark distance, and \(2\epsilon _0(a_0+1)\le a_0/2\), every point within \(2\epsilon _0t\) of a true vertex of either homogenized sheet of an exchange lies at distance at least \(\tfrac {a_0}2\max \{ t,\min (n,d_{\mathcal V}(x))\} \) from the lens boundary. This is the step of [ 248 , Section 7 ] , lines 400–415.
With \(\delta =|x-c|\) one has \(Q_x\le Q_c+\delta \) and \(g(x)\ge g(c)-\delta \), so \(g(x)\ge aQ_x-(a+1)\delta \ge aQ_x-2\epsilon _0(a+1)t\ge aQ_x-\tfrac a2Q_x\). A true vertex of either sheet lies in the closure of the noncommon-\(C_e\) positions, so it has the floor of Lemma 27.126.21.
A closed sup square every point of which is at distance at least \(b{\gt}0\) from the boundary of a set \(Y\) lies in \(Y\) or in its complement. In particular, let \(0{\lt}t\le n\) and \(2\epsilon _0(a_0+1)\le a_0/2\); then the outer hole square of radius \(2\epsilon _0t\) about a true vertex of either homogenized sheet of an exchange lies in the lens \(Y\) or in its complement. If a point \(x\) is at distance at least \(b\) from every point of the boundary of \(Y\), then it is at distance at least \(b\) from every point on the other side of \(Y\). These are the statements of [ 248 , Section 7 ] , lines 403–405 and 421–429.
A preconnected set missing the boundary of \(Y\) is covered by the disjoint open sets \(\operatorname {int}Y\) and the complement of \(\overline Y\), so it lies in one of them; closed squares and segments are convex. By Lemma 27.126.23 every point of an outer hole square is at distance at least \(\tfrac {a_0}2t{\gt}0\) from \(\partial Y\). The segment from \(x\) to a point \(z\) on the other side meets \(\partial Y\) at some \(w\) with \(|x-w|+|w-z|=|x-z|\).
The labels incident at a point \(c\) of a guide are those whose region has \(c\) in its closure; a true vertex is a point with three distinct incident labels. A label may hold the tag of a true vertex if it is incident there and is not the placeholder. A tag rule chooses one such label at every true vertex, as in [ 248 , Section 7 ] , lines 109–125.
Two guides that agree near \(c\) have the same incident labels at \(c\), and a point with at most two incident labels is not a true vertex.
Let the closure of the set where two guides differ lie in an open set \(U\) on which both take only the values \(P\) and \(Q\). Then the two guides have the same true vertices; each lies outside \(U\) and has the same incident labels in both guides.
Let two guides \(h_1,h_2\) both equal \(C\) on an open set \(U\) containing \(\partial Y\), and let \(h\) be \(h_2\) on \(Y\) and \(h_1\) off \(Y\). Then \(h\) has a true vertex at \(c\) if and only if \(c\in Y\) and \(h_2\) has one there, or \(c\notin Y\) and \(h_1\) has one there; it then has the incident labels of that guide at \(c\).
A true vertex of a guide homogenized to \(P\) on an open set \(U\) is either a true vertex of the guide off \(\overline U\), or a point of \(\partial U\) at which two distinct labels other than \(P\) of the guide meet.
Closures at \(c\) depend only on a neighborhood of \(c\). In the first case a point outside \(\overline{\{ f\ne g\} }\) has a neighborhood where the guides agree, and a point of \(U\) has one where both take only two values. In the second, a point of \(U\) has only the incident label \(C\), and a point off \(\partial Y\) lies in the interior of \(Y\) or of its complement. In the third, a point of \(U\) has only the incident label \(P\), a point off \(\overline U\) has a neighborhood where homogenization does nothing, and homogenization adds at most the label \(P\) to the incident labels.
Let \(0{\lt}t\le n\). Each of the seven births and deaths of an edge construction introduces a single label: at every changed normal ratio the words before and after take only the surrounding label and one further label. Homogenize the guides of an elementary birth or death, the central birth included, to its surrounding label on the treated squares of radius \(t\). Then the guides before and after have the same true vertices, each at distance at least \(a_0t\) from the closure of the remaining changed region and with the same incident labels, so with the same tag holders; its outer hole square, of radius \(2\epsilon _0t\), misses the closed change once \(2\epsilon _0{\lt}a_0\). Homogenize both sheets of a lens exchange to \(C_e\) on the treated squares. Both are \(C_e\) on the open neighborhood of radius \(a_0t\) of \(\partial Y\), and each exchanged guide has inside \(Y\) exactly the true vertices and incident labels of the sheet it takes there, and outside \(Y\) those of the other sheet. These are the statements of [ 248 , Section 7 ] , lines 375–382 and 389–398.
By the logarithmic floor of Lemma 27.126.21, the homogenized surrounding guide is \(P_\circ \) on the open neighborhood of radius \(a_0t\) of the remaining changed region. On that neighborhood both guides take only the values \(P_\circ \) and the inserted label: off the changed region they agree with the surrounding guide, and on it they carry the operation’s words, outside the treated squares. Apply the first part of Lemma 27.126.26. For the exchange, a point within \(a_0t\) of \(\partial Y\) that is not common \(C_e\) would contradict the floor; apply the second part. A point within \(2\epsilon _0t{\lt}a_0t\) of a retained true vertex is closer to it than any point of the closed change.
The source’s further conclusions at lines 383–387, that the raw target of the bulk change lies outside all holes, satisfies the tapering hypothesis and leaves a single-owner raw buffer, so that the birth lemma implements it, belong to the protocol and are not formalized here.
A true vertex has at least two incident labels other than the placeholder, so a tag holder exists, and guides with the same incident labels at a point have the same tag holders there. Tag holders are values of the guide other than the placeholder. Hence at the start of the pass to level \(n\) only labels of \(2n\)-blocks hold main tags; at every repainting boundary during it, and in every main or auxiliary guide of a repainting, only labels of \(2n\)- and \(n\)-blocks do; and at its end only labels of \(n\)-blocks: a party of side \(2n\) has disappeared from every main tag by the end of level \(n\). During the edge construction along an edge \(e\), the label \(C_e\) with which its auxiliary sheet is initialized is a value of the main guide at that moment. These are the tag part of the invariant on active labels, with a tag held by an incident label other than the placeholder, and the selection of \(C\) in the statement on frozen auxiliary frames, of [ 248 , Section 7 ] , lines 610–612 and 625–627.
Of three distinct incident labels at most one is the placeholder. An incident label is a value of the guide, and the block labels of a level are described by Lemma 27.126.14. The current main guide agrees with the starting guide outside the closed square and outside the outer band of \(e\), and the starting guide is \(C_e\) on the neighboring block across \(e\), which reaches outside that band.
Here the tag of a true vertex is identified with an incident label other than the placeholder chosen by a tag rule. The source’s statement that the protocol updates the stored tag at a junction by the corner rewrite of its last incident child square (lines 613–614), and the retirement of frozen frames (lines 620–631), concern the registers of the protocol of Proposition 7.1 and are not formalized here.
Let finitely many compact sets \(F_c\) in a metric space be such that no three with distinct labels have a common point. Then there is \(\delta {\gt}0\) such that no closed ball of radius \(\delta \) meets three of them with distinct labels. If the \(F_c\) are compact sets of pairs of a parameter and a point, one \(\delta \) works for all values of the parameter. In particular, let a guide take finitely many values, let \(W\) be a compact working neighborhood, and let every true vertex in \(W\) lie in one of the open inner hole squares. Then there is \(\delta {\gt}0\) such that every closed footprint of radius \(\delta \) meets at most two labels at its points of \(W\) outside the open inner holes. This is the abstract compactness argument of [ 248 , Section 7 ] , lines 453–462 and 475–479, and its parametric form, lines 464–469, for compact joint sets.
For three compact sets without a common point, \(\max (|y_1-y_2|,|y_1-y_3|)\) attains a positive minimum on their product; take a third of the smallest of these minima over the finitely many triples. For the parameters, apply this in the product space, where points with the same parameter are at the distance of their second coordinates. For the guide, take \(F_c\) the closure of the region of \(c\) in \(W\) with the open inner holes removed; a common point of three of them would be a true vertex outside every inner hole.
Say that a guide satisfies the two-owner condition with working set \(W\), hole centers \(H\), inner radius \(r\) and footprint radius \(\delta \) if every closed footprint of radius \(\delta \) meets at most two labels at its points of \(W\) outside the open inner holes of radius \(r\) about the points of \(H\). This condition survives shrinking \(\delta \), identifying labels, and enlarging some of the inner holes, so it holds as well when the holes have different radii, all at least \(r\). It is carried by the similarity \(u\mapsto v+su\), \(s{\gt}0\), to the image guide, working set and hole centers with the radii \(sr\) and \(s\delta \). For a guide with finitely many values and a compact \(W\), the pairs of a radius \(r\in [r_0,r_1]\) and a point of \(W\) in the closure of the region of one label and outside the open inner holes of radius \(r\) form a compact set. Hence, if every true vertex in \(W\) lies in an open inner hole of radius \(r_0{\gt}0\), one footprint radius works for every \(r\in [r_0,r_1]\). Consequently, for finitely many template guides with this property there is one ratio \(\nu {\gt}0\) such that every relabelled image of a template under a similarity of ratio \(s{\gt}0\) satisfies the two-owner condition with inner radius \(sr\), \(r\in [r_0,r_1]\), and footprint radius \(\nu s\). This is the uniformity argument of [ 248 , Section 7 ] , lines 464–479, with the joint sets of lines 465–468 and the minimum over finitely many label-equality types of lines 469–471.
Being outside an open square is the non-strict inequality \(r\le |y-h|\), which is closed in the pair \((r,y)\); with \(r\in [r_0,r_1]\) and \(y\in W\) the joint sets are compact. Three of them with distinct labels have no common point, since a common point \(y\) would be a true vertex with \(r_0\le r\le |y-h|{\lt}r_0\) for some \(h\). Apply the parametric form of Lemma 27.126.29, take the minimum over the finitely many templates, and transport by the similarity, which multiplies all sup distances by \(s\).
Let \(u{\gt}0\). Every point is within distance less than \(u\) of the net \(u\mathbb Z^2\). A set inside a closed ball of radius \(\rho \) is covered by the inner squares of radius \(u\) about at most \((2(\rho +u)/u+1)^2\) net points, each within \(u\) of the set, and the outer square of radius \(2u\) about such a net point lies within \(3u\) of the set. For a fixed ratio \(\nu {\gt}0\), with \(u=\nu t\) and \(\rho =O(t)\), or \(u=\nu n\) and \(\rho =O(n)\), the number of patches is bounded independently of the scale, as in [ 248 , Section 7 ] , lines 491–499 and 520–525.
Round each coordinate down to the net. The net coordinates within \(\rho +u\) of a coordinate of the center form an interval of at most \(2(\rho +u)/u+1\) integers.
Let \(\epsilon _0\ge 0\) and \(h_n=\epsilon _0\min (n,t)\). If \(0{\lt}n\le K_0t\) with \(t{\gt}0\), then \(\epsilon _0/\max (K_0,1)\le h_n/n\le \epsilon _0\). If \(n,t\ge 0\), then \(h_n\le \epsilon _0\min (2n,t)\le 2h_n\), and the two radii differ only when \(n{\lt}t\). These are [ 248 , Section 7 ] , lines 511–516 and 529–533.
If \(n\le t\) the ratio is \(\epsilon _0\). Otherwise it is \(\epsilon _0t/n\), which lies between \(\epsilon _0/K_0\) and \(\epsilon _0\). For the resizing, compare \(\min (2n,t)\) with \(\min (n,t)\) in the two cases \(n\le t\) and \(t\le n\).
Let \(n{\gt}0\). The true vertices of a block guide are grid corners \((na,nb)\). Changing the label of one block \(S\) changes the true-vertex status, and the incident labels at a true vertex, only at the four corners of \(S\); in particular this holds for the direct repainting of a block of the root square during a level. Every other grid corner is at distance at least \(n\) from the closed block \(S\). These are the facts on true vertices and untouched holes used for the direct repainting in [ 248 , Section 7 ] , lines 507–527.
Near a point whose first coordinate is not a multiple of \(n\), the first block index is constant and the second takes at most two values, so at most two labels occur; similarly for the second coordinate. Off the closed block \(S\) the two guides agree near the point. A grid corner in the closed block is one of its corners, and a grid corner differing by at least \(2\) or by \(-1\) in a block index is at distance at least \(n\) in that coordinate.
Distinct grid corners are at distance at least \(n\). Hence for \(n{\gt}20t\) the squares of radius \(10t\) about distinct grid corners are disjoint and contain no other grid corner. Let \(\sigma \) be the parallel distance from an endpoint of an edge, so \(\sigma =s\) at the first endpoint and \(\sigma =n-s\) at the second. In the open square of radius \(n/2\) about the endpoint, a band \(\{ \alpha {\lt}x{\lt}\beta \} \) is the cone \(\{ \sigma {\gt}0,\ \alpha \sigma /1000{\lt}d{\lt}\beta \sigma /1000\} \), so its interfaces are straight rays. A point of the curve \(x=\alpha \), \(|\alpha |\le 8\), with \(0{\lt}\sigma \le n/2\), on the rim of the square of radius \(t\) about the endpoint is the point \(\sigma =t\), \(d=\alpha t/1000\); rays of one edge with \(|\alpha -\alpha '|\ge 1/2\) meet that rim at points at distance at least \(t/2000\), and the closed squares of radius \(2\epsilon _0t\) about them are disjoint once \(\epsilon _0{\lt}1/8000\). These are among the separations used for the choice of \(K_0\) and \(\epsilon _0\) in [ 248 , Section 7 ] , lines 341–351.
Distinct grid corners differ by at least \(1\) in some block index. On \(\sigma \le n/2\) one has \(w=\sigma /1000\). On the curve \(|d|=|\alpha |\sigma /1000{\lt}\sigma \), so the distance from the endpoint is \(\sigma \). The two rim points differ only in \(d\), by \(|\alpha -\alpha '|t/1000\).
Call a guide of the repainting of a block from a block guide unmodified if it is the starting block guide, the guide after the central birth, a guide after completed edges, or a main or auxiliary guide along an edge whose band carries one of the normal words of that sheet. The auxiliary words are the uniform word, the words after the three auxiliary births, and the word \(C\mid A\mid B\mid C\) with interfaces \(-3,-2,-1\) left by the lens exchange, to which the point treatment of the exchange is removed before the auxiliary sheet is retired. Every unmodified guide takes at most two labels near every point that is not a grid corner, so all its true vertices are grid corners. Consequently, for \(n{\gt}20t\), the closed square of radius \(10t\) about a grid corner contains no true vertex of an unmodified guide other than that corner. This is the observation of [ 248 , Section 7 ] , lines 321–326, and the clause of the choice of \(K_0\), lines 341–343, that the tenfold enlargements contain no other unmodified true vertex.
A normal word is constant just below every point and from the point on, so near a point it takes at most two values. Inside the open band of an edge a guide carrying a word therefore takes at most two labels near each point. On the curve \(x=-8\), away from the endpoints, the word is \(C_e\) and the guide beyond the band is the label \(C_e\) of the neighboring block. On the curve \(x=2\) the word of the main sheet is \(B\), and the curve lies in the open central region and off the closed bands of the other edges, so the guide beyond the band is \(B\) as well; the auxiliary sheet is \(C_e\) on both sides. Off the closed band nothing changes. The closure of the central region meets the boundary of the square only at its corners, so near any other point of that closure the guide takes only \(A\) and \(B\). The corners of the square are grid corners, a block guide takes at most two labels near every other point, and distinct grid corners are at distance at least \(n\) by Lemma 27.126.34.
For a point \(v\) and \(t{\gt}0\), the rim pattern about \(v\) consists of the points \(v+(\sigma t,jt/1000)\) and \(v+(jt/1000,\sigma t)\) with \(\sigma =\pm 1\) and \(j\in \mathbb Z\), \(|j|\le 6\): on each side of the rim of the square of radius \(t\) about \(v\), thirteen points at spacing \(t/1000\) about the midpoint of the side. It is finite, its points are at distance \(t\) from \(v\), and distinct points are at distance at least \(t/1000\). Off the grid lines and off the curves \(x=k\), \(k\in \{ -6,\dots ,1\} \), of the bands of the four edges, every unmodified guide is constant near each point. Let \(2t{\lt}n\). A point at distance \(t\) from a grid corner that lies on a grid line, or on one of these curves of an edge of the block, lies in the rim pattern of that corner: the grid line passes through the corner, and the curve belongs to one of the two edges ending at the corner. Hence every point of the rim about a grid corner at which an unmodified guide has two distinct incident labels lies in the rim pattern. This is the finite pattern of rim intersection points of [ 248 , Section 7 ] , lines 344–349, with the rays of both adjacent edges and the grid lines. The source’s illustration of a point treatment for three distinct labels is redrawn below in units of \(t\): homogenizing the open square of radius \(t\) about the true vertex to \(P\) moves the meeting point of \(P\), \(A\) and \(B\) to the midpoint of the right side of the rim, a point of the rim pattern.
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\(\longrightarrow \) ![\begin{tenkz}[lattice={9x9}, bonds=none, size=s] % chktex 29
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\end{tenkz}](tenkz_svg/tenkz-71bb2e394a98473f.svg)
A word is constant near every point other than its interfaces, and the interfaces of the main and auxiliary words are integers from \(-6\) to \(1\); the boundary of the central region lies on the curves \(x=1\). For a point on a grid line, the line is within \(t{\lt}n/2\) of the corner, so it passes through the corner, and the point is the midpoint of a side of the rim. For a point on a curve of an edge, its normal coordinate is at most \(6\cdot 10^{-3}\min (s,n-s)\) in absolute value, so the corner has normal coordinate \(0\) and parallel coordinate \(0\) or \(n\); the straight-ray chart of Lemma 27.126.34 then places the point at parallel distance \(t\) and normal coordinate \(kt/1000\). Distinct pattern points on one side differ by at least \(t/1000\) along it, and points on different sides differ by at least \((1-6/1000)t\) in one coordinate.
Let \(20t{\lt}n\), let \(\mathcal M\) be a finite set of grid corners and \(P\) a label, and homogenize an unmodified guide to \(P\) on the open squares of radius \(t\) about the corners of \(\mathcal M\). Every true vertex of the resulting guide is a grid corner or a point of the rim pattern of a corner of \(\mathcal M\), and distinct true vertices are at distance at least \(t/1000\). Hence for \(\epsilon _0{\lt}1/4000\) the closed outer hole squares of radius \(2\epsilon _0t\) about distinct true vertices are disjoint. The same holds for every reading of the homogenized guide after a displacement, and such a reading exists after every displacement. This is the conclusion of [ 248 , Section 7 ] , lines 347–351, that the outer holes are disjoint in the temporary guides, for the separations of the true vertices; the further smallness of \(\epsilon _0\) compared with \(a_0\) enters through Lemma 27.126.22.
By Lemma 27.126.26 a true vertex of the homogenized guide is an old true vertex off the closed treated squares, a grid corner by Lemma 27.126.35, or a point of a treated rim with two incident labels, in the rim pattern by Lemma 27.126.36. Two rim points about one corner are separated by the pattern. Otherwise the two points belong to distinct grid corners, at distance at least \(n\), and each lies within \(t\) of its corner, so they are at distance at least \(n-2t{\gt}t/1000\). A point of the region of a label in a reading is a limit of points of that region in the guide, so a true vertex of a reading is a true vertex of the guide; readings exist by Lemma 27.126.17.
Fix \(\epsilon _0{\gt}0\). There is a ratio \(\nu {\gt}0\) with the following property. Let \(8t{\lt}n\), consider the repainting of a block \(S\) of side \(n\) from a block guide, an unmodified guide of it, a corner \(v\) of \(S\), and either no homogenization or the homogenization, on the open square of radius \(t\) about \(v\), to the final label or to the label of one of the nine blocks around \(S\). Then the resulting guide satisfies the two-owner condition with working set the closed square of radius \(2t\) about \(v\), holes at its true vertices within \(3t\) of \(v\), inner radius \(\epsilon _0t\) and footprint radius \(\nu t\). The same holds, about each corner of \(S\) in a finite set \(\mathcal M\) of grid corners, for the guide homogenized to one such label on the open squares of radius \(t\) about all corners of \(\mathcal M\) at once. This is the uniformity of the patch ratio near a mark in [ 248 , Section 7 ] , lines 464–479 and 481–505: in units of \(t\) the guides come from finitely many ray patterns, with or without the square clearing. Its patches have inner radius \(u=\nu t/2\), since their outer squares have radius at most \(2u\). The same ratios hold for the readings of these guides after a small displacement (Theorem 27.126.40).
Scaling by \(1/n\) carries the bands, their normal ratios and the central region of the block of side \(n\) to those of the unit square, so every unmodified guide is an unmodified guide of the repainting of the unit square composed with the scaling. Relabelling commutes with replacing a band by a word and with the central birth, so near the corner the guide is the relabelling by the window labels of an unmodified guide of one reference repainting, whose labels are the nine blocks around \((0,0)\) and the final label. On the unit square the bands, the normal ratios, the central region and the block guide are conical about each corner within distance \(1/2\), by the straight-ray charts of Lemma 27.126.34; so are the unmodified guides, and the scale \(t/n{\lt}1/8\) can be replaced by \(1/32\). The reference repainting has finitely many unmodified guides, since a guide after completed edges depends only on which edges are completed. Choosing for each label-equality type a representative map of the finitely many slots, the guide near \(v\) is an injective relabelling of one of finitely many templates, and injective relabellings and similarities preserve true vertices. Apply Lemma 27.126.30. Near one corner of \(\mathcal M\) the other treated squares are at distance more than \(n-5t{\gt}3t\), so homogenizing on all of them agrees with homogenizing on its square alone within distance \(4t\) of the corner, which contains the working set and the holes.
Fix \(\rho \) and \(0{\lt}r_0\le r_1\). There is a ratio \(\nu {\gt}0\) such that every guide uniform on the blocks of side \(s{\gt}0\), about every grid corner \(v\) and for every inner radius \(r\in [r_0s,r_1s]\), satisfies the two-owner condition with working set the closed square of radius \(s\rho \) about \(v\), holes at its true vertices within \(s(\rho +1)\) of \(v\), and footprint radius \(\nu s\). In particular, for fixed \(\epsilon _0{\gt}0\) and \(K_0\), one ratio serves every direct repainting of a square of side \(0{\lt}n\le K_0t\), before and after, with inner radius \(h_n\) and working set the closed square of radius \(2n\) about a corner of the square; and one ratio serves every resizing with \(0{\lt}n{\lt}t\), with both radii \(h_n\) and \(h_{2n}\) on the guide uniform on the blocks of side \(2n\) and working set of radius \(2n\), and also when some holes have radius \(h_n\) and the others \(h_{2n}\). These are the compactness arguments of [ 248 , Section 7 ] , lines 515–520 and 534–538, in units of \(n\), with the intermediate configurations of lines 538–540.
In units of \(s\) about a grid corner, a block guide is the block guide of side one whose labels are those of a fixed square window of blocks, so it is an injective relabelling of one of finitely many templates, one for each label-equality type of the window. Apply Lemma 27.126.30 and Lemma 27.126.32 for the radius ratios; when \(n{\lt}t\) the radius \(h_n=\epsilon _0n\) and \(h_{2n}=\epsilon _0\min (2n,t)\) lie between \(\epsilon _0n\) and \(2\epsilon _0n\). Since \(h_n\le h_{2n}\), a configuration with some holes resized has larger holes than the one with all radii \(h_n\).
There is a ratio \(\nu {\gt}0\), depending only on the fixed constants, such that for every generic displacement \(v\) the conclusions of Theorems 27.126.38 and 27.126.39 hold for every labelling that reads the guides concerned after \(v\), with holes at its own true vertices. These are the source’s sampled guides, by Lemma 27.126.19; the point-treated guides are also constant off finitely many lines with normals in \(\mathcal N\) near every point, so the same holds for them. As in [ 248 , Section 7 ] , lines 73–77 and 559–563, the ratio is chosen before the displacement, which is chosen only after all the guides are specified.
A reading takes only values of the labelling. If a guide agrees on an open square about \(c\) with \(u\mapsto \sigma (T((u-c)/s))\) for a template \(T\), then every reading of the guide after \(v\) agrees there with \(u\mapsto \sigma (T'((u-c)/s))\) for the reading \(T'\) of \(T\) after \(v\), since \((u+\varepsilon v-c)/s=(u-c)/s+(\varepsilon /s)v\). The templates of the point treatment and of the block guides are eventually constant along rays, so they have readings after \(v\). They are also constant off walls on lines with normals in the finite set \(\mathcal N\). Let \(v\) and \(w\) lie strictly on the same side of each line through the origin with normal in \(\mathcal N\). For a point \(p\) and small \(\varepsilon {\gt}0\), the segment from \(p+\varepsilon v\) to \(p+\varepsilon w\) stays near \(p\) and avoids the walls that miss \(p\); a wall through \(p\) with normal \(N\) meets it only where \(N\cdot ((1-a)v+aw)=0\) for some \(a\in [0,1]\), which is impossible because \(N\cdot v\) and \(N\cdot w\) have the same sign. So the template is locally constant along the segment, hence constant on it, and its readings after \(v\) and after \(w\) agree. A template therefore has at most one reading for each choice of sides of the finitely many wall directions, and only finitely many readings after generic displacements. Apply the compactness argument of Lemma 27.126.30 to these finitely many readings of the finitely many templates.
At level \(n=1\), a closed square of radius less than \(1/2\) about a grid corner contains no cell center. With \(\epsilon _0{\lt}1/4\) the outer hole squares, of radius \(2\epsilon _0\), contain no lattice site, so their encodings are trivial, as in [ 248 , Section 7 ] , lines 669–672.
A half-odd number is at distance at least \(1/2\) from every integer.
For \(N=2^k\) the hierarchy consists of the aligned squares of \([0,N]^2\) of side \(2^j\), \(0\le j\le k\), indexed by \(j\) and two block indices below \(2^{k-j}\); the junctions at scale \(j\) are the grid corners, indexed by two integers at most \(2^{k-j}\). A square of side \(2^j\) with block index \(r\) has address coordinate \(2^jr+2^{j-1}\) for \(j\ge 1\) and \(r\) for \(j=0\), the anchor coordinate of Definition 27.131.1.
The hierarchy has \((4^{k+1}-1)/3\) squares, and three times the number of junctions over all scales is at most \(4^{k+2}\). If \(2^k{\lt}2L\), the hierarchy has fewer than \(16L^2/3\) squares, and \(M\) times the number of squares plus junctions is at most \(80ML^2/3\), and for \(L\ge 1\) at most \(80ML^3/3\). This is the count behind a schedule using at most \(M\) changes for each repainting of a square and each junction resize; the schedule of changes and gates of the protocol is not constructed here.
The scale \(j\) contributes \(4^{k-j}\) squares and \((2^{k-j}+1)^2\le 4\cdot 4^{k-j}\) junctions, and \(3\sum _{i{\lt}k+1}4^i+1=4^{k+1}\) by induction. Finally \(4^k=(2^k)^2{\lt}4L^2\).
In each coordinate: an address coordinate lies in \([2^jr,2^jr+2^j)\), hence below \(2^k\). If the intervals \([2^jr,2^j(r+1)]\) and \([2^{j'}r',2^{j'}(r'+1)]\) are within distance \(g\) and \(j'\le j+1\), the address coordinates differ by less than \(g+3\cdot 2^j\). In particular, for two blocks of side \(2^j\) with equal or adjacent indices, each block or its parent of side \(2^{j+1}\) has an address coordinate within \(3\cdot 2^{j+1}\) of the other’s. The intervals of side \(b\) meeting \([V,U]\) number at most \(\lfloor U/b\rfloor +2-\lfloor V/b\rfloor \). With Lemma 27.126.14, these are ingredients of item 3 and of the bounded lifetime participation in [ 248 , Proposition 7.1 ] ; the gates and parties of the protocol are not treated here.
The first two claims follow from the position of the address in its block and \(2^{j'}\le 2\cdot 2^j\). For adjacent blocks, the interval of a block or of its parent contains the block’s own interval, and the two blocks’ intervals touch, so the second claim applies with \(g=0\). For the count, \(r\mapsto r+1\) maps the meeting blocks injectively into \([\lfloor V/b\rfloor ,\lfloor U/b\rfloor +1]\).
Fix a block of side \(2^j\) with block indices \((r,s)\) and a constant \(c\). A square of the hierarchy of scale \(j'\) is near the block if \(j'\in \{ j-1,j\} \) and, in each coordinate, its closed interval is within distance \(c\, 2^j\) of the closed interval of the block. A junction of scale \(j'\) is near the block if \(j'\in \{ j-1,j\} \) and each of its two coordinates is within distance \(c\, 2^j\) of the closed interval of the block in that coordinate.
For every depth \(k\), every block and every constant \(c\), at most \(2(4c+4)^2\) squares and at most \(2(4c+4)^2\) junctions of the hierarchy are near the block. Consequently, if every repainting of a square and every junction resize consists of at most \(G\) gates, a party all of whose gates belong to repaintings and junction resizes near its block participates in at most \(4G(4c+4)^2\) gates. This is the final count of the bounded lifetime participation in [ 248 , Proposition 7.1, item 2 ] (lines 611–618): a party of side \(m\) occurs in linked gates only at levels \(m\) and \(m/2\), and at these scales there are only a fixed number of repaintings or junctions within distance \(O(m)\) of its block. That a block party occurs only in such gates is the hypothesis here; the gates of the protocol are not constructed.
In one coordinate, with \(b=2^{j'}\ge 2^{j-1}\) and \(g\le c\, 2^j\), the blocks of side \(b\) meeting \([2^jr-g,2^j(r+1)+g]\) number at most \(\lfloor U/b\rfloor +2-\lfloor V/b\rfloor \) by Lemma 27.126.44, and \(b\lfloor U/b\rfloor \le U\le V+2^j+2g{\lt}b(\lfloor V/b\rfloor +1)+b(4c+2)\) bounds this by \(4c+4\). A junction coordinate within distance \(g\) of the interval lies in such a block. There are at most two scales \(j'\in \{ j-1,j\} \), so the near squares and the near junctions number at most \(2(4c+4)^2\) each. Finally, the gates of the party are distributed over the at most \(4(4c+4)^2\) near repaintings and junction resizes, at most \(G\) to each.
Index the block parties by their scale and block indices \((j,r,s)\). At level \(2^{j'}\) the block of side \(2^{j'}\) with indices \((q,t)\) is owned by its own party \((j',q,t)\) and by the party \((j'+1,\lfloor q/2\rfloor ,\lfloor t/2\rfloor )\) of its parent; these are its new and its old label. The repainting of the square of scale \(j'\) with indices \((r',s')\) uses the owners of the blocks whose indices differ from \((r',s')\) by at most one in each coordinate. The resize at the grid corner \((2^ju,2^jv)\) uses the parties of the four blocks of side \(2^j\) at the corner.
Label the blocks by an injective assignment of parties, none of them the placeholder. At the start and end of each repainting of level \(2^{j'}\), when every block carries its old or its new label, a block party that labels a block of side \(2^{j'}\) labels a block of the root square and owns it; in particular it has scale \(j'\) or \(j'+1\). At the start of the level only parties of scale \(j'+1\) occur, and once every block of the root square is repainted only parties of scale \(j'\). At these times the block party labelling a block adjacent to a square, or the square itself, is used by the repainting of that square; this covers the labels \(A\), \(B\), \(C_e\) of its edge constructions and the labels at its corners. At the end of the pass to level \(2^j\), the block parties incident at a grid corner of the block guide, among them its tag holder, are used by the resize of the hole there. This is the invariant on active labels in [ 248 , Section 7 ] , lines 597–609, at the transition times between repaintings, where the guide is a block guide, together with the block-guide part of lines 585–594. The labels and tags inside a repainting (the edge and point diagrams, the auxiliary initial owners, the tag updates by corner rewrites) are not covered, because the repainting gates are not constructed.
A block label of the level is the placeholder, the new label of the block, or the old label of its parent, and the new labels do not occur at the start nor the old ones at the end of the level; injectivity of the party assignment identifies the party. Near a grid corner \((nu,nv)\), within distance \(n/2\), every point lies in one of the four blocks with indices \(u-1\) or \(u\) and \(v-1\) or \(v\), so every incident label is one of their labels.
A block party used by a repainting or by a junction resize has the scale of the event or the next larger one, and the event is near its block with constant \(c=0\). Consequently, if every repainting and every junction resize consists of at most \(G\) gates and each gate involves only parties its event uses, every block party participates in at most \(64G\) gates. This is the bounded lifetime participation for block parties in [ 248 , Proposition 7.1, item 2 ] , lines 597–618, under the stated hypothesis on the gates; the gates of the protocol are not constructed.
A party of side \(2^j\) owning a block adjacent to a square of scale \(j\) has index within one of the square’s in each coordinate, so the closed intervals touch. For a square of scale \(j-1\), a block adjacent to it has a parent with index \(\lfloor q/2\rfloor \), and \(2\lfloor q/2\rfloor \le q\le 2\lfloor q/2\rfloor +1\) gives touching intervals again. A corner of a block lies in its closed interval. Theorem 27.126.46 with \(c=0\) bounds the number of gates by \(4G\cdot 4^2=64G\).
The patch data of a candidate small-patch rewrite of one sheet consists of its old and new guide, the centres of finitely many patches with a common inner radius \(u\), and the centres of its affected old and new holes with one outer radius covering each of them. Its footprint is the union of the outer patch squares, of radius \(2u\), and of the outer squares of the affected holes. Its parties are the labels of the old and of the new guide on the footprint, and the tag holders of both guides at the affected holes; these are the raw owners and the affected tag owners named in condition 4 of the small-patch rewrite lemma in [ 248 , Section 6 ] . Conditions 1–3 of that lemma, the two-owner tests outside the old and new inner holes and the avoidance of the untouched hole footprints, are not part of the patch data.
The patch data is that of the resize at the grid corner \((2^ju,2^jv)\) when both guides are the block guide uniform on blocks of side \(2^j\) at the end of the pass to level \(2^j\), its only affected hole is at the corner, and its footprint lies within sup distance \(2^j\) of the corner. It is that of the direct repainting of the square \(S\) with indices \((r,s)\) and side \(2^{j'}\) when its old guide is the block guide of level \(2^{j'}\) at a transition time, its new guide is the one in which \(S\) also carries its new label, its affected holes lie at corners of \(S\), and its footprint lies within sup distance \(2^{j'}\) of the closed square \(S\).
Let \(n{\gt}0\). A point within sup distance \(n\) of the grid corner \((nu,nv)\) lies in a block with indices \(u-1\) or \(u\) and \(v-1\) or \(v\). A point within sup distance \(n\) of the closed block with indices \((r,s)\) lies in a block whose indices differ from \((r,s)\) by at most one in each coordinate.
If \(na-n{\lt}y{\lt}nb+n\) then \(a-1{\lt}y/n{\lt}b+1\), so \(a-1\le \lfloor y/n\rfloor \le b\). Apply this to each coordinate, with \(a=b=u\) near a corner and with \(a=r\), \(b=r+1\) near the closed block.
Label the blocks by an injective assignment of parties, none of them the placeholder. If the outer radius of the affected holes is positive, the parties of the patch data are labels of its old or new guide on its footprint. Every block party of the patch data of the resize at a grid corner is one of the four block parties at the corner, and every block party of the patch data of the direct repainting of a square is used by the repainting of that square. This is “Resizing uses the current block labels near its junction” and the locality of the direct repaintings in [ 248 , Section 7 ] , lines 500–541 and 585–594.
An outer hole square of positive radius is a neighbourhood of its centre, so an incident label there, in particular a tag holder, is a label on the footprint. A point of the footprint of a resize lies in one of the four blocks at the corner, and a point of the footprint of a direct repainting lies in a block adjacent to the square or in the square itself, by Theorem 27.126.51. The party labelling such a block is identified by Theorem 27.126.48; at the end of a pass only the parties of the blocks themselves occur.
Let \(\epsilon _0,\nu ,t{\gt}0\) with \(2\epsilon _0+3\nu {\lt}1\). The resize at the grid corner \(c=(2^ju,2^jv)\) of the block guide at the end of the pass to level \(2^j\), from the hole radius \(\epsilon _0\min (2^j,t)\) to \(\epsilon _0\min (2^j/2,t)\), has patch data whose patches form a square net of mesh \(u=\nu 2^j\). There are at most \((2(2\epsilon _0+\nu )/\nu +1)^2\) patches, and their inner squares cover both outer squares of the hole. The direct repainting of a square \(S\) of side \(n=2^{j'}\) at a transition time of level \(n\), with affected holes at the four corners of \(S\) and hole radius \(\epsilon _0\min (n,t)\), has patch data whose patches form a square net of mesh \(\nu n\). There are at most \((2(1/2+2\epsilon _0+\nu )/\nu +1)^2\) patches, and their inner squares cover the closed square \(S\) and the outer squares of the four holes. Both counts are independent of the scale. These are the fine nets of the direct repainting and of the hole resizing in [ 248 , Section 7 ] , lines 500–541. Conditions 1–3 of the small-patch rewrite lemma are not verified for these nets. The source repaints directly only for \(n\le K_0t\); that restriction enters the two-owner condition on the patches, not the nets or their parties.
The hole radius at scale \(m\) is at most \(\epsilon _0m\), and the new radius of a resize is at most the old one. The set to be covered lies in the closed square of radius \(2\epsilon _0\, 2^j\) about the corner, or of radius \((1/2+2\epsilon _0)n\) about the centre of \(S\), and Lemma 27.126.31 gives a net of mesh \(u\) with the stated number of patches and with centres within \(u\) of the covered set. An outer patch square is then within \(3u\) of a point of the covered set, and every point of the covered set is within \(2\epsilon _0m\) of the corner or of the closed square \(S\). Since \(2\epsilon _0+3\nu {\lt}1\), the footprint lies within sup distance \(2^j\) of the corner, or \(n\) of the closed square \(S\).
Let the gates of a schedule on the hierarchy belong to repaintings and junction resizes, at most \(G\) to each event, and let each gate carry the patch data of its event: a direct repainting of its square or the resize at its junction, at a level \(j\) whose root square \([0,2^k]^2\) has \(2^{k-j}\) blocks per side. If a block party participates in a gate only when it is a party of the gate’s patch data, it participates in at most \(64G\) gates. The bound is conditional on this hypothesis, which is the conclusion of the small-patch rewrite lemma and is not derived here from it. This is a conditional form of the bounded lifetime participation of [ 248 , Proposition 7.1, item 2 ] for schedules consisting of direct repaintings and junction resizes; the large-scale repaintings, with their births, deaths, exchanges and point treatments, are not covered.
27.127 Finite incidence in distributed PEPS compression
The finite counting statements below are those used in the proof of [ 248 , Theorem 5.2 ] , September 24, 2026, 04-compression.tex, lines 342–381 and 565–588.
For an index set of source positions, let \(e\mapsto (a_e,b_e)\) assign an ordered pair of parties. Define \(E_e=\{ a_e,b_e\} \). Thus \(p\in E_e\) if and only if \(p=a_e\) or \(p=b_e\); coincident endpoints are counted once.
For a finite set \(S\) of source positions, define \(A_S=\bigcup _{e\in S}E_e\). Membership means that \(p\) is an endpoint of some position \(e\in S\). No monomial label is selected by \(S\).
Let \(G\) be a finite set of nonprivate gate occurrences and let \(P_g\) be the finite participant set of occurrence \(g\). Identifiers distinguish repeated occurrences of the same operation. Define \(G(p)=\{ g\in G:p\in P_g\} \) using the entire gate list.
For a finite party set \(A\), define \(\Gamma (A)=\bigcup _{p\in A}G(p)\). Thus \(g\in \Gamma (A)\) precisely when \(g\in G\) and some \(p\in A\) participates in \(g\).
Every source position satisfies \(|E_e|\le 2\).
A set containing two specified elements has cardinality at most two.
For every finite correction set \(S\), \(|A_S|\le 2|S|\).
The cardinality of a finite union is at most the sum of the constituent cardinalities, so \(|A_S|\le \sum _{e\in S}|E_e|\le 2|S|\).
If \(|G(p)|\le b\) for every \(p\in A\), then \(|\Gamma (A)|\le b|A|\).
Use \(|\Gamma (A)|\le \sum _{p\in A}|G(p)|\le b|A|\).
If \(|G(p)|\le b\) for every \(p\in A_S\), then \(|\Gamma (A_S)|\le 2b|S|\). This includes touching gates with no selected position of their own.
The two finite-union bounds give \(|\Gamma (A_S)|\le b|A_S|\le 2b|S|\).
Let \(o(e)\) be the owner of a source position. If \(e\in S\), \(o(e)\in G\), and \(a_e\in P_{o(e)}\), then \(o(e)\in \Gamma (A_S)\).
The party \(a_e\) belongs to \(A_S\) and participates in \(o(e)\), giving the required witness.
If \(g\in G\setminus \Gamma (A)\), then \(P_g\cap A=\varnothing \).
A common party would witness \(g\in \Gamma (A)\), a contradiction.
For a finite party set \(P\), let \(U(P)\) be the unordered pairs of distinct elements of \(P\). Equivalently, an unordered pair belongs to \(U(P)\) when both its endpoints lie in \(P\) and it is not diagonal. The pair \(\{ p,q\} \) belongs to \(U(P)\) precisely when \(p,q\in P\) and \(p\ne q\).
For a specified root \(r\), define the disjoint label set \(T(P,r)=(P\setminus \{ r\} )\sqcup U(P)\). The first summand indexes star links and the second indexes sample links.
For a star label \(q\) put \(F(r,q)=\{ r,q\} \). For a sample label \(\{ p,q\} \) put \(F(r,\{ p,q\} )=\{ p,q\} \). The two cases retain their disjoint summand labels.
For chosen roots \(r_g\), define \(\mathcal L_g=\{ (g,t):t\in T(P_g,r_g)\} \). The gate tag preserves distinct occurrences even if their participant sets and operations agree.
Define \(\mathcal L=\bigcup _{g\in G}\mathcal L_g\). A pair \((g,t)\) belongs to \(\mathcal L\) exactly when \(g\in G\) and \(t\in T(P_g,r_g)\). Parallel links remain distinct.
For a tagged link \(\ell =(g,t)\) define \(V(\ell )=F(r_g,t)\).
Define \(I(p)=\{ \ell \in \mathcal L:p\in V(\ell )\} \). Parallel links are counted separately.
Define \(T_p(P,r)=\{ t\in T(P,r):p\in F(r,t)\} \).
Define \(\mathcal L_g(p)=\{ \ell \in \mathcal L_g:p\in V(\ell )\} \).
If \(|P|=n\), then \(|U(P)|=\binom n2\).
The off-diagonal ordered pairs map two-to-one onto unordered pairs, giving \(|U(P)|=n(n-1)/2=\binom n2\).
If \(p\in P\), then the sample labels containing \(p\) are exactly the image of \(P\setminus \{ p\} \) under \(q\mapsto \{ p,q\} \). Consequently their number is \(|P|-1\).
The map is injective. Every distinct unordered pair containing \(p\) has a unique other endpoint in \(P\setminus \{ p\} \).
If \(r\in P\) and \(|P|=n\), then \(|T(P,r)|=n-1+\binom n2\). If \(n\le b\), this is at most \(b-1+\binom b2\).
Add the cardinalities of the disjoint star and sample summands. The two cardinality terms are monotone in \(n\).
If \(r,p\in P\), then \(|T_p(P,r)|\le 2(|P|-1)\).
The star contribution is at most the whole star size \(|P|-1\). The sample contribution is exactly \(|P|-1\).
If \(r\in P\) and \(t\in T(P,r)\), then \(F(r,t)\subseteq P\).
A star label lies in \(P\setminus \{ r\} \), and a sample label has both endpoints in \(P\).
For every \(t\in T(P,r)\), \(|F(r,t)|=2\). Root membership in \(P\) is not needed for this count.
Star labels exclude the root; sample labels are off-diagonal.
For every gate occurrence \(g\) and party \(p\), \(|\mathcal L_g|=|T(P_g,r_g)|\) and \(|\mathcal L_g(p)|=|T_p(P_g,r_g)|\).
The injection \(t\mapsto (g,t)\) preserves cardinality, including after filtering by party incidence.
Suppose \(r_g\in P_g\) for every \(g\in G\). If \(\ell =(g,t)\in \mathcal L\), then \(V(\ell )\subseteq P_g\). In particular any two endpoints \(p,q\in V(\ell )\) participate in a common original occurrence \(g\in G\).
The gate tag witnesses \(g\in G\), and gate-local endpoint locality supplies both participant memberships.
Every \(\ell \in \mathcal L\) satisfies \(|V(\ell )|=2\).
Apply the gate-local count to the label certified by membership in \(\mathcal L\).
Suppose \(r_g\in P_g\) for every \(g\in G\). Then \(I(p)\subseteq \bigcup _{g\in G(p)}\mathcal L_g(p)\).
For an incident link the original gate tag belongs to \(G\), and locality puts \(p\) in that gate’s participant set.
Suppose \(r_g\in P_g\) and \(|P_g|\le b\) for every \(g\in G\). For a party \(p\) with \(|G(p)|\le b\), the constructed link family satisfies \(|I(p)|\le 2b(b-1)\).
Each participating gate contributes at most \(2(b-1)\) links incident to \(p\). Taking their union gives \(|I(p)|\le \sum _{g\in G(p)}|\mathcal L_g(p)|\le 2b(b-1)\).
For a finite label set \(\Xi \) and nonnegative coefficient weights \(a_\xi \), put \(W(\Xi ,a)=(\sum _{\xi \in \Xi }a_\xi )^2\).
The coefficient cost satisfies \(W(\Xi ,a)=\sum _{\xi \in \Xi }\sum _{\zeta \in \Xi }a_\xi a_\zeta \).
Expand the product of the two finite sums.
For finite gate label sets \(\Xi _g\), nonnegative weights \(a_{g,\xi }\) and intended physical dimensions \(n_p\in \mathbb N\), define
Exterior gate coefficients and exterior physical dimensions are absent from these products.
Suppose \(B,d\ge 1\), \(|G(p)|\le b\) for every \(p\in A_S\), \(\sum _{\xi \in \Xi _g}a_{g,\xi }\le B\) for every \(g\in \Gamma (A_S)\), and \(n_p\le d\) for every \(p\in A_S\). Then \(\mathcal C(S)\le (B^{4b}d^4)^{|S|}\).
The coefficient product is at most \(B^{2|\Gamma (A_S)|}\le B^{4b|S|}\), and the physical-entry product is at most \(d^{2|A_S|}\le d^{4|S|}\). Multiply the bounds and collect the common exponent \(|S|\).
Let \(L,B,d,C,D\) be nonnegative real numbers and \(a,c,b\in \mathbb N\). If \(B\le CL^a\) and \(d\le DL^c\), then \(B^{4b}d^4\le C^{4b}D^4L^{4ba+4c}\).
Raise each monomial bound to its nonnegative integer power and multiply; the powers of \(L\) add.
27.128 Exact operator contraction for distributed PEPS compression
This is the finite operator representation used in [ 248 , Theorem 5.2 ] , September 24, 2026, 04-compression.tex, lines 565–579. Let \(P\) be a finite party set, \(G\) a finite set of gate occurrences, \(P_g\subseteq P\) the participant set of \(g\), and \(r_g\in P_g\) its chosen root. Let \(\Lambda _g\) be any finite branch alphabet, possibly empty, and \(J_k=\{ 0,\ldots ,k-1\} \) for \(k\in \mathbb N\). The physical input and output alphabets \(Y_p,X_p\) may be arbitrary sets; the finite-dimensional density application uses finite ones.
The active gate occurrences are the elements of the finite set \(G\). Occurrences are distinguished even when their operations and participant sets coincide.
The link family \(\mathcal L\) is the disjoint union over \(g\in G\) of the star labels \(q\in P_g\setminus \{ r_g\} \) and the unordered distinct pairs in \(P_g\). Every link retains both its gate occurrence and its star or sample kind.
For \(p\in P\), let \(G_p=\{ g\in G:p\in P_g\} \). These are precisely the gate labels visible at \(p\).
Let \(S=\bigsqcup _{g\in G}\{ \{ p,q\} \subseteq P_g:p\ne q\} \), retaining the gate occurrence in each position. Put \(S_p=\{ s\in S:p\in s\} \). Positions do not depend on chosen monomial branches.
For \(g\in G\) and \(q\in P_g\setminus \{ r_g\} \), the star edge is \((g,\mathrm{star},q)\in \mathcal L\). For \(s=(g,\{ p,q\} )\in S\), the sample edge is \((g,\mathrm{sample},\{ p,q\} )\in \mathcal L\).
Orient a star from \(r_g\) to \(q\). Choose an ordered representative \((s^-,s^+)\) of each sample pair and orient its edge accordingly. For every \(\ell \in \mathcal L\), write \(u_\ell ,v_\ell \) for these endpoints. Then \(\{ u_\ell ,v_\ell \} \) is exactly the already specified party endpoint set of \(\ell \).
Put \(I_p=\{ \ell \in \mathcal L:p\in \{ u_\ell ,v_\ell \} \} \). The star \((g,\mathrm{star},q)\) belongs to \(I_p\) when \(p=r_g\) or \(p=q\); either case gives \(g\in G_p\). Every \(g\) belongs to \(G_{r_g}\). A sample position in \(S_p\) supplies its sample edge in \(I_p\).
Assign the alphabet \(D_\ell =\Lambda _g\) to a star edge of \(g\) and \(D_\ell =J_k\) to a sample edge. These alphabets are finite, including when empty. Private memory and garbage coordinates are absent from \(D_\ell \).
A global gate assignment is \(\kappa \in \prod _{g\in G}\Lambda _g\), and a local assignment at \(p\) is \(\xi _p\in \prod _{g\in G_p}\Lambda _g\). The restriction of \(\kappa \) to \(G_p\) is denoted \(\kappa _p\).
A global virtual assignment is \(\eta \in \prod _{\ell \in \mathcal L}D_\ell \). Its restriction \(\eta _p\) to \(I_p\) is a party-local virtual assignment.
A global sample assignment is \(j\in J_k^S\), whose restriction to \(S_p\) is \(j_p\). A local virtual assignment \(\eta _p\) supplies the sample assignment \(j(\eta _p)_s=\eta _p(g,\mathrm{sample},s)\) for \(s\in S_p\).
The pair \((\eta _p,\xi _p)\) is consistent if \(\eta _p(g,\mathrm{star},q)=\xi _p(g)\) for every star edge incident to \(p\). No condition is imposed on sample or physical indices.
At each incident star both labels are \(\kappa _g\).
Let \(c_g:\Lambda _g\to \mathbb C\) be arbitrary deterministic coefficients. For a local assignment \(\xi _p\), put \(C_p(\xi _p)=\prod _{g:r_g=p}c_g(\xi _p(g))\). A gate with one participant has no star edge, but its label and coefficient are still present at its root.
Let \(X_p,Y_p\) be the output and input alphabets. Supply arbitrary rectangular matrices \(T_p(\xi _p,j_p):Y_p\to X_p\). Define the local tensor by
No positivity, Hermiticity, normalization or bound on the entries of \(T_p\) is required.
A globally consistent configuration is a pair \((\eta ,(\xi _p)_{p\in P})\) such that \((\eta _p,\xi _p)\) is consistent at every party.
For a globally consistent configuration, define \(\kappa _g=\xi _{r_g}(g)\) and let \(j_s\) be the virtual label on the actual sample edge of \(s\).
In a globally consistent configuration, \(\xi _p(g)=\kappa _g\) whenever \(g\in G_p\).
If \(p=r_g\) this is the definition of \(\kappa _g\). Otherwise \((g,\mathrm{star},p)\) is an actual star edge, and consistency at its two endpoints gives \(\xi _p(g)=\eta _{(g,\mathrm{star},p)}=\xi _{r_g}(g)\).
If \((\kappa ,j)\) is recovered from a globally consistent configuration \((\eta ,(\xi _p))\), then \(\eta (\kappa ,j)=\eta \).
A star edge is determined by consistency at its root, and a sample edge is determined by the definition of \(j_s\).
The map \((\kappa ,j)\mapsto (\eta (\kappa ,j),(\kappa _p)_{p\in P})\) is a bijection from \((\prod _{g\in G}\Lambda _g)\times J_k^S\) onto the globally consistent configurations. Its inverse is recovery at the participating roots and sample links. The two inverse identities follow from root agreement and recovery of every virtual label. Singleton gates and empty alphabets are included.
Give each edge independent labels at its two endpoints. At a party \(p\), read the first endpoint when \(u_\ell =p\) and the second otherwise. This assigns a local virtual label to every \(\ell \in I_p\).
If both endpoints of each edge carry the same global label \(\eta _\ell \), then the labels read at \(p\) are exactly \(\eta _p\).
Either endpoint choice reads \(\eta _\ell \).
By the definition of the shared assignment, the sample edge of \(s\) reads \(j_s\).
For every global gate assignment \(\kappa \), \(\prod _{p\in P}C_p(\kappa _p)=\prod _{g\in G}c_g(\kappa _g)\).
Regroup the gate product by the fibers of \(g\mapsto r_g\). Every active gate has precisely one root.
Apply the endpoint reader to a party’s independent endpoint configuration and evaluate \(A_p\) on the resulting local virtual labels. The physical argument remains the output-input pair \((x_p,y_p)\).
Use identity bond matrices on every \(D_\ell \), the endpoint form of \(A_p\), and physical output-input pairs. The resulting contraction defines the rectangular operator \(\sigma :\prod _pY_p\to \prod _pX_p\).
The actual identity-bond contraction satisfies
The identity matrix on each edge forces its two endpoint labels to coincide. Reindex the remaining diagonal assignments by one label per edge; the endpoint reader then yields \(\eta _p\). No dimension factor is introduced.
The constructed operator has the exact entries
The identity permits empty branch alphabets, empty link families, singleton gates and \(k=0\).
The operator identity corresponding to (884) is
The tensor product here has entry \(\prod _pT_p(\kappa _p,j_p)_{x_p,y_p}\) and allows different input and output alphabets at each party.
Evaluate both sides at \((x,y)\), distribute the finite matrix sums and scalar multiplication, and apply the entrywise identity.
A star edge belonging to \(g\) has dimension \(|\Lambda _g|\), and every sample edge has dimension \(k\). These dimensions contain no private-memory or source-support factor.
Compute the cardinalities of the two cases in the definition of \(D_\ell \).
If \(|\Lambda _g|\le M\) for every \(g\in G\), then every virtual edge satisfies \(|D_\ell |\le \max \{ M,k\} \).
A star edge inherits the supplied bound at its tagged active gate; a sample edge has dimension exactly \(k\).
Let \(\Lambda _g=K_g\times B_g\) be the complete ordered ket/bra label pair. For coefficient functions \(a_g,b_g\) and a gate-owned weight \(w_g\), put \(c_g(\xi ,\zeta )=w_g a_g(\xi )\overline{b_g(\zeta )}\). Any averaging factor for a sample position is included once in its gate’s \(w_g\).
For arbitrary finite \(K_g,B_g\), the star shares the entire pair \((\xi _g,\zeta _g)\). The contracted operator is
Each gate-owned weight appears once, including at a singleton gate.
Specialize the rectangular matrix identity to the complete ket/bra alphabet and the stated coefficient function.
Multiplying the tensor \(A_p\) at one vertex \(p\) by \(s\in \mathbb {C}\) and leaving the other tensors unchanged multiplies the contraction \(\mathcal N_A(B)(\sigma )\) of Definition 27.32.7.3 by \(s\). The links and their dimensions are unchanged.
Every summand of the contraction in Definition 27.32.7.3 contains exactly one factor \(A_p\).
27.129 Whole-group tensor bound and simultaneous truncation
The results below formalize the whole-group tensor bound [ 248 , Lemma 6.2 ] , September 24, 2026, 05-frames.tex, lines 125–179, together with the two dimension-free matrix estimates used in its proof. All tensors have finitely many coordinates. For a matrix \(M\), write \(\lVert M\rVert _1\) for the sum of its singular values and \(\lVert M\rVert _2\) for its Hilbert–Schmidt norm.
For a finite rectangular complex matrix \(M\), the nuclear norm \(\lVert M\rVert _1\) is the sum of the singular values of \(M\).
For finite complex matrices \(A\) and \(B\) with the same column index set, \(\lVert AB^{\mathsf T}\rVert _1\le \lVert A\rVert _2\lVert B\rVert _2\).
Let \(w_i\) be an orthonormal basis of right singular vectors of \(M=AB^{\mathsf T}\), and let \(u_i\) be the normalized images \(Mw_i\) when these are nonzero. The \(i\)th singular value equals \(\langle u_i, A(B^{\mathsf T}w_i)\rangle =\langle A^{*}u_i, B^{\mathsf T}w_i\rangle \), so it is at most \(\lVert A^{*}u_i\rVert \, \lVert B^{\mathsf T}w_i\rVert \). Summing over \(i\) and applying the Cauchy–Schwarz inequality gives \(\lVert M\rVert _1\le (\sum _i\lVert A^{*}u_i\rVert ^2)^{1/2}(\sum _i\lVert B^{\mathsf T}w_i\rVert ^2)^{1/2}\). Since the \(u_i\) and \(w_i\) are orthonormal families, the two sums are at most \(\lVert A\rVert _2^2\) and \(\lVert B\rVert _2^2\).
Let \(M\) be a finite complex matrix and \(k\ge 0\) an integer. There is an orthonormal family of at most \(k\) vectors in the column space of coordinates of \(M\) whose orthogonal projector \(Q\) satisfies \((k+1)\lVert M-MQ\rVert _2^2\le \lVert M\rVert _1^2\). If the number of columns is at most \(k\), then \(MQ=M\).
Take \(Q\) to be the projector onto the first \(\min (k,n)\) right singular vectors, where \(n\) is the number of columns. Then \(\lVert M-MQ\rVert _2^2\) is the sum \(\sum _{j{\gt}k}s_j^2\) of the squares of the remaining singular values \(s_1\ge s_2\ge \cdots \ge 0\). Since the sequence decreases, \((k+1)s_{k+1}\le \sum _j s_j\) and \(\sum _{j{\gt}k}s_j^2\le s_{k+1}\sum _j s_j\). Multiplying gives \((k+1)\sum _{j{\gt}k}s_j^2\le (\sum _j s_j)^2=\lVert M\rVert _1^2\).
Let \((V,E)\) be a finite graph whose edges \(e\) have endpoints \(t(e),h(e)\in V\) and finite label sets \(D_e\). At each vertex \(v\) let \(A_v\) be a tensor with one leg for each edge incident to \(v\) and one group of open legs with finite configuration set \(P_v\). The coordinate contraction is
where \(\beta \) runs over labellings of all edges and \(\beta |_v\) is the restriction of \(\beta \) to the edges incident to \(v\). The norm of \(A_v\) is its Hilbert-space norm over all its legs.
Suppose that no edge joins a vertex to itself and every vertex tensor has norm at most one. Then the contraction \(Z\) has norm at most one. The same bound holds for the contraction of every vertex subset, with the open legs of the subset and the edges leaving it kept open.
Argue by induction on the vertex subset, adding one vertex at a time. Merging two disjoint vertex sets contracts all edges between them at once. After grouping their common indices, the merged tensor is the product of the two partial contractions, read as matrices from the remaining legs to the common edges, up to a transpose. Hence its norm is at most the product of the two norms, by \(\lVert AB\rVert _2\le \lVert A\rVert _2\lVert B\rVert _2\). A single vertex has the norm of its tensor, since it has no self-contraction. The empty set has norm one. Graphs with cycles are allowed.
Under the hypotheses of Theorem 27.129.5, for every set \(T\) of vertices, the matrix reshaping of \(Z\) with rows the open-leg groups of \(T\) and columns those of the complement has nuclear norm at most one.
Contract \(T\) and its complement internally, giving tensors \(A\) and \(B\) of norm at most one by Theorem 27.129.5. Group all edges crossing the bipartition into one index. The reshaping of \(Z\) is then \(AB^{\mathsf T}\), so Lemma 27.129.2 gives \(\lVert AB^{\mathsf T}\rVert _1\le \lVert A\rVert _2\lVert B\rVert _2\le 1\). No dimension enters.
Let \(Z\) be a tensor on \(\bigotimes _{v\in V}\mathbb C^{P_v}\) of norm at most one whose reshaping across each single factor \(v\) and its complement has nuclear norm at most one. Let \(G\subseteq V\) contain every \(v\) with \(|P_v|{\gt}1\), and let \(k\ge 1\). Then there are orthonormal families of at most \(k\) vectors in the respective factors whose tensor-product projection sends \(Z\) to \(Z_k\) with \(\lVert Z-Z_k\rVert \le |G|/\sqrt{k}\). In those families \(Z_k\) has at most \(k^{|G|}\) product terms, with coefficients of absolute value at most one.
For each factor \(v\), Lemma 27.129.3 applied to the single-factor reshaping gives a projection \(Q_v\) of rank at most \(k\) with \(\lVert Z-Q_v Z\rVert \le 1/\sqrt{k+1}\le 1/\sqrt{k}\), and with \(Q_v Z=Z\) when \(|P_v|\le 1\). The projections act on disjoint tensor factors, so they commute and each has operator norm at most one. Telescoping their product over \(G\) bounds \(\lVert Z-Z_k\rVert \) by the sum of the separate errors. The tensor \(Z_k\) has norm at most one, so each of its coefficients in the orthonormal product basis has absolute value at most one.
Let a finite graph carry at each vertex a tensor of norm at most one, contract its internal edges by the coordinate pairing, and suppose there are no edges from a vertex to itself. Group the open legs according to their original vertices. The resulting tensor \(Z\) has norm at most one and has nuclear norm at most one across every bipartition into whole open-leg groups. If there are \(g\) nonempty open-leg groups, then for every integer \(k\ge 1\) there are subspaces of dimension at most \(k\) in the respective group spaces whose tensor-product projection sends \(Z\) to a tensor \(Z_k\) with \(\lVert Z-Z_k\rVert \le g/\sqrt{k}\). In orthonormal bases of those subspaces, \(Z_k\) has at most \(k^g\) product terms, with coefficients of absolute value at most one.
The norm and nuclear bounds are Theorems 27.129.5 and 27.129.6; in particular every single-group reshaping has nuclear norm at most one. A vertex with no open legs has a one-point configuration set, so the set of \(g\) vertices owning a nonempty open-leg group contains every factor with more than one configuration. Lemma 27.129.7 then gives the truncation.
Consider a tensor network in which each vertex carries one leg for each endpoint incidence and each edge carries a matrix. If every edge matrix is the identity, no edge joins a vertex to itself, and every vertex tensor has norm at most one, then the conclusions of Theorem 27.129.8 hold for the network contraction.
With identity edge matrices the network contraction is the coordinate contraction. Without edges from a vertex to itself, the endpoint incidences at a vertex are exactly its incident edges, so the vertex norms agree, and the whole-group tensor bound applies.
27.130 Extracting one ket column from an operator network
The statements below formalize [ 248 , Lemma 8.1 ] , September 24, 2026, 07-assembly.tex, lines 67–94 (statement lines 67–78, proof lines 79–94). Throughout, \(\Omega \) is a unit vector in a finite-dimensional space with a distinguished orthonormal basis \(\{ |z\rangle \} \), and \(\sigma \) is an arbitrary operator on that space; neither positivity nor Hermiticity of \(\sigma \) is assumed. The trace norm is independent of the order in which the basis is enumerated, so it may be evaluated along any enumeration.
For every operator \(E\) on a finite-dimensional space with orthonormal basis \(\{ |z\rangle \} \),
The left side is the sum of the squared moduli of all matrix entries of \(E\), which is the squared Hilbert–Schmidt norm \(\lVert E\rVert _2^2\). The Hilbert–Schmidt norm is at most the trace norm.
Let \(f_z\) be vectors and \(a_z\) complex numbers indexed by a finite set, with \(\sum _z\lVert f_z\rVert ^2\le \eta ^2\) and \(\sum _z|a_z|^2=1\). Then some index \(z\) has \(a_z\ne 0\) and \(\lVert f_z\rVert ^2\le \eta ^2|a_z|^2\). No lower bound on \(|a_z|\) is required, and \(\eta =0\) is allowed.
Let \(S\) be the set of indices with \(a_z\ne 0\); it is nonempty and \(\sum _{z\in S}|a_z|^2=1\). If every \(z\in S\) had \(\lVert f_z\rVert ^2{\gt}\eta ^2|a_z|^2\), summing over \(S\) would give \(\eta ^2{\lt}\sum _{z\in S}\lVert f_z\rVert ^2\le \eta ^2\), a contradiction.
Let \(\lVert \Omega \rVert =1\) and \(\eta {\lt}1\). If \(\lVert w-\Omega \rVert \le \eta \), then \(w\ne 0\) and \(\lVert w/\lVert w\rVert -\Omega \rVert \le 2\eta \). Consequently, if \(v=f+a\Omega \) with \(a\ne 0\) and \(\lVert f\rVert \le \eta |a|\), then \(v\ne 0\) and some real \(\theta \) satisfies \(\lVert v/\lVert v\rVert -e^{i\theta }\Omega \rVert \le 2\eta \).
The reverse triangle inequality gives \(|\lVert w\rVert -1|\le \eta {\lt}1\), so \(w\ne 0\). Then \(\lVert w/\lVert w\rVert -w\rVert =|1-\lVert w\rVert |\le \eta \), and the triangle inequality through \(w\) gives the bound \(2\eta \). For the second statement put \(w=v/a\), so that \(w-\Omega =f/a\) has norm at most \(\eta \). Since \(v/\lVert v\rVert =(a/|a|)\, w/\lVert w\rVert \), the phase \(e^{i\theta }=a/|a|\) transfers the bound.
Let \(\Omega \) be a unit vector and let \(\sigma \) be an operator with \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\). Then some basis column \(v=\sigma |z\rangle \) is nonzero, and some real \(\theta \) satisfies
This is the vector part of [ 248 , Lemma 8.1 ] , with exactly its hypotheses. The source bounds the minimum over \(\theta \); an explicit \(\theta \) is equivalent, since the minimum over the circle is attained.
Put \(E=\sigma -|\Omega \rangle \langle \Omega |\), \(f_z=E|z\rangle \) and \(a_z=\langle \Omega | z \rangle \). By Theorem 27.130.1, \(\sum _z\lVert f_z\rVert ^2\le \eta ^2\), and \(\sum _z|a_z|^2=\lVert \Omega \rVert ^2=1\). Theorem 27.130.2 gives \(z\) with \(a_z\ne 0\) and \(\lVert f_z\rVert \le \eta |a_z|\). The column satisfies \(\sigma |z\rangle =f_z+a_z\Omega \), and Theorem 27.130.3 concludes.
Consider a contraction on a directed multigraph with edge-dependent virtual alphabets \(X_e\) and bond matrices \(B_e\), whose physical index at each vertex \(v\) is an output-input pair \((s_v,r_v)\). Its contraction defines an operator \(\sigma \) with entries \(\sigma _{\tau \rho }\) indexed by output and input configurations. For an input configuration \(z\), the fixed-input network has, at each vertex \(v\), the tensor \((\eta ,s)\mapsto A_v(\eta ,(s,z_v))\), on the same multigraph with the same alphabets and bond matrices. Its contraction at \(\tau \) equals \(\sigma _{\tau z}\), so its contracted vector is the column \(\sigma |z\rangle \).
Let \(\sigma \) be the operator contracted from an operator network as in the preceding definition, with the same local physical space for output and input at each vertex, and let \(\Omega \) be a unit vector with \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\). Then for some product-basis input \(z\) the fixed-input network, on the same multigraph with the same virtual alphabets and bond matrices, contracts to the column \(\psi =\sigma |z\rangle \), which is nonzero and satisfies \(\lVert \psi /\lVert \psi \rVert -e^{i\theta }\Omega \rVert \le 2\eta \) for some real \(\theta \). This is [ 248 , Lemma 8.1 ] for this network model.
Apply the vector statement to \(\sigma \) and identify the selected column with the fixed-input contraction, coordinate by coordinate.
Let \(A\) be a tensor on a finite simple graph with physical dimension \(d^2\), each physical index read as an output-input pair in \(\{ 0,\dots ,d-1\} ^2\). Its coefficients define an operator \(\sigma \) on \((\mathbb {C}^d)^{\otimes V}\). For an input configuration \(z\), the fixed-input tensor has physical dimension \(d\) and, at each vertex \(v\), the components of \(A\) with input index \(z_v\). Its bond dimensions are those of \(A\) on every edge, and its state is the column \(\sigma |z\rangle \).
Let \(\sigma \) be the operator of an operator tensor \(A\) on a finite simple graph, and let \(\Omega \) be a unit vector with \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\). Then for some product-basis input \(z\) the fixed-input tensor has the bond dimensions of \(A\), and its state \(\psi =\sigma |z\rangle \) is nonzero and satisfies \(\lVert \psi /\lVert \psi \rVert -e^{i\theta }\Omega \rVert \le 2\eta \) for some real \(\theta \). This is [ 248 , Lemma 8.1 ] for this tensor model.
Apply the vector statement to \(\sigma \); fixing inputs leaves every bond dimension unchanged, and the state of the fixed-input tensor is the selected column.
The assembly of [ 248 , §8 ] (07-assembly.tex, lines 9–99) applies the column lemma to an operator network and then realizes the resulting ket network as a PEPS on the open \(L\times L\) square in the sense of [ 248 , §1.1 ] (00-introduction.tex, lines 20–34). The statements below instantiate the column lemma on both presentations of square-grid tensors from the exact square-grid contraction. An operator tensor on the open square is a square-grid tensor with physical dimension \(q^2\), each physical index read as an output-input pair in \(\{ 0,\dots ,q-1\} ^2\).
Let \(A_v\) be operator tensors on the open square with edge dimensions \(D_e\). Their contraction defines an operator \(\sigma \) with entries \(\sigma _{\tau \rho }\) indexed by output and input configurations. For an input configuration \(z\), the fixed-input tensors are \(A_v((s,z_v),\, \cdot \, )\) in the physical-first presentation and \(A_v(\, \cdot \, ,(s,z_v))\) in the virtual-first presentation, with the same edge dimensions \(D_e\) (in the virtual-first presentation, the same positive dimensions and hence the same maximum bond dimension). Their contraction is exactly the column \(\sigma |z\rangle \). On a \(3\times 3\) square, fixing the inputs closes the input leg of every site tensor with the basis vector \(|z_v\rangle \) and leaves every bond in place.
![\begin{tenkz}[rows={op,op,op}, cols=3, frame=plane,
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\(\quad \longmapsto \quad \) ![\begin{tenkz}[rows={op,op,op}, cols=3,
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\end{tenkz}](tenkz_svg/tenkz-db08096545db02bb.svg)
Fixing the inputs commutes with the transport of square-grid tensors to tensors on the open square graph, and the operator of a square-grid operator network equals the operator of its transported graph tensor.
Both sides of the first identity have the same dimension on every edge and the same components, coordinate by coordinate. The second identity is the equality of square-grid contractions applied at each pair of output and input configurations.
Let \(\sigma \) be the operator of an operator network on the open \(L\times L\) square with edge dimensions \(D_e\), and let \(\Omega \) be a unit vector with \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\). Then for some product-basis input \(z\) the fixed-input tensors, with the same edge dimensions \(D_e\), contract to the column \(\Phi =\sigma |z\rangle \), which is nonzero and satisfies \(\lVert \Phi /\lVert \Phi \rVert -e^{i\theta }\Omega \rVert \le 2\eta \) for some real \(\theta \). This is [ 248 , Lemma 8.1 ] on the open square, with exactly its hypotheses; \(\sigma \) need not be positive or Hermitian.
Apply the vector statement to \(\sigma \) and identify the selected column with the contraction of the fixed-input tensors.
Let an operator network on the open \(L\times L\) square have edge dimensions \(D_e\le B\), and let its operator \(\sigma \) satisfy \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\) for a unit vector \(\Omega \). Then for every \(\varepsilon \ge 2\eta \) there is a nonzero PEPS \(\Phi \) on the same grid with edge dimensions \(0{\lt}D_e\le B\) and
With \(B=CL^c\) and \(\varepsilon =L^{-1}\) the conclusion is that of [ 248 , Theorem 1.1 ] (00-introduction.tex, lines 36–57) for one lattice size. The source applies the column lemma on the party graph and then routes the ket network onto the grid (07-assembly.tex, lines 96–177). To use the corollary instead, the routing must be applied to the operator network before the column is fixed. The two steps commute: fixing the inputs acts site by site on physical indices, and the routing leaves physical coordinates unchanged (07-assembly.tex, lines 125–126 and 159–160). Sizes \(L{\lt}L_0\) are treated separately in the source by an exact spanning-tree PEPS (07-assembly.tex, lines 205–212) and do not pass through the corollary.
Take the fixed-input tensors of the preceding theorem; their edge dimensions are those of the operator network. These dimensions are positive because the column is nonzero: if some \(D_e\) vanished, the contraction would be an empty sum.
27.130.1 Error and normalization of the protocol
The following estimates are the calculation of [ 248 , Section 8.1 ] . The reference trajectory, its step errors, the number of changes, the private factor and the compressed operator are supplied; they are the outputs of the protocol proposition, the small-patch rewrite lemma and the compression theorem of that work, which are not proved here.
Let \(E_0,E_1,\dots \) be normed spaces, let \(S_t\colon E_t\to E_{t+1}\) be linear maps with \(\lVert S_tv\rVert \leq \lVert v\rVert \) for \(t{\lt}n\), and let \(r_t\in E_t\). Put \(x_0=r_0\) and \(x_{t+1}=S_tx_t\). Then
Induct on \(n\). Since \(x_{n+1}-r_{n+1}=S_n(x_n-r_n)+(S_nr_n-r_{n+1})\) and \(S_n\) is a contraction, the error grows by at most \(\lVert S_nr_n-r_{n+1}\rVert \). The norm bound is the same induction.
Let \(L\geq 1\) and \(C_{\rm ch},C_{\rm step},C_{\rm hole},C_{\rm comp}\) be constants with \(C_{\rm step},C_{\rm hole}\geq 0\). Suppose that, in the notation of Lemma 27.130.1.1, there are \(n\leq C_{\rm ch}L^3\) contractions, \(\lVert r_0\rVert \leq 1\), and \(\lVert S_tr_t-r_{t+1}\rVert \leq C_{\rm step}L^{-20}\) for \(t{\lt}n\). Suppose that \(E_n\) is identified isometrically with the physical and private coordinates, so that \(r_n=\Omega \otimes \Xi \) with
Then the actual vector \(\Psi =x_n\) satisfies \(\lVert \Psi \rVert \leq 1\),
If moreover \(\lVert \sigma -\operatorname{Tr}_{\rm private}|\Psi \rangle \langle \Psi |\rVert _1 \leq C_{\rm comp}L^{-5}\), then
The operator \(\sigma \) need not be positive or Hermitian, and no vector is renormalized.
By Lemma 27.130.1.1, the vector error is at most \(n\, C_{\rm step}L^{-20}\leq C_{\rm ch}C_{\rm step}L^{-17}\), and \(\lVert \Psi \rVert \leq \lVert r_0\rVert \leq 1\). The partial trace of the product reference is \(\lVert \Xi \rVert ^2|\Omega \rangle \langle \Omega |\), so the pure-density error bound gives a density error of at most \(2C_{\rm ch}C_{\rm step}L^{-17}+2C_{\rm hole}L^{-57}\), which is at most the stated multiple of \(L^{-10}\) because \(L\geq 1\). The last bound is the triangle inequality through the actual physical density.
27.130.2 Column selection after discarding a product reference
Let \(\Omega \) be a unit vector on the open \(L\times L\) square. Let \(\Psi \) be a joint vector on its physical and private coordinates, and \(\Xi \) a private vector such that
Let \(\sigma \) be the contracted operator of a supplied operator network on the same square, with edge dimensions \(D_e\), satisfying
Then some product-basis input \(z\) gives fixed-input tensors whose contraction is the nonzero column \(\Phi =\sigma |z\rangle \), with
for some real \(\theta \). The edge dimensions remain exactly \(D_e\); they are positive and preserve any supplied bounds \(D_e\leq B\). The operator \(\sigma \) need not be positive or Hermitian. This combines the normalization calculation of [ 248 , Section 8.1 ] with its Lemma 8.1. The hypothesis on \(\Psi \) and the compression estimate concern the same joint vector. The network is already on the square, so the routing order is as in Corollary 27.130.12. The supplied estimates and the assumption of a network already on the square are distinguished from the protocol construction in [ 92 ] .
The partial trace of the product-reference density is \(\lVert \Xi \rVert ^2|\Omega \rangle \langle \Omega |\). The pure-density error bound and contraction under partial trace give
The triangle inequality through the actual physical density gives \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\leq \alpha {\lt}1\). Apply Theorem 27.130.11. Fixing the input changes only physical entries of the supplied tensors, leaving every virtual dimension unchanged. A zero virtual dimension would make the selected contraction vanish, so all \(D_e\) are positive.
27.131 Dyadic routing of a party network into a square-lattice PEPS
This section treats the routing step of [ 248 , Lemma 8.2 ] , September 24, 2026, 07-assembly.tex, lines 102–177: the links of a finite tensor network whose parties sit at anchors of dyadic blocks are routed on the square grid with constant edge congestion, folded into the genuine \(L\times L\) square, and the routed network is contracted into a PEPS on that square. The source derives three properties of its party network (boundedly many parties per block, bounded degree, link separation bounded by a fixed multiple of the endpoint scales) from the distribution protocol of [ 248 , Proposition 7.1 ] , and a fourth (all anchors in the padded square) from attaching each party to the anchor of its dyadic block; its lemma has no hypotheses. Here these four properties are hypotheses on an arbitrary party network, so the results below are a conditional form of the lemma; the lemma for the network of the source follows once the protocol and the attachment are shown to give these properties [ 83 ] . The source also records that the two endpoint scales of every link are equal or adjacent; the congestion bound below does not use this, so it is not assumed.
For \(L\ge 1\) let \(N=2^{\lceil \log _2L\rceil }\) be the least power of two with \(N\ge L\). A dyadic block of side \(2^j\) with block index \(r\) in one coordinate has anchor coordinate
A party attached to the block of scale \(j\) and block indices \((r,s)\) sits at the anchor \((a_j(r),a_j(s))\).
For \(L\ge 1\) one has \(L\le N{\lt}2L\).
By minimality of the exponent, \(2^{\lceil \log _2L\rceil -1}{\lt}L\) whenever the exponent is positive, so \(N{\lt}2L\); the case \(L=1\) gives \(N=1\).
For \(j\ge 1\) the two-adic valuation of \(a_j(r)=2^{j-1}(2r+1)\) is exactly \(j-1\). Consequently, if \(j,j'\ge 1\) and \(a_j(r)=a_{j'}(r')\), then \(j=j'\) and \(r=r'\).
The factor \(2r+1\) is odd, so the valuation of \(2^{j-1}(2r+1)\) is \(j-1\). Equal anchors therefore have equal scales, and dividing by \(2^j\) recovers the block index.
On padded coordinates define \(f(x)=x\) for \(0\le x\le L-1\) and \(f(x)=2(L-1)-x\) for \(x\ge L\).
Let \(L\ge 1\). The fold fixes \(\{ 0,\ldots ,L-1\} \) and takes values there. For \(x+2{\lt}2L\) the coordinates \(f(x)\) and \(f(x+1)\) are adjacent. For \(x+1{\lt}2L\) every value \(u=f(x)\) forces \(x\in \{ u,\, 2(L-1)-u\} \), and for \(x+2{\lt}2L\) every unit interval \(\{ u,u+1\} \) that is the image of \(\{ x,x+1\} \) forces \(x\in \{ u,\, 2L-3-u\} \).
Each statement is a case distinction on whether \(x\) and \(x+1\) lie below \(L\), followed by linear arithmetic.
A party network with physical dimension \(d\) consists of finite sets of parties and links, a dimension \(D_\ell \) for each link \(\ell \), two endpoint parties for each link, and for each party \(p\) a tensor reading one label of every link incident to \(p\) and one physical index. When the physical index of party \(p\) is read at a site \(\iota (p)\) of a finite simple graph \(G\), the coefficient at a physical configuration \(\sigma \) is the sum over all link labellings of the product of the party tensors evaluated at \(\sigma (\iota (p))\). A routing of the network through \(G\) assigns to every link a walk in \(G\) from the site of its first party to the site of its second party.
Let a party network be routed through a finite simple graph \(G\). If every link has dimension at most \(D\) and every edge of \(G\) is traversed at most \(\chi \) times by the walks, counted with repetition, then there is a PEPS on \(G\) with exactly the coefficients of the party network and every bond dimension at most \({\max (D,1)}^\chi \).
Every step of a walk carries an identity tensor on the label of its link. At a vertex, the labels of all walks passing through it are kept independently, so crossings are permitted, and the party tensors located there are multiplied; zero-length links and coincident parties are contracted inside the vertex tensor. All traversals of one edge are grouped into a product alphabet, whose size is the product of the link dimensions, hence at most \(D^m\) for \(m\) traversals. Consistent configurations of the routed network are in bijection with link labellings, since the identity tensors force one label along each walk, so the contraction reproduces the coefficients of the party network.
Attach every party to a dyadic block, and so to its anchor. Each link, oriented arbitrarily, is routed from the anchor of its first party horizontally to the column of the anchor of its second party, then vertically to that anchor. The load of a unit edge of the padded square is the number of steps of all these walks that traverse it, counted with repetition. On the padded square of side \(N=8\), with \(L=5\), the square, the rounded marks and the blue dots below are the anchors of scales \(3\), \(2\) and \(1\), the arrow is one routed link from \(P\) to \(Q\), and the shaded square is the genuine \(L\times L\) square, onto which the fold \(f\) of Definition 27.131.4 maps the endpoint \(Q\).
![\begin{tenkz}[rows={ket,ket,ket,ket,ket,ket,ket,ket}, cols=8,
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{(1,1) .. (5,5)}{$L\times L$}
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Suppose at most \(B\) parties are attached to any dyadic block, every party has at most \(\Delta \) incident links, and the two anchors of every link differ in each coordinate by at most \(c\) times the side of either endpoint block. Then every horizontal and every vertical unit edge of the padded square has load at most \(2(2c+1)B\Delta \).
A horizontal step lies on the row of the anchor of the first party of its link, and each link traverses a given edge at most once. For a scale \(j\ge 1\) the row \(2^{j-1}(2s+1)\) determines \(j\) and \(s\) by its two-adic valuation. Anchors of one scale on one row are \(2^j\) apart, so at most \(2c+1\) blocks of that scale lie within horizontal distance \(c2^j\) of the edge; scale zero gives a second bound of the same form. Each block carries at most \(B\) parties, each the first endpoint of at most \(\Delta \) links. Vertical steps are treated in the same way using the column of the second party.
Let \(L\ge 1\) and let every dyadic block lie in the padded square of side \(N\). Under the hypotheses of the previous theorem, applying the fold in both coordinates turns the padded walks into walks of the open \(L\times L\) square lattice, the party at an anchor is placed at its fold, and every edge of the genuine square is traversed at most \(8(2c+1)B\Delta \) times, counted with repetition. This is the congestion part of [ 248 , Lemma 8.2 ] for every party network satisfying the hypotheses stated above; the source states it for its own network without hypotheses.
Since \(N{\lt}2L\), every padded walk point has coordinates below \(2L-1\), so adjacent padded points fold to adjacent genuine points. A horizontal edge of the genuine square has at most two padded preimages in each coordinate, hence at most four padded unit-edge preimages, and likewise for vertical edges. Summing the padded bound over these preimages gives \(4\cdot 2(2c+1)B\Delta \).
Let \(L\ge 1\) and consider a party network on dyadic anchors of the padded square with at most \(B\) parties per block, at most \(\Delta \) links per party, link separation at most \(c\) times either endpoint scale, and link dimensions at most \(D\). Then there is a PEPS on the open \(L\times L\) square lattice whose coefficients equal those of the party network, with the physical index of each party read at the fold of its anchor, and whose bond dimensions are all at most \({\max (D,1)}^{\chi }\) with \(\chi =8(2c+1)B\Delta \). The exponent \(\chi \) depends neither on \(L\) nor on the number of dyadic scales.
Apply the routed contraction to the folded routing, whose congestion is at most \(\chi \) by the previous theorem. The fold fixes genuine coordinates, so a party whose anchor is a genuine site is read at that site.
27.131.1 Fixing a column before routing
The source fixes one input column of the compressed operator network on the party graph and routes the resulting ket network afterwards [ 248 , Section 8.2–8.3 ] (07-assembly.tex, lines 96–177). The statements below follow this order. The operator network, its placement, the four routing properties, and the trace-norm estimate remain hypotheses, as described at the beginning of this section and in [ 92 ] .
An operator party network is a party network whose physical index at each party is an output-input pair in \(\{ 1,\ldots ,q\} ^2\). When the pair of party \(p\) is read at the site \(\iota (p)\), the operator \(\sigma \) of the network has the entry \(\langle \tau |\sigma |\rho \rangle \) equal to the coefficient of the network at the configuration \(v\mapsto (\tau (v),\rho (v))\). For a product-basis input \(z\), fixing the input of every party \(p\) to \(z(\iota (p))\) gives a party network with physical dimension \(q\), the same links, and the same link dimensions.
The coefficient vector of the network with input fixed to \(z\) is the column \(\sigma |z\rangle \). If \(\Omega \) is a unit vector and \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\), then for some \(z\) this column \(\psi \) is nonzero and \(\lVert \psi /\lVert \psi \rVert -e^{i\theta }\Omega \rVert \le 2\eta \) for some real \(\theta \). The operator \(\sigma \) need not be positive or Hermitian. This is [ 248 , Lemma 8.1 ] on the party graph.
Fixing the input of every party selects the entries of \(\sigma \) in the column of \(z\) without changing any link. The vector form of the column lemma, Theorem 27.130.4, applied to \(\sigma \) gives the column.
Let \(L\ge 1\) and let an operator party network on dyadic anchors of the padded square satisfy the hypotheses of Theorem 27.131.11, with link dimensions at most \(D\) and \(\chi =8(2c+1)B\Delta \). Read the physical pair of each party at the fold of its anchor, and suppose that the resulting operator satisfies \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \eta {\lt}1\) for a unit vector \(\Omega \). Then there is a nonzero PEPS \(\Phi \) on the open \(L\times L\) square, with every bond dimension in \(\{ 1,\ldots ,{\max (D,1)}^\chi \} \), and a real \(\theta \) such that \(\lVert \Phi /\lVert \Phi \rVert -e^{i\theta }\Omega \rVert \le 2\eta \). The same conclusion holds with \(\eta =\kappa +2\varepsilon +2\delta \) when the hypotheses of Theorem 27.130.2.1 hold for this operator.
Select a column of the party network by Theorem 27.131.1.2, and route the resulting ket network by Theorem 27.131.11. The routed PEPS has the coefficients of the input-fixed network, hence it is the selected column. A zero bond dimension would make the sum over virtual configurations empty and the contraction zero, so every bond dimension is positive. In the second form, the estimate in the proof of Theorem 27.130.2.1 gives \(\lVert \sigma -|\Omega \rangle \langle \Omega |\rVert _1\le \kappa +2\varepsilon +2\delta \).
In the setting of Theorem 27.131.1.3, suppose that every link dimension is at most \(C_1L^k\) with \(C_1\ge 1\) and \(k\ge 0\), and that \(2\eta \le L^{-1}\). Then the nonzero PEPS has every bond dimension at most
and normalized error at most \(L^{-1}\), up to a phase. This is the final bond bound of [ 248 , Section 8.3 ] (07-assembly.tex, lines 164–170) for the routed column.
Since \(L\ge 1\) and \(k\ge 0\), one has \(1\le C_1L^k\), so every link dimension is at most \(D=\lfloor C_1L^k\rfloor \) and \(\max (D,1)\le C_1L^k\). Apply Theorem 27.131.1.3 and raise this bound to the power \(\chi \).
27.132 Encoded frames and small-patch rewrites
The statements below formalize Definition 6.1 and the first assertion of Lemma 6.3 of [ 248 ] , September 24, 2026, 05-frames.tex, lines 11–97 and 182–252, the registers of a frame with one register per site and per tag, together with the steps of the proof of the second assertion listed below. The expansion of the branches into allowed monomials on these registers is in Section 27.137, and the second assertion with its polynomial count is Theorem 27.132.25.
Throughout, \(\Lambda \) is a finite set of sites with positions in the plane, each site carries a raw register \(\mathbb {C}^q\) with \(q\ge 1\), and \(\mathcal H_\Lambda =\bigotimes _{x\in \Lambda }\mathbb {C}^q\). For \(D\subseteq \Lambda \) and an operator \(X\) on \(\mathcal H_D\), write \(X_D\otimes \operatorname{id}\) for the operator acting as \(X\) on the sites of \(D\) and as the identity elsewhere. Operators supported on disjoint sets of sites commute. A vector \(\Omega \) is fixed, and \(\lVert \cdot \rVert \) denotes the Euclidean norm of vectors and the operator norm of matrices, rectangular ones included.
For a center \(c\in \mathbb {R}^2\) and a scale \(h{\gt}0\), the inner and outer squares \(Q^-(c,h)\) and \(Q^+(c,h)\) are the closed sup-norm balls of radii \(h\) and \(2h\); their physical samples are the sites whose positions lie in them. Square cylinder data at \((c,h)\) consist of radii \(h\le r_1\le \cdots \le r_n\le 2h\), the samples \(D_j\) of the squares of radius \(r_j\), and unit vectors \(v_{j\ell }\in \mathcal H_{D_j}\) for \(\ell \le d_j\), such that the cylinder projectors \(\bigl(|v_{j\ell }\rangle \langle v_{j\ell }|\bigr)_{D_j}\otimes \operatorname{id}\) have mutually orthogonal ranges. Their sum is
Every \(D_j\) contains the inner sample and is contained in the outer sample. If the outer sample is empty, then \(P_{c,h}=\operatorname{id}\), whatever the cylinder data. This is the projector of 05-frames.tex, lines 26–48, with the cylinder data of [ 248 , Proposition 4.1 ] ; the accuracy \(\lVert (\operatorname{id}-P_{c,h})\Omega \rVert \le \varepsilon \) enters as a hypothesis where it is used. The polynomial bound on \(\sum _jd_j\) is not recorded: the count of branches below takes a uniform bound on the number of cylinder terms of each patch instead.
Let \(\mathcal J_{c,h}\) be a tag register with orthonormal basis \(\{ |j,\ell \rangle \} \) and let \(|0\rangle \) be a fixed one-site unit vector. The hole encoder is
It keeps a raw register at every site, inside the hole as well. If the outer sample is empty, the tag register is one-dimensional and \(K_{c,h}\) is the identity encoding, as in 05-frames.tex, lines 67–69. On a square grid of unit spacing with \(c\) a site and \(h=3/2\), the inner sample, one sample \(D_j\) of radius \(r_j=2\), and the outer sample are nested squares of sites, and the encoder maps the raw registers to a tag together with the raw registers.
![\begin{tenkz}[rows={ket,ket,ket,ket,ket,ket,ket}, cols=7,
bonds=none]
\tnmark[form=enclosure, species=selected, tint]{(3,3) .. (5,5)}{}
\tnmark[form=enclosure, species=secondary]{(2,2) .. (6,6)}{}
\tnmark[form=enclosure, species=complement]{(1,1) .. (7,7)}{}
\tnmark[form=label, label pos=n]{midway (4,3) and (4,4)}{$Q^-$}
\tnmark[form=label, label pos=n]{midway (3,2) and (3,3)}{$D_j$}
\tnmark[form=label, label pos=n]{midway (2,1) and (2,2)}{$Q^+$}
\tnmark[form=label, label pos=e]{(4,4)}{$c$}
\end{tenkz}](tenkz_svg/tenkz-89834dfce8b3e922.svg)
![\begin{tenkz}[rows={wire,wire}, cols=1, bonds=none]
\tn[at=(1,1), skin=box, wires=2,
ports={180@2:physical:$\mathcal H_\Lambda$,
0@1:physical:$\mathcal J_{c,h}$,
0@2:physical:$\mathcal H_\Lambda$}]{K_{c,h}}
\end{tenkz}](tenkz_svg/tenkz-bb095fb4a44904a6.svg)
The operator \(P_{c,h}\) is an orthogonal projection, \(K_{c,h}^\dagger K_{c,h}=P_{c,h}\), and \(\lVert K_{c,h}\rVert \le 1\). This is 05-frames.tex, equation (encoder-contraction), lines 62–64.
If the outer sample is empty, \(K_{c,h}\) is the identity encoding and all three claims are immediate. Otherwise, each cylinder projector is Hermitian and idempotent because \(v_{j\ell }\) is a unit vector, and products of distinct cylinder projectors vanish by orthogonality; hence \(P_{c,h}\) is an orthogonal projection. Orthogonality of the tag basis gives \(K_{c,h}^\dagger K_{c,h}=\sum _{j,\ell } (|v_{j\ell }\rangle \langle 0^{\otimes D_j} | 0^{\otimes D_j} \rangle \langle v_{j\ell }|)_{D_j} \otimes \operatorname{id}=P_{c,h}\). Finally \(\lVert K_{c,h}\rVert ^2=\lVert K_{c,h}^\dagger K_{c,h}\rVert =\lVert P_{c,h}\rVert \le 1\).
An encoded frame \(\mathcal F\) consists of a raw ownership assignment on one sheet, a list of holes \((c,h)\) with square cylinder data and pairwise disjoint outer samples, and a specified party holding each hole’s tag. The order of the list is the fixed ordering of the appended tags. The encoding \(K_{\mathcal F}\) is the product of the hole encoders, with the tags appended in that order, and the reference vector is \(\Omega _{\mathcal F}=K_{\mathcal F}\Omega \). A frame without holes is a raw sheet. This is [ 248 , Definition 6.1 ] , 05-frames.tex, lines 71–82. The source requires the outer squares to be disjoint in the plane; only the disjointness of their physical samples, which follows, is used.
For two lists of holes with disjoint outer footprints, the raw parts of their encodings commute on every pair of tag configurations. The tags of a concatenated list are the pairs of tags of the two parts, and on such a pair the raw part of the encoding of the concatenation is the product of the raw parts of the two encodings. Hence the encodings commute after the canonical identification of tag orderings (05-frames.tex, lines 84–85).
The raw part on a tag configuration is a product of operators supported in the outer footprints of the holes of the list, so the two products are supported on disjoint sets of sites. The factorization follows by induction on the first list.
For a frame with hole projectors \(P_1,\dots ,P_r\), \(K_{\mathcal F}^\dagger K_{\mathcal F}=P_1\cdots P_r\), this product is an orthogonal projection, and \(\lVert K_{\mathcal F}\rVert \le 1\).
By induction on the list. Write the raw part on the tags \((s,t)\) as \(B_sR_t\), with \(B_s\) from the first hole. Summing \(R_t^\dagger B_s^\dagger B_sR_t\) over \(s\) gives \(R_t^\dagger P_1R_t\), and \(P_1\) commutes with \(R_t^\dagger \) by disjointness of supports; the sum over \(t\) is then \(P_1\) times the identity for the remaining holes. The projectors commute pairwise, so their product is an orthogonal projection of norm at most one.
For contractions \(A_1,\dots ,A_s\) and every vector \(\psi \),
In particular, if each of \(r\) projectors has error at most \(\varepsilon \) on \(\psi \), so does their product up to \(r\varepsilon \); no commutation is assumed (05-frames.tex, lines 237–248).
Induction on \(s\), from \((A_1R-\operatorname{id})\psi =A_1(R-\operatorname{id})\psi +(A_1-\operatorname{id})\psi \) and \(\lVert A_1\rVert \le 1\).
For a frame with \(r\) holes, \(\lVert \Omega _{\mathcal F}\rVert =\lVert P_1\cdots P_r\Omega \rVert \). If \(\lVert \Omega \rVert =1\) and every hole projector satisfies \(\lVert (\operatorname{id}-P_a)\Omega \rVert \le \varepsilon \), then
This is 05-frames.tex, equation (frame-norm), lines 86–94.
The two vectors have the same squared norm \(\langle \Omega | P_1\cdots P_r\Omega \rangle \), because the product of the projectors is an orthogonal projection. The upper bound is contractivity, and the lower bound is the reverse triangle inequality with the telescoping lemma.
The registers of a frame \(\mathcal F\) are one register \(\mathbb {C}^{\mathcal J_a}\) for each hole \(a\), held by the party holding its tag, listed in the order of the holes, followed by one raw register \(\mathbb {C}^q\) for each site \(x\in \Lambda \), held by the raw owner of \(x\), listed in a fixed order of the sites. This list contains every site exactly once, so taking the site at a position gives a bijection from its positions to \(\Lambda \). More generally, if all sites in any list have raw owner \(p\), every raw register of that list is owned by \(p\). The tensor product of the frame’s registers is its memory in the sense of Theorem 5.2 of [ 248 ] . This is the sheet of 05-frames.tex, lines 13–20, with one raw register at each site and a specified owner for each register, together with the tags of Definition 6.1, lines 71–82.
The tensor product of the registers of a frame is identified, by a unitary, with \(\mathbb {C}^{\mathcal J\times [q]^\Lambda }\), the space of the tags and the sheet on which the encoding \(K_{\mathcal F}\) and its operators act: the product basis vector \(|\tau \rangle \otimes \bigotimes _x|c(x)\rangle \) is sent to \(|\tau ,c\rangle \). More generally, the raw registers of a list of distinct sites enumerating a set \(A\) are identified with \(\mathbb {C}^{[q]^A}\) in the same way. These are the canonical site and sheet coordinates of 05-frames.tex, lines 94–96, in which vectors are compared while the owners of their registers are specified separately. For any list \(\ell \) of sites, including lists with repetitions, write \(|c\rangle _\ell \) for its product basis vector. If \(c(x)=c'(x)\) for every site in \(\ell \), then \(|c\rangle _\ell =|c'\rangle _\ell \).
By induction on the list of registers, identifying \(\mathbb {C}^\alpha \otimes \mathbb {C}^\beta \) with \(\mathbb {C}^{\alpha \times \beta }\) at each step; product basis vectors are sent to standard basis vectors, and a list of distinct sites enumerating \(A\) gives a bijection between its positions and \(A\). Configurations agreeing on a list give the same basis vector at every position, hence the same product vector, even if sites repeat.
Old and new one-sheet frames share their untouched holes, with the same encodings and tag owners; the remaining holes are the affected old holes and the affected new holes, \(r_{\rm old}\) of the former. There are \(m\) additional square patches \((c_i,u_i)\) with cylinder data, and a specified finite list of parties; the source requires this list to be bounded, and its size enters, with \(m\) and the numbers of affected holes, the polynomial count of Theorem 27.132.25. Write \(H^\mp (\mathcal F)\) for the unions of the physical samples of the inner and outer hole squares. The four conditions of [ 248 , Lemma 6.3 ] (05-frames.tex, lines 193–207) are:
every patch’s outer square, and every affected old or new hole, avoids the physical outer footprints of the untouched holes;
the union of the patch inner squares contains every site whose raw owner changes and the entire outer footprint of every affected old or new hole;
in each patch’s outer square, the sites outside \(H^-(\mathcal F_{\rm old})\) have at most two old owners, and the sites outside \(H^-(\mathcal F_{\rm new})\) have at most two new owners;
all old and new raw owners on the patch footprints and the affected-hole footprints, and all affected tag owners, belong to the specified list.
Patches may overlap, and affected old holes may overlap affected new holes. In the canonical coordinates used below, the tags of the untouched holes come first in both frames; Theorem 27.132.16 also covers arbitrary tag orderings.
With \(K_{\rm old}\) and \(K_{\rm new}\) the encodings of the affected old and new holes alone and \(P_1,\dots ,P_m\) the patch projectors in a fixed order,
acting as the identity on the tags of the untouched holes. Its input layout is that of \(\mathcal F_{\rm old}\) and its output layout that of \(\mathcal F_{\rm new}\) (05-frames.tex, lines 222–232).
For complex matrices \(A\) and \(B\), not necessarily square, and identity matrices \(I\) of any size, \(\lVert A\otimes I\rVert \le \lVert A\rVert \) and \(\lVert I\otimes B\rVert \le \lVert B\rVert \).
For a vector \(x=\sum _k x_k\otimes e_k\) one has \((A\otimes I)x=\sum _k Ax_k\otimes e_k\), so \(\lVert (A\otimes I)x\rVert ^2=\sum _k\lVert Ax_k\rVert ^2 \le \lVert A\rVert ^2\lVert x\rVert ^2\). The second bound follows by exchanging the two tensor factors, which reindexes rows and columns.
\(\lVert M\rVert \le 1\).
Every factor is a contraction. Tensoring with the identity on the untouched tags and relabelling coordinates do not increase the operator norm; the first estimate sums squared norms over the untouched tag slices and introduces no dimension factor.
Under condition (1), \(MK_{\mathcal F_{\rm old}}=K_{\mathcal F_{\rm new}}P_m\cdots P_1 Q_1\cdots Q_{r_{\rm old}}\), where \(Q_1,\dots ,Q_{r_{\rm old}}\) are the projectors of the affected old holes.
On untouched tags \(u\), affected old tags \(b\) and affected new tags \(a\), write \(U_u\), \(A_b\) and \(B_a\) for the raw parts of the encodings. By condition (1) and disjointness of the old frame, \(U_u\) commutes with \(A_b^\dagger \), with every patch projector, with \(B_a\) and with the affected old projectors. Summing over \(b\), \(\sum _bA_b^\dagger U_uA_b=U_u\sum _bA_b^\dagger A_b =U_uQ_1\cdots Q_{r_{\rm old}}\) by the frame Gram identity; then \(U_u\) commutes to the left of \(B_aP_m\cdots P_1\).
Under the four conditions, if every patch projector and every affected old-hole projector has error at most \(\varepsilon \) on \(\Omega \), the canonical map \(M\) is a contraction and
The same holds for frames whose tags are listed in any order, with \(M\) relabelled along the identification of the two tag orderings. This is the first assertion of [ 248 , Lemma 6.3 ] , with its hypotheses; the error estimate uses condition (1) only. No smallness of the errors of the untouched holes is assumed, and the projectors of overlapping patches need not commute.
By the intertwining lemma, \(M\Omega _{\mathcal F_{\rm old}}-\Omega _{\mathcal F_{\rm new}} =K_{\mathcal F_{\rm new}}(P_m\cdots P_1Q_1\cdots Q_{r_{\rm old}}-\operatorname{id})\Omega \). The encoding is a contraction, and the telescoping lemma bounds the remaining vector by \((m+r_{\rm old})\varepsilon \). For another ordering of the tags, the raw parts of the encodings of holes with disjoint outer footprints commute, so moving every tag to the new position of its hole is a relabelling of the tag basis that carries one encoding to the other. Relabelling the coordinates of \(M\) along it keeps \(M\) a contraction and changes neither norm in the estimate.
A complete branch \(\beta =(a,s,b)\) chooses a tag \(a\) of the affected new holes, one cylinder term of every patch, and a tag \(b\) of the affected old holes. With \(C_s\) the product of the chosen cylinder projectors, \(M=\sum _\beta M_\beta \), where \(M_\beta \) is \(|a\rangle \langle b|\otimes B_aC_sA_b^\dagger \) extended by the identity on the untouched tags. If every affected hole and every patch has at most \(D\) cylinder terms, there are at most \(D^{\, r_{\rm new}+m+r_{\rm old}}\) branches (05-frames.tex, lines 254–257).
Expand every patch projector as the sum of its cylinder projectors and read off the entries of \(K_{\rm new}P_m\cdots P_1K_{\rm old}^\dagger \) on the tag basis. The number of tag configurations of a list is the product of the numbers of cylinder terms of its members.
Assume the four conditions and fix a branch \((a,s,b)\). The selected square of a term \((j,\ell )\) is \(D_j\), and that of the identity term of a square with empty outer sample is empty; the selected squares are read off from the terms of the branch position by position in the lists, so a patch listed twice may carry two different terms. Every site of a selected affected old-hole square, and of a selected affected new-hole square, lies in a selected patch square. The sites of a selected patch square not consumed by a selected old-hole square have at most two old owners, and those not received by a selected new-hole square have at most two new owners. A site in no selected patch square has the same old and new owner. All old and new owners of sites in selected squares belong to the specified list (05-frames.tex, lines 267–305).
Every selected square contains its inner square and lies in its outer square (for an empty outer sample both are empty), so condition (2) gives the first statement and the statement on direct sites. A site of a patch outer square in \(H^-(\mathcal F_{\rm old})\) is not in an untouched hole by condition (1), hence lies in the inner square of an affected old hole and so in its selected square; condition (3) bounds the owners of the remaining sites. The new case is symmetric, and condition (4) gives the last statement.
Let \(M=\sum _\beta M_\beta \) be a contraction with \(N\) branches and let \(\lVert M_\beta -\widetilde M_\beta \rVert \le \eta \) for every branch. If \(N\eta \le \tfrac 12\delta \), then \(M_\delta =(1+\tfrac 12\delta )^{-1}\sum _\beta \widetilde M_\beta \) satisfies \(\lVert M_\delta \rVert \le 1\) and \(\lVert M_\delta -M\rVert \le \delta \). For \(N,g\ge 0\) and \(\delta {\gt}0\) there is an integer \(k\ge 1\) with \(Ng/\sqrt k\le \tfrac 12\delta \) (05-frames.tex, lines 331–341).
The sum \(\widetilde M\) is within \(\tfrac 12\delta \) of \(M\), so \(\lVert \widetilde M\rVert \le 1+\tfrac 12\delta \). Splitting \(M_\delta -M=(1+\tfrac 12\delta )^{-1}(\widetilde M-M) -\tfrac {\delta /2}{1+\delta /2}M\) bounds the error by \(\delta /(1+\tfrac 12\delta )\le \delta \). Take \(k=\lceil (2Ng/\delta )^2\rceil +1\).
Under condition (1), if \(\lVert \Omega \rVert \le 1\) and \(\lVert M_a-M\rVert \le \delta \), then \(\lVert M_a\Omega _{\mathcal F_{\rm old}}-\Omega _{\mathcal F_{\rm new}}\rVert \le \delta +(m+r_{\rm old})\varepsilon \). With \(\varepsilon =L^{-60}\), \(\delta =L^{-30}\) and \(m+r_{\rm old}+1\le L^{10}\), this is at most \(L^{-20}\) (05-frames.tex, lines 217–218 and 343–344).
Split \(M_a\Omega _{\mathcal F_{\rm old}}-\Omega _{\mathcal F_{\rm new}}\) as \((M_a-M)\Omega _{\mathcal F_{\rm old}}\) plus the error of the theorem, and use \(\lVert \Omega _{\mathcal F_{\rm old}}\rVert \le 1\). For the specialization, \(L^{-60}\le L^{-30}\).
Let \(L_0,\dots ,L_{n-1}\) be operators \(L_j=|k_j\rangle \langle b_j|_{S_j}\otimes \operatorname{id}\) on the raw registers, where each \(k_j\) and \(b_j\) is a vector on the configurations of the finite set \(S_j\) or the product zero vector. Form the network whose vertices are the bras \(\langle b_j|\) and the kets \(|k_j\rangle \), kept distinct, and whose internal edges join the output of a ket at a site to the input of the next bra containing that site, in the order \(L_{n-1},\dots ,L_0\) of application. The input of a bra at a site contained in no earlier factor is an open input leg, and the output of a ket at a site contained in no later factor is an open output leg, except that the legs of a product zero vector are fixed to \(0\). Then the entry of \(L_0\cdots L_{n-1}\) at output configuration \(\sigma \) and input configuration \(\tau \) vanishes unless \(\sigma \) and \(\tau \) agree at the sites contained in no factor and carry \(0\) at the fixed legs, and otherwise equals the contraction of the network evaluated at the open legs read off from \(\sigma \) and \(\tau \). No edge joins a vertex to itself (05-frames.tex, lines 274–281).
Expand the product as a sum over paths of intermediate configurations. A path of nonzero weight keeps the value of a site across factors that do not contain it, so it is determined by the values on the internal edges, and conversely every labelling of the internal edges determines such a path; along it the weight is the product of the vertex tensors.
In the setting of the previous lemma, let every \(k_j\) and \(b_j\) have norm at most one, and let \(G\) be a set of vertices containing every vertex with an open leg. For every \(k\ge 1\) there are orthonormal families of at most \(k\) vectors on the open-leg groups and coefficients of modulus at most one such that \(L_0\cdots L_{n-1}\) differs in operator norm by at most \(|G|/\sqrt k\) from the sum of the corresponding product terms, of which there are at most \(k^{|G|}\). Each product term is the tensor product of one normalized vector on the open output legs of each ket, one normalized covector on the open input legs of each bra, the product zero vectors at the fixed legs, and the identity on the sites contained in no factor. No constant depends on the dimensions of the sites (05-frames.tex, lines 306–329).
The vertex tensors are normalized, so the whole-group truncation lemma applies to the contraction \(Z\). An operator read off from a tensor \(W\) on the open legs is \(W\), reshaped as a matrix from inputs to outputs, tensored with the identity on the remaining sites, so its operator norm is at most the largest Hilbert–Schmidt norm of its diagonal blocks, hence at most \(\lVert W\rVert \). Apply this to \(W=Z-Z_k\).
Assume the four conditions and fix a branch \((a,s,b)\). Its raw operator \(B_aC_sA_b^\dagger \) is the ordered product of the factors \(|0\rangle \langle u'_r|\) of the affected new holes, \(|w_i\rangle \langle w_i|\) of the patch terms, and \(|u_t\rangle \langle 0|\) of the affected old holes. In its network only the bras and kets of the patch terms have open legs, so at most \(2m\) vertices do; the open input legs of a patch bra have at most two old owners, the open output legs of a patch ket have at most two new owners, and all these owners belong to the specified list (05-frames.tex, lines 257–305).
The factorization follows from the definitions of the hole encodings and the cylinder projectors. A decoded old-hole ket meets a later patch bra at every site of its square, and a new-hole bra meets an earlier patch ket at every site of its square, by the wire classification; the zero vectors carry no open legs. An open input of a patch bra is consumed by no selected old-hole square, so the wire classification bounds its old owners; the output case is symmetric.
Assume the four conditions, and let \(N\) be the number of branches and \(k=\lceil (4Nm/\delta )^2\rceil +1\) for \(\delta {\gt}0\), for instance \(\delta =L^{-a}\). Every branch is approximated in operator norm within \(2m/\sqrt k\) by at most \(k^{2m}\) product terms with coefficients of modulus at most one, and the rescaled sum \(M_\delta =(1+\tfrac 12\delta )^{-1}\sum _\beta |a\rangle \langle b|\otimes \bigl(\sum _ic_{\beta ,i}T_{\beta ,i}\bigr)\), extended by the identity on the untouched tags, is a contraction with \(\lVert M_\delta -M\rVert \le \delta \). Each product term \(T_{\beta ,i}\) is the tensor product of normalized vectors on the open output groups of the patch kets, normalized covectors on the open input groups of the patch bras, product zero vectors on the selected hole squares, and identities; by Lemma 27.132.23 each open group lies on at most two parties of the specified list. This formalizes the branch-network, truncation and rescaling steps of the proof of the second assertion of [ 248 , Lemma 6.3 ] . The reading of the product terms as allowed monomials is Theorem 27.137.8, and the polynomial bound on the number of terms is Theorem 27.132.25.
Apply the truncation of a chain to the network of each branch with the patch vertices as groups, and assemble with the rescaling lemma, using \(N\cdot 2m/\sqrt k\le \tfrac 12\delta \).
In the setting of the frame construction, where the projector of every hole and every additional patch is a sum of \(\sum _jd_j\le CL^c\) cylinder projectors for fixed \(C\) and \(c\) (05-frames.tex, lines 34–46), assume the four conditions with boundedly many affected holes and additional patches and a bounded specified list of parties. Then for every fixed \(a{\gt}0\) the canonical map \(M\) admits a contractive approximation \(M_a\) with \(\lVert M_a-M\rVert \le L^{-a}\) and a polynomial sum expansion into monomials allowed by [ 248 , Theorem 5.2 ] , whose participating parties all lie in the specified list, of bounded size. In particular, when \(\lVert \Omega \rVert \le 1\) and every additional patch projector and every affected old-hole projector has error at most the accuracy \(\varepsilon =L^{-60}\) fixed for the frame construction (05-frames.tex, lines 21–25), the choice \(a=30\) gives \(\lVert M_{30}\Omega _{\mathcal F_{\rm old}}-\Omega _{\mathcal F_{\rm new}} \rVert \le (2b+1)L^{-20}\), where \(b\) bounds \(m\), \(r_{\rm old}\), \(r_{\rm new}\) and the number of specified parties; the rewrite is a bounded change with reference-vector error \(O(L^{-20})\). This is the second assertion of [ 248 , Lemma 6.3 ] .
Expand \(M\) into its polynomially many branches. In a branch \(\beta =(a,s,b)\), let the old affected holes carry the terms with selected squares \(D^{\rm old}_t\) and unit vectors \(u_t\), the new affected holes the terms with selected squares \(D^{\rm new}_r\) and unit vectors \(u'_r\), and the patches the terms with selected squares \(E_i\) and unit vectors \(w_i\). Then \(M_\beta =|a\rangle \langle b|\otimes B_aC_sA_b^\dagger \) with
the factors of \(C_s\) taken in the order of the patches. Factor out the tag ket \(|a\rangle \) and bra \(\langle b|\), the zero bras on the squares \(D^{\rm old}_t\), the zero kets on the squares \(D^{\rm new}_r\), and the identity on the sites outside \(\bigcup _iE_i\), whose old and new owners agree. The residual network \(Z\) has the vertices \(|u_t\rangle \), \(\langle w_i|\), \(|w_i\rangle \) and \(\langle u'_r|\), the bra and the ket of each patch term being distinct vertices; each site wire joins a ket to the next bra containing that site, so there are no self-contractions. By the wire classification, \(|u_t\rangle \) has no open output legs and \(\langle u'_r|\) has no open input legs. The open legs of \(Z\) are the input legs of \(\langle w_i|\) at sites of \(E_i\) outside \(\bigcup _tD^{\rm old}_t\) not reached by an earlier ket, which have at most two old owners, and the output legs of \(|w_i\rangle \) at sites of \(E_i\) outside \(\bigcup _rD^{\rm new}_r\) not met by a later bra, which have at most two new owners. The whole-group truncation lemma (Theorem 27.129.8) approximates its tensor within \(g/\sqrt k\) by at most \(k^g\) product terms with coefficients of modulus at most one, each an allowed monomial, and the Hilbert-space error bounds the operator error of the branch. The rescaling lemma assembles \(M_a\), and the reference estimate with \(a=30\) gives the final assertion. The branch network, its truncation and the assembly are formalized in the approximation by product terms (Theorem 27.132.24), and the reading of each product term as an allowed monomial on the registers of the frames, with at most \(Nk^{2m}\) monomials, in Theorem 27.137.8. With \(\sum _jd_j\le CL^c\), \(m,r_{\rm old},r_{\rm new}\le b\) and at most \(b\) specified parties, the number of branches is at most \(\bigl((C+1)L^c\bigr)^{3b}\), and for \(\delta =L^{-a}\), \(L\ge 1\) the count is at most \((C+1)^{3b}\bigl(16b^2(C+1)^{6b}+1\bigr)^{2b}L^{3bc+4b(3bc+\lceil a\rceil )}\), with at most \(2b\) normalized pair sources and \(2b\) normalized pair effects in each monomial, and every monomial uses only the at most \(b\) specified parties. The bound on the participating parties is the boundedness of the list in the fourth condition and does not follow from the other three: every site of a selected hole square receives a zero bra at its old owner and a zero ket at its new owner, and the third condition does not restrict the owners inside the inner holes [ 141 ] .
27.133 Changes of ownership in encoded frames
The statements below formalize Lemmas 6.5 and 6.6 of [ 248 ] , September 24, 2026, 05-frames.tex, lines 347–570, for the canonical maps constructed in their proofs, with the information bounds kept as hypotheses, together with their reading as bounded changes: allowed monomials of [ 248 , Theorem 5.2 ] (Section 27.138) on the registers of the frames, one per site and per tag (Definition 27.132.9). The monomials are first constructed on registers in which the raw registers of a region held by one party are grouped into one register, and then placed on the registers of the frames by reorderings of tensor factors and grouping unitaries.
We keep the setting of Section 27.132. For a set \(S\subseteq \Lambda \) write \(\mathcal H_S=\bigotimes _{x\in S}\mathbb {C}^q\). For a vector \(\Omega \in \mathcal H_\Lambda \) and disjoint sets \(T,E\subseteq \Lambda \), the mutual information \(I_\Omega (T:E)\) is \(S(\rho _T)+S(\rho _E)-S(\rho _{TE})\) with natural logarithms, where the \(\rho \) are the reduced density matrices of \(|\Omega \rangle \langle \Omega |\).
27.133.1 Splitting a sheet
Identify \(\mathcal H_\Lambda \) with \(\mathcal H_S\otimes \mathcal H_{\Lambda \setminus S}\), and, for disjoint \(T\) and \(E\) with \(U=\Lambda \setminus (T\cup E)\), with \(\mathcal H_T\otimes \mathcal H_E\otimes \mathcal H_U\). The latter identification reads a configuration \(c\) as \(((c|_T,c|_E),c|_U)\). An operator acting on the sites of \(S\) is \(A'\otimes \operatorname{id}\), an operator acting on a set disjoint from \(S\) is \(\operatorname{id}\otimes A'\), an operator acting on \(T\cup E\) is \(A'\otimes \operatorname{id}_U\), and an operator acting on a subset of \(E\) is \(\operatorname{id}_T\otimes A'\otimes \operatorname{id}_U\).
A product operator \(\bigotimes _xm_x\) is, in these coordinates, the tensor product of its restrictions to the parts, and \(m_x=\operatorname{id}\) off the support. Both sides are linear in the operator.
Let \(T,E\subseteq \Lambda \) be disjoint, \(U=\Lambda \setminus (T\cup E)\), \(\lVert \Omega \rVert =1\) and \(b=I_\Omega (T:E)\). There are finite-dimensional spaces \(\mathcal B_T,\mathcal B_E\), an isometry \(V\colon \mathcal H_U\to \mathcal B_T\otimes \mathcal B_E\) and unit vectors \(s\in \mathcal H_T\otimes \mathcal B_T\), \(s'\in \mathcal H_E\otimes \mathcal B_E\) with
In particular, if \(b\le L^{-60}\) the error is at most \(L^{-30}\). This is [ 248 , Lemma 6.4 ] (05-frames.tex, lines 352–368) on the sheet.
The relative entropy of \(\rho _{TE}\) with respect to \(\rho _T\otimes \rho _E\) is \(b\). The splitting estimate for a unit vector on three systems, with \(\mathcal B_T=\mathcal H_T\) and \(\mathcal B_E=\mathcal H_E\oplus \mathcal H_U\), applies in the coordinates \(\mathcal H_T\otimes \mathcal H_E\otimes \mathcal H_U\).
27.133.2 Homogeneous birth and death
With the data of the splitting lemma put \(\mathcal V=\operatorname{id}_{TE}\otimes V\) and let \(S\colon \mathcal H_E\otimes \mathcal B_E\to \mathcal H_{TE}\otimes \mathcal B_T\otimes \mathcal B_E\) insert \(s\) on \(T\mathcal B_T\). The canonical map of the birth is
(05-frames.tex, equation (birth-map), lines 414–426). It is the effect \(S^\dagger \mathcal V\), which applies \(V\) on \(U\) and contracts \(T\mathcal B_T\) with \(\langle s|\), followed by its adjoint, which supplies \(s\) and applies \(V^\dagger \).
The map \(B\) is a Hermitian contraction, it acts as the identity on \(\mathcal H_E\), so it commutes with every operator acting on a subset of \(E\), and for every vector \(\psi \)
Since \(s\) is a unit vector, \(S^\dagger S=\operatorname{id}\), so \(\Pi =SS^\dagger \) is an orthogonal projection with \(\Pi (s\otimes s')=s\otimes s'\). As \(\mathcal V^\dagger \mathcal V=\operatorname{id}\), \((B-\operatorname{id})\psi =\mathcal V^\dagger (\Pi -\operatorname{id})(\mathcal V\psi -s\otimes s')\), and \(\operatorname{id}-\Pi \) is a projection. In the coordinates \(\mathcal H_E\otimes (\mathcal H_T\otimes \mathcal H_U)\) the effect \(S^\dagger \mathcal V\) is \(\operatorname{id}_E\otimes K\) with \(K=\langle s|(\operatorname{id}_T\otimes V)\), so \(B=\operatorname{id}_E\otimes K^\dagger K\).
If a raw operator \(B\) commutes with the raw part of the encoding of a frame on every tag configuration, then \((\operatorname{id}_{\rm tags}\otimes B)K_{\mathcal F}=K_{\mathcal F}B\) and \(\lVert (\operatorname{id}\otimes B)\Omega _{\mathcal F}-\Omega _{\mathcal F}\rVert \le \lVert B\Omega -\Omega \rVert \).
Compare the two sides on each tag; the encoding is a contraction.
For an encoded frame \(\mathcal F\), a party \(P_\circ \) and a set \(T\) of raw sites, let \(\mathcal F'\) be the frame in which the raw registers of \(T\) are owned by \(Q_\circ \), with every hole, encoding and tag owner kept, and
(05-frames.tex, equation (birth-partition), lines 400–405).
Every register outside \(E\) is owned by \(P_\circ \) in \(\mathcal F\), and by \(P_\circ \) or \(Q_\circ \) in \(\mathcal F'\). If \(T\) is owned by \(P_\circ \) and avoids every outer hole footprint, then \(T\) and \(E\) are disjoint (05-frames.tex, lines 397–405).
A register outside \(E\) has old owner \(P_\circ \) by the definition of \(E\); in \(\mathcal F'\) its owner is \(Q_\circ \) if it lies in \(T\) and its old owner otherwise. A register of \(T\) has old owner \(P_\circ \) and lies in no outer hole footprint, so it is in neither part of \(E\).
In an encoded frame \(\mathcal F\), let \(T\) be owned by \(P_\circ \) and disjoint from every outer hole footprint, let \(E\) be as in the birth partition, and let \(\lVert \Omega \rVert =1\). If \(I_\Omega (T:E)\le L^{-60}\), the canonical map \(B\) of splitting data with error at most \(L^{-30}\) is a Hermitian contraction, acts as the identity on \(E\) and on all tags, satisfies \((\operatorname{id}\otimes B)K_{\mathcal F}=K_{\mathcal F'}B\), and
Since \(\mathcal F\) and \(\mathcal F'\) differ only in raw owners, \(K_{\mathcal F'}=K_{\mathcal F}\) and \(\Omega _{\mathcal F'}=\Omega _{\mathcal F}\), so the intertwining identity says that \(B\) commutes with the encoding; the change of owner is the content of the bounded-change assertion below. The death is the same map with the reverse input and output ownership; its conditions are those of the frame after the death. This is the content of [ 248 , Lemma 6.5 ] for the canonical map, with its hypotheses.
Every hole encoder acts on \(H^+(\mathcal F)\subseteq E\), so it commutes with \(B\). Hence \(\lVert (\operatorname{id}\otimes B)\Omega _{\mathcal F}-\Omega _{\mathcal F}\rVert \le \lVert (B-\operatorname{id})\Omega \rVert \le \lVert \mathcal V\Omega -s\otimes s'\rVert \le L^{-30}\), and \(\Omega _{\mathcal F'}=\Omega _{\mathcal F}\) in canonical coordinates, since only raw owners change.
An allowed monomial on parties uses only the parties of a set \(S\) when each of its local maps acts at a party of \(S\) and each of its pair sources and pair effects joins two parties of \(S\); reorderings of tensor factors use no party. Its numbers of pair sources and of pair effects are counted along the composition.
Let \(M\) be an allowed monomial using only parties in \(S\). Let \(W_0\) and \(W_1\) be allowed compositions containing no pair effects, also using only parties in \(S\), whose output and input registers respectively match the input and output registers of \(M\). Let \(M_0\) be the monomial formed by preceding \(M\) by \(W_0\), and \(M_1\) the monomial formed by following \(M\) by \(W_1\). Both are allowed and use only parties in \(S\). If \(F_N\) denotes the operator of a monomial \(N\), then \(F_{M_0}=F_M\circ F_{W_0}\) and \(F_{M_1}=F_{W_1}\circ F_M\). Writing \(s(N)\) and \(e(N)\) for its numbers of pair sources and pair effects, the counts satisfy
For the preceding composition, combine \(W_0\) with the first segment containing no pair effects. For the following composition, induct along the effects of \(M\) and combine its final segment with \(W_1\). The local contraction and normalized-source conditions are preserved by composition, the source counts add, and every original effect is kept exactly once. The operators compose in the stated order.
Identify one register \(\mathbb {C}^\alpha \) with \(\mathbb {C}^\alpha \), and two registers \(\mathbb {C}^\alpha \), \(\mathbb {C}^\beta \) with \(\mathbb {C}^{\alpha \times \beta }\). A matrix \(A\) from \(\mathbb {C}^{\alpha \times \beta }\) (or \(\mathbb {C}^\alpha \)) to \(\mathbb {C}^{\gamma \times \delta }\) (or \(\mathbb {C}^\gamma \)) defines a local map of a party \(p\) on registers of \(p\) of these dimensions, of norm at most \(\lVert A\rVert \). Placed in front of an arbitrary list of further registers, its operator is \(A\otimes \operatorname{id}\) in these identifications. For any finite row and column index sets, the action of a matrix and its associated continuous linear operator agree: both send \(\psi \) to the vector with coordinate \(\sum _i A_{ji}\psi _i\) at \(j\).
Check the identity on pure tensors, using linearity.
Let \(K=\langle s|(\operatorname{id}_T\otimes V)\colon \mathcal H_T\otimes \mathcal H_U\to \mathcal B_E\), with the data of the splitting lemma. Consider a register \(\mathcal H_U\) of \(P_\circ \) and further registers left untouched. For \(Q_\circ \neq P_\circ \), the birth monomial starts from a register \(\mathcal H_T\) of \(P_\circ \): it applies the private contraction \(K\) at \(P_\circ \) from \(\mathcal H_T\otimes \mathcal H_U\) to a register \(\mathcal B_E\) of \(P_\circ \), prepares the normalized pair vector \(s\) on a new register \(\mathcal H_T\) of \(Q_\circ \) and a new register \(\mathcal B_T\) of \(P_\circ \), and applies the private contraction \(V^\dagger \) at \(P_\circ \) from \(\mathcal B_T\otimes \mathcal B_E\) to \(\mathcal H_U\) (05-frames.tex, lines 413–419). The death monomial starts from a register \(\mathcal H_T\) of \(Q_\circ \): it applies the private isometry \(V\) at \(P_\circ \) from \(\mathcal H_U\) to \(\mathcal B_T\otimes \mathcal B_E\), contracts \(\mathcal H_T\) and \(\mathcal B_T\) with the normalized pair effect \(\langle s|\), and applies the private contraction \(K^\dagger \) at \(P_\circ \) from \(\mathcal B_E\) to new registers \(\mathcal H_T\), \(\mathcal H_U\) of \(P_\circ \) (lines 439–446). For \(Q_\circ =P_\circ \) both are the private contraction \(K^\dagger K\) at \(P_\circ \) (lines 447–449).
The birth and death monomials are allowed and use only \(P_\circ \) and \(Q_\circ \); for \(Q_\circ \neq P_\circ \) the birth has one pair source and no pair effect, and the death one pair effect and no pair source. With the registers \(\mathcal H_T\) and \(\mathcal H_U\) identified with \(\mathcal H_T\otimes \mathcal H_U\), the operator of each is \(K^\dagger K\otimes \operatorname{id}\) on the untouched registers, and on the sheet \(B=\operatorname{id}_E\otimes K^\dagger K\).
The contraction \(K\) is the effect \(S^\dagger \mathcal V\) with \(E\) trivial, so \(\lVert K\rVert \le 1\), and \(\lVert V\rVert \le 1\). For the birth, on \(t\otimes u\otimes w\) the first contraction gives \(k=K(t\otimes u)\); preparing \(s=\sum t'\otimes b\) and applying \(V^\dagger \) to \(b\otimes k\) gives \(\sum t'\otimes V^\dagger (b\otimes k)=K^\dagger k\), by comparing coordinates and using linearity in the prepared vector. For the death, \(V u=\sum b\otimes e\) and the effect gives \(\sum \langle s,t\otimes b\rangle e=K(t\otimes u)\), after which \(K^\dagger \) is applied.
In the setting of the canonical birth map, with its hypotheses, there are splitting data with the reference error of Theorem 27.133.2.6 such that \(B=\operatorname{id}_E\otimes K^\dagger K\) with \(K=\langle s|(\operatorname{id}_T\otimes V)\), the registers of \(U\) are owned by \(P_\circ \) before and after the birth, and those of \(T\) by \(P_\circ \) before and by \(Q_\circ \) after it. Group the raw registers of \(T\) into one register \(\mathcal H_T\) and those of \(U\) into one register \(\mathcal H_U\), followed by an arbitrary list of further registers. On these registers there is an allowed monomial using only \(P_\circ \) and \(Q_\circ \), with at most one normalized pair source, from \(\mathcal H_T\) held by \(P_\circ \) to \(\mathcal H_T\) held by \(Q_\circ \), whose operator is \(K^\dagger K\otimes \operatorname{id}\); for \(Q_\circ \neq P_\circ \) it is a private contraction at \(P_\circ \), one pair source between \(Q_\circ \) and \(P_\circ \), and a private contraction at \(P_\circ \). The death is the same with no pair source and at most one normalized pair effect. These grouped monomials are the middle steps of the bounded changes on the registers of the frame in Theorem 27.133.2.17.
Take the splitting data of Theorem 27.133.2.6. The registers of \(U\) lie outside \(E\), so they are owned by \(P_\circ \) before and after, and those of \(T\) are owned by \(P_\circ \) before and by \(Q_\circ \) after. The birth and death monomials have operator \(K^\dagger K\) on \(\mathcal H_T\otimes \mathcal H_U\), and \(B=\operatorname{id}_E\otimes K^\dagger K\).
For a list of sites \(\ell \) and a condition \(f\), let \(\ell _+\) and \(\ell _-\) retain the sites satisfying \(f\) and not satisfying \(f\), respectively, in their original order. Thus \(x\in \ell _+\) exactly when \(x\in \ell \) and \(f(x)\), and \(x\in \ell _-\) exactly when \(x\in \ell \) and \(\neg f(x)\). If \(\ell \) has no repeated site, neither part has one. The following compositions of exchanges of adjacent tensor factors are allowed monomials that use no party and no pair resource, and each sends product basis vectors to the indicated product basis vectors: moving the raw registers of the sites satisfying a condition in front of the others, keeping the order within both parts, and back; reassociating and interchanging blocks of registers; splitting the tag registers of a concatenated list of holes into those of the two parts, and merging them. These reorderings may be performed before arbitrary untouched registers. One step of the site partition starts with a register \(x\) followed by the already separated lists \(A,B\): it leaves \(x\) first in \(A\) if selected, and otherwise moves it past \(A\) to the front of \(B\). The inverse step reverses this move. Adjoining or removing an empty trailing list is an identity and uses no party or pair resource. Naming a register with an equal owner label moves nothing. If a second list of holes is a reordering of the first, with pairwise disjoint outer footprints, there are a relabelling of the tags that preserves the raw part of the encoding and such a composition sending \(|\tau \rangle \) to the relabelled \(|\tau \rangle \) (the canonical identification of tag orderings, 05-frames.tex, lines 84–85).
More generally, adjoining untouched registers before any composition containing no pair effects preserves whether it is allowed, whether it uses only the parties of a given set, and its number of pair sources. Its operator is the identity on the added registers tensored with the original operator, under the prescribed tensor identifications.
Each composition is built by induction on the lists. For the partition step, let \(A,B\) be two site lists and \(C\) an arbitrary untouched register list. In concatenations below, site lists and sites denote their raw register lists and individual raw registers. Write \(J_{A,C}\) for the isometry separating concatenated lists, and let \((A,C)\) denote their concatenation. For the product basis vectors \(u=|c(x)\rangle \), \(a=|c\rangle _A\), \(b=|c\rangle _B\) and any \(y\) in the memory of \(C\), put \(t=J_{B,C}^{-1}(b\otimes y)\) and \(z=J_{A,(B,C)}^{-1}(a\otimes t)\). The selected and unselected steps satisfy
The inverse steps read (908) from right to left. They are the identity and the move of \(x\) back past \(A\), respectively. This proves the partition identities by induction; the two lists are obtained by retaining sites with the indicated condition, which gives the membership and non-repetition assertions. For a reordering of the holes, induct on the reordering: an exchange of two adjacent holes is an exchange of their tag registers, and the raw parts of the encodings of the two holes commute by disjointness of supports. Adjoining untouched registers changes none of the original operations, so induction on the added list preserves each of the stated properties.
Let \(\ell \) be any list of sites, \(T,E\subseteq \Lambda \), and \(U=\Lambda \setminus (T\cup E)\). Retain the sites of \(T\) in \(\ell _T\) and the remaining sites in \(\ell _R\); then retain the sites of \(U\) in \(\ell _R\) to obtain \(\ell _U\), with the remaining sites forming \(\ell _E\). All these lists keep their original order. If \(\ell \) contains every site of \(\Lambda \), then
If also \(T\cap E=\varnothing \), then \(x\in \ell _E\) if and only if \(x\in E\). If \(\ell \) has no repetitions, neither \(\ell _T\) nor \(\ell _U\) has any; this assertion does not require \(\ell \) to contain every site.
When \(\ell \) contains every site exactly once, taking the site at a position gives bijections from the positions of \(\ell _T\) to \(T\) and of \(\ell _U\) to \(U\). For every configuration \(c\) and any assignment of owners to the sites, the corresponding grouping isometries \(G_T,G_U\) satisfy
Filtering gives the membership assertions and preserves the absence of repetitions. Outside \(T\), membership in \(T\cup E\) is membership in \(E\); disjointness ensures that all sites of \(E\) remain. Completeness and non-repetition make the position maps bijective. The canonical coordinate identification sends a product basis vector to the configuration read along that position map, giving (910).
Let \(T,E\subseteq \Lambda \) be disjoint and \(U=\Lambda \setminus (T\cup E)\). For \(a\in [q]^T\), \(b\in [q]^U\), and \(c\in [q]^\Lambda \), define \(m(a,b;c)\) by the inverse of the three-part coordinate identification applied to \(((a,c|_E),b)\). Its restrictions are
For every configuration \(c'\), these restrictions characterize it: \(c'=m(a,b;c)\) if and only if \((c'|_T,c'|_U)=(a,b)\) and \(c'|_E=c|_E\). In particular, \(c^u=m(c|_T,u;c)\) equals \(c\) on \(T\cup E\) and equals \(u\) on \(U\), for any \(u\in [q]^U\).
The inverse coordinate map assigns the prescribed value on each of the three disjoint regions, proving (911). Every site belongs to exactly one region, so equality of the restrictions is equivalent to equality of the configurations. Taking \(a=c|_T\) gives the last assertion.
Let \(T,E\subseteq \Lambda \) be disjoint and \(U=\Lambda \setminus (T\cup E)\), let the raw registers of \(T\) be held by one party and those of \(U\) by one party, before a change of raw owners that keeps the owners on \(E\), and similarly after it. Moving the raw registers of \(T\) and of \(U\) in front by exchanges of tensor factors and grouping each block into one register by a unitary at its owner is an allowed monomial using only these owners and no pair resource; ungrouping has the same properties. Naming the untouched raw registers of \(E\) by their equal owner labels also uses no party or pair resource. If a map acts on the two grouped registers, followed by the tag registers and the raw registers of \(E\), as \(M\otimes \operatorname{id}\), then grouping at the owners before, applying it, and ungrouping at the owners after acts on the registers of the frame, in canonical coordinates, as \(\operatorname{id}_{\rm tags}\otimes (\operatorname{id}_E\otimes M)\).
By (910), on the product basis vector of a configuration \(c\), grouping gives \(|c|_T\rangle \otimes |c|_U\rangle \) in front of the remaining registers; the map gives \(\sum _{a,b}M_{(a,b),(c|_T,c|_U)}|a\rangle \otimes |b\rangle \); ungrouping turns each term into the basis vector of the configuration equal to \(a\) on \(T\), to \(c\) on \(E\) and to \(b\) on \(U\), namely \(m(a,b;c)\) by (911). These are the columns of \(\operatorname{id}_{\rm tags}\otimes (\operatorname{id}_E\otimes M)\), and both sides are linear.
In an encoded frame \(\mathcal F\), let \(T\) be a set of raw sites owned by \(P_\circ \) and disjoint from every outer hole footprint, let \(E\) and \(U\) be as in the birth partition, let \(\lVert \Omega \rVert =1\), and suppose \(I_\Omega (T:E)\le L^{-60}\). Then the birth is a bounded change involving only \(P_\circ \) and \(Q_\circ \), with reference-vector error at most \(L^{-30}\): there are splitting data and an allowed monomial from the registers of \(\mathcal F\) to those of the frame in which \(T\) is owned by \(Q_\circ \), using only \(P_\circ \) and \(Q_\circ \) and at most one normalized pair source, whose operator in canonical coordinates is \(\operatorname{id}_{\rm tags}\otimes B\) for the canonical map \(B\), and which maps the reference vector of \(\mathcal F\) within \(L^{-30}\) of the reference vector of the new frame. The reversed death obeys the same bound when these conditions hold in the frame after the death, with one normalized pair effect instead and no pair source. This is [ 248 , Lemma 6.5 ] , 05-frames.tex, lines 396–410.
Take the splitting data of Theorem 27.133.2.6. The registers of \(T\) and \(U\) are held by \(P_\circ \) before the birth, and those of \(T\) by \(Q_\circ \) and those of \(U\) by \(P_\circ \) after it; the owners on \(E\) are unchanged. Compose the grouping at \(P_\circ \), the birth or death monomial on grouped registers, and the ungrouping at the owners after the change. By Lemma 27.133.2.16 its operator is \(\operatorname{id}_{\rm tags}\otimes (\operatorname{id}_E\otimes K^\dagger K)=\operatorname{id}_{\rm tags}\otimes B\), and the reference error is that of Theorem 27.133.2.6, read through the unitary identification of the registers. For \(Q_\circ \neq P_\circ \) the monomial so built consists of a private contraction at \(P_\circ \), one normalized pair source between \(Q_\circ \) and \(P_\circ \) (one pair effect for the death), and a private contraction at \(P_\circ \), together with reorderings and grouping unitaries.
27.133.3 Exchanging assignments between two sheets
For \(S\subseteq \Lambda \), \(F_S\) is the unitary swapping the two raw sheets at the positions in \(S\) (05-frames.tex, lines 476–477).
For a permutation \(\sigma \) of a finite set, its complex permutation matrix \(P_\sigma \) satisfies \(P_\sigma P_\sigma ^\dagger =P_\sigma ^\dagger P_\sigma =I\).
The adjoint of \(P_\sigma \) is \(P_{\sigma ^{-1}}\), and \(P_\sigma P_{\sigma ^{-1}}=P_{\sigma \sigma ^{-1}}=I\).
\(\| F_S\| \leq 1\).
\(F_S\) is the matrix of a permutation of the two-sheet configurations, so it maps the standard basis bijectively onto itself and preserves the Euclidean norm.
If \(Y\) is the disjoint union of \(T\) and \(A\), then \(F_AF_Y=F_T\). The swap \(F_S\) commutes with \(R_1\otimes R_2\) whenever \(R_1\) and \(R_2\) act on a set disjoint from \(S\) (05-frames.tex, lines 512–520).
The swaps are permutation matrices of the two-sheet configurations. On a configuration, swapping at \(Y\) and then at \(A\subseteq Y\) swaps exactly at the sites of \(Y\) outside \(A\), that is, at \(T\). For the second claim, \(R_1\) and \(R_2\) are the identity on \(S\) tensored with operators \(R_1'\) and \(R_2'\) on the complement, so in the coordinates splitting off the two copies of \(S\) both \(F_S\) and \(R_1\otimes R_2\) act on different factors.
Let \(R\subseteq \Lambda \), and let \(A\) and \(B\) be operators on \(\mathcal H_\Lambda \) supported on \(R\). Then
This identity gives the exchange of replica terms supported inside the swapped region in [ 248 ] , 02-information.tex, lines 416–424.
Split each sheet into \(R\) and its complement. The support assumptions give \(A=A_R\otimes I\) and \(B=B_R\otimes I\). The swap interchanges the two copies of \(R\) and fixes the complementary factors. Writing each configuration as \(x=(x_R,x_{R^c})\), both sides have the entry
in row \((y_1,y_2)\) and column \((x_1,x_2)\).
Let \(q\geq 0\) be a local dimension and \(R,S\subseteq \Lambda \). Let \(A\) be an operator supported on \(S\), and let \(F_R\) exchange two copies at the sites in \(R\). If \(S\subseteq R\) or \(S\cap R=\varnothing \), then
This is the noncrossing cancellation used in the boundary perturbation argument of [ 248 ] .
If \(S\subseteq R\), the exchange identity applied with \(B=I\) on both sides gives
If \(S\cap R=\varnothing \), the swap commutes with \(A\otimes I\) and with \(I\otimes A\). In either case \(F_R(A\otimes I+I\otimes A)=(A\otimes I+I\otimes A)F_R\), and multiplying on the right by \(F_R^\dagger =F_R^{-1}\) gives the stated identity.
Let \(q\geq 0\) be a local dimension and \(R,S\subseteq \Lambda \). Let \(A\) be an operator supported on \(S\). Write \(T(A)=A\otimes I+I\otimes A\), and let \(F_D\) exchange the two copies at the sites in \(D\). Then
This is the support reduction in the boundary perturbation argument of [ 248 ] .
Put \(B=R\cap S\) and \(O=R\setminus S\). The composition of the swaps gives \(F_BF_R=F_O\), hence \(F_R=F_B^\dagger F_O\) by unitarity. Since \(S\cap O=\varnothing \), cancellation outside \(S\) gives
Let \(H=\sum _v h_v+\sum _e h_e\) be an open-square Hamiltonian with the original bounded site and edge interactions, and let \(R\) be any set of vertices. For an operator \(A\), write \(T(A)=A\otimes I+I\otimes A\). Let \(F_R\) exchange the two copies at the sites of \(R\), and let \(\partial R\) be the set of edges with exactly one endpoint in \(R\). Then
This includes zero local dimension and squares of side zero or one. Individual interactions need not be Hermitian. The identity is the crossing-edge reduction in the boundary perturbation argument of [ 248 ] .
Write \(\delta (A)=F_RT(A)F_R^\dagger -T(A)\), which is linear in \(A\). By Theorem 27.133.3.6, \(\delta (h_v)=0\) for every site \(v\), since \(\{ v\} \subseteq R\) or \(\{ v\} \cap R=\varnothing \), and \(\delta (h_e)=0\) for every edge \(e\notin \partial R\), since both endpoints of \(e\) lie in \(R\) or neither does. Hence
Let \(q\geq 0\) be a local dimension and \(R\subseteq \Lambda \). For an arbitrary operator \(A\) on \(\mathcal H_\Lambda \), write \(T(A)=A\otimes I+I\otimes A\). Then
This is the single-interaction estimate for the boundary perturbation in [ 248 ] .
The two identity-factor estimates give \(\| T(A)\| \leq \| A\otimes I\| +\| I\otimes A\| \leq 2\| A\| \). Since \(\| F_R\| \leq 1\) and \(\| F_R^\dagger \| =\| F_R\| \), submultiplicativity gives \(\| F_RT(A)F_R^\dagger \| \leq \| T(A)\| \). The triangle inequality proves the assertion.
Let \(q\geq 0\) be a local dimension and \(R\subseteq \Lambda \). Identify two configurations of local dimension \(q\) with one configuration of local dimension \(q^2\) by pairing the two values at each site. In these coordinates, the sheet swap \(F_R\) is supported on \(R\). The assertion includes empty regions and empty configuration spaces. This is the local support property used for the doubled Hamiltonian in [ 248 ] .
In paired coordinates the swap is the permutation \(\sigma \) of configurations \(x=(x_i)_{i\in \Lambda }\), \(x_i=(a_i,b_i)\), given by
Thus \(\sigma \) fixes every coordinate outside \(R\), and its coordinates in \(R\) depend only on the coordinates of \(x\) in \(R\). The permutation matrix of such a permutation is supported on \(R\).
Let \(\Lambda \) be a finite site set, let \(q\geq 0\) be an integer, and let \(R,S\subseteq \Lambda \). Suppose that \(A\) is supported on \(S\). Express the two-copy operators in paired-site coordinates, and denote the resulting matrices of \(F_R\) and \(A\otimes I+I\otimes A\) by \(\widetilde F_R\) and \(\widetilde T(A)\), respectively. Then \(\widetilde F_R\widetilde T(A)\widetilde F_R^\dagger \) is supported on \(S\). Empty regions and zero local dimension are included. This is the interaction-support assertion used for the boundary perturbation in [ 248 ] .
Set \(B=R\cap S\). By Theorem 27.133.3.7, (912) gives \(F_RT(A)F_R^\dagger =F_B^\dagger T(A)F_B\). In paired-site coordinates, \(F_B\) is supported on \(B\subseteq S\), and each summand of \(T(A)\) is supported on \(S\). Operators supported on \(S\) are closed under adjoints, sums and products, so the conjugated operator is supported on \(S\).
Let \(V=\{ 0,\ldots ,L-1\} ^2\), where \(L\geq 0\) is an integer, and let the local dimension be an integer \(q\geq 0\). Let \(H=\sum _{v\in V}h_v+\sum _e h_e\) be a Hermitian open-square Hamiltonian. Each original site interaction is supported on its site, each original edge interaction is supported on its endpoints, and all these interactions have norm at most \(J\in \mathbb {R}\). For \(R\subseteq V\), let \(\partial R\) consist of the edges with exactly one endpoint in \(R\), and put
In paired-site coordinates, let \(K\) be the matrix of \(H\otimes I+I\otimes H\), and let \(F\) be the matrix of the operator exchanging the two copies at the sites in \(R\). Then \(FKF^\dagger -K\) is supported on \(Z\), with local dimension \(q^2\), and
Individual interactions need not be Hermitian. No further assumption on the sign of \(J\) is required. The statement includes empty grids and zero local dimension. This is the boundary perturbation estimate used in [ 248 ] .
The crossing-edge expansion expresses the perturbation as a sum over \(e\in \partial R\). Each doubled interaction, before and after the regional swap, is supported on the endpoints of \(e\). Closure of the supported operators under subtraction and finite sums proves the support assertion.
If \(L=0\), both original interaction sums vanish and the estimate is \(0\leq 0\). Otherwise, any site \(v\in V\) gives \(0\leq \| h_v\| \leq J\). Each crossing-edge contribution has norm at most \(4\| h_e\| \leq 4J\). The triangle inequality and the square-edge count give
Relabelling the two-copy configurations by paired-site configurations preserves the operator norm.
Consider two encoded frames \(\mathcal F_1,\mathcal F_2\) and the physical sample of a plane region \(Y\), such that every hole’s outer square lies on one side of \(\partial Y\); that side classifies the hole as outside or inside \(Y\). The holes of each frame are listed as those outside \(Y\) followed by those inside \(Y\), each in its original order, which differs from the given listing by the canonical identification of tag orderings (05-frames.tex, lines 84–85). The exchanged frames \(\mathcal F_1',\mathcal F_2'\) carry, inside \(Y\), the raw ownership assignment and the holes, with their tag owners, of the other frame (05-frames.tex, lines 461–464). The renaming \(\mathcal R\) exchanges the sheet names of the raw registers inside \(Y\) and of the tags of the holes inside \(Y\); every register keeps its party.
With \(K_{\rm in}=K_{\mathcal F_1}\otimes K_{\mathcal F_2}\) and \(K_{\rm out}=K_{\mathcal F_1'}\otimes K_{\mathcal F_2'}\),
(05-frames.tex, equation (exchange-intertwining), lines 490–498).
On every pair of tags the raw part of each encoding is the product of an operator acting outside \(Y\) and an operator acting on \(Y\), which is a tensor product in the coordinates \(\mathcal H_Y\otimes \mathcal H_{\Lambda \setminus Y}\). Swapping the sheets at \(Y\) exchanges the \(Y\)-parts, and the four factors match.
With the data of the splitting lemma, the buffer correction on the two copies of \(U\) is
where \(F_{\mathcal B_T}\) swaps the two copies of \(\mathcal B_T\) (05-frames.tex, equation (exchange-buffer-correction), lines 503–510). It acts as the identity on the two copies of \(T\cup E\).
With \(\mathcal V=\operatorname{id}_{TE}\otimes V\),
(05-frames.tex, equation (exchange-net-map), lines 516–526).
In the coordinates in which the two copies of \(\mathcal H_T\otimes \mathcal H_E\) come first, \(\mathcal V^{\otimes 2}=\operatorname{id}\otimes V^{\otimes 2}\), \(F_{T\mathcal B_T}=F_T\otimes F_{\mathcal B_T}\), \(F_T=F_T\otimes \operatorname{id}\) and \(D_U=\operatorname{id}\otimes D_U\), so both sides equal \(F_T\otimes D_U\).
For a unit vector \(\omega \) and \(\delta =\lVert \mathcal V\omega -s\otimes s'\rVert \),
Both \(\mathcal V\omega \) and \(\zeta =s\otimes s'\) are unit vectors, so \(\lVert (\mathcal V\omega )^{\otimes 2}-\zeta ^{\otimes 2}\rVert \le 2\delta \). The swap \(F_{T\mathcal B_T}\) exchanges the two identical factors \(s\) of \(\zeta ^{\otimes 2}\) and fixes it, so by the net map the difference is \((\mathcal V^{\otimes 2})^\dagger (F_{T\mathcal B_T}-\operatorname{id})\bigl((\mathcal V\omega )^{\otimes 2}-\zeta ^{\otimes 2}\bigr)\), of norm at most \(2\cdot 2\delta \) (05-frames.tex, lines 534–553).
For a party \(P_\circ \) put
\(T=Z\cap Y\), \(E=Z\setminus Y\), \(U=\Lambda \setminus Z\) and \(A=Y\cap U\) (05-frames.tex, equations (exchange-Z) and (exchange-partition), lines 466–473). The implemented map is \(C=D_UF_A\mathcal R\), with \(D_U\) and \(F_A\) acting on raw registers and as the identity on all tags.
Every register of \(U\) is raw and held by \(P_\circ \) on both sheets, before and after the exchange (05-frames.tex, lines 500–501).
A register of \(U\) is outside \(Z\), so both its old raw owners are \(P_\circ \) and it lies in no outer hole footprint. The exchanged frames take their raw owners at each site from one of the two old frames, so both new raw owners are \(P_\circ \) as well.
In the setting of the exchange partition, let \(\lVert \Omega \rVert =1\). If \(I_\Omega (T:E)\le L^{-60}\), then for splitting data with error at most \(L^{-30}\) the map \(C\) is a contraction, \(CK_{\rm in}=K_{\rm out}D_UF_T\), and
The encoded reference vectors are not assumed normalized. This is the content of [ 248 , Lemma 6.6 ] for the implemented map, with its hypotheses, when the holes of each frame are listed as those outside \(Y\) followed by those inside \(Y\).
The corrections act on \(U\), away from every hole of both sheets, so they commute with \(K_{\rm out}\); with \(F_AF_Y=F_T\) and the intertwining of the renaming, \(CK_{\rm in}=K_{\rm out}D_UF_T\). The encoding \(K_{\rm out}\) is a contraction, and the exchange error on the reference with \(\delta \le L^{-30}\) gives the bound.
Let \(\mathcal F_1,\mathcal F_2\) be encoded frames whose holes are listed in any order, each hole classified as outside or inside \(Y\) as in the exchange data, and let \(\lVert \Omega \rVert =1\) and \(I_\Omega (T:E)\le L^{-60}\). There are a relabelling \(G\) of the tags of both frames with \(G(K_{\mathcal F_1}\otimes K_{\mathcal F_2})=K_{\rm in}\), the encodings of the regrouped listings, and splitting data with error at most \(L^{-30}\), such that \(CG\) is a contraction, \(CG(K_{\mathcal F_1}\otimes K_{\mathcal F_2})=K_{\rm out}D_UF_T\), and
Encodings of distinct holes commute after the canonical identification of tag orderings (05-frames.tex, lines 84–85), which gives \(G\). Apply the exchange to the regrouped listings and compose with \(G\), a contraction.
In the setting of the exchange partition, the corrections are \(D_UF_A=\operatorname{id}\otimes W\) in the coordinates in which the two copies of \(\mathcal H_T\otimes \mathcal H_E\) come first, where \(W=(V^{\otimes 2})^\dagger F_{\mathcal B_T}V^{\otimes 2}F_A\) acts on the two copies of \(\mathcal H_U\) and \(\lVert W\rVert \le 1\). The registers of \(U\) are held by \(P_\circ \) on both sheets before and after the exchange. On two registers of \(P_\circ \), each grouping the raw registers of \(U\) on one sheet, followed by an arbitrary list of further registers, \(W\otimes \operatorname{id}\) is the operator of one allowed private contraction at \(P_\circ \) (05-frames.tex, lines 503–515).
Since \(A\subseteq U\), the swap \(F_A\) exchanges only the \(A\)-parts of the two copies of \(\mathcal H_U\), so \(F_A=\operatorname{id}\otimes F'_A\), and \(D_U=\operatorname{id}\otimes (V^{\otimes 2})^\dagger F_{\mathcal B_T}V^{\otimes 2}\). Each factor of \(W\) is a contraction.
The registers of two frames, those of the first followed by those of the second, are identified with the product of the canonical coordinates of the two frames, product basis vectors going to standard basis vectors. Two operators on these registers agree as soon as they agree on product basis vectors.
Combine the identifications of the two frames, and expand a vector in the standard basis.
The renaming \(\mathcal R\) is the operator of a composition of exchanges of tensor factors from the registers of the two frames to the registers of the two exchanged frames, using no party and no pair resource. It splits the tags of each frame into those of the holes outside and inside \(Y\), moves the raw registers of the sites of \(Y\) next to the inside tags, exchanges these blocks between the sheets, and merges back; every register keeps its party (05-frames.tex, lines 485–490).
Follow a product basis vector through the reorderings. Inside \(Y\) the configurations and the inside tags of the two sheets change places, and the owners of the moved raw registers are the new owners at those sites, so the result is the basis vector of the renamed configuration.
Let \(T,E\subseteq \Lambda \) and \(U=\Lambda \setminus (T\cup E)\), with two arbitrary assignments of raw owners. Retain the sites of \(U\) in the fixed site order; every site in the complementary list belongs to \(T\cup E\). Let \(|c\rangle _{U,j}\) be the product basis vector on these raw registers on sheet \(j\), and let \(\iota _U\) be the inverse of the canonical isometry separating their two consecutive lists. Grouping both lists gives an isometry \(Q_U\) to \(\mathbb {C}^{[q]^U\times [q]^U}\) satisfying
If \(T\cap E=\varnothing \), take any configurations \(c_1,c_2\) and \(u_1,u_2\in [q]^U\), and set \(c_j^{u_j}=m(c_j|_T,u_j;c_j)\). Then
Group the two lists by their position bijections and identify the tensor product of their coordinate spaces with the space indexed by pairs of configurations. This proves (922). The complementary list contains only sites outside \(U\). For the inverse, \(c_j^{u_j}|_U=u_j\) by (911); substitute into (922) and apply \(Q_U^{-1}\).
Let \(T,E\subseteq \Lambda \) be disjoint and let the raw registers of \(U=\Lambda \setminus (T\cup E)\) be held by \(P_\circ \) on both frames. Moving the raw registers of \(U\) of both sheets to the front, applying a contraction \(W\) of the two copies of \(\mathcal H_U\) as one private contraction at \(P_\circ \), and moving them back is an allowed monomial using only \(P_\circ \) and no pair resource, whose operator in canonical coordinates is the identity on the tags and on the two copies of \(T\cup E\), tensored with \(W\).
By (922) and (923), on a product basis vector the moved registers carry the two configurations on \(U\), the contraction gives the column of \(W\), and moving back gives the basis vectors with these configurations replaced. The reconstructed configurations agree with the originals on \(T\cup E\), so the remaining raw registers and all tags are unchanged.
Consider two encoded frames and a region \(Y\) such that every hole’s outer square lies on one side of \(\partial Y\), a party \(P_\circ \) with \(Z\), \(T\), \(E\), \(U\) as in the exchange partition, and \(\lVert \Omega \rVert =1\) with \(I_\Omega (T:E)\le L^{-60}\). The exchange is implemented by private contractions and register renaming, with reference-vector error at most \(4L^{-30}\): there are splitting data and an allowed monomial from the registers of the two frames to those of the two exchanged frames, consisting of exchanges of tensor factors and private contractions at \(P_\circ \), so using only \(P_\circ \) and no pair resource, whose operator in canonical coordinates is the implemented map \(C=D_UF_A\mathcal R\) (composed with the relabelling \(G\) of the tags when the holes are listed in any order), and which maps the product of the two reference vectors within \(4L^{-30}\) of the product of the new ones. In particular the exchange is a bounded change in the sense of [ 248 , Theorem 5.2 ] . This is [ 248 , Lemma 6.6 ] , 05-frames.tex, lines 460–479; the source allows also the owners of the renamed registers and tags, which exchanges of tensor factors do not involve.
Compose the renaming as reorderings with the corrections on the registers of the exchanged frames, whose raw registers of \(U\) are held by \(P_\circ \) on both sheets. The operators compose to \(D_UF_A\mathcal R\), since \(D_UF_A\) is the identity on the tags and on the two copies of \(T\cup E\) tensored with one contraction of the two copies of \(\mathcal H_U\). For holes in any order, precede by the canonical identification of tag orderings as reorderings, whose operator is \(G\); these exchanges of tag registers move the tag of each hole to the position of that hole in the new list, and \(G\) is the corresponding relabelling of the tags, which preserves the raw part of each encoding. The reference error is that of Theorems 27.133.3.20 and 27.133.3.21, read through the unitary identification of the registers.
27.134 Local interactions after a partial swap
Fix integers \(L,q\ge 0\) and a real number \(J\ge 0\). Let \(H=\sum _i A_i\) be a Hermitian square Hamiltonian, with each \(A_i\) supported on its designated site or edge \(S_i\) and \(\| A_i\| \le J\). Let \(R\) be any set of sites. Put \(A_i^{\mathrm h}=(A_i+A_i^*)/2\) and \(D(A)=A\otimes I+I\otimes A\).
Pair the two replicas at every site, so that the local dimension becomes \(q^2\), and write \(F_R\) for the partial swap in these paired coordinates. A hat will denote this pairing of coordinates. Let \(\sigma \) be the canonical bijection from the square-grid sites to the integer rectangle, and let \(U\) be its induced unitary on configuration spaces. There is a range-one Hermitian interaction family \(g\) on the rectangle, with interaction bound \(2J\), whose terms are
At every other admissible support the term is zero. Its sum is exactly
This is the local interaction construction for the partially swapped doubled Hamiltonian in [ 248 ] .
The Hermitian terms \(A_i^{\mathrm h}\) have their original supports and norm bounds, and their sum is \(H\). After doubling and the partial swap, each term remains supported on \(S_i\). Its norm is at most \(2J\), by the triangle inequality for the two Kronecker factors and the norm bound one for the swap. These estimates also hold in zero-dimensional spaces. Transport the resulting interactions to the integer rectangle and assign zero to unused supports. The supports of distinct original labels are distinct. Linearity of doubling and conjugation gives the asserted exact sum.
27.135 Buffer operators in paired physical coordinates
Let \(V\) be a finite set of sites, let \(q\ge 0\) be an integer, and let \(R,C\subseteq V\) be disjoint. Put \(Q=(R\cup C)^c\) and write \(\mathcal X_D=[q]^D\) for the configurations on a region \(D\). Use the canonical finite enumerations of \(\mathcal X_R\) and \(\mathcal X_C\), and put \(P=[|\mathcal X_R|]\times [|\mathcal X_C|]\). Let \(s:\mathcal X_V\to P\times \mathcal X_Q\) be the resulting coordinate bijection. Define
Let \(p_D:\mathcal X_D\times \mathcal X_D\to [q^2]^D\) pair the two configuration values at each site, and put \(e=p_V\circ g^{-1}\). For every complex matrix \(W\) on \(\mathcal X_Q\times \mathcal X_Q\),
Here reindexing transports both matrix indices, and \(X_Q\otimes \operatorname{id}\) acts as \(X\) on the sites of \(Q\) and as the identity elsewhere. This identifies the physical buffer operator used in the localization argument of [ 248 ] . The identity includes empty regions and zero local dimension.
Compare matrix entries at two paired configurations \(z,z'\), and write \(e^{-1}(z)=(u,x)\) and \(e^{-1}(z')=(u',x')\) with \(u,u'\in P\times P\). Then
The identity factor on \(P\times P\) requires equality of both copies of the restrictions to \(R\) and \(C\). Since the finite enumerations are bijective,
When this condition holds, both entries are the same entry of \(W\) on the two restrictions to \(Q\); otherwise both entries vanish.
27.136 A supported Gaussian operator for the physical buffer
Fix an integer \(q\ge 1\) and a real number \(J\ge 0\). There are positive constants \(C,v,c\) with the following property. Let \(L\ge 0\) be an integer and let \(H=\sum _x h_x+\sum _e h_e\) be a Hermitian Hamiltonian on the open square \(\mathcal V=\{ 0,\ldots ,L-1\} ^2\). Each original interaction is supported on its designated site or nearest-neighbour edge and has norm at most \(J\). Let \(R,C_0\subseteq \mathcal V\) be disjoint, and put \(Q=(R\cup C_0)^c\).
Write \(\mathcal X_D=[q]^D\). The canonical finite enumerations of \(\mathcal X_R\) and \(\mathcal X_{C_0}\) give a coordinate bijection \(s:\mathcal X_{\mathcal V}\to P\times \mathcal X_Q\), where \(P=[|\mathcal X_R|]\times [|\mathcal X_{C_0}|]\). Define the doubled regrouping by
Let \(p\) pair the two configuration values at each physical site. Let \(a:\mathcal V\to \Lambda \) be the canonical bijection to the integer rectangle, and let \(a_*\) be its induced bijection on paired configurations. Put \(u=a_*\circ p\). For a coordinate bijection \(b\), denote matrix transport by \(\mathcal R_b(A)(x,y)=A(b^{-1}x,b^{-1}y)\).
Let \(F_R\) exchange the two copies at the sites in \(R\). For an arbitrary complex matrix \(W\) on \(\mathcal X_Q\times \mathcal X_Q\), define
Let \(\partial R\) be the original nearest-neighbour edges with exactly one endpoint in \(R\), and put
For each integer \(\ell \ge 0\), set \(B=N_\ell (a(Q)\cup a(Z_R))\), with distance measured in the induced graph of \(\Lambda \).
There is a differentiable path \(U\) such that
For every real \(t\), the operator \(U(t)\) is unitary and supported on \(B\). This same path has the following property for every variance \(\eta \ge 0\) and every cutoff \(T\ge 0\). With \(\gamma _\eta \) the centered Gaussian measure of variance \(\eta \), define
Then \(M_{\eta ,T}\) is supported on \(B\) and
The integrals retain the original restricted Gaussian measure. Empty grids and regions, zero variance and zero cutoff are included. This is the spatial Gaussian approximation in [ 248 ] .
Choose \(C,v,c\) from Theorem 36.9.1 at local dimension \(q^2\), range one and interaction bound \(2J\). Theorem 27.134.1 constructs a local interaction family with these parameters and total \(K'\). It uses the Hermitian parts of the original interactions, whose sum is the given Hermitian operator \(H\).
Theorem 27.135.1 identifies \(V\) with the transported lift of \(W\) from \(Q\), so \(V\) is supported on \(a(Q)\). Theorem 27.133.3.12 gives support of \(K'-K\) on \(a(Z_R)\) and \(\| K'-K\| \le 8JL^2\). Coordinate transport preserves norms, while the tensor-product norm inequality gives \(\| V\| \le \| W\| \).
Apply Theorem 36.9.1 to these actual operators and the common neighbourhood \(B\). It constructs the stated path before \(\eta ,T\), and hence the literal integral \(M_{\eta ,T}\). Its support and norm conclusions give the first assertion. Since \(|\Lambda |=L^2\) and \(T\ge 0\), its error estimate yields
27.137 Small-patch rewrites as allowed monomials
The statements below formalize the last step of the proof of the second assertion of Lemma 6.3 of [ 248 ] , September 24, 2026, 05-frames.tex, lines 262–273 and 306–316: every product term of the truncated branch expansion (Theorem 27.132.24) is a monomial allowed by [ 248 , Theorem 5.2 ] on the registers of the frames, one register per site and per tag (Definition 27.132.9). The number of monomials is bounded by the number of branches times \(k^{2m}\); it is polynomial in \(L\) once the number of cylinder terms of each patch and each affected hole is and \(m\), \(r_{\rm old}\), \(r_{\rm new}\) and the specified list of parties are bounded, which is Theorem 27.132.25.
Throughout, \(\Lambda \) is a finite set of sites, each with a raw register \(\mathbb {C}^q\), and operators on \(\mathcal H_\Lambda \) are written in the canonical coordinates of Lemma 27.132.10.
Two allowed monomials on parties applied in succession form an allowed monomial; if both use only parties of a set \(S\), so does their composition, and the numbers of pair sources and of pair effects add up. Let \(v,w\in \mathbb {C}^\alpha \otimes \mathbb {C}^\beta \) be unit vectors. On a register \(\mathbb {C}^\alpha \) of a party \(p_A\) and a register \(\mathbb {C}^\beta \) of a party \(p_B\), the map \(|v\rangle \langle w|\), with output registers held by \(p_A'\) and \(p_B'\) and every other register untouched, is an allowed monomial using only \(p_A,p_B,p_A',p_B'\): the bra \(\langle w|\) is a normalized pair effect if \(p_A\ne p_B\) and a private contraction at \(p_A\) otherwise, and the ket \(|v\rangle \) is a normalized pair source if \(p_A'\ne p_B'\) and a private preparation at \(p_A'\) otherwise. It has at most one pair source, none if \(p_A'=p_B'\), and at most one pair effect, none if \(p_A=p_B\).
The composition of two monomials appends the second to the last source-only composition of the first. The bra applied to \(y\otimes t\) gives \(\langle w,y\rangle t\), and the ket applied to \(t\) gives \(v\otimes t\); their composite is \(\langle w,y\rangle v\otimes t=(|v\rangle \langle w|\, y)\otimes t\). A private contraction or preparation by a unit vector has norm at most one.
A rank-one block consists of disjoint sets \(T,E\subseteq \Lambda \), with \(U=\Lambda \setminus (T\cup E)\), and two vectors \(v,w\) on \([q]^T\times [q]^U\). Its operator is \(|v\rangle \langle w|\) on the sites of \(T\cup U\) tensored with the identity on \(E\); its entry at configurations \(c',c\) is \(v(c'|_T,c'|_U)\, \overline{w(c|_T,c|_U)}\) when \(c'\) and \(c\) agree on \(E\), and \(0\) otherwise. The one-site zero block of a site \(x\) has \(T=\{ x\} \), \(U=\emptyset \) and \(v=w=|0\rangle \), so its operator is \(|0\rangle \langle 0|\) at \(x\). The trivial block has \(T=U=\emptyset \), and its operator is the identity.
Let \(b_1,\dots ,b_r\) be rank-one blocks such that every site lies in \(E_{b_i}\) for all \(i\) but at most one. The entry of the product of their operators at configurations \(c',c\) is \(\prod _iv_{b_i}(c')\, \overline{w_{b_i}(c)}\) when \(c'\) and \(c\) agree on \(\bigcap _iE_{b_i}\), and \(0\) otherwise, where \(v_b(c)\) and \(w_b(c)\) denote the vectors of \(b\) read on the restrictions of \(c\).
By induction on \(r\). In the entry of \(\mathrm{op}(b_1)\, X\), where \(X\) is the product of the remaining operators, only the intermediate configuration equal to \(c'\) on \(E_{b_1}\) and to \(c\) elsewhere contributes, since every site outside \(E_{b_1}\) lies in every \(E_{b_i}\) with \(i\ge 2\); the vectors of each block depend only on the configuration outside its set \(E\).
Let \(b\) be a rank-one block with unit vectors \(v,w\), let the raw registers of \(T\) be held by \(p_A\) before and by \(p_A'\) after, those of \(U\) by \(p_B\) and \(p_B'\), and those of \(E\) by the same party before and after. Then \(\operatorname{id}_{\rm tags}\otimes \mathrm{op}(b)\) is the operator of an allowed monomial between the registers of the frame with the owners before and those after, using only \(p_A,p_B,p_A',p_B'\), with at most one pair source and at most one pair effect. Consequently:
if a nonempty set \(O\) of sites has raw owners in a set of at most two parties of \(S\), then for unit vectors \(v,w\) on \([q]^O\) the operator \(|v\rangle \langle w|\) on \(O\), tensored with the identity, is such a monomial using only parties of \(S\), with the same owners before and after;
for distinct sites \(x_1,\dots ,x_r\) outside of which two owner assignments agree, the product of the one-site zero blocks of the \(x_i\) is the operator of an allowed monomial from the registers with the first owners to those with the second, using only the owners of the \(x_i\) and no pair resource;
a product of operators of allowed monomials on the same registers, each using only parties of \(S\), is the operator of an allowed monomial using only parties of \(S\), with the pair sources and pair effects added up.
If \(O\) is empty, a covector or vector on \([q]^O\) is a scalar of modulus at most one.
Group the registers of \(T\) and of \(U\), apply the rank-one map of Lemma 27.137.1 and ungroup, by Lemma 27.133.2.16. For a set \(O\) with owners in \(\{ p,p'\} \), take \(T\) the sites of \(O\) held by the owner of a chosen site and \(E=\Lambda \setminus O\); a vector on \([q]^O\) is a vector of the same norm on \([q]^T\times [q]^U\). For one-site zero blocks, proceed site by site, contracting with \(\langle 0|\) at the old owner and preparing \(|0\rangle \) at the new owner. Products are compositions.
In the network of a chain \(L_0\cdots L_{n-1}\) of rank-one site operators, an open input leg at a site belongs to one bra vertex, the bra of the factor applied first among those containing the site, and an open output leg to one ket vertex. A site has a fixed input leg exactly when some factor contains it and it is an open input leg of no bra, and dually for outputs. Choose one vector of the truncation on every open-leg group, and let \(W\) be the set of sites contained in some factor. Then the product term equals \(\sigma \, K_1\cdots K_n\, Z\, B_1\cdots B_n\), where \(B_j\) is the block \(|0\rangle \langle w_j|\) on the open input legs of the \(j\)-th bra, \(w_j\) being the conjugate of its covector, \(K_j\) is the block \(|v_j\rangle \langle 0|\) on the open output legs of the \(j\)-th ket, \(Z\) is the product of the one-site zero blocks of \(W\), and \(\sigma \) is the product of the values of the vectors on the vertices without open legs.
By Lemma 27.137.3 the blocks \(B_j\) have entry \(\prod _j\overline{w_j(c)}\) when \(b\) equals \(c\) off the open input legs and vanishes on them; \(Z\) has entry \(1\) when its configurations agree off \(W\) and vanish on \(W\); the blocks \(K_j\) have entry \(\prod _jv_j(c')\) when \(c'\) agrees with the intermediate configuration off the open output legs and the latter vanishes on them. Summing over the two intermediate configurations leaves one term, which is nonzero exactly when \(c'\) and \(c\) agree off \(W\), \(c\) vanishes at the fixed input legs and \(c'\) at the fixed output legs; these are the conditions of the product term.
For a branch \((a,s,b)\), contracting the tag registers of the affected old holes with \(\langle b|\) and preparing those of the affected new holes in \(|a\rangle \), each at its tag owner, is an allowed monomial using only the affected tag owners and no pair resource. In canonical coordinates it acts as \(|a\rangle \langle b|\otimes \operatorname{id}\) on the affected tags and the raw registers, extended by the identity on the untouched tags, and placing a matrix \(Y\) on the raw registers in this tag block equals this operator after \(\operatorname{id}\otimes Y\).
On the basis vector \(|\tau \rangle \otimes |c\rangle \) the monomial gives \(\delta _{\tau _{\rm aff},b}|\tau _{\rm unt},a\rangle \otimes |c\rangle \); an operator between the registers of two frames is determined by its values on basis vectors.
Assume the four conditions of Lemma 6.3, fix a branch \((a,s,b)\), and choose a unit vector on every open-leg group of its network. Then the product term placed in the tag block, \(|a\rangle \langle b|\otimes T\otimes \operatorname{id}\), is \(\sigma \) times the operator of an allowed monomial from the registers of \(\mathcal F_{\rm old}\) to those of \(\mathcal F_{\rm new}\), with \(\lvert \sigma \rvert \le 1\). The monomial uses only the specified parties and has at most \(2m\) normalized pair sources and at most \(2m\) normalized pair effects (05-frames.tex, lines 262–273 and 306–316).
Write the product term as in Lemma 27.137.5. The open input legs of a patch bra have at most two old owners and the open output legs of a patch ket at most two new owners, all in the specified list (Lemma 27.132.23); every site contained in a selected square lies in the physical sample of a patch or an affected hole, so its old and new owners belong to the list; a site in no selected square keeps its owner (Lemma 27.132.18). Apply Lemma 27.137.4 to the blocks \(B_j\) with the old owners, to \(Z\) changing the old owners into the new ones, and to the blocks \(K_j\) with the new owners, and follow with the tag maps of Lemma 27.137.6. Only patch vertices have open legs, and there are at most \(2m\) of them.
Assume the four conditions of Lemma 6.3, let \(\delta {\gt}0\), let \(N\) be the number of branches and \(k=\lceil (4Nm/\delta )^2\rceil +1\). There are a contraction \(M_\delta \) with \(\lVert M_\delta -M\rVert \le \delta \) and an expansion \(M_\delta =\sum _tc_tM_t\) into at most \(Nk^{2m}\) allowed monomials \(M_t\) from the registers of \(\mathcal F_{\rm old}\) to those of \(\mathcal F_{\rm new}\), using only the specified parties, each with at most \(2m\) normalized pair sources and at most \(2m\) normalized pair effects, and with \(\lvert c_t\rvert \le 1\). If every affected hole and every additional patch has at most \(D\) cylinder terms, the number of monomials is at most \(D^Ek'^{2m}\) with \(E=r_{\rm new}+m+r_{\rm old}\) and \(k'=\lceil (4D^Em/\delta )^2\rceil +1\).
This formalizes the second assertion of [ 248 , Lemma 6.3 ] (05-frames.tex, lines 214–217) except for the polynomial count, which is stated here with \(N\) or \(D\) as a parameter. The count is polynomial in \(L\) for \(\delta =L^{-a}\) when \(D\le CL^c\) and \(m\), \(r_{\rm old}\) and \(r_{\rm new}\) are bounded (Theorem 27.132.25). The expansion is stated in the canonical coordinates of the two frames, the tags of the untouched holes first.
Take \(M_\delta =(1+\tfrac 12\delta )^{-1}\sum _\beta |a\rangle \langle b|\otimes \bigl(\sum _ic_{\beta ,i}T_{\beta ,i}\bigr)\otimes \operatorname{id}\) from Theorem 27.132.24 and replace each placed product term by \(\sigma _{\beta ,i}\) times its monomial from Theorem 27.137.7; the coefficients \((1+\tfrac 12\delta )^{-1}c_{\beta ,i}\sigma _{\beta ,i}\) have modulus at most one. Each branch carries at most \(k^{2m}\) terms. The branch count is at most \(D^E\), and \(k\) increases with \(N\).
27.138 Elimination of normalized pair effects
This section treats [ 248 , Lemma 5.1 ] , September 24, 2026, 04-compression.tex, lines 53–127, together with the choice of the stack length and the rescaling in the proof of [ 248 , Theorem 5.2 ] , lines 201–212. The estimates are first proved for monomials written as chains of recorded pair effects between arbitrary contractions. The statements about parties are then proved after assigning every register to a party. Preparing the sources before the local operations and combining them by unordered pairs gives the full conclusion of the source lemma.
Let \(G\) be a contraction with an expansion \(G=\sum _\xi c_\xi M_\xi \) into \(K\) allowed monomials on a fixed set of parties, each with at most \(r\) pair effects. For every integer \(m\ge 1\) there are a gate \(\widetilde G_m\) with additional garbage registers and a fixed normalized garbage vector \(\Gamma _m\) with \(\| \widetilde G_m-G\otimes |\Gamma _m\rangle \| \le (r/\sqrt m)\sum _\xi |c_\xi |\). The gate \(\widetilde G_m\) has an expansion using only local contractions and normalized pair sources, with at most \(Km^r\) monomials and absolute coefficient sum at most \(\sum _\xi |c_\xi |\). Its additional registers are owned by the original participating parties, and all sources on the same pair of parties may be combined into one normalized pair source. This is [ 248 , Lemma 5.1 ] .
The error, normalization, expansion, and ownership conclusions follow from Theorem 27.138.2. Apply Theorem 27.139.7 to each monomial in this expansion. It replaces all sources on a given unordered pair by one normalized source and recovers the original registers by local isometries and exchanges. The operator of each monomial is unchanged, as are the number of monomials, their coefficients, and their common output space.
Let \(G=\sum _\xi c_\xi M_\xi \) be an expansion into \(K\) allowed monomials on parties, each with at most \(r\) pair effects, and let \(m\ge 1\). Then \(\Gamma _m\) is a unit vector prepared by normalized pair sources, and \(\| \widetilde G_m-G\otimes |\Gamma _m\rangle \| \le (r/\sqrt m)\sum _\xi |c_\xi |\). Read on the registers of the parties, \(\widetilde G_m\) is a weighted sum of at most \(Km^r\) operators of allowed source-only compositions, with absolute coefficient sum \(\sum _\xi |c_\xi |\). Its additional registers are the stacks of the effect occurrences, and the stack of an effect on the parties \((p,q)\) consists of \(m\) registers of \(p\) and \(m\) registers of \(q\). This is [ 248 , Lemma 5.1 ] without its last clause, that all sources on one pair of parties may be combined into one, and without the hypothesis that \(G\) is a contraction, which is not needed. The participating parties are the endpoint parties of the effects of the monomials.
Throughout, \(\eta \) is a unit vector of a finite-dimensional Hilbert space \(\mathcal W\), typically a pair space \(\mathcal W=\mathcal U\otimes \mathcal V\), and \(m\ge 1\).
For a position \(k\in \{ 0,\dots ,m-1\} \), let \(I_k:\mathcal W\to \mathcal W^{\otimes m}\) place its input in register \(k\) and a copy of \(\eta \) in every other register. The cyclic insertion map is the average
and the ideal insertion map is \(P_m=|\eta ^{\otimes m}\rangle \langle \eta |\).
For a permutation \(\sigma \) of the registers, \(I_{\sigma (k)}\) is \(I_k\) followed by the corresponding permutation of the tensor factors of \(\mathcal W^{\otimes m}\). Hence \(T_m\) is the average of the \(m\) cyclic register permutations applied after the insertion in the first register. For a pair space \(\mathcal W=\mathcal U\otimes \mathcal V\), the index map of a register permutation permutes the \(\mathcal U\) indices and the \(\mathcal V\) indices by the same permutation.
Both sides agree on product vectors, coordinate by coordinate. The pair statement is the corresponding identity of index maps.
The map \(T_m\) is a contraction, and \(T_m\eta =\eta ^{\otimes m}\).
Each \(I_k\) is an isometry because \(\| \eta \| =1\), so the average of the \(m\) maps has norm at most one. Each \(I_k\) sends \(\eta \) to \(\eta ^{\otimes m}\), and the average of \(m\) equal terms is that term.
For every \(v\perp \eta \) one has \(\| T_mv\| =\| v\| /\sqrt m\). Consequently \(\| T_m-P_m\| \le m^{-1/2}\), with equality when the orthogonal complement of \(\eta \) is nonzero, and \(T_m=P_m\) when it is zero. The same upper bound holds for \((T_m-P_m)\otimes \mathbf1_{\mathcal S}\) on any spectator space \(\mathcal S\).
For \(v\perp \eta \), the vectors \(I_jv\) and \(I_kv\) with \(j\ne k\) are orthogonal, since their inner product contains the factor \(\langle v,\eta \rangle =0\), and each has norm \(\| v\| \). Hence \(\| \sum _kI_kv\| ^2=m\| v\| ^2\) and \(\| T_mv\| =\| v\| /\sqrt m\). For a general input \(v\), the difference \((T_m-P_m)v\) equals \(T_m\) applied to the component \(v-\langle \eta ,v\rangle \eta \) orthogonal to \(\eta \), whose norm is at most \(\| v\| \). This gives the upper bound. A nonzero vector orthogonal to \(\eta \) attains it, and when no such vector exists the difference vanishes. Tensoring with the identity does not increase the operator norm.
A monomial with \(n\) recorded pair effects is a composition
where each \(F_i\) ends with a pair register \(\mathcal W_i\) beside a spectator space \(\mathcal S_i\), and \(E_i=\langle \eta _i|\otimes \mathbf1_{\mathcal S_i}\) contracts that register with a pair bra. The monomial is allowed when every \(F_i\) is a contraction and every \(\eta _i\) is a unit vector. Here the pieces \(F_i\) are arbitrary contractions; no division of registers among parties is recorded.
Give each effect occurrence \(E_i\) of a monomial \(M\) its own output stack \(\mathcal W_i^{\otimes m}\). The replaced monomial \(\widetilde M_m\) replaces each \(E_i\) by \(T_m\) for \(\eta _i\), writing into the stack of that occurrence, which stays idle afterwards. Its ideal stack content is \(\bigotimes _i\eta _i^{\otimes m}\). For a gate expansion \(G=\sum _\xi c_\xi M_\xi \), the stack space of \(G\) is the tensor product of the stacks of all effect occurrences of all monomials, and \(\Gamma _m=\bigotimes _o\eta _o^{\otimes m}\) is the product of all ideal stack contents. The replaced gate \(\widetilde G_m\) is \(\sum _\xi c_\xi \) times the branch in which \(\widetilde M_{\xi ,m}\) acts and the stacks of every other monomial are prepared in their ideal content. Selecting one insertion position \(\kappa _i\) for each occurrence in place of the average defines one term of the expansion of \(\widetilde M_m\), with weight \(m^{-n}\).
For an allowed monomial \(M\) with \(n\) recorded pair effects, \(M\) and \(\widetilde M_m\) are contractions, the ideal stack content is a unit vector, and
By induction on \(n\). Write \(M=M'\circ E_1\circ F_0\), where \(M'\) is the monomial formed by the remaining \(n-1\) effects, and let \(I_{M'}=M'\otimes |\bigotimes _{i\ge 2}\eta _i^{\otimes m}\rangle \) be its ideal replacement. Appending the ideal content of the first stack after \(E_1\) is the same as applying \(P_m\otimes \mathbf1\) in place of \(E_1\). Up to rearranging the stacks, the difference therefore splits as
where \(\mathbf1\) in the first factor of each term is the identity on the new first stack. All other factors are contractions, so the two terms contribute \((n-1)/\sqrt m\) and \(1/\sqrt m\).
The replaced monomial with \(n\) recorded pair effects equals \(m^{-n}\sum _\kappa \widetilde M_{m,\kappa }\), the sum over all choices \(\kappa \) of one insertion position per occurrence. Every term \(\widetilde M_{m,\kappa }\) is a contraction.
Expand each average \(T_m\) by linearity and collect the \(m^n\) choices of positions, by induction on \(n\). Each term composes the contractions \(F_i\) with the isometries \(I_{\kappa _i}\) and tensor rearrangements.
Let \(G=\sum _\xi c_\xi M_\xi \) be an expansion into \(K\) allowed monomials, each with at most \(r\) recorded pair effects, and let \(m\ge 1\). Then \(\Gamma _m\) is a unit vector,
and \(\widetilde G_m\) is a weighted sum of at most \(Km^r\) contractions whose absolute coefficient sum equals \(\sum _\xi |c_\xi |\).
These are the estimates of [ 248 , Lemma 5.1 ] for monomials given as chains of recorded pair effects: the error bound, the common garbage vector, the monomial count and the coefficient sum. The statements about local contractions, normalized pair sources and ownership of the additional registers by the parties are proved in Theorem 27.138.2. The hypothesis that \(G\) is a contraction is not needed.
By induction on the list of monomials. Let \(M_\xi \) be the first monomial, with ideal stack content \(\eta _{(\xi )}\), and let \(G_L\) be the gate of the remaining monomials, with replaced gate \(\widetilde G_{L,m}\) and garbage vector \(\Gamma _{L,m}\). Up to rearranging tensor factors,
The first term is bounded by Theorem 27.138.9 and the second by induction; appending a unit vector and rearranging tensor factors are contractions. Expanding each replaced monomial by Theorem 27.138.10 gives \(m^{n_\xi }\le m^r\) terms of coefficient \(c_\xi m^{-n_\xi }\) for the monomial \(\xi \), whose absolute values sum to \(|c_\xi |\).
For pair vectors \(\eta \in \mathcal U\otimes \mathcal V\) and \(\eta '\in \mathcal U'\otimes \mathcal V'\), regard \(\eta \otimes \eta '\) as a pair vector in \((\mathcal U\otimes \mathcal U')\otimes (\mathcal V\otimes \mathcal V')\). Its norm is \(\| \eta \| \, \| \eta '\| \); in particular, normalized pair vectors combine into a normalized pair vector.
The squared norm is a sum of products of squared coordinates, which factorizes after regrouping the index set.
A register is a Hilbert space together with the party that owns it. The memory of a finite list of registers is the tensor product of their spaces. A source-only composition is a composition of the following operations: a local map from some registers of one party \(p\) to new registers of \(p\), tensored with the identity on all other registers; the preparation of a vector \(\eta \in \mathcal U\otimes \mathcal V\) on two new registers \(\mathcal U\) of a party \(p\) and \(\mathcal V\) of a party \(q\ne p\); and reorderings of the tensor factors. It is allowed when its local maps are contractions and its prepared pair vectors are unit vectors. An operator between two memories is source-only when it is the operator of an allowed source-only composition.
An allowed monomial on parties is a chain of source-only compositions separated by pair effects. Each pair effect contracts a register \(\mathbb C^\alpha \) of a party \(p\) and a register \(\mathbb C^\beta \) of a party \(q\ne p\) with a bra \(\langle \eta |\), where \(\eta \in \mathbb C^\alpha \otimes \mathbb C^\beta \); the monomial is allowed when its source-only compositions are allowed and every \(\eta \) is a unit vector. An effect on arbitrary finite-dimensional registers of \(p\) and \(q\) takes this form after local isometric identifications with \(\mathbb C^\alpha \) and \(\mathbb C^\beta \) and a reordering of tensor factors, absorbed into the preceding source-only composition. As an operator it is a monomial with recorded pair effects in the sense of Definition 27.138.7, whose pieces between effects are source-only. This is the notion of allowed monomial of [ 248 , Section 5.1 ] , lines 32–35.
Let \(\eta \in \mathbb C^\alpha \otimes \mathbb C^\beta \) be a unit vector on the parties \(p\ne q\), and consider a stack of \(m\) pair registers, each made of a register \(\mathbb C^\alpha \) of \(p\) and a register \(\mathbb C^\beta \) of \(q\). Preparing pair sources of \(\eta \) in the positions other than \(k\), with the input pair register in position \(k\), is a source-only composition, and it equals the insertion map \(I_k\). Preparing \(\eta \) in all \(m\) positions gives \(\eta ^{\otimes m}\).
By induction on \(m\): for \(k\) the first position, the remaining \(m-1\) positions are prepared; otherwise a source of \(\eta \) is prepared in the first position and the insertion at position \(k-1\) is applied to the remaining registers.
Let \(G=\sum _\xi c_\xi M_\xi \) be an expansion into allowed monomials on parties. Identify the stack of each effect occurrence on \((p,q)\) with \(m\) pair registers whose \(\mathbb C^\alpha \) halves belong to \(p\) and whose \(\mathbb C^\beta \) halves belong to \(q\). Then every term of the expansion of \(\widetilde G_m\) is source-only, and the garbage vector \(\Gamma _m\) is prepared by normalized pair sources on the endpoint parties of the effects.
In a term, each effect is replaced by an insertion, which is source-only by Theorem 27.138.15, and the remaining pieces are the source-only compositions of the monomial, now acting next to the idle stacks. The stacks of the other monomials are prepared in their ideal content \(\eta ^{\otimes m}\), again by pair sources. Both statements follow by induction on the monomial and on the list of monomials.
The output registers of \(\widetilde G_m\) are the output registers of \(G\) together with the stacks of all effect occurrences of all monomials. The stack of an effect on the parties \((p,q)\) consists of \(m\) pair registers, each made of one register of \(p\) and one register of \(q\).
By induction on the monomials, collecting the stacks.
Let \(\eta \in \mathcal U\otimes \mathcal V\) and \(\eta '\in \mathcal U'\otimes \mathcal V'\) be pair vectors on the parties \(p\ne q\). Preparing \(\eta '\) and then \(\eta \), and merging the two new registers of \(p\) into \(\mathcal U\otimes \mathcal U'\) and the two new registers of \(q\) into \(\mathcal V\otimes \mathcal V'\), is the preparation of the regrouped vector \(\eta \otimes \eta '\in (\mathcal U\otimes \mathcal U')\otimes (\mathcal V\otimes \mathcal V')\), whose norm is \(\| \eta \| \, \| \eta '\| \). For coordinate spaces the norm identity is Theorem 27.138.12.
Both operators agree on product vectors, and the regrouping is an isometry.
In the setting of Theorem 27.138.11, suppose moreover that \(G\) is a contraction, and let \(\delta {\gt}0\) and \(S\ge \sum _\xi |c_\xi |\). With \(m=\lceil (rS/\delta )^2\rceil +1\), the replaced gate satisfies \(\| \widetilde G_m-G\otimes |\Gamma _m\rangle \| \le \delta \) and \(\| \widetilde G_m\| \le 1+\delta \), and \(\widetilde G_m/(1+\delta )\) is a contraction within \(2\delta \) of \(G\otimes |\Gamma _m\rangle \). The number \(m\) depends only on \(r\), \(S\) and \(\delta \).
The choice of \(m\) gives \(\sqrt m\ge rS/\delta \), so the bound of Theorem 27.138.11 is at most \(\delta \). Since \(G\otimes |\Gamma _m\rangle \) is a contraction, the triangle inequality gives \(\| \widetilde G_m\| \le 1+\delta \). Dividing by \(1+\delta \) yields a contraction, and its distance to \(G\otimes |\Gamma _m\rangle \) is at most \(\delta /(1+\delta )+\delta \le 2\delta \).
27.139 Preparing and combining pair sources
Let \(\mathcal H_A\), \(\mathcal H_B\), and \(\mathcal H_C\) be the memories of three ordered lists of registers. Adjacent exchanges give an allowed composition that sends \(a\otimes (b\otimes c)\) to \(b\otimes (a\otimes c)\). The order within each list is preserved. This composition contains no pair-source preparations. Taking the second list to consist of one register gives the corresponding move of that register to the front.
Write \((A,C)\) for the list \(A\) followed by \(C\), and let \(J_{A,C}\) identify its memory with \(\mathcal H_A\otimes \mathcal H_C\). For a register \(r\) with vector \(x\) and vectors \(a\in \mathcal H_A\), \(c\in \mathcal H_C\), the move \(M_{r;A,C}\) of \(r\) past \(A\) satisfies
Induct on \(A\): after moving \(r\) through the tail, exchange it with the first register. The empty-list case is the scalar unit identity. Next induct on \(B\), composing this move for its first register with the exchange of the remaining lists, while leaving that register untouched. The resulting exchange \(X_{A,B;C}\) satisfies, for \(b\in \mathcal H_B\),
The identities extend from elementary tensors by linearity. Only adjacent exchanges and identities are used, so the composition is allowed and has no sources.
A pair source consists of distinct parties \(p,q\), their Hilbert spaces \(\mathcal U,\mathcal V\), and a vector \(\eta \in \mathcal U\otimes \mathcal V\). Its two registers remain assigned to \(p\) and \(q\), respectively. For a finite list \(S=(s_1,\ldots ,s_m)\) of pair sources, let \(\mathcal H_S\) be the tensor product of their registers in list order. The list is normalized when \(\| \eta _j\| =1\) for every source in \(S\); the empty list is normalized. Its preparation on a memory \(\mathcal H\) is the map
Tensor products are identified by the prescribed associativity maps. The source list of a composition records each occurrence separately, including repetitions of the same vector on the same pair of parties. This is the source data used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–251.
Let \(F:\mathcal H_{\rm in}\to \mathcal H_{\rm out}\) be an allowed composition of local contractions, pair-source preparations, and changes of tensor-factor order. Let \(S\) be its source list. There is an allowed composition \(B:\mathcal H_S\otimes \mathcal H_{\rm in}\to \mathcal H_{\rm out}\) consisting of local contractions and changes of tensor-factor order, with no source preparations, such that
Every vector in \(S\) is normalized. The source occurrences and their endpoint parties are those of the original composition.
Induct on the composition. A local operation, an exchange, or the identity contributes no source. For a pair-source preparation, take that source as the entire list and take the remaining operation to be the identity. In the composition case, move a later preparation before an earlier operation \(A:\mathcal H\to \mathcal H'\) by adjoining its untouched registers. The required identity is
It follows first on elementary tensors and then by linearity. When an untouched register precedes the composition, exchanges move it in front of the newly prepared source registers before the remaining local operations act. These exchanges are isometries. Tensoring each original local map with an identity preserves its contraction bound.
For any composition \(W\), write \(\mathcal P(W)\) for the parties of its registers and named local operations, and let \(B_W\) be the remaining composition after its pair preparations are moved to the beginning. Then \(\mathcal P(B_W)=\mathcal P(W)\). If \(f:P\to Q\) relabels the owners, retaining a pair whose new owners coincide as a local preparation, then \(\mathcal P(W^f)=f(\mathcal P(W))\). In particular, for \(q\notin f(P)\), the restriction of \(B_{W^f}\) to \(q\) acts as the identity on the empty memory \(\mathbb {C}\). These statements require neither normalization nor nonzero source vectors. A scalar local operation retains its named party.
Induct on the composition. Moving a pair preparation to the beginning replaces it by the identity on its two endpoint registers and the original spectators, with the same owners. In a concatenation, the extra spectators are source registers of the later composition, so their owners already occur there, and
The exchanges in the framed case involve only the framed owner and owners of the original registers and sources. Thus no party is added or removed. Relabelling preserves register owners and named local maps, and the party sets transform through
When two endpoints acquire the same owner, \(\{ f(p),f(q)\} =\{ f(p)\} \) and that owner is still recorded in the local preparation. Finally, for \(q\notin f(P)\),
and the no-party clause of Theorem 27.140.5 gives the identity on \(\mathbb {C}\).
A source list \(G\) expands to \(S\) when, beside every spectator memory \(\mathcal H\), there is an allowed composition \(E\) with no source preparations such that \(E\circ P_{G,\mathcal H}=P_{S,\mathcal H}\). The unordered pair of a source with endpoints \(p,q\) is \(\{ p,q\} \). Reversing that source exchanges its endpoints and sends its vector through the isometry \(\mathcal U\otimes \mathcal V\to \mathcal V\otimes \mathcal U\); its unordered pair and norm are unchanged.
Expansion is reflexive and transitive, and remains valid upon placing a common source first in both lists. A list beginning with two sources \(t,s\) expands to the list beginning with \(s,t\) and the same remaining sources. A list beginning with the reversal of \(s\) expands to the list beginning with \(s\) and the same remaining sources.
Identity maps and composition prove reflexivity and transitivity. Tensor an expansion \(E\) with the identity on a common first source’s registers, using the prescribed associativity maps, to obtain \(E_s\). Writing \((s,G)\) for \(s\) followed by \(G\), the preparation identity in Theorem 27.139.3 gives \(E_sP_{(s,G),\mathcal H}=P_{(s,S),\mathcal H}\) whenever \(EP_{G,\mathcal H}=P_{S,\mathcal H}\). For successive sources, exchange their entire register lists using (931). For a reversed source, exchange its two registers; applying the tensor exchange twice is the identity. Each construction is allowed and contains no sources.
Let \(S\) be a finite list of normalized pair sources. There is a list \(G\) of normalized pair sources containing exactly one source for each unordered pair of parties occurring in \(S\), and an allowed composition \(E:\mathcal H_G\otimes \mathcal H\to \mathcal H_S\otimes \mathcal H\) with no source preparations, such that
Consequently every allowed composition admits (933) with at most one normalized source on each participating pair.
Insert the sources successively. Exchange tensor factors to bring together the sources with a prescribed unordered pair of owners; reverse the order of the two half-spaces when necessary. Replace two such vectors, on \(U\otimes V\) and \(U'\otimes V'\), by the image of their tensor product under the canonical isometry \((U\otimes V)\otimes (U'\otimes V') \to (U\otimes U')\otimes (V\otimes V')\). Its norm is the product of the two original norms and is therefore one. Splitting the two local tensor factors and exchanging the middle registers recovers the original preparations by isometries. Continuing through the finite list proves (934); no other pair of owners is introduced. Compose this expansion with the local operations in Theorem 27.139.3.
27.140 Collecting local operations by party
For an ordered list of registers \(\ell \) and a party \(p\), let \(\ell _p\) retain precisely the registers owned by \(p\), in their original order. Write \(\mathcal H_{\ell ,p}\) for their joint memory. The memory of an empty list is \(\mathbb C\). A finite ordered list of distinct parties \(p_1,\ldots ,p_s\) gives the grouped memory \(\bigotimes _{j=1}^s\mathcal H_{\ell ,p_j}\). Local maps between these memories act by their ordered tensor product; for an empty party list this is the identity on \(\mathbb C\). Equal register lists are identified by the identity on their memories, preserving operator norms.
The parties of a composition include every register owner and the party assigned to each local operation. In particular, a scalar operation \(\mathbb C\to \mathbb C\) retains its assigned party when both its input and output register lists are empty. These are the local spaces and operations in [ 248 , Theorem 5.2 ] , 04-compression.tex, lines 233–251.
For a choice of parties \(A\), there is a prescribed isometric identification
which retains the order within each selected list. It is obtained solely from associativity, exchanges of tensor factors, and the unit identifications with \(\mathbb C\).
If \(p_1,\ldots ,p_s\) are distinct and contain every owner in \(\ell \), successive partitions give a prescribed isometric identification \(U_\ell :\mathcal H_\ell \to \bigotimes _{j=1}^s\mathcal H_{\ell ,p_j}\).
Induct on \(\ell \). For the empty list, \(U_{A,\emptyset }(z)=1\otimes z\) on \(\mathbb C\). Let \(r\) be a register with vector \(x\), let \(y\in \mathcal H_\ell \) with \(U_{A,\ell }(y)=a\otimes b\), where \(a\in \mathcal H_{\ell |_A}\) and \(b\in \mathcal H_{\ell |_{A^c}}\). For the list \((r,\ell )\) consisting of \(r\) followed by \(\ell \), define \(U_{A,(r,\ell )}\) by linearity from
The first case is the inverse associativity map and the second the exchange of \(x\) past \(a\), each applied after \(\operatorname {id}_r\otimes U_{A,\ell }\). For the second assertion, successively separate the registers of \(p_1,\ldots ,p_s\). Distinctness prevents any register from being repeated, and coverage makes the final remaining register list empty.
Let \(F\) be a composition containing no pair-source preparations. For any choice of parties \(A\), retaining their registers and local operations gives a composition \(F_A\) containing no pair-source preparations. It is allowed whenever \(F\) is allowed. Its local operations occur in their original chronological order, and its parties are precisely those of \(F\) that belong to \(A\).
Retain a local operation exactly when its assigned party belongs to \(A\). Restrict every exchange to the retained registers, and restrict compositions and untouched registers recursively. The local contraction bounds are unchanged. Induction on the composition proves the assertion about its parties, including scalar local operations.
If \(F:\mathcal H_\ell \to \mathcal H_{\ell '}\) contains no pair-source preparations, then for any choice of parties \(A\) its restrictions satisfy
This identity does not require bounds on the norms of the local maps.
Induct on the composition. Let a local operation \(L\) at party \(p\) act on registers \(a\), all owned by \(p\), in front of a list \(\ell \), and let \(J_{a,\ell }\) identify the memory of \(a\) followed by \(\ell \) with \(\mathcal H_a\otimes \mathcal H_\ell \). For \(s\in \mathcal H_a\), \(x\in \mathcal H_{\ell |_A}\), and \(y\in \mathcal H_{\ell |_{A^c}}\),
Thus \(L\) acts on \(s\) inside the first factor when \(p\in A\) and inside the second otherwise, while the other restriction is the identity. Let \(S_{r,t,\ell }\) exchange the first two registers \(r,t\), whose owners are \(p_r,p_t\), and write \(S_A,S_{A^c}\) for its restrictions. For \(u\in \mathcal H_r\) and \(v\in \mathcal H_t\), put \(w=u\otimes (v\otimes U_{A,\ell }^{-1}(x\otimes y))\). The four cases give
When both registers belong to one part, its restriction exchanges them and the other restriction is the identity. When their owners belong to different parts, both restrictions are identities on the selected register lists. Elementary tensors span the memories, so (946) proves the exchange case. Tensoring with an untouched register and composition preserve the identity.
Let \(F:\mathcal H_\ell \to \mathcal H_{\ell '}\) be an allowed composition containing no pair-source preparations. Let \(p_1,\ldots ,p_s\) be distinct parties containing every party of this composition. There are contractions \(B_{p_j}:\mathcal H_{\ell ,p_j}\to \mathcal H_{\ell ',p_j}\) such that
The tensor product is itself a contraction. The list of parties may be chosen to consist exactly of the parties of \(F\). Each map is obtained by composing the local operations at its party. A composition without parties has empty input and output register lists and is the identity on \(\mathbb C\); this is the case \(s=0\). This is the collection of local maps in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–251.
Apply (938) successively with \(A=\{ p_j\} \). The remaining composition has no parties once all \(p_j\) have been removed and is the identity on \(\mathbb C\). The restriction to each party is allowed, so its operator has norm at most one. The tensor product of the resulting contractions gives (947).
Let \(P\) be the prescribed finite set of parties of a gate, and use a fixed ordering of \(P\). Let \(F:\mathcal H_{\rm in}\to \mathcal H_{\rm out}\) be an allowed composition whose register owners and assigned local parties all belong to \(P\). There is a normalized source list \(S\), containing exactly one source for each unordered source pair occurring in \(F\), and a contraction \(B_p\) at each \(p\in P\) such that
Here \(U_{S,\rm in}\) groups the prepared source registers together with the original input registers by their owners; \(U_{\rm out}\) groups the output registers in the same party order. The input and output of \(B_p\) are precisely the corresponding memories of party \(p\).
Let \(W:\mathcal H_\ell \to \mathcal H_m\) be a source-free composition, and let \(p_1,\ldots ,p_s\) be a list of distinct parties containing every party named by \(W\). This includes parties assigned scalar operations on empty memories. Let \(U_\ell ,U_m\) be the canonical isometries grouping registers in this party order. Then
Thus each factor is the operator of the actual singleton restriction. No norm bound, nonvanishing condition or finite-dimensionality assumption on the register spaces is required. Zero maps and the empty party list are included. This is an auxiliary identity for the local-map construction in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–251 and 351–381.
Induct on the ordered party list. For an empty list, the composition has no named parties, hence has empty input and output lists and is the identity on \(\mathbb {C}\), by the no-party clause of Theorem 27.140.5. For a list beginning with \(p\), split the registers into those owned by \(p\) and the complementary registers. Specializing (938) with \(A=\{ p\} \) splits \(W\) into \(W|_{\{ p\} }\) and \(W|_{P\setminus \{ p\} }\). Apply the induction hypothesis to the latter composition. For each remaining party \(q\ne p\), its singleton factor agrees with \(W|_{\{ q\} }\) by (994). The canonical grouping isometries identify this split with the stated party order. Tensoring and composing the resulting operator identities proves (949). No tensor factor is cancelled.
27.140.1 Physical bases and party grouping
Let \(P\) be a finite party set and let \(p_1,\ldots ,p_m\) be an ordering containing each party exactly once. For a register list \(a\), write \(M(a)\) for its right-associated tensor memory, including a terminal factor \(\mathbb {C}\), and \(a_p\) for the registers owned by \(p\), in their original order. The canonical isometry \(G_a\) successively partitions off \(a_{p_1},\ldots ,a_{p_m}\), identifying \(M(a)\) with \(M(a_{p_1})\otimes \cdots \otimes M(a_{p_m})\otimes \mathbb {C}\).
For each \(p\), let \(b_p\) be an orthonormal basis of \(M(a_p)\), indexed by a finite set \(I_p\). The ordered party basis has column \(b_{p_1}(x_{p_1})\otimes \cdots \otimes b_{p_m}(x_{p_m})\otimes 1\), indexed by assignments \(x_p\in I_p\). For an empty ordering, its single column is the scalar \(1\).
Transport the ordered party basis by \(G_a^{-1}\). Its column on the original register memory is
Let the physical space at \(p\) be \(H_p=\mathbb {C}^{d_p}\), where \(d_p\) is a nonnegative integer, and denote its standard orthonormal basis by \(e_{p,j}\). The physical register list has one register \(H_{p_i}\) owned by \(p_i\) in each position. Its labelled basis has column \(E_x=e_{p_1,x_{p_1}}\otimes \cdots \otimes e_{p_m,x_{p_m}}\otimes 1\) for each assignment \(x_p\in \{ 0,\ldots ,d_p-1\} \). The physical output list may be followed by any specified list of private registers.
Let \(P,Q\) be arbitrary sets, let \(f:P\to Q\), and let \(\delta _q\) be nonnegative integers. Let \(\mathbf p=(p_0,\ldots ,p_{m-1})\) be any ordered list in \(P\), with repetitions permitted. Fix a prescribed configuration \(y_q\in \{ 0,\ldots ,\delta _q-1\} \) for every \(q\in Q\). At position \(j\), take the physical space \(\mathbb {C}^{\delta _{f(p_j)}}\). Changing its owner from \(p_j\) to \(f(p_j)\) preserves its space and its position. Write \(U_f\) for the resulting canonical memory isometry. Then
Here the column on the left belongs to the original register list \(\mathbf p\), whose party \(p\) carries the physical dimension \(\delta _{f(p)}\), and the column on the right belongs to the relabelled register list \((f(p_0),\ldots ,f(p_{m-1}))\). No injectivity of \(f\), distinctness of the listed owners, or coverage of \(P\) or \(Q\) is required. The empty list has column \(1\in \mathbb {C}\).
By Theorem 27.140.2.7, changing owners leaves each tensor factor unchanged, and \(U_f\) is the tensor product of the identities on these factors. The prescribed basis column is the ordered tensor product of its local standard basis vectors. Induction on the list therefore gives (951); the empty tensor product is the scalar \(1\).
Let an ordered list of parties have no repetitions. For any party \(p\) in this list, selecting its physical registers gives exactly its one register \(H_p\), with memory \(H_p\otimes \mathbb {C}\).
Selecting a party filters the given list by equality with \(p\). Since \(p\) occurs exactly once, the filtered register list is the singleton containing its physical register.
Let \(p\) occur in an ordered physical list with no repetitions. On its physical memory \(H_p\otimes \mathbb {C}\), take the orthonormal basis with columns \(b_p(j)=e_{p,j}\otimes 1\). This is the standard basis transported by the inverse of the right scalar-unit isometry.
Let the first register have owner \(p\) and Hilbert space \(H\), and let every remaining register have an owner different from \(p\). Write \(K\) for the memory of the remaining registers. Under the canonical partition into the registers of \(p\) and their complement, every \(e\in H\) and \(y\in K\) satisfy
Partition the remaining registers first. Their selected memory is \(\mathbb {C}\), so \(y\) is sent to \(1\otimes y\). Adjoining the first register and applying the associativity isometry gives \(e\otimes y\mapsto (e\otimes 1)\otimes y\).
Let \(P\) be an arbitrary set of parties, let \(d_p\) be arbitrary nonnegative integers, and let \(\mathbf p=(p_0,\ldots ,p_{m-1})\) be an ordered list without repetitions. Put \(a=a_{\mathbf d}(\mathbf p)\), with register \(\mathbb {C}^{d_{p_j}}\) at the party in position \(j\). For each party \(p\), let \(b_p\) be an orthonormal basis of the complete memory \(M(a_p)\), indexed by a finite set \(I_p\). Let \(x\in \prod _{p\in P}\{ 0,\ldots ,d_p-1\} \) and \(y\in \prod _{p\in P} I_p\).
Suppose that, at every party in \(\mathbf p\), the canonical singleton-memory identification sends \(b_p(y_p)\) to \(e_{x_p}\otimes 1\). Then grouping the prescribed physical tensor column gives the ordered column of these local bases:
Here \(b_{\mathbf d,\mathbf p}\) is the ordered Euclidean tensor basis, \(G_{\mathbf p,a}\) successively separates the complete memories in the listed order, and \(b_{a,\mathbf p}\) is the ordered tensor basis formed from \(b_{p_0},\ldots ,b_{p_{m-1}}\). The local index sets need not be the physical index sets. No coverage of all of \(P\) is required: the list covers the owners of \(a\) by construction. The empty list and zero local dimensions are included, with quantification over the indicated assignments. Both ordered memories retain their terminal scalar factor \(\mathbb {C}\).
Induct on the length of the ordered list. For the empty list, both columns are \(1\in \mathbb {C}\). At a nonempty list, its first party does not occur in the tail. The selected-register partition therefore separates its physical column as \(e_{x_p}\otimes 1\). Apply the induction hypothesis to the tail. The prescribed physical column thus groups as the ordered tensor product of its singleton columns. By the given identifications, these are precisely \(b_{p_j}(y_{p_j})\) at every listed party. The ordered tensor basis has the same column,
Let the physical register list \(a\) and the singleton bases \(b_p\) be as above. For every physical assignment \(x\),
Thus the labelled physical basis equals the party basis on the original register order. No positivity of \(d_p\) is required. If some \(d_p=0\), the assignment set and both bases are empty. If \(P\) is empty, both tensor products are the scalar \(1\).
First consider any ordered list without repetitions, retaining only its physical registers. Induct on this list. For the empty list, both memories are \(\mathbb {C}\) with basis vector \(1\). In the induction step, no remaining register is owned by the first party. Theorem 27.140.1.7 separates its basis vector as \(e_{p,j}\otimes 1\), leaving the ordered physical basis vector on the tail. Apply the induction hypothesis to that tail. When the list exhausts \(P\), transporting the resulting equality by \(G_a^{-1}\) gives equality of the two bases.
For register lists \(a,c\), choose orthonormal bases \(b_p\) of \(M(a_p)\) and \(f_p\) of \(M(c_p)\), indexed respectively by finite sets \(X_p\) and \(F_p\). Let \(J_{u,v}:M(u\mathbin {+\! +}v) \longrightarrow M(u)\otimes M(v)\) be the canonical concatenation isometry. The joint basis at party \(p\) has column \(J_{a_p,c_p}^{-1}(b_p(x_p)\otimes f_p(y_p))\), under the identification \((a\mathbin {+\! +}c)_p=a_p\mathbin {+\! +}c_p\).
With the joint local bases just defined, let \(B^{a+c}\) denote their party basis on \(M(a\mathbin {+\! +}c)\), and let \(B^a,B^c\) be the separately transported party bases. Then for all assignments \(x_p\in X_p\) and \(y_p\in F_p\),
The index \((x,y)\) denotes the assignment \(p\mapsto (x_p,y_p)\); equivalently, this is equality of the two bases after the canonical separation of the two assignment families.
Induct on the ordered party list. Partitioning a concatenation first partitions each list and then regroups its four factors, joining the two selected memories and the two complementary memories. The selected vector is the stated joint local column. Apply the induction hypothesis to the two complementary lists, and transport by \(G_{a+c}^{-1}\). For the empty list all memories are scalar and the assertion reduces to multiplication in \(\mathbb {C}\).
These coordinate identities apply to the local contractions in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 137–151 and 565–588.
Let \(P\) be any party set, let \(Q=P\sqcup \{ *\} \), and let \(\iota :P\to Q\) assign each original party its selected owner. For any register list \(a\), the registers of \(\iota _*a\) at \(*\) form the empty list. At \(\iota (p)\), their list is \(\iota _*(a_p)\), in the original order. This is the all-party selection in the proof of [ 248 , Theorem 5.2 ] .
No original party maps to \(*\), so restriction to \(*\) is empty. An original register maps to \(\iota (p)\) exactly when its owner is \(p\). Restriction after relabelling therefore selects exactly \(a_p\).
Let \(d:P\to \mathbb N\) specify the original physical dimensions, and let \(\mathbf p\) list every original party exactly once. Set \(X_*=\{ \bullet \} \) and \(X_{\iota (p)}=\{ 0,\ldots ,d(p)-1\} \). On the physical memory at \(*\), take the scalar basis with column \(1\). At \(\iota (p)\), transport the original singleton physical basis \(b_p\) along the canonical owner isometry and the register-list equality of Theorem 27.140.1.12. Denote the resulting basis by \(b^{\mathrm{sel}}_{\iota (p)}\). Dimensions are retained individually, including \(d(p)=0\); the original party set may be empty.
Under the canonical memory identifications, \(b^{\mathrm{sel}}_{*,\bullet }=1\) and \(b^{\mathrm{sel}}_{\iota (p),j}=b_{p,j}\) for every \(j\in X_{\iota (p)}\). No equality between independently chosen bases is required.
The scalar singleton basis has column \(1\). Equality identifications of register lists preserve vectors. The canonical owner isometry also preserves each vector under the equality of the underlying memory spaces, so the transported original columns have the stated values.
Write \(a\) for the original physical register list and \(c\) for the private output list. For each selected owner \(q\), choose an orthonormal basis \(f_q\) of \(M((\iota _*c)_q)\), indexed by a finite set \(F_q\). The local basis on \(M((\iota _*(a\mathbin {+\! +}c))_q)\) has indices \((x_q,e_q)\in X_q\times F_q\) and column
Here the two memories are identified using \((\iota _*(a\mathbin {+\! +}c))_q =(\iota _*a)_q\mathbin {+\! +}(\iota _*c)_q\). Thus the physical coordinate has its original dimension at each \(\iota (p)\) and a single scalar coordinate at \(*\). The private bases are arbitrary; their columns are retained in (957).
The canonical bijection \(e:\prod _{q\in Q}X_q\to \prod _{p\in P}\{ 0,\ldots ,d(p)-1\} \), given by \(e(x)_p=x_{\iota (p)}\), removes the single exterior coordinate. Its inverse assigns \(x_*=\bullet \) and \(x_{\iota (p)}=y_p\). It neither enlarges a physical space nor requires that a party exist.
Let \(P\) be finite, and let \(M_q\) be arbitrary complex matrices indexed by \(X_q\). In the original physical configurations,
The exterior scalar may vanish. If \(P\) is empty, the product on the right is \(1\).
Separate the exterior factor from the finite product over \(Q=P\sqcup \{ *\} \), and relabel the remaining factors by their original owners. No division is used.
From the original order \(\mathbf p=(p_0,\ldots ,p_{m-1})\), form \(\mathbf q=(*,\iota (p_0),\ldots ,\iota (p_{m-1}))\). If \(\mathbf p\) lists every original party exactly once, then \(\mathbf q\) lists every selected owner exactly once. If there is no original party, \(\mathbf q=(*)\).
Let \(\mathbf p\) list every original party exactly once. Group the relabelled physical memory in the selected order \(\mathbf q\), using the local bases \(b_q^{\mathrm{sel}}\). Write \(B\) for the resulting basis on the original register order, and \(U_\iota \) for the canonical owner isometry. In each original physical configuration \(x\),
This identifies the two constructed bases. The exterior party contributes the scalar column \(1\), and no additional physical register is inserted. Empty party sets and zero physical dimensions are included.
First group the relabelled physical column along \((\iota (p_0),\ldots ,\iota (p_{m-1}))\). The owner isometry preserves the ordered physical factors. At each listed owner the selected local column is the original singleton column, so the grouped vector is their ordered tensor product. Prepending \(*\) contributes only \(1\), since it owns no register. Transport back to the register memory to obtain (959).
Choose any order containing every selected owner exactly once. Let \(C\) be the grouped basis obtained from the actual local columns in (957), and let \(B^{\iota _*a}\) and \(B^{\iota _*c}\) be the separately grouped physical and private bases in the same order. Separating the two coordinate families gives
The output memories are identified through \(\iota _*(a\mathbin {+\! +}c)=\iota _*a\mathbin {+\! +}\iota _*c\). All private basis choices are retained, and the physical indices have their original dimensions with the exterior singleton coordinate.
Apply the concatenation identity to the separately relabelled register lists. Transport each local column and the grouped column by the displayed equality of output lists. Equality identifications preserve these columns, yielding (960).
27.140.2 Register identifications and tensor coordinates
A register has an owner in a set \(P\) and a complex inner product space. For a finite ordered register list \(a\), let \(M(a)\) be its right-associated tensor memory, with \(M(\varnothing )=\mathbb {C}\) and a terminal factor \(\mathbb {C}\). Write \(ab\) for concatenation, and \(J_{a,b}:M(ab)\longrightarrow M(a)\otimes M(b)\) for its canonical isometry. For a selection \(f:P\to \{ 0,1\} \), let \(a_f\) retain the registers with \(f(p)=1\), in their original order, and write \(\bar f=1-f\). The canonical partition isometry is \(U_f^a:M(a)\longrightarrow M(a_f)\otimes M(a_{\bar f})\). Equal register lists are identified by their canonical isometries, denoted \(E_{a=b}\). All tensor products below retain these conventions.
For equal register lists \(a=a'\), the identification \(E_{a=a'}\) preserves each vector after identifying its underlying space, and \(E_{a'=a}=E_{a=a'}^{-1}\). For \(a=a'\) and \(b=b'\), the tensor \(E_{a=a'}x\otimes E_{b=b'}y\) is the same tensor as \(x\otimes y\) under the induced identification. More generally, if \(x'\) and \(y'\) agree with \(x\) and \(y\) under these identifications, then \(J_{a,b}^{-1}(x\otimes y)\) and \(J_{a',b'}^{-1}(x'\otimes y')\) agree under the identification \(M(ab)=M(a'b')\).
Identify the equal register lists and their equal vectors. Each identification becomes the identity, so all assertions are equalities of the same vectors or isometries.
Let \(r\) be a register with space \(U\), let \(a=b\) be equal register lists, and let \(u\in U\) and \(x\in M(a)\). Then \(E_{ra=rb}(u\otimes x)=u\otimes E_{a=b}x\).
Identify the equal tail lists. Both sides are \(u\otimes x\).
Let \(r\) be a register with space \(U\), let \(a=a'\) and \(b=b'\) be equal register lists, and suppose \(z\in M(a)\otimes M(b)\) and \(z'\in M(a')\otimes M(b')\) agree after these identifications. Write \(\alpha \) for the tensor associativity isometry and \(L\) for the isometry exchanging the first two factors. For every \(u\in U\), after identifying the memories, \(\alpha ^{-1}(u\otimes z)=\alpha ^{-1}(u\otimes z')\) and \(L(u\otimes z)=L(u\otimes z')\). More generally, for equal complex inner product spaces \(A=B\) and vectors \(x,y\in A\) and \(x',y'\in B\) agreeing under this equality, \(x+y\) agrees with \(x'+y'\).
Identify the equal lists or spaces and their equal vectors. Each pair is then the same tensor or sum.
For every register list \(a\), vector \(x\in M(a)\) and scalar \(z\in \mathbb {C}\), \(E_{a\varnothing =a}J_{a,\varnothing }^{-1}(x\otimes z)=zx\). In particular, \(J_{a,\varnothing }^{-1}(x\otimes 1)=E_{a\varnothing =a}^{-1}x\).
Induct on \(a\), expanding the concatenation isometry on elementary tensors. Move the scalar through each tensor factor. Linearity gives the identity for arbitrary vectors; set \(z=1\) for the second assertion.
Let \(q:P\to Q\) be any map. Write \(q_*a\) for the register list obtained by replacing each owner \(p\) by \(q(p)\), retaining the ordered register spaces, and let \(T_q^a:M(a)\to M(q_*a)\) be the tensor product of their identity maps.
For any selection \(g\) on \(Q\), one has \((q_*a)_g=q_*(a_{g\circ q})\).
Selecting a register after changing its owner retains it precisely when its original owner satisfies \(g\circ q\). The ordered spaces on the two sides therefore agree.
The selected-memory isometry \(T_{q,g}^a:M(a_{g\circ q})\to M((q_*a)_g)\) is \(T_q^{a_{g\circ q}}\) followed by the identification of the equal register lists in Theorem 27.140.2.5.
For any \(q:P\to Q\) and register list \(a\), the ordered tensor spaces satisfy \(M(q_*a)=M(a)\). Under this equality, \(T_q^a x=x\) for every \(x\in M(a)\).
Induct on the register list. Relabelling leaves each head space unchanged; apply the induction hypothesis to its tail. On the resulting tensor product the isometry is the identity, first on elementary tensors and then by linearity.
For every \(q:P\to Q\), selection \(g\) on \(Q\), register list \(a\), and \(x\in M(a)\),
After identifying the selected and complementary tensor spaces, this is the same vector as \(U_{g\circ q}^a x\).
Induct on \(a\). The first register has the same selection value before and after relabelling. Use the corresponding associativity or exchange isometry, and apply the induction hypothesis to the tail. The selected-memory isometries preserve the underlying vectors, giving (961).
If \(f\) vanishes at the owner of every register in \(a\), then \(a_f=\varnothing \), \(a_{\bar f}=a\), and, under these list identifications, \(U_f^a x=1\otimes x\) for every \(x\in M(a)\).
The filtered lists have the stated forms. Induct on \(a\), using the complementary-head partition formula. Exchange the scalar unit with each head factor; linearity extends the identity from elementary tensors to every vector.
Let \(\mathbf p\) be an ordered list without repetitions containing every owner of registers in \(a\) and \(b\). Suppose \(a=b\) and that \(x\in M(a)\) and \(y\in M(b)\) agree under this identification. Write \(G_{\mathbf p,a}\) for the isometry that successively collects the registers of each party in the prescribed order. Then \(G_{\mathbf p,a}x\) and \(G_{\mathbf p,b}y\) agree under the equality of their grouped register lists. The list need not contain parties absent from the register memories.
Identify \(a=b\) and \(x=y\). The two isometries give the same successive partitions of the same ordered registers.
Let \(P\) be any party set and let \((p,\mathbf p)\) be an ordered list without repetitions containing every owner in the ordered register list \(a\). Write \(a_p\) and \(a_{\bar p}\) for its registers owned by \(p\) and its remaining registers, respectively. Let \(U_p^a\) be the partition isometry and \(G_{\mathbf p,a}\) the grouping isometry in the prescribed order. Each grouped memory has one memory factor per listed party, with a terminal factor \(\mathbb {C}\).
Since \(p\) is absent from \(\mathbf p\), removing its registers leaves the registers of each party in \(\mathbf p\) unchanged. Let \(E\) identify the resulting equal grouped tail memories. For every \(x\in M(a_p)\) and \(y\in M(a_{\bar p})\),
The list need not contain parties absent from \(a\); its tail may be empty. No dimension or nonvanishing condition is imposed.
By the recursive grouping identity, \(G_{(p,\mathbf p),a}=(1\otimes E G_{\mathbf p,a_{\bar p}})U_p^a\). Cancel \(U_p^a(U_p^a)^{-1}\) and apply the tensor product map to \(x\otimes y\).
Let \((q,\mathbf q)\) be an ordered list of distinct parties, and let \(a\) be a register list whose owners all belong to \(\mathbf q\). Under the canonical identifications \(a_q=\varnothing \) and \(a_{\bar q}=a\), every \(x\in M(a)\) satisfies
The register spaces are arbitrary, the list \(\mathbf q\) may be empty, and the identity also holds for the zero vector. Both grouped memories retain their terminal scalar factor; for an empty tail the right side is \(1\otimes x\) with \(x\in \mathbb {C}\). This is the empty exterior factor used in the proof of Theorem 5.2 of [ 248 ] .
Since the order has no repetitions and all owners lie in the tail, the first party owns no register. The partition identity therefore gives \(U_q^a x=1\otimes x\). Apply the first-party grouping identity to this tensor and identify the complementary register list with \(a\). The resulting tail isometry is \(G_{\mathbf q,a}\).
For register lists \(a_1,a_2,b_1,b_2\), let \(K\) be the isometry that exchanges the middle two factors and concatenates the two resulting pairs. For every \(x_i\in M(a_i)\) and \(y_i\in M(b_i)\),
For any selection \(f\), register lists \(a,b\), and vectors \(x\in M(a)\), \(y\in M(b)\), identify \((ab)_f=a_fb_f\) and \((ab)_{\bar f}=a_{\bar f}b_{\bar f}\). With \(K\) from Definition 27.140.2.13 for these four lists,
Induct on \(a\) and extend by linearity in \(x\). A selected head adjoins its factor to the first joined memory; an excluded head adjoins it to the second. In each case, associativity and the exchange of the middle factors reduce the assertion to the tail.
Choose orthonormal bases \(b_A,b_E\) of \(M(a_f),M(a_{\bar f})\), and \(b_D,b_F\) of \(M(b_f),M(b_{\bar f})\), indexed by finite sets \(X,Y,I,L\), respectively. Their partition bases on the original memories are \(D^a_{x,y}=(U_f^a)^{-1}(b_A(x)\otimes b_E(y))\) and \(D^b_{i,j}=(U_f^b)^{-1}(b_D(i)\otimes b_F(j))\). On the selected and complementary concatenated memories, take \(C_{x,i}=J_{a_f,b_f}^{-1}(b_A(x)\otimes b_D(i))\) and \(F_{y,j}=J_{a_{\bar f},b_{\bar f}}^{-1}(b_E(y)\otimes b_F(j))\), transported along the corresponding list equalities.
With the bases of Definition 27.140.2.15, every \(x\in X\), \(y\in Y\), \(i\in I\), \(j\in L\) satisfies
The second identity holds for every \(z\in M(ab)\). It also holds for a register list \(c\) with a specified equality \(c=ab\) and \(z\in M(c)\): apply \(E_{c=ab}\) before \(J_{a,b}\) on the left, and transport \(C\) and \(F\) to \(M(c_f)\) and \(M(c_{\bar f})\) on the right, where the partition is \(U_f^c\).
For any register lists \(a,b\) and selection \(f\), let \(b_A\) be an orthonormal basis of \(M(a_f)\) indexed by a finite set \(X\). Denote its coordinate isometry by \(b_A^\# \). The selected output isometry is \(O=(b_A^\# \otimes 1)J_{a_f,b_f}\), after identifying \((ab)_f=a_fb_f\).
With the selected output isometry just defined, for any orthonormal basis \(b_D\) of \(M(b_f)\) indexed by a finite set \(I\), and any \(x\in X\), \(i\in I\),
Here \(e_x\) is the standard basis of \(\mathbb {C}^X\), \(C_{x,i}=J_{a_f,b_f}^{-1}(b_A(x)\otimes b_D(i))\) in the identified selected memory, and \(z\in M((ab)_f)\) is arbitrary.
The concatenation isometry sends \(C_{x,i}\) to \(b_A(x)\otimes b_D(i)\). Taking the coordinates of the first factor gives \(e_x\otimes b_D(i)\). Preservation of inner products gives the second identity.
Let \(q:P\to Q\), let \(a,b\) be register lists, and let \(b_A,b_D\) be orthonormal bases of \(M(q_*a),M(q_*b)\) indexed by finite sets \(X,I\). For every \(z\in M(ab)\), \(x\in X\), \(i\in I\),
with \(q_*(ab)=(q_*a)(q_*b)\) identified on the right.
Changing owners commutes with concatenation. Apply that identity to the two transported basis columns, then use preservation of inner products by the owner and concatenation isometries.
Physical register lists and regional dimensions
For a nonnegative integer \(d\), write \([d]=\{ 0,\ldots ,d-1\} \). If \(d=0\), this set is empty. Let \(\mathbf p=(p_0,\ldots ,p_{m-1})\) be an ordered list of parties. Repetitions are permitted unless excluded explicitly.
Let \(d_p\) be arbitrary nonnegative local dimensions, and write \(a_{\mathbf d}(\mathbf p)\) for the register list with space \(\mathbb {C}^{d_{p_i}}\) in position \(i\). The owner sequence of \(a_{\mathbf d}(\mathbf p)\) is precisely \(\mathbf p\). Restriction by any selection \(f\) retains exactly the corresponding subsequence:
Taking owners reads the same ordered party list. Filtering the registers commutes with assigning their spaces, giving the restriction identity.
Suppose \(\mathbf p\) has no repetitions and contains every party of \(P\), and let \(A\subseteq P\) be finite. Reading the party at each position of the filtered list \(\mathbf p_A\) is a bijection from its positions to \(A\). Reindex the ordered standard tensor basis of these registers by this bijection. Its indices are assignments \(x_p\in [d_p]\) for \(p\in A\). Transport along the physical-list restriction identity to obtain bases on the actual memories selected by \(A\). The common-dimension case is obtained by setting \(d_p=d\).
For arbitrary nonnegative local dimensions \(d_p\) and finite \(A\subseteq P\),
If \(\mathbf p\) has no repetitions and contains every party of \(P\), then \(|\mathbf p_A|=|A|\) and
In particular, for a common dimension \(d_p=d\), both the number of regional configurations and the selected memory dimension are \(d^{|A|}\). If \(D\) is a nonnegative integer and \(d_p\le D\) for every \(p\in A\), the selected memory dimension is at most \(D^{|A|}\). Empty products have value \(1\), including when \(A\) is empty. If \(d_p=0\) for some \(p\in A\), the regional dimension is zero.
Count the assignments indexing each tensor basis. An exhaustive ordering without repetitions contains each element of \(A\) once, so its filtered list has length \(|A|\). Dimension equals the number of basis elements. Multiply the inequalities \(d_p\le D\) over \(A\) to obtain the bound.
For any set \(P\) and finite \(A\subseteq P\), splitting an assignment into its restrictions to \(A\) and \(P\setminus A\) is a bijection, for both constant dimension \(d\) and arbitrary local dimensions \(d_p\). Now suppose \(\mathbf p\) lists \(P\) without repetitions, and let \(b\) be a private register list with finite-dimensional memory \(M(b)\) of dimension \(r\). Fix its standard orthonormal basis \(\gamma _0,\ldots ,\gamma _{r-1}\). Let \(E_x\) denote the labelled physical standard basis, and define the coordinate isometry \(R:M(a b)\to \mathbb {C}^{\mathcal X\times [r]}\) by
where \(a\) is the chosen physical list and \(\mathcal X\) its assignment set. The regional coordinate isometry \(R_A:M(ab)\to \mathbb {C}^{\mathcal X_A\times (\mathcal X_{P\setminus A}\times [r])}\) is obtained by separating the physical assignment into its regional and exterior restrictions, with the private index retained in the latter coordinate pair.
Under the hypotheses of Definition 27.140.2.23, let \(s,t\) be assignments on \(A,P\setminus A\), respectively, and let \(s\sqcup t\) be their unique common assignment on \(P\). For every \(v\in M(ab)\) and \(i\in [r]\), both choices of physical dimensions satisfy \(R_A(v)(s,(t,i))=R(v)(s\sqcup t,i)\) and \(\| R_A\| _{\rm op}\le 1\).
Evaluate the reindexed product-basis coordinates at \((s,(t,i))\). This is the original product-basis coordinate at \((s\sqcup t,i)\). The coordinate map is an isometry, which gives the operator-norm bound, also for a zero-dimensional domain.
Ordered party tensor coordinates
Let \(a_p\) select all registers of \(a\) owned by \(p\). For every \(p\), choose an orthonormal basis \(b_p\) of \(M(a_p)\) indexed by a finite set \(I_p\). For any ordered party list \(\mathbf p=(p_0,\ldots ,p_{m-1})\), let \(B^{\mathbf p}\) be its right-associated product basis on \(M(a_{p_0})\otimes \cdots \otimes M(a_{p_{m-1}})\otimes \mathbb {C}\).
The empty list has one basis index, with column \(1\), and the coordinate of \(z\in \mathbb {C}\) at this index is \(z\). If a party \(p\) is adjoined to the head of an arbitrary list \(\mathbf p\), its basis column with index \((x_0,x_{\rm tail})\) is \(b_p(x_0)\otimes B^{\mathbf p}_{x_{\rm tail}}\). For every \(u\in M(a_p)\) and tail vector \(v\),
The list may have repetitions and may omit parties.
The empty basis is the scalar singleton basis. The recursive tensor basis has the stated columns. Its coordinate on an elementary tensor is the product of the two factor coordinates.
Let \(a,c\) be register lists, and for every \(p\) choose finite orthonormal bases \(b_p,c_p\) of \(M(a_p),M(c_p)\) and a continuous linear map \(A_p:M(a_p)\to M(c_p)\). For any ordered party list, the product map \(A_{p_0}\otimes \cdots \otimes A_{p_{m-1}}\otimes 1\) has matrix entries
If \(P\) is finite and \(\mathbf p\) lists all of \(P\) without repetitions, reindex the columns by party assignments: the labelled column at \(y\) is the ordered column at \((y(p_0),\ldots ,y(p_{m-1}))\). In these labelled bases the matrix entry at \((x,y)\) equals \(\prod _{p\in P}\langle c_p(x_p),A_pb_p(y_p)\rangle \). No contraction bound on the local maps is imposed.
Induct on the party list. Apply the first local map to the first basis column and use the product coordinate formula to separate it from the tail. The empty product is \(1\). For labelled coordinates, reindex both bases by the exhaustive ordering and replace the product over its positions by the product over \(P\).
These are the tensor-coordinate identities used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 137–151, 233–267, 383–450 and 565–588.
Let \(P\) be finite, fix an ordering of all parties without repetitions, and let \(W:\mathcal H_a\to \mathcal H_z\) be a source-free composition. For each \(p\), choose finite orthonormal bases \((b_p(i))_{i\in I_p}\) and \((c_p(j))_{j\in J_p}\) of its input and output memories. Let \(B_y,C_x\) be the corresponding product basis columns, transported to the original register orders by the inverse canonical grouping isometries. Then
Thus the matrix of \(W\) is the product of the matrices of its actual singleton restrictions. The two families of bases may be chosen independently. Zero maps, scalar operations on empty memories, and empty coordinate sets are included.
In the same bases and party order, let \(W,V:\mathcal H_a\to \mathcal H_z\) and \(U:\mathcal H_a\to \mathcal H_a\) be three independently chosen source-free compositions. Denote their matrices by \(M_W,M_V,M_U\), and the matrices of their restrictions at \(p\) by \(A_p,B_p,X_p\), respectively. Then
The middle matrix is the matrix of the actual composition \(U\); its product form is a consequence of its restrictions. No positivity, normalization, norm bound or nonvanishing hypothesis is imposed on these compositions.
These are auxiliary identities for [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 409–434 and 565–588.
27.140.3 Restrictions with excluded registers
Let \(P,Q\) be arbitrary sets of party labels. A register consists of an owner and a complex inner product space. For an ordered finite register list \(\ell \), let \(\mathcal H_\ell \) be its right-associated tensor memory, including a terminal factor \(\mathbb {C}\); the empty list has memory \(\mathbb {C}\). For a selection \(s:Q\to \{ 0,1\} \), let \(A=\{ p\in Q:s(p)=1\} \) and write \(\ell |_A\) for the ordered sublist of registers whose owners lie in \(A\). An owner map \(q:P\to Q\) changes the labels in \(\ell \), retaining its spaces and order; denote the resulting list by \(\ell ^q\).
Let \(q:P\to Q\) be any map and \(s:Q\to \{ 0,1\} \) any selection. Suppose \(s(q(p))=0\) for every \(p\in P\). Then, for every register list \(\ell \) with owners in \(P\), the selected sublist \((\ell ^q)|_A\) is empty. No injectivity, finiteness, or nonemptiness assumption on the party sets is required.
If a register belonged to the selected sublist, it would come from a register with some original owner \(p\in P\) and would satisfy \(s(q(p))=1\). This contradicts the hypothesis.
A source-free composition is a finite composite of local linear operations, register exchanges, and identities, containing no pair-source preparations. Its restriction to selected parties retains their local operations and registers and restricts the exchanges accordingly. For a source-free composition \(W\) from register list \(\ell \) to register list \(m\), write \(W|_A:\mathcal H_{\ell |_A}\to \mathcal H_{m|_A}\) for its selected operator. If \(\rho \) is a spectator register list, write \(\widetilde W\) for the composition obtained by adjoining \(\rho \) unchanged to every step of \(W\), from \(\rho \ell \) to \(\rho m\). Concatenation of register lists is denoted by juxtaposition.
Let \(s:Q\to \{ 0,1\} \) be any selection, let \(\rho \) be a register list with \(\rho |_A=\varnothing \), and let \(W:\ell \to m\) be any source-free composition with owners in \(Q\). Under the canonical identifications \((\rho \ell )|_A=\ell |_A\) and \((\rho m)|_A=m|_A\), \(\widetilde W|_A=W|_A\). The retained input and output lists may be arbitrary. No contraction bound or nonzero-dimension assumption on the registers is imposed.
Induct on the spectator list \(\rho \). For the empty list the two compositions coincide. Write a nonempty list as \(r\sigma \), and let \(W_\sigma \) denote \(W\) with the remaining spectators \(\sigma \). The owner of \(r\) is excluded because \(\rho |_A=\varnothing \). If \(H_r\) is the register space, adjoining \(r\) acts as \(\operatorname {id}_{H_r}\otimes W_\sigma \), and restriction gives
Apply the induction hypothesis to the remaining spectator registers.
These identities concern the register selections used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–267 and 351–381.
Let \(P\) be any set of parties and let \(W:\ell \to m\) be a source-free composition. Let \(A,B\subseteq P\) with \(B\subseteq A\). Under the canonical identifications of the selected register lists, restricting first to \(A\) and then to \(B\) gives the same composition as restricting directly to \(B\):
In particular, their linear operators coincide. Local maps may vanish, and a named scalar operation retains its party even when its input and output register lists are empty.
A register retained by \(B\) is also retained by \(A\), so its position and space are identical after either selection: \((\tau |_A)|_B=\tau |_B\) for every register list \(\tau \). Induct on the composition. Identities and composition respect this equality of selected lists. Write \(\Lambda _p(M;\tau )\) for a local map \(M\) at party \(p\) beside the unchanged registers \(\tau \). Its restriction to \(C\) is \(\Lambda _p(M;\tau |_C)\) when \(p\in C\) and \(\operatorname {id}_{\tau |_C}\) otherwise. Since \(p\in B\) implies \(p\in A\), and \((\tau |_A)|_B=\tau |_B\),
Write \(S_{r,s}(\tau )\) for the exchange of registers \(r,s\) with owners \(o(r),o(s)\) before \(\tau \). Its restriction to \(C\) keeps the exchange when both owners lie in \(C\), and otherwise is the identity on the retained registers among \(r,s\) followed by \(\tau |_C\). Hence
Write \(F_r(w)\) for a composition \(w\) beside an unchanged register \(r\). Its restriction to \(C\) is \(F_r(w|_C)\) when \(o(r)\in C\) and \(w|_C\) otherwise. Since \(o(r)\in B\) implies \(o(r)\in A\), the induction hypothesis \((w|_A)|_B=w|_B\) gives
This proves (980) as an identity of compositions and hence of their operators.
Let \(p,q\in P\) be distinct and let \(W\) be any source-free composition. Under the canonical register identifications,
as linear operators. No contraction or nonvanishing hypothesis is required.
Since \(q\ne p\), the set \(\{ q\} \) is contained in \(P\setminus \{ p\} \). Apply (980).
Let \(W:\ell \to m\) and \(V:m\to n\) be chronological compositions, possibly containing pair preparations. Fix a selection for which \(\ell ,m,n\) and both source-register lists restrict to the empty list. Suppose the corresponding remaining operations restrict to \(\alpha \operatorname {id}_{\mathbb C}\) and \(\beta \operatorname {id}_{\mathbb C}\). The remaining operations of \(VW\) then restrict to \(\alpha \beta \operatorname {id}_{\mathbb C}\).
If \(W\) instead is extended by an untouched register whose owner is excluded, its remaining operations have the same scalar restriction \(\alpha \operatorname {id}_{\mathbb C}\). These statements allow zero scalars. They are auxiliary identities for [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 246–251 and 351–381.
Write \(A\) for the selected registers and \(W^{\mathrm r},V^{\mathrm r}\) for the remaining operations of \(W,V\). In the remaining operations of \(VW\), the sources of \(V\) are untouched registers beside those of \(W\), and every one of them is excluded. By Theorem 27.140.3.2 they disappear under restriction, and restriction of a composition is the composition of the restrictions, so
Let \(\widetilde W\) be \(W\) extended by an untouched register \(r\) with excluded owner and register space \(H_r\). Its remaining operations first move \(r\) past the prepared sources by an exchange \(E\) and then adjoin \(r\) unchanged, so \(\widetilde W^{\mathrm r}=(\operatorname {id}_{H_r}\otimes W^{\mathrm r})\circ E\). Every register in \(E\) is excluded, so \(E|_A=\operatorname {id}_{\mathbb C}\), and (979) gives
27.141 One-dimensional sources on unused pairs
For distinct parties \(p,q\), the trivial pair source has the complex line at each endpoint and vector \(1\otimes 1\in \mathbb C\otimes \mathbb C\). Both endpoint spaces have dimension one, and the source vector is normalized. This is the one-dimensional source assigned to an unused pair in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 242–246.
A trivial pair source can be removed by an allowed composition of local maps containing no source preparations. For any remaining source list \(S\) and any original memory \(\mathcal H\), this composition recovers \(P_{S,\mathcal H}\) exactly from the preparation with the trivial source placed before \(S\).
At each endpoint, apply the canonical isometry \(\lambda _{\mathcal K}:\mathbb C\otimes \mathcal K\to \mathcal K\), \(\lambda _{\mathcal K}(1\otimes x)=x\), as a local map at that endpoint; here \(\mathcal K\) is the memory of the registers following it. These maps are isometries and therefore contractions. On the prepared vector the two removals give
for every \(x\) in the remaining memory \(\mathcal K\). Taking \(x=P_{S,\mathcal H}(y)\) for \(y\in \mathcal H\) recovers \(P_{S,\mathcal H}\).
Let \(P\) be a finite set of parties. Suppose that \(S\) is a normalized source list on \(P\) with no repeated unordered pair. There is a list \(U\) of sources with one-dimensional endpoint spaces such that the list \(T\) obtained by placing \(U\) before \(S\) is normalized and contains exactly one source for every unordered pair of distinct parties in \(P\).
Every pair of a source in \(U\) is absent from \(S\). Thus the original sources are retained with their vectors, spaces, endpoint order, and list order unchanged. For every original memory, an allowed composition containing no sources recovers its preparation by \(S\) from its preparation by \(T\).
List the unordered pairs of distinct parties that are absent from \(S\). For each such pair, choose either endpoint order and adjoin the corresponding trivial source. The pairs are distinct, and every pair occurs in exactly one of the original and added lists. Repeatedly apply Theorem 27.141.2 to remove the added sources. This also covers empty and singleton party sets, for which no distinct pair exists.
Let \(P\) be the prescribed finite set of gate parties, with a fixed ordering. Let \(F:\mathcal H_{\rm in}\to \mathcal H_{\rm out}\) be an allowed composition whose register owners and assigned local parties all belong to \(P\). There is a normalized source list \(T\) containing exactly one source on every unordered pair of distinct parties in \(P\), and a contraction \(B_p\) at each \(p\in P\), such that
The identifications group registers by party in the prescribed order. The input of \(B_p\) consists of that party’s original input registers and its halves of the sources in \(T\); its output consists of that party’s original output registers. A source in \(T\) whose pair does not occur among the original sources of \(F\) has one-dimensional endpoint spaces.
First prepare and combine the actual sources of \(F\) to obtain \(F=V\circ P_{S,\mathcal H_{\rm in}}\), with \(V\) allowed and containing no sources. Complete \(S\) to \(T\) by Theorem 27.141.3. The local maps removing the added sources, followed by \(V\), form another allowed composition containing no sources. Collect its local maps by party using Theorem 27.140.5.
27.142 Common private source spaces
Fix an ordered list of pairs \((p_e,q_e)\) with \(p_e\ne q_e\). Assign complex inner product spaces \(U_e,V_e\) to the respective parties and a vector \(\eta _e\in U_e\otimes V_e\) to each pair. These data define an ordered source list. Its register layout depends only on the parties and halfspaces, and its unordered pairs are the prescribed pairs. The list is normalized when each \(\eta _e\) is normalized. The empty list prepares no registers; separating the first pair separates its two registers from those of the remaining sources.
For any original memory \(\mathcal H\), write \(Q_{\eta ,\mathcal H}\) for preparation of these sources, using the fixed register layout and the canonical associativity identifications. Thus \(Q_{\eta ,\mathcal H}x=(\bigotimes _e\eta _e)\otimes x\). These are the pair slots of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267.
Let \(R\) be a reference source list with distinct unordered pairs, and let \(S\) be a normalized source list with precisely the same distinct unordered pairs. There are normalized sources in the pair order and endpoint orientations of \(R\) whose preparation can be transformed into the preparation of \(S\) by allowed operations containing no sources, for every spectator memory. Permuting whole source blocks also gives such a transformation.
Induct on the reference list. Select the actual source on its first unordered pair and move this source block to the front. If its endpoint order is reversed, apply the tensor-factor exchange \(U\otimes V\to V\otimes U\), which preserves the vector norm. Remove the matched pair and continue. Equal pair sets and distinctness ensure that every source is used exactly once.
Let \(\Xi \) be a finite set and let \(U_\xi \) be complex inner product spaces. Their common space is the orthogonal sum \(\widehat U=\bigoplus _{\xi \in \Xi }U_\xi \). Denote its coordinate inclusion by \(j_\xi :U_\xi \to \widehat U\) and its coordinate projection by \(\pi _\xi :\widehat U\to U_\xi \). The norm is given by \(\| (u_\xi )_{\xi \in \Xi }\| ^2=\sum _{\xi \in \Xi }\| u_\xi \| ^2\). This is the common-space construction in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267.
Each \(j_\xi \) is isometric, \(\| \pi _\xi \| \le 1\), and \(\pi _\xi j_\xi =\mathbb {1}\). For a second family \(V_\xi \), use inclusions \(k_\xi \) and projections \(\rho _\xi \) on its orthogonal sum. If \(\eta _\xi \in U_\xi \otimes V_\xi \), set \(\widehat\eta _\xi =(j_\xi \otimes k_\xi )\eta _\xi \). Then
In particular, normalized sources remain normalized in the common spaces.
A coordinate inclusion has exactly one nonzero coordinate, so it preserves norms. Every coordinate norm is bounded by the norm of the orthogonal sum, proving the projection bound. Tensor products of linear isometries preserve norms. The recovery identity follows first on elementary tensors from \(\pi _\xi j_\xi =\mathbb {1}\) and \(\rho _\xi k_\xi =\mathbb {1}\), then by linearity.
Let \(A:U\to U'\) and \(B:V\to V'\) be continuous linear maps at the respective parties of a pair source. Applying these maps locally, with identities on a spectator memory, sends \(u\otimes (v\otimes x)\) to \(Au\otimes (Bv\otimes x)\) and satisfies
When \(A\) and \(B\) are contractions, these two local operations form an allowed composition containing no sources.
Act first at the first party and then at the second party, keeping the remaining registers untouched. Equation (1000) follows on elementary tensors, hence on every source vector by linearity. Tensoring each local contraction with an identity preserves its contraction bound.
Let \(p\ne q\) be parties, \(A:U\to U'\) and \(B:V\to V'\) be contractions, \(\eta \in U\otimes V\), and \(T\) be a finite source list. For every spectator memory \(\mathcal H\), there is an allowed composition \(D\) containing no sources such that \(D P_{S,\mathcal H}=P_{S',\mathcal H}\), where \(S\) and \(S'\) are obtained by adjoining \(s\) and \(s'\), respectively, before \(T\). The source \(s\) has endpoint spaces \(U,V\) and vector \(\eta \), and \(s'\) has the same parties, endpoint spaces \(U',V'\) and vector \((A\otimes B)\eta \). Neither the pair vector nor the tail is required to be normalized. This is an auxiliary consequence of the local contraction step in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267.
Apply \(A\) and \(B\) only to the two registers of the first pair, with identities on the prepared tail and the spectator memory. By (1000), \(D P_{s,\mathcal H_T\otimes \mathcal H} =P_{s',\mathcal H_T\otimes \mathcal H}\). Composing on the right with \(P_{T,\mathcal H}\) gives the asserted identity. The two local maps are contractions and create no source.
Let \((S_\xi )_{\xi \in \Xi }\) be a finite family of normalized source lists, each with the same distinct unordered pair set as a fixed reference list. There are common halfspaces in the reference pair order and normalized vectors \(\widehat\eta _{\xi ,e}\) such that, for every spectator memory \(\mathcal H\), an allowed composition \(D_\xi \) containing no sources satisfies
The domain of \(D_\xi \) is the fixed source register layout together with \(\mathcal H\); it does not depend on the vectors.
Put every list in the reference order using Theorem 27.142.2. At each half of each pair, form the orthogonal sum of its branch spaces and embed the branch vector. Apply the coordinate projections to each pair in turn. Equation (999) and Theorem 27.142.6 recover its original source vector. Compose these operations with the exchanges recovering the original source order. All local maps remain contractions.
Let \(P\) be the prescribed finite set of gate parties, with a fixed ordering, and let \((F_\xi )_{\xi \in \Xi }\) be a finite family of allowed compositions with the same input and output register layouts. There is one common source slot for every unordered pair of distinct parties, with fixed halfspaces, normalized vectors \(\widehat\eta _{\xi ,e}\), and local contractions \(B_{\xi ,p}\) such that
Here \(U_{\rm in}\) groups the common source registers and the original input registers by party, and \(U_{\rm out}\) groups the original output registers in the same party order. Both identifications and the domain and codomain of each party’s local maps are independent of \(\xi \). Equivalently, preparation in the common source layout can be followed by an allowed composition containing no sources to recover \(F_\xi \) exactly.
These are the common private spaces in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267.
Choose a reference ordering of all unordered pairs by completing the empty source list. Complete the actual sources of each \(F_\xi \) by Theorem 27.141.4, and then apply Theorem 27.142.7. Compose the resulting recovery maps with the remaining operations of \(F_\xi \) and collect the local maps by party. This gives (1002). The reference list is chosen independently of \(\xi \), so the construction also applies when \(\Xi \) is empty.
Let \(F=\sum _{\xi \in \Xi }c_\xi F_\xi \), where \(\Xi \) is finite, each \(c_\xi \in \mathbb {C}\), and the \(F_\xi \) are allowed compositions with the same input and output register layouts on a prescribed finite set of parties \(P\). There are fixed halfspaces for all unordered pairs of distinct parties, normalized pair sources \(\widehat\eta _{\xi ,e}\), and local contractions \(B_{\xi ,p}\) such that
The coefficients are those of the original sum. The input and output identifications and all private spaces are common to every summand. This is the common-space form of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–267.
Let \(U,V\) be complex inner product spaces, let \(\Xi \) be finite, and let \(\eta _\xi \in U\otimes V\) for each \(\xi \in \Xi \), where the tensor product is algebraic. There exist \(a,b\in \mathbb {N}\), linear isometries \(f:\mathbb {C}^a\to U\) and \(g:\mathbb {C}^b\to V\), and vectors \(\eta ^0_\xi \in \mathbb {C}^a\otimes \mathbb {C}^b\) such that \((f\otimes g)\eta ^0_\xi =\eta _\xi \) and \(\| \eta ^0_\xi \| =\| \eta _\xi \| \) for every \(\xi \). The integers \(a,b\) may be zero. This gives finite coordinates for the source vectors used in the Schmidt decompositions of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 279–299.
A finite set of algebraic tensors is contained in \(U_0\otimes V_0\) for some finite-dimensional subspaces \(U_0\subseteq U\) and \(V_0\subseteq V\): express each tensor as a finite sum of elementary tensors and span the finitely many factors on each side. Choose orthonormal coordinates on \(U_0,V_0\) and compose their coordinate isometries with the subspace inclusions. The induced tensor map is isometric, so the coordinate vectors have exactly the original norms.
Fix a finite source list, common halfspaces \(U_e,V_e\), and a finite family of vectors \(\eta _{\xi ,e}\in U_e\otimes V_e\). There are dimensions \(a_e,b_e\), isometries \(f_e:\mathbb {C}^{a_e}\to U_e\) and \(g_e:\mathbb {C}^{b_e}\to V_e\), and coordinate vectors \(\eta ^0_{\xi ,e}\) such that \((f_e\otimes g_e)\eta ^0_{\xi ,e}=\eta _{\xi ,e}\) and \(\| \eta ^0_{\xi ,e}\| =\| \eta _{\xi ,e}\| \). For every \(\xi \) and every spectator memory \(\mathcal H\), an allowed composition \(D_\xi \) containing no sources satisfies \(D_\xi Q_{\eta ^0_\xi ,\mathcal H}=Q_{\eta _\xi ,\mathcal H}\). More generally, any prescribed isometric inclusions of the two halves of each source induce such a recovery of the included sources.
Apply Theorem 27.142.10 at each pair slot. Apply \(f_e\) and \(g_e\) locally to the two halves of that slot, then proceed through the source list. Theorem 27.142.6 recovers the corresponding source vector at each step. The inclusions are isometries, hence contractions, and create no further sources.
Let \(F=\sum _{\xi \in \Xi }c_\xi F_\xi \), where \(\Xi \) is finite, \(c_\xi \in \mathbb {C}\), and each \(F_\xi \) is an allowed composition between fixed register layouts on a prescribed finite set of parties \(P\). There are common dimensions \(a_e,b_e\) for every unordered pair of distinct parties, normalized sources \(\eta ^0_{\xi ,e}\in \mathbb {C}^{a_e}\otimes \mathbb {C}^{b_e}\), and local contractions \(B_{\xi ,p}\) such that
Here \(U_{\rm in}\) groups the common finite source registers and original input registers by party, while \(U_{\rm out}\) groups the unchanged output registers in the same order. Each branch is also the exact composition of its coordinate source preparation with allowed operations containing no sources.
This is an exact finite-coordinate form of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–299. No bound on the private dimensions \(a_e,b_e\) is asserted.
Theorem 27.142.8 prepares every branch’s sources in common private spaces, followed by allowed operations containing no sources. Apply Theorem 27.142.11 to the finitely many common vectors at each slot. Compose the coordinate inclusions with the remaining operations of each branch. All these operations are local contractions. Collect them by party, giving one identity (1002) for each branch. Multiply each by \(c_\xi \) and sum over \(\Xi \) to obtain (1005).
27.143 Restricting the source-register operators
Let \(\mathcal P\) be the set of parties, and let \(R=((p_i,q_i))_{i=0}^{n-1}\) be an ordered list of pairs of distinct parties. For each \(i\), let \(U_i,V_i\) be complex inner product spaces. The ordered register list is
An empty register list has memory space \(\mathbb {C}\). Register memories are tensored in the order (1006), with a final scalar factor.
Let \(A_i:U_i\to U_i\) and \(B_i:V_i\to V_i\) be continuous linear maps. The word \(Z_R\) applies \(A_i\) to the register \((p_i,U_i)\) and \(B_i\) to \((q_i,V_i)\), leaving every other register unchanged, in increasing order of \(i\). It contains no source preparations. In particular, \(Z_{\varnothing }=\mathbb {1}_{\mathbb {C}}\). These are the local matrices used in the random source replacement and the contraction within each party in [ 248 , proof of Theorem 5.2 ] .
Let \(f:\mathcal P\to \{ 0,1\} \) select parties, and suppose that \(f(p_i)=f(q_i)=0\) for every \(i\). Then the selected register list is empty, and the restricted word \(Z_R|_f\) acts as \(\mathbb {1}_{\mathbb {C}}\). The register spaces and their local maps are arbitrary, and \(n\) may be zero.
Every party occurring in \(Z_R\) owns a register in \(\mathcal L_R\). Indeed, adjoining the first pair introduces only its two owners through its local maps and unchanged registers; the claim for the remaining pairs follows by induction. Thus the assumptions exclude every participating party. Restriction retains exactly the selected participating parties, so none remain. A source-free word with no participating party has empty input and output lists and acts as \(\mathbb {1}_{\mathbb {C}}\).
Let \(u_i\in U_i\) and \(v_i\in V_i\). In the original register order, the actual endpoint word satisfies
The vectors and maps are arbitrary. An empty list of slots gives the scalar identity, with both empty tensors equal to \(1\).
Induct on the ordered source list. At the first pair, the two recorded endpoint operations send \(u_0\otimes v_0\) to \(A_0u_0\otimes B_0v_0\), leaving the remaining registers unchanged. Apply the induction hypothesis to those registers. The canonical association and register identifications preserve the original order.
Choose finite orthonormal bases \(b_i,c_i\) of \(U_i,V_i\) and let \(e_x\) be their ordered joint source basis. For arbitrary \(\eta _i\in U_i\otimes V_i\),
Thus each coordinate is the product of the corresponding individual source coordinates. The assertion includes empty source lists and zero-dimensional register spaces.
Expand each factor in its orthonormal basis and distribute the finite tensor product over the finite sums. Taking the coordinate of \(e_x\) leaves the coefficient in (1007).
Let \(b=(b_j)_{j\in I}\) be an orthonormal basis of a finite-dimensional complex inner product space. For a matrix \(M\), define \(T_b(M)\) by \(T_b(M)b_j=\sum _i M_{ij}b_i\). Its matrix in \(b\) is exactly \(M\). The matrix may be arbitrary, and the coordinate set may be empty.
In the joint source basis, the endpoint operator satisfies
In particular, constructing the endpoint maps from matrices \(L_i,H_i\) gives \([Z_R]_{x,y}=\prod _i (L_i)_{x_i^{(1)},y_i^{(1)}} (H_i)_{x_i^{(2)},y_i^{(2)}}\).
Apply the endpoint action to the basis vector \(e_y\), then take its coordinate at \(e_x\). Inner products of elementary tensors multiply.
Let \(L_{i,s},H_{i,s}\) be endpoint matrices for \(s\in \{ 0,\ldots ,k-1\} \). Then, entrywise in the joint source basis,
where \(Z_j\) is the actual endpoint operator with maps constructed from \(L_{i,j(i)},H_{i,j(i)}\). The same assignment \(j(i)\) is used at both endpoints. The algebraic identity includes \(k=0\), with \(0^{-1}=0\), and \(n=0\), with the empty assignment contributing the scalar identity.
Substitute the endpoint-coordinate formula and distribute the product of finite sums. One choice of a summand at every original slot gives exactly one global assignment \(j\).
Let \(b,c\) be finite orthonormal bases and let \(A\) be a continuous linear map between their spaces. Its linear-map matrix is the coordinate-entry matrix \(M_{ij}=\langle c_i,A b_j\rangle \). Empty index sets and the zero map are included.
The matrix of a linear map is defined by the coordinates of the images of its basis vectors. Evaluate that definition at each pair of indices.
Fix the ordered source-register list and a party selector \(f\). Consider endpoint maps \(A_i,B_i\) and \(A'_i,B'_i\) on the same spaces. Suppose that \(A_i=A'_i\) whenever \(f(p_i)=1\), and \(B_i=B'_i\) whenever \(f(q_i)=1\). Then the two restricted source-register operators agree:
No equality is required at an unselected endpoint. The register order and spectator memories are unchanged; an empty inventory and zero-dimensional spaces are included. This is the pointwise endpoint locality used in [ 248 , proof of Theorem 5.2 ] .
Induct on the ordered source inventory. An unselected endpoint map is omitted in either restriction. At a selected endpoint the two maps agree by hypothesis. Tensoring with the unchanged spectator identities and composing with the inductive equality preserves equality. The canonical identifications of the fixed register lists are the same on both sides.
Let a source-free chronological operator act between two ordered register lists. Canonically identify these with equal input and output lists, and retain the registers and local operations selected by \(f\). The result is the same as first making the restriction and then canonically identifying the selected input and output lists. The equalities of register lists supply the identifications; no new isometry is chosen. This is the canonical transport used in [ 248 , proof of Theorem 5.2 ] .
Identify the equal input lists and equal output lists. Both constructions then retain the same registers and local operations in the same chronological order.
Fix independent ket and bra choices \(\xi ,\zeta \) and a selector \(f\) of the grouped owners. A sample assignment on the original source occurrences consists of a sample index \(j(e)\) and a family of Gaussian coordinates \(\omega _{e,\alpha ,\beta ,s,p,q}\) (Definition 27.152.4) at each source \(e\); write \(U_{\xi ,\zeta ,j,\omega }\) for the actual sampled source-input word of Definition 27.164.17 built from it. Let \((j,\omega )\) and \((j',\omega ')\) be two sample assignments. Suppose that, whenever \(f\) selects at least one endpoint of a source \(e\), with actual branch pair \((\alpha ,\beta )\) chosen by \((\xi ,\zeta )\) at \(e\),
Then the actual sampled source-input words have equal restrictions:
At each source the two endpoints still use the same sample index and the same ordered ket–bra Gaussian array. No condition is imposed on samples at sources whose endpoints are both unselected. This is the sample dependence used in [ 248 , proof of Theorem 5.2 ] .
Unfold both words by Definition 27.164.17. The formulas defining the endpoint matrices agree at every selected endpoint, since by (1010) their sample index and Gaussian array agree there. Apply Theorem 27.143.9. The canonical identification with the actual source-input register list preserves this equality by Theorem 27.143.10.
Let \(H\) be the original physical memory and \(G\) the original retained private memory. Use the prescribed selected-party order and the actual grouped physical/private basis \(B\). Let \(\chi \) separate the two coordinate families and remove the exterior singleton coordinate. Identify the original output memory with \(H\otimes G\) by its canonical append isometry. For \(v,t\in H\otimes G\), let \(b_v,b_t\) denote their actual grouped coordinates after relabelling owners. Fix the original physical basis and any finite orthonormal reference basis of \(G\), with product readout \(J\). Then
Both sides have rows indexed by the original configurations \(\prod _{p\in \mathcal P}\{ 0,\ldots ,d_p-1\} \). The two vectors are independent and need not be normalized. No physical register is adjoined at the exterior owner. This identifies the discarded trace in [ 248 , proof of Theorem 5.2 ] .
The two grouping identities identify \(B\) after \(\chi \) with the product of the original physical basis and the grouped private basis transported to \(G\). The append and owner isometries preserve these coordinates. Thus each mixed outer product has the claimed product coordinates. Changing only the private basis by its actual unitary matrix \(V\) conjugates that product by \(I\otimes V\); the partial trace removes this conjugation. This gives (1012).
For a distributed construction, let \(v\) be its actual original circuit output and let \(b_v\) be its actual grouped coordinates. In the notation of Theorem 27.143.12, its original physical density satisfies \(\rho =\operatorname {tr}_{\mathrm{private}}((b_vb_v^\dagger )^\chi )\). The physical dimensions are the original dimensions \(d_p\); the output is not divided by its norm.
Apply Theorem 27.143.12 with \(t=v\) and the reference private basis used in the construction’s original readout. The right side is its defining physical density.
Let \(a,b,t\) be arbitrary ordered register lists, and let \(f\) select owners. Write \(\ell |_f\) for the retained list, in its original order. Let \(S_{a,b;t}\) exchange the successive blocks \(a,b\) in front of the tail \(t\), preserving the order within each block and leaving the tail unchanged. After the canonical identifications of the selected input and output register lists, its restricted operator satisfies
No retained block or spectator memory is required to be empty. Empty lists and zero-dimensional register spaces are included. This is the register-order identity used when grouping sources in [ 248 , proof of Theorem 5.2 ] .
First move one register through a prefix by successive exchanges. If that register is retained, exchanges with retained prefix registers give the corresponding move through the retained prefix; exchanges with erased registers act as identities. If the moving register is erased, every exchange restricts to an identity on the retained memory. Framing preserves these identities with the retained spectators. Now induct on the second block. Moving its first retained register through the first block, followed by the exchange of the remaining blocks, gives (1013). An erased first register instead contributes the identity before the inductive exchange. The canonical list identifications preserve each composition.
Let \(W:a\to b\) be a source-free chronological composition, let \(t\) be a spectator list, and let \(f\) select owners. Write \(W_t\) for extension by spectators as in Definition 27.148.4.1. Under the canonical identifications of the retained concatenations, \((W_t)|_f=(W|_f)_{t|_f}\). The retained spectator memory is thus unchanged beside the restricted composition. The input, output and spectator memories may be arbitrary, including empty lists and zero-dimensional spaces. This is the spectator identity used in [ 248 , proof of Theorem 5.2 ] .
Induction on the spectator list shows that restriction commutes with framing, retaining precisely the selected spectator registers. Apply this identity and Theorem 27.143.14 to the initial exchange, the framed composition and the final exchange defining \(W_t\). The canonical identifications of the concatenated register lists preserve their composition.
Let \(U:a\to b\) be a source-free chronological composition with owners in \(C\), and relabel its owners by \(\gamma :C\to \mathcal P\). Fix \(p\in \mathcal P\) and suppose that \(\gamma (c)\ne p\) for every \(c\in C\). Extend the relabelled composition by any spectator list \(t\) with owners in \(\mathcal P\). Its restriction to \(p\) is the identity on the memory of the registers of \(t\) owned by \(p\). The retained spectator memory may be arbitrary, including a zero-dimensional space. This is the owner-exclusion identity used in [ 248 , proof of Theorem 5.2 ] .
The restricted core has empty input and output memories and acts as the scalar identity by Theorem 27.147.3. Restriction commutes with spectator extension by Theorem 27.143.15. The operator formula in Theorem 27.148.4.2 then tensors the scalar identity with the retained spectator identity. Canonical concatenation identifications give the identity on the retained spectator memory.
Let \(a=a'\), \(b=b'\) and \(c=c'\) be equalities of ordered register lists. Let \(U:a\to b\), \(V:b\to c\), \(U':a'\to b'\) and \(V':b'\to c'\) be chronological compositions whose operators satisfy \(U=U'\) and \(V=V'\) under the canonical memory identifications. Then \(V\circ U=V'\circ U'\) under those identifications. Likewise, if \(U:a\to b\) and \(U':a'\to b'\) have equal operators under \(a=a'\) and \(b=b'\), adjoining the same register \(r\) in front preserves this equality: \(I_{\mathcal H_r}\otimes U=I_{\mathcal H_r}\otimes U'\). More generally, for equal spectator lists \(t=t'\), framing \(U\) by \(t\) and \(U'\) by \(t'\) gives equal operators under the induced concatenation identifications. These assertions do not require the compositions to be source-free or contractive. They are elementary compatibility identities for the register identifications used in [ 248 , proof of Theorem 5.2 ] .
Identify the equal input, intermediate and output register lists. The first assertion follows by composing equal operators in the indicated chronological order. The second follows by tensoring their equality with \(I_{\mathcal H_r}\). For spectator lists, identify \(t=t'\) and induct on \(t\), applying the single-register assertion at each step.
Let \(W:a\to b\) be a chronological composition with empty source inventory, and let \(f\) select owners. Denote by \(L(W)\) its local part, obtained by moving its source preparations before the other operations as in Theorem 27.139.3. Under the canonical identification of the empty source prefix, \((L(W))|_f=W|_f\). The selected input and output memories are arbitrary; no contraction or normalization assumption is imposed. This is an auxiliary register identity for [ 248 , proof of Theorem 5.2 ] .
Induct on \(W\). Identities, local operations and exchanges agree directly, and a source preparation is excluded by the empty inventory. In a composition, each component has empty inventory; its source prefix therefore contributes no spectator register. Apply the inductive identities and compatibility with composition. For a framed register, moving it through the empty source prefix is the identity. If it is selected, tensor the inductive identity with its identity operator; otherwise erase the register.
Let \(r\) be a register, let \(s,t\) be ordered register lists, and let \(f\) select owners with \(s|_f\) empty. Move \(r\) through the prefix \(s\), from \(srt\) to \(rst\), leaving \(t\) unchanged. After restriction and the canonical identification of both retained lists with \((rt)|_f\), this move is \(\mathbb {1}_{\mathcal H_{(rt)|_f}}\). The moving register may be retained or erased, and the retained tail is arbitrary. This is the register-order identity used in [ 248 , proof of Theorem 5.2 ] .
If \(r\) is retained, its restricted move is through the empty retained prefix, hence is the identity on \(r(t|_f)\). If \(r\) is erased, its restricted move is the identity on \(s|_f\, t|_f=t|_f\). The canonical list identifications give the asserted identity in both cases.
Let \(W:a\to b\) be an arbitrary chronological composition, with ordered source-register list \(s\). Let \(f\) select owners such that \(s|_f\) is empty, and let \(t\) be any spectator list. Write \(F_t(W)\) for \(W\) framed in front by the registers of \(t\), and \(L(W):sa\to b\) for its local part. Under the canonical identifications of the retained register lists,
The original source inventory need not be empty. The input, output and retained spectator memories may be arbitrary, including empty lists and zero-dimensional spaces. This is an auxiliary identity used in [ 248 , proof of Theorem 5.2 ] .
Induct on \(t\). The empty list gives the identity directly. For \(t=rt'\), forming the local part first moves \(r\) through the source prefix, then frames the local part for \(t'\) by \(r\). The source registers are erased, so the restricted move is the identity by Theorem 27.143.19. If \(r\) is retained, frame the inductive identity by \(r\); otherwise erase it. Compatibility with composition and the canonical list identifications gives (1014).
Let \(a=a'\) be an equality of ordered register lists. The identity operators on \(\mathcal H_a\) and \(\mathcal H_{a'}\) agree under the canonical memory identification. Moreover, let \(W:a\to b\) be source-free and let \(f\) select owners. Replacing its input list by the equal list \(a'\) before restriction gives the same selected operator as restricting \(W\) directly, under the induced identification \(a|_f=a'|_f\). The register spaces and selected memories are arbitrary. These are elementary compatibility identities for the register identifications in [ 248 , proof of Theorem 5.2 ] .
Identify the equal register lists. Both assertions reduce to equality of an operator with itself.
Let \(U:a\to b\) be an arbitrary chronological composition whose owners lie in \(C\), and let \(\gamma :C\to \mathcal P\) relabel them. Fix \(p\in \mathcal P\) with \(\gamma (c)\ne p\) for every \(c\in C\). Extend \(U^\gamma \) by an arbitrary spectator list \(t\), obtaining \((U^\gamma )_t\), and move its source preparations to the beginning. The local part then satisfies, under the canonical retained-list identifications,
The source inventory of \(U\) may be nonempty, its source vectors are arbitrary, and the spectator memory at \(p\) may be arbitrary. Empty lists and zero-dimensional spaces are included. This is an auxiliary owner-exclusion identity for [ 248 , proof of Theorem 5.2 ] .
No core input, output or source register is owned by \(p\). By Theorem 27.139.4, the restricted core local part is the identity on the empty memory. In the spectator extension, each of the initial and final exchanges therefore exchanges the retained spectators with an empty block, giving their identity. For the middle framed composition, Theorem 27.143.20 tensors the scalar identity with the retained spectator identity. When taking the local part of either chronological composition, the additional source prefix is erased. Its framing contributes no selected register. Compose the three identity operators under the canonical list identifications to obtain (1015).
Let \(U:a\to b\) be an arbitrary chronological composition and let \(p\notin \mathcal P(U)\), where \(\mathcal P(U)\) includes the owners of its registers and the named parties of its local operations. Let \(t\) be any spectator list. Under the canonical retained-list identifications, the local part of the spectator extension satisfies \((L(U_t))|_p=\mathbb {1}_{\mathcal H_{t|_p}}\). The source inventory may be nonempty, and the retained spectator memory may be arbitrary. Empty lists and zero-dimensional spaces are included. This is an auxiliary restriction identity for [ 248 , proof of Theorem 5.2 ] .
Every core input, output and source-register owner belongs to \(\mathcal P(U)\), so their restrictions to \(p\) are empty. The local part has the same participating parties as \(U\); its restriction to \(p\) has none and acts as the identity on \(\mathbb {C}\). The initial and final exchanges of the spectator extension restrict to exchanges with empty core blocks, hence to identities on \(t|_p\). The framed middle local part tensors its scalar identity with the retained spectator identity. In each composition, the additional source prefix is erased and contributes no retained register. Composing these identities gives the asserted operator.
Let \(G\) be a prepared gate with finite participant set \(C\), finite branch set \(\Xi \) and participant injection \(\gamma :C\hookrightarrow P\). Let \(t\) be any spectator list and fix \(p\in P\) with \(\gamma (c)\ne p\) for every \(c\in C\). Select all original parties, and let \(g:P\to Q=P\sqcup \{ *\} \) be the resulting owner map; thus \(g(p)\) is the retained copy of \(p\). For any partial choice \(\xi \) for this gate as in Definition 27.149.5, let \(V_\xi \) be its actual partial composition, including the spectators and the owner map \(g\). Its local part satisfies
under the canonical retained-list identifications. If \(C\) is nonempty, \(\xi \) is an original branch label. If \(C\) is empty, the gate is untouched and has its unique partial choice, even when \(\Xi \) is empty. The spectator memory is arbitrary, including empty lists and zero-dimensional spaces. This is an auxiliary identity for the chronological expansion in [ 248 , proof of Theorem 5.2 ] .
If \(C\) is nonempty, the literal partial composition is the chosen prepared branch relabelled by \(\gamma \), extended by \(t\), and then relabelled by \(g\). Move the last relabelling through the spectator extension using Theorem 27.148.4.3. Its core has participating parties in \(g(\gamma (C))\), which excludes \(g(p)\). Apply Theorem 27.143.23. If \(C\) is empty, the literal partial composition retains the complete gate as a local operation at the exterior owner \(*\). It has no sources. Its local part preserves its restriction, and restriction to \(g(p)\) erases this exterior operation while retaining precisely the spectators at \(g(p)\). This gives (1016) also when the complete gate vanishes.
Let \(W:a\to b\) be a source-free chronological composition, let \(t\) be an arbitrary spectator list, and let \(f\) select owners. Write \(F_t(W)\) for \(W\) framed in front by the registers of \(t\). Under the canonical identifications of the retained concatenations, \((F_t(W))|_f=F_{t|_f}(W|_f)\). The input, output and retained spectator memories are arbitrary, including empty lists and zero-dimensional spaces. This is an auxiliary restriction identity for [ 248 , proof of Theorem 5.2 ] .
Induct on \(t\). The empty list gives \(W|_f\) directly. If the first register is retained, restriction frames the inductive identity by that register. If it is erased, restriction removes the framing and gives the inductive identity. The retained prefix is respectively the first register followed by the retained tail, or just the retained tail.
Let \(W:a\to b\), \(V:b\to c\), \(W':a'\to b'\) and \(V':b'\to c'\) be arbitrary chronological compositions, and let \(f\) select owners. Suppose the selected input, intermediate and output register lists agree: \(a|_f=a'|_f\), \(b|_f=b'|_f\) and \(c|_f=c'|_f\). Denote their ordered source-register lists by \(s_W,s_V,s_{W'},s_{V'}\), respectively. Suppose \(s_W|_f=s_{W'}|_f\) and \(s_V|_f=s_{V'}|_f\), and suppose that \((L(W))|_f=(L(W'))|_f\) and \((L(V))|_f=(L(V'))|_f\) under the induced memory identifications. Then the selected local parts of the chronological compositions satisfy \((L(V\circ W))|_f=(L(V'\circ W'))|_f\). Original and source registers outside the selection need not agree. No source-free, normalization or nonzero-operator assumption is imposed. This is an auxiliary composition identity for [ 248 , proof of Theorem 5.2 ] .
The local part of \(V\circ W\) frames \(L(W)\) by the later source-register list \(s_V\), then applies \(L(V)\), with the canonical input identification. Restriction retains exactly \(s_V|_f\) in that framing. Equality of the selected source lists, original input and intermediate lists, and selected earlier local parts therefore equates the framed operators. Compose this equality with the equality of the selected later local parts. Compatibility with the input identification gives the asserted identity.
Let \(r\) be a register, let \(s,s',a,a'\) be ordered register lists, and let \(f\) select owners. Suppose \(s|_f=s'|_f\) and \(a|_f=a'|_f\). The move of \(r\) through \(s\) in front of the tail \(a\), from \(sra\) to \(rsa\), and the corresponding move from \(s'ra'\) to \(rs'a'\) have equal restricted operators under the induced retained-list identifications. The retained prefix may be arbitrary, and \(r\) may be retained or erased. No equality of the unselected registers is required. This is an auxiliary register-order identity for [ 248 , proof of Theorem 5.2 ] .
If \(r\) is retained, both restrictions move it through the same retained prefix before the same retained tail. Identify these equal lists to equate the operators. If \(r\) is erased, both restrictions are identities on the equal memories of \(s|_f\, a|_f=s'|_f\, a'|_f\).
Let \(W:a\to b\) and \(W':a'\to b'\) be arbitrary chronological compositions. Let \(f\) select owners, suppose \(a|_f=a'|_f\) and \(b|_f=b'|_f\), and let \(r\) be a common spectator register. Write \(s_W,s_{W'}\) for the ordered source-register lists. Suppose \(s_W|_f=s_{W'}|_f\) and \((L(W))|_f=(L(W'))|_f\) under the induced memory identifications. Framing by \(r\) then gives
The selected source prefix may be nonempty. Original and source registers outside the selection may differ. No source-free, normalization or nonzero assumption is imposed. This is an auxiliary framing identity for [ 248 , proof of Theorem 5.2 ] .
Forming the local part of \(F_r(W)\) first moves \(r\) through \(s_W\), then frames \(L(W)\) by \(r\). The corresponding moves for \(W\) and \(W'\) have equal restricted operators by Theorem 27.143.27. If \(r\) is retained, frame the assumed equality of the selected local parts by its identity operator; if it is erased, use that equality directly. Compose the move and framed-core identities under the induced memory identifications to obtain (1017).
Let \(W:a\to b\) be a chronological composition of prepared gates and local operations on the original parties \(P\), and fix \(p\in P\). Select all original parties and let \(g:P\to P\sqcup \{ *\} \) be the resulting owner map. Let \(V_\xi \) and \(V_\zeta \) be the actual partial compositions corresponding to two supplied choices \(\xi ,\zeta \in \Xi _P\). If their incident projections agree, \(\pi _{\{ p\} }\xi =\pi _{\{ p\} }\zeta \), then \((L(V_\xi ))|_{g(p)}=(L(V_\zeta ))|_{g(p)}\) under the canonical identifications of the retained input and output register lists. Both full choices are supplied; no choice of a label at an unused occurrence is required. This is an auxiliary locality identity for the chronological contraction in [ 248 , proof of Theorem 5.2 ] .
Every supplied all-party branch has the same ordered source-register list in the canonical slot spaces. Restricting this list therefore gives the same selected source prefix for \(\xi \) and \(\zeta \). Induct on the chronological composition. Identity operations, local maps and register exchanges carry no branch choice. For a composition, the incident projections agree separately on its earlier and later operations. Apply the inductive identities and Theorem 27.143.26, using equality of the two selected source lists. Framing is handled by Theorem 27.143.28. At a gate incident to \(p\), projected choices preserve its original label, so both supplied choices specify the same branch. At a nonincident gate, both selected local parts are the identity on precisely the retained spectators by Theorem 27.143.24. The latter case includes empty participant sets, for which the aggregate gate requests no branch label.
Let \(r\) be a register, let \(s,t\) be arbitrary ordered register lists, and let \(f\) select owners. Write \(M_r(s,t):srt\to rst\) for the chronological move of \(r\) through the prefix \(s\). If \(f\) selects the owner of \(r\), then the restricted operator is \(M_r(s|_f,t|_f)\) under the canonical retained-list identifications. If \(f\) erases the owner of \(r\), then the restricted operator is \(\mathbb {1}_{\mathcal H_{(s|_f)(t|_f)}}\). The retained prefix and tail are arbitrary, including nonempty prefixes and zero-dimensional memories. This is an auxiliary register-order identity for [ 248 , proof of Theorem 5.2 ] .
Induct on the prefix \(s\). For the empty prefix the move is the identity on \(rt\). For a first prefix register \(u\), first perform the move through the remaining prefix while framing by \(u\), and then exchange \(u\) and \(r\). If \(r\) and \(u\) are both selected, the restriction gives these same operations on the retained lists. If only \(r\) is selected, the framing by \(u\) is erased and the last exchange restricts to the identity. These two cases give the retained move. If \(r\) is erased, the inductive move is the identity; the final exchange is also the identity on whichever of \(u\) and the tail is retained. Composing yields the identity on the retained prefix followed by the retained tail.
Let \(W:a\to b\) be an arbitrary chronological composition with ordered source-register list \(s_W\), let \(r\) be a register, and let \(f\) select owners. Suppose \(f\) erases the owner of \(r\). Then the selected local parts satisfy
Both operators have input memory \(\mathcal H_{(s_W|_f)(a|_f)}\) and output memory \(\mathcal H_{b|_f}\) under the canonical register-list identifications. The selected source prefix is retained and may be nonempty. Pair sources and arbitrary local maps are permitted. This is an auxiliary framing identity for [ 248 , proof of Theorem 5.2 ] .
The local part of \(F_r(W)\) first moves \(r\) through \(s_W\) and then frames \(L(W)\) by \(r\). Since \(r\) is erased, the restricted move is the identity on the memory of \((s_W|_f)(a|_f)\) by Theorem 27.143.30. Restriction also removes the framing of \(L(W)\). Identifying the retained concatenated lists and composing these two restricted operators gives \((L(W))|_f\circ \mathbb {1}=(L(W))|_f\).
27.144 Density expansions of source gates
Fix the ordered pair slots of Definition 27.142.1, with halfspaces \(U_e,V_e\), and write \(\mathcal S\) for their joint register space. All tensor products below use this order and its canonical associativity identifications. The slot set and all finite index sets may be empty. These are the source-coordinate identities used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 279–355.
The joint source vector is the multilinear map \(\Phi :\prod _e(U_e\otimes V_e)\to \mathcal S\) defined recursively by \(\Phi _{\varnothing }=1\in \mathbb {C}\) and \(\Phi _{e::E}(\eta )=\eta _e\otimes \Phi _E(\eta |_E)\), with the first pair separated into its two registers. Thus \(\Phi (\eta )\) is the ordered tensor product of the pair vectors, including the terminal scalar factor.
For every original memory \(\mathcal H\), the actual source preparation satisfies \(Q_{\eta ,\mathcal H}x=\Phi (\eta )\otimes x\). In particular, let \(J_e\) be finite, let \(d_e:J_e\to \mathbb {C}\), and let \(\theta _{e,j}\in U_e\otimes V_e\). Then
Induction on the source list identifies \(\Phi (\eta )\) with its actual joint vector: the empty list contributes \(1\), and the induction step prepares \(\eta _e\) in front of the remaining sources. Changing to the fixed register layout preserves this identity and its tensor product with \(x\). Expand each argument of the multilinear map \(\Phi \) as a finite sum. Its value on a scalar multiple in every argument is \(\Phi ((d_e\theta _e)_e)=(\prod _e d_e)\Phi ((\theta _e)_e)\). Tensoring with \(x\) gives (1018).
Let \(T:\mathcal S\otimes \mathcal H_{\rm in} \to \mathcal H_{\rm out}\) be the operator of a source-only composition as in Definition 27.138.13, without imposing normalization or contraction bounds. Choose finite orthonormal bases \(f_j\) and \(g_i\) of the input and output memories. Denote the matrix of the prepared branch by
The input and output bases are fixed throughout the density expansions below. No finite-dimensionality assumption on the private spaces is required for this definition.
For finite families as in Theorem 27.144.2,
In particular, choose finite orthonormal bases \((u_{e,r})_r\) of \(U_e\) and \((v_{e,s})_s\) of \(V_e\), and put \(\eta _e(r,s)=\langle u_{e,r}\otimes v_{e,s},\eta _e\rangle \). For \(a_e=(r_e,s_e)\), set \(\theta _{e,a_e}=u_{e,r_e}\otimes v_{e,s_e}\). Formula (1020) then holds with \(d_e(a_e)=\eta _e(a_e)\) and left-hand side \(K_T(\eta )\). The identical expansion holds for \(Q_{\eta ,\mathcal H}\).
Compose (1018) with \(T\) and take matrices in the prescribed bases. Both operations are linear. For the basis specialization, use \(\eta _e=\sum _{r,s}\eta _e(r,s)u_{e,r}\otimes v_{e,s}\) in each slot.
Choose vector families \(u_{e,r}\in U_e\), \(v_{e,s}\in V_e\) for a ket branch \(T\), and independent families \(\widetilde u_{e,r}\in U_e\), \(\widetilde v_{e,s}\in V_e\) for a bra branch \(T'\). Their index sets may differ. Write \(K_T(a)=K_T((u_{e,a_e^{(1)}}\otimes v_{e,a_e^{(2)}})_e)\) and define \(K_{T'}(b)\) using the second families. For any complex input matrix \(\rho \), define
The families need not span the private spaces or be orthogonal.
Suppose the families in Definition 27.144.5 have finite index sets. Choose complex coefficients \(d_e(r,s)\) and \(\widetilde d_e(r,s)\), and put
For every input matrix \(\rho \),
This identity also applies to vector families contained in proper subspaces of the private halfspaces.
Expand the two prepared matrices using (1020). The adjoint conjugates each scalar coefficient on the bra side. Distributing both finite sums yields (1024), since
Let \(\Xi \) be finite. For each \(\xi \in \Xi \), choose a branch \(T_\xi \), source vectors \(\eta _{\xi ,e}\), and finite orthonormal bases \((u_{\xi ,e,r})_r\), \((v_{\xi ,e,s})_s\) of the two halfspaces. The bases and their index sets may depend on \(\xi \). Set \(K_\xi =K_{T_\xi }(\eta _\xi )\) and \(\kappa _\xi (a)=\prod _e\langle u_{\xi ,e,a_e^{(1)}}\otimes v_{\xi ,e,a_e^{(2)}},\eta _{\xi ,e}\rangle \). For any input matrix \(\rho \) and any two branches,
If \(G=\sum _\xi c_\xi K_\xi \), then
Neither identity requires positivity or normalization of \(\rho \).
If \(T\) is an allowed composition and \(\| \eta _e\| =1\) in every slot, then \(\| K_T(\eta )\| \le 1\) in the Euclidean operator norm. In particular, if every \(u_{e,r}\) and \(v_{e,s}\) has norm one, then \(\| K_T(a)\| \le 1\) for every coordinate assignment \(a\). The unit-vector families need not span their halfspaces. These are the coefficient contractions in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 409–427.
The normalization of each source makes its preparation allowed, hence contractive. Composing with \(T\) gives \(\| TQ_{\eta ,\mathcal H_{\rm in}}\| \le 1\). Orthonormal coordinate maps are isometries, so the matrix in (1019) has the same operator norm. For the last assertion, use \(\| u_{e,r}\otimes v_{e,s}\| =\| u_{e,r}\| \| v_{e,s}\| =1\).
Let \(P\) and \(\Xi \) be finite. For every \(\xi \in \Xi \), let \(F_\xi \) be an allowed source-only composition on the parties \(P\), with fixed input and output memories. Fix finite orthonormal bases of these memories and complex coefficients \(c_\xi \), and let \(M_\xi \) be the matrix of \(F_\xi \) in these bases.
There is one ordered slot for each unordered pair of distinct parties, with common finite halfspaces \(U_e=\mathbb {C}^{d_e}\) and \(V_e=\mathbb {C}^{d'_e}\), normalized vectors \(\eta _{\xi ,e}\in U_e\otimes V_e\), and allowed compositions \(T_\xi \) containing no sources such that \(F_\xi =T_\xi Q_{\eta _\xi ,\mathcal H_{\rm in}}\). Use the standard orthonormal bases of these halfspaces to define \(K_\xi (a)\) as in Definition 27.144.5, and put \(\kappa _\xi (a)=\prod _e\eta _{\xi ,e}(a_e)\). Each \(K_\xi (a)\) has Euclidean operator norm at most one.
These choices are independent of the input matrix \(\rho \). For every complex input matrix \(\rho \), writing \(G=\sum _\xi c_\xi M_\xi \), one has
The halfspaces, source vectors, and coefficient matrices are obtained from the original compositions. This is the exact source-coordinate density identity preceding the random replacement in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 233–355.
Theorem 27.142.12 constructs the common finite source spaces, their normalized vectors, and the exact source-free compositions \(T_\xi \). Replacing every source half by a standard basis vector gives normalized elementary preparations; Theorem 27.144.8 therefore proves \(\| K_\xi (a)\| \le 1\). Taking matrices of \(F_\xi =T_\xi Q_{\eta _\xi ,\mathcal H_{\rm in}}\) identifies \(M_\xi \) with the prepared branch matrix. Apply Theorem 27.144.7 to obtain (1027) for every \(\rho \).
27.145 Joint contractions of the free source inputs
Fix ordered source slots with halfspaces \(U_e,V_e\) and joint register space \(\mathcal S\), as in Definition 27.142.1. All source and physical input coordinates in this section remain free. This gives the contractions used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 383–468.
Choose finite orthonormal bases \((u_{e,r})_{r\in I_e}\) of \(U_e\) and \((v_{e,s})_{s\in J_e}\) of \(V_e\). Let \(\mathcal A=\prod _e(I_e\times J_e)\). The orthonormal basis of \(\mathcal S\) indexed by \(a\in \mathcal A\) consists of
The empty source list has its single basis vector \(1\in \mathbb {C}\). If \((f_j)_{j\in N}\) is an orthonormal basis of the original input memory \(\mathcal H\), the resulting orthonormal basis of \(\mathcal S\otimes \mathcal H\) is indexed by \((a,j)\) and satisfies
Both formulas use the canonical associativity identifications.
Induct on the ordered slots. Tensor the bases of the first two halfspaces with the basis already constructed for the remaining slots, and identify a full assignment with its first component and its restriction to the remaining slots. Tensor products and isometric identifications preserve orthonormal bases. This gives (1028); tensoring with the input basis and applying the preparation identity gives (1029).
Let \(T:\mathcal S\otimes \mathcal H\to \mathcal K\) be the operator of a composition, and choose an orthonormal output basis \((g_k)_{k\in M}\). Its complete matrix \(A_T\) has row index \(k\in M\) and column index \((a,j)\in \mathcal A\times N\):
Here \(K_T(a)\) is the actual prepared matrix of Definition 27.144.5.
For a prescribed vector \(\psi \in \mathcal H\), let \(A_{T,\psi }\) have row index \(k\in M\) and column index \(a\in \mathcal A\), with
No normalization of \(\psi \) is required.
Let \(T,T':\mathcal S\otimes \mathcal H\to \mathcal K\) be two compositions, expressed in the same output basis. For independently chosen \(a,b\in \mathcal A\), their prepared matrices satisfy
Neither normalization nor allowedness is required. The identity includes the zero input vector and the empty source list.
Let \(K_e\) be sets indexed by the ordered source positions, and put \(\mathcal A_K=\prod _e K_e\). For matrices \(X_e\in \mathbb {C}^{K_e\times K_e}\), define \(\mathcal X\in \mathbb {C}^{\mathcal A_K\times \mathcal A_K}\) by
The product over an empty source list is one.
Suppose each \(K_e\) is finite. Let \(\mathcal R,\mathcal T\) be arbitrary sets, and let \(L\in \mathbb {C}^{\mathcal R\times \mathcal A_K}\) and \(M\in \mathbb {C}^{\mathcal T\times \mathcal A_K}\). For \(H(\alpha ,\beta )=L(:,\alpha )M(:,\beta )^\dagger \), the contraction of (1086) satisfies
The matrices \(X_e\) are arbitrary complex matrices; no positivity or nonvanishing condition is required.
Pairing the ket and bra assignments at each source position identifies them with \((\alpha ,\beta )\in \mathcal A_K^2\). The \((r,s)\) entry of the contraction is
The sums are finite, so their order may be interchanged.
In the source bases of Definition 27.145.1, take \(K_e=I_e\times J_e\) and let \(X_e\) be arbitrary complex matrices on \(K_e\). For the actual prepared coefficients \(H(\alpha ,\beta )=K_T(\alpha )|\psi \rangle \langle \psi | K_{T'}(\beta )^\dagger \), one has
The two compositions are independent, and \(\psi \) need not be normalized. Empty source lists and zero input vectors are included.
Suppose the output consists of a physical memory followed by a discarded memory. Choose finite orthonormal bases \((h_r)_{r\in X}\) and \((t_z)_{z\in D}\) of these memories, and use their tensor product as the output basis in (1031). For independent source assignments \(\alpha ,\beta \in \mathcal A\), put \(H_{\alpha ,\beta }=K_T(\alpha )|\psi \rangle \langle \psi | K_{T'}(\beta )^\dagger \). Then
No normalization or positivity hypothesis is imposed.
Take the \(((r,z),(s,z))\) entry of (1032) and sum over the common discarded index \(z\).
Let \(P\) be finite, let \(I_p\) be finite input alphabets, and let \(O_p\) be output alphabets. For matrices \(A_p,B_p\in \mathbb {C}^{O_p\times I_p}\) and \(X_p\in \mathbb {C}^{I_p\times I_p}\), one has
The local matrices are arbitrary; the middle global matrix is their prescribed product \(\bigotimes _p X_p\).
Complex conjugation preserves finite products, giving the adjoint identity entrywise. Coordinatewise matrix multiplication of Theorem 27.32.7.7, applied twice, gives
where the second line also uses the adjoint identity.
Let \(P\) be finite, with physical alphabets \(H_p\) and finite private alphabets \(D_p\). For arbitrary matrices \(M_p\in \mathbb {C}^{(H_p\times D_p)\times (H_p\times D_p)}\), identify \(\prod _p(H_p\times D_p)\) with \((\prod _pH_p)\times (\prod _pD_p)\). Then
In particular, if \(O_p=H_p\times D_p\) in (1038), tracing the mixed product gives \(\bigotimes _p\operatorname{tr}_{D_p}(A_pX_pB_p^\dagger )\). These identities include empty index sets.
The \((a,b)\) entry of the partial trace is
by finite distributivity. For the mixed product, first apply (1038) and then this identity.
Let \(P\) be finite and choose \(p_0\in P\). For any family of rectangular matrices \(M_p\) and any \(\mu \in \mathbb {C}\), put \(c_p=\mu \) when \(p=p_0\) and \(c_p=1\) otherwise. Then
The scalar is absorbed once, including when it vanishes.
At each matrix entry, \(\prod _pc_pM_p(x_p,y_p)=\bigl(\prod _pc_p\bigr)\prod _pM_p(x_p,y_p) =\mu \prod _pM_p(x_p,y_p)\), since \(c_{p_0}=\mu \) and \(c_p=1\) for \(p\ne p_0\).
If \(T\) is an allowed composition, then \(\| A_T\| \le 1\) in the Euclidean operator norm. This bounds its action on arbitrary superpositions of the source and physical input coordinates, independently of the number of slots and their dimensions.
The complete matrix is the matrix of the same operator \(T\) in orthonormal input and output bases. Coordinate maps preserve norms, so \(\| A_T\| =\| T\| \le 1\).
Let \(T\) and \(T'\) be compositions with a common output memory and output basis. Their source lists, source spaces, source bases, and physical input memories and bases may differ. Write \(A=A_T\) and \(B=A_{T'}\), and let \(a,b\) denote the respective source assignments. Then the rectangular matrix \(O=B^\dagger A\) satisfies
Thus the bra source assignment and its physical input index form the row index of \(O\). If both compositions are allowed, then \(\| O\| \le 1\).
With the ket and bra input coordinates as above, let \(\rho \in \mathbb {C}^{N\times N'}\) be any rectangular matrix. No contraction assumption on \(T\) or \(T'\) is needed for the identity
When the two branches have common source spaces and physical input coordinates, this is the trace of \(C_{T,T'}(\rho ;a,b)\) from Definition 27.144.5, with independent endpoint bases on the two sides.
Cyclicity of the rectangular trace gives \(\operatorname{tr}(K_T(a)\rho K_{T'}(b)^\dagger ) =\operatorname{tr}(\rho K_{T'}(b)^\dagger K_T(a))\). Expand the trace and substitute the block identity (1043).
Let \(R\) be a source list whose unordered pair labels are distinct and contain every unordered pair of distinct parties exactly once. Sending a slot to the unordered pair of its actual source is a bijection from its slot indices to \(\{ \{ p,q\} :p,q\in P,\ p\ne q\} \). This identification depends only on the parties of each source.
Count the unordered pairs of distinct elements under the stated bijection. Their number is \(\binom {|P|}{2}\), which is at most \(|P|^2\).
For each slot \(e\), let \(L_e:U_e\to U'_e\) and \(M_e:V_e\to V'_e\) be continuous linear maps. Apply these maps to the corresponding source registers and leave the original input memory \(\mathcal H\) unchanged. Denote the resulting composition by \(F_{L,M}\). This composition depends on the maps and the ordered slots, but not on the prepared source vectors.
For every family \(\eta _e\in U_e\otimes V_e\),
If \(\| L_e\| ,\| M_e\| \le 1\) for every slot, the composition \(F_{L,M}\) is allowed and contains no source preparations. The identity requires neither these norm bounds nor normalization of the vectors \(\eta _e\).
Induct on the ordered source slots. Apply the two endpoint maps of the first source, and then apply the construction to the remaining sources while leaving those first two registers fixed. Moving the remaining preparation past these fixed registers gives (1045). Each constituent is a local map, and its allowedness follows from the corresponding endpoint norm bound.
Let \(T:\mathcal S'\otimes \mathcal H\to \mathcal K\) be a composition on the target halfspaces \(U'_e,V'_e\). Use the finite orthonormal bases of \(U_e,V_e\) and the physical input and output bases already specified. Define
This is the complete free-input matrix of \(T F_{L,M}\):
The images of the endpoint maps need not span the target halfspaces.
Assume that \(T\) is allowed and that each \(L_e\) and \(M_e\) is a contraction. Then \(\| A_T^{L,M}\| \le 1\) in the Euclidean operator norm. In particular, this applies to isometric inclusions of proper Schmidt subspaces. No orthonormal basis of the target halfspaces is required.
Let \(\psi \) be the recorded private product input, let \(\xi ,\zeta \) be independent choices of gate monomials, and choose finite orthonormal bases of the private input and complete output memories. For source labels \(x,y\in I_W\), let \(C_{\xi ,\zeta }(x,y)\) be the prepared output coefficient of \(|\psi \rangle \! \langle \psi |\) for the canonical remaining operations \(V_\xi ,V_\zeta \) and elementary source vectors \(b_x,b_y\). For arbitrary complex matrices \(X_e\) on the endpoint-coordinate pairs, put \(\mathcal X_{x,y}=\prod _e(X_e)_{x_e,y_e}\). The literal column matrices satisfy
This is the actual-source specialization of the coordinate contraction in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 137–139, 279–355 and 409–434.
The action of the recorded preparation identifies the column indexed by \(x\) with the output coordinates of \(V_\xi (b_x\otimes \psi )\). The pure-input coefficient is therefore the outer product of column \(x\) of \(M_\xi \) and column \(y\) of \(M_\zeta \). The finite column-contraction identity gives (1048).
For each gate occurrence, its recorded source slots are in bijection with the unordered pairs of distinct participants of that occurrence. The slot corresponding to a pair has the original two endpoint owners. The order of these endpoints, their two Euclidean spaces, and the source vector assigned to the slot are retained independently of this unordered pair index. This is an auxiliary identification in [ 248 , proof of Theorem 5.2 ] .
The recorded slots contain each unordered pair of distinct local participants exactly once. The embedding of the participants into the original party set is injective, and therefore gives a bijection onto the pairs in its image. Composing these two bijections gives the assertion. Under chronological composition, the occurrence tags distinguish the two constituent circuits; framing leaves the occurrence and its slot unchanged.
Identify an original source occurrence with the sample position consisting of its nonempty-participant gate occurrence and its unordered endpoint pair. A gate with no participants or only one participant contributes no sample position. The inverse identification recovers the original recorded slot, with its original endpoint order and dimensions. This is an auxiliary identification in [ 248 , proof of Theorem 5.2 ] .
Let \(e\) be an original source position with ordered endpoint owners \(p_e,q_e\), and let \(p\) be any party. The sample position corresponding to \(e\) is incident to \(p\) if and only if \(p_e=p\) or \(q_e=p\). This is the incidence identification for the sample links in [ 248 , proof of Theorem 5.2 ] .
The pair attached to \(e\) is \(\{ p_e,q_e\} \). Membership in this two-element set is equivalent to \(p_e=p\) or \(q_e=p\).
Fix a party \(p\). Retain the original ordered source slots whose endpoint owners include \(p\), and write \(r_p\) for their number. Enumerating these slots in their original order gives a bijection from \(\{ 0,\ldots ,r_p-1\} \) to the sample positions incident to \(p\). The corresponding original slot retains its ordered endpoint spaces. For any sample alphabet \(J\), a local assignment therefore consists of a map \(\{ 0,\ldots ,r_p-1\} \to J\); no value at a sample position not incident to \(p\) is required. This includes the empty alphabet and the empty list of incident positions. This is the local indexing used for the sample links in [ 248 , proof of Theorem 5.2 ] .
Let \(w\) be a chronological source composition on an arbitrary party set \(P\), with original source-occurrence set \(\mathcal E\) and \(r\) ordered source slots. Let \(\mathcal S\) be its gate-tagged unordered participant-pair sample positions. Write \(\rho :[r]\to \mathcal E\) for the ordered source enumeration and \(\tau :\mathcal E\to \mathcal S\) for the source-to-sample bijection, where \([r]=\{ 0,\ldots ,r-1\} \). For each \(k\in \mathbb N\), write \([k]=\{ 0,\ldots ,k-1\} \). The bijection \(\sigma =\tau \circ \rho \) induces the assignment bijection
The original source occurrence retains its ordered endpoint spaces and Gaussian array. The same sample value is used at its two endpoints. Empty sample alphabets and empty source sets are included. This is an auxiliary identification for [ 248 , proof of Theorem 5.2 ] .
Compose the two occurrence bijections to obtain \(\sigma \). Precomposition with \(\sigma ^{-1}\) is a bijection of assignment sets, with inverse \(j\mapsto j\circ \sigma \).
For each assignment \(j:\mathcal S\to [k]\) and each original source occurrence \(e\in \mathcal E\), \((T^{-1}j)(\rho ^{-1}e)=j(\tau e)\). This identity reads the sample at the original occurrence and chooses no orientation for its unordered pair. It is an auxiliary identity for [ 248 , proof of Theorem 5.2 ] .
The inverse is precomposition with \(\sigma =\tau \circ \rho \). Thus \((T^{-1}j)(\rho ^{-1}e)=j(\tau (\rho (\rho ^{-1}e)))=j(\tau e)\).
For arbitrary index sets \(I,J\) and any matrix family \(F:([r]\to [k])\to \mathbb {C}^{I\times J}\),
If \(k=0\) and \(r{\gt}0\), both sums are empty. If \(r=0\), both assignment sets are singletons, also when \(k=0\). No normalization or positivity condition is imposed on \(F\). This is an auxiliary reindexing of the sample sum in [ 248 , proof of Theorem 5.2 ] .
Reindex the finite sum by the inverse bijection \(T^{-1}\).
27.146 Grouping parties into two sides
Let \(f:P\to Q\) be an arbitrary map of party labels. We group all parties with the same image while retaining every register and every operation. This is the owner identification used for the affected and exterior systems in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 383–427.
If \(\ell =((p_1,H_1),\ldots ,(p_n,H_n))\) is a register list, let \(\ell ^f=((f(p_1),H_1),\ldots ,(f(p_n),H_n))\). Empty lists and concatenation are preserved. The canonical isometry \(R_{f,\ell }:\mathcal H_\ell \to \mathcal H_{\ell ^f}\) is the tensor product of the identities on the original register spaces. On the empty list it is the identity of \(\mathbb {C}\), and on an elementary tensor it satisfies
No injectivity or finiteness assumption on \(f\), \(P\), or \(Q\) is imposed.
Write \(C_{\ell ,m}:\mathcal H_{\ell m}\to \mathcal H_\ell \otimes \mathcal H_m\) for the canonical concatenation isometry. Under the identification \((\ell m)^f=\ell ^f m^f\), for \(x\in \mathcal H_\ell \) and \(y\in \mathcal H_m\) one has
Equivalently, \(R_{f,\ell m}C_{\ell ,m}^{-1} =C_{\ell ^f,m^f}^{-1}(R_{f,\ell }\otimes R_{f,m})\).
Induct on \(\ell \). For the empty list the assertion is scalar linearity. For a nonempty list, first consider a tensor with its leading register separated. Apply (1052) and the induction hypothesis to the remaining registers. Additivity extends the identity to arbitrary vectors. Equality on elementary tensors gives the operator identity.
A source \(s=(p,q;U,V;\eta )\) remains between distinct owners exactly when \(f(p)\ne f(q)\). Its new record is then \((f(p),f(q);U,V;\eta )\), with the same spaces and vector. For an ordered source list \(S\), let \(S_f\) be the list of these surviving records in their original order. This operation preserves the empty list and concatenation.
The unordered pair labels of \(S_f\) are obtained by applying \(f\) to both endpoints of each original pair and removing the diagonal images. Their multiplicities are retained: different original pairs may acquire the same new label. Thus a pair \(k\) occurs in \(S_f\) exactly when it is non-diagonal and is the image of some pair occurring in \(S\).
Induct on \(S\). A source with endpoints \(p,q\) contributes the label \(\{ f(p),f(q)\} \) precisely when \(f(p)\ne f(q)\), and nothing otherwise; concatenation of lists corresponds to concatenation of label lists.
If the vectors of \(S\) are normalized, then those of \(S_f\) are normalized. Moreover, \(S_f\) is empty if and only if \(f(p)=f(q)\) for the two endpoints of every source in \(S\).
Surviving records have unchanged vectors. A source is removed precisely when its two new owners agree, which proves the second assertion.
For \(q\in Q\) and \(\eta \in U\otimes V\), prepare two fresh registers \(U,V\) at the single owner \(q\) by the local map \(c\mapsto \eta \otimes c\) from \(\mathbb {C}\) to \(U\otimes (V\otimes \mathbb {C})\), with the canonical associativity identification. Tensor this map with the identity on all spectator registers.
The operation in Definition 27.146.6 maps a spectator vector \(x\) to \(\eta \otimes x\), under canonical associativity. If \(\| \eta \| =1\), it is an allowed local operation and creates no pair source between distinct owners.
The preparation map has norm at most \(\| \eta \| \), and the associativity identification is isometric. Tensoring with the spectator identity preserves the contraction bound. For the operator identity, separate an elementary tensor in \(\eta \) and apply the concatenation isometries; then extend by linearity.
Let \(F:\mathcal H_\ell \to \mathcal H_{\ell '}\) be an actual composition of local maps, pair preparations, exchanges, and operations beside untouched registers. Its composition \(F^f\) on the new owners is defined in the same chronological order. A local map \(A:\mathcal H_a\to \mathcal H_b\) becomes \(R_{f,b}AR_{f,a}^{-1}\), acting at its new owner. Exchanges and untouched registers retain their original spaces. A pair source retains its vector and spaces; it remains a pair source when its new owners differ, and becomes the internal local preparation when they agree.
The operators of \(F\) and \(F^f\) satisfy
This identity does not require norm bounds. If \(F\) is allowed, then \(F^f\) is allowed. Its pair-source list is exactly \(S_f\), where \(S\) is the pair-source list of \(F\).
Induct on the chronological composition. For a local map, (1053) moves the owner identifications through its local and spectator factors; the conjugating isometries cancel. An internal source has the same preparation operator by Theorem 27.146.7, while a surviving source has the same vector and endpoint spaces. Exchanges and operations beside untouched registers commute with the tensor product of identities. Composition preserves the equality. Isometric conjugation preserves each local contraction bound, and an internal normalized source is an allowed local preparation. The description of the source list follows by the same induction.
Let \(f:P\to \{ 0,1\} \), and suppose \(F\) is allowed. Assume that \(f(p)=f(q)\) for the endpoints of every pair source occurring in \(F\). Let \(\mathcal H_{\ell ,+}\) and \(\mathcal H_{\ell ,-}\) be the memories of the registers with owner images \(1\) and \(0\), respectively, in their original relative order. Let \(J_{f,\ell }:\mathcal H_\ell \to \mathcal H_{\ell ,+}\otimes \mathcal H_{\ell ,-}\) first change the owner labels and then partition the two owners. There are two allowed compositions \(A_+,A_-\) on the respective memories, each containing no pair sources and each having operator norm at most one, such that
The two compositions are the actual restrictions of \(F^f\) to the two owners.
The internal-source condition makes \(S_f\) empty. Hence \(F^f\) contains no pair preparations, while remaining an allowed composition. Apply Theorem 27.140.4 to its two owners and compose the resulting identity with (1054). Its two restricted compositions are allowed and therefore have norm at most one.
27.147 Party restrictions after changing owner labels
Let \(P\) and \(Q\) be sets of owner labels. A finite composition \(W\) contains local operations, exchanges, and operations beside untouched registers. Its participating parties are the owners of all occurring registers and the assigned party of every local operation, including a scalar operation on an empty register list. For \(f:P\to Q\), write \(W^f\) for the composition obtained by changing each owner \(p\) to \(f(p)\) while preserving the register spaces and chronological order.
For \(q\in Q\), define \(\chi _q:Q\to \{ \mathrm{false},\mathrm{true}\} \) by \(\chi _q(x)=\mathrm{true}\) exactly when \(x=q\).
If \(W\) contains no pair-source preparation, then the participating parties of \(W^f\) are the images under \(f\) of the participating parties of \(W\). No injectivity assumption on \(f\) is required.
Induct on the composition. The owners of a relabelled register list are the image of its original owners. For a composition, the induction hypotheses show that the participating parties of \((W_1;W_2)^f\) are \(f(S_1)\cup f(S_2)=f(S_1\cup S_2)\), where \(S_1\) and \(S_2\) are the participating parties of \(W_1\) and \(W_2\). Local maps, exchanges, and operations beside untouched registers use \(f(\{ p\} \cup S)=\{ f(p)\} \cup f(S)\). The pair-source case is excluded by hypothesis. Named scalar local operations are included by the same identity, even when their register lists are empty.
Suppose that \(W\) contains no pair-source preparation and that \(f(p)\ne q\) for every \(p\in P\). Let \(V\) be the restriction of \(W^f\) to the owner \(q\). Then \(V\) has no participating party, and its operator is \(\operatorname {id}_{\mathbb {C}}\) under the canonical identification of the empty input and output register memories with \(\mathbb {C}\).
The pair-source list of \(W^f\) is that of \(W\) with relabelled owners, so \(W^f\) also contains no pair-source preparation and its restriction \(V\) is defined. If \(S\) denotes the participating parties of \(W\), those of \(V\) are \(\{ x\in f(S):x=q\} =\varnothing \). A composition with no participating party has empty input and output register lists and acts as their identity. In particular, it cannot contain a scalar local operation with a named party.
These auxiliary identities concern the separated local operations in [ 248 , proof of Theorem 5.2 ] . For the all-party private preparation, every original party has its own owner in \(P\sqcup \{ \bot \} \). Its restriction to \(\bot \) is therefore the scalar identity. Adjoining fixed source registers requires, in addition, that none of those registers has exterior owner \(\bot \). The subsequent residual composition can contribute a separate scalar from operations on empty memories; that scalar is not determined by the preparation identity.
27.148 Free source inputs and untouched exterior gates
The separation argument in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 342–434, uses corrected source registers as free inputs. Exact sources internal to either side are prepared there, while gates wholly in the exterior retain their complete operators. We give the corresponding exact constructions.
27.148.1 Finite gate expansions
Fix a finite branch set \(\Xi \), coefficients \(c_\xi \in \mathbb {C}\), and input and output register lists \(a,b\) on the gate’s participating party set \(P\). Common finite source data consist of an ordered list \(E\) containing each unordered pair of distinct parties exactly once, spaces \(U_e=\mathbb {C}^{u_e}\) and \(V_e=\mathbb {C}^{v_e}\), normalized vectors \(\eta _{\xi ,e}\in U_e\otimes V_e\), and allowed compositions \(T_\xi :\mathcal S\otimes \mathcal H_a\to \mathcal H_b\) containing no pair preparations. Here \(\mathcal S\) is the memory of all source registers in their prescribed order. Write \(F_\xi =T_\xi Q_{\eta _\xi ,\mathcal H_a}\) and
and require \(\| G\| \leq 1\). The coefficients are fixed before these data are chosen. Completion by unused pairs is taken only over the participating parties \(P\) of this gate.
Each \(F_\xi \) is an allowed composition. Its ordered source list is precisely the list of prescribed pairs with spaces \(U_e,V_e\) and vectors \(\eta _{\xi ,e}\). The complete sum \(G\), with its original scalar coefficients, is a contraction.
Normalization makes the source preparation allowed. Compose it with \(T_\xi \), whose source list is empty. Thus the only sources are the prescribed preparations. The bound on \(G\) is part of the gate data.
Suppose \(P\) and \(\Xi \) are finite, each \(W_\xi :a\to b\) is an allowed composition, and \(\| \sum _{\xi \in \Xi }c_\xi W_\xi \| \leq 1\). There are common finite source data with the same coefficients \(c_\xi \) for which \(G=\sum _{\xi \in \Xi }c_\xi W_\xi \) exactly. No nonemptiness assumption on \(\Xi \) is needed.
Apply Theorem 27.142.12 to the original branch compositions. It gives common finite spaces, normalized vectors, and allowed compositions without sources, with \(T_\xi Q_{\eta _\xi ,\mathcal H_a}=W_\xi \) for each \(\xi \). Substitution in (1056) preserves the original sum and therefore its contraction bound.
27.148.2 Affected parties and the exterior
Let \(\mathcal A\subseteq P\) have decidable membership. Set \(Q=\mathcal A\sqcup \{ *\} \), where \(*\) denotes the exterior, and define \(g:P\to Q\) by
Thus \(g(p)=*\) exactly when \(p\) is exterior.
One has \(g(p)=g(q)\) exactly when \(p=q\) or both parties are exterior. Consequently, for distinct source endpoints \(p,q\), \(g(p)\neq g(q)\) exactly when at least one endpoint belongs to \(\mathcal A\). Both corrected pairs within \(\mathcal A\) and pairs crossing to its exterior remain between distinct owners.
Separate the four possibilities for membership of \(p,q\) in \(\mathcal A\). Two affected images agree only when their original parties agree; an affected image cannot equal \(*\).
A source occurs in \(S_g\) precisely when it is obtained from an occurrence \(s=(p,q;U,V;\eta )\) in \(S\) with at least one affected endpoint, by replacing its owners with \(g(p),g(q)\). Its spaces and vector are unchanged. The ordered pair labels are obtained by filtering \(S\) by this same condition and then applying \(g\) to both endpoints. Order and multiplicity are retained, even when different original pairs acquire the same image. In particular, \(S_g\) is empty exactly when both endpoints of every original source are exterior.
Substitute the distinct-owner criterion of Theorem 27.148.2.2 into the occurrence description of \(S_g\). Induction on the original source list gives the ordered pair-label identity. The empty-list criterion follows from Theorem 27.146.5.
Let \(f:P\to Q\), and suppose every register in \(a\) and \(b\) has owner image \(q\in Q\). For \(D:\mathcal H_a\to \mathcal H_b\), regard \(R_{f,b}DR_{f,a}^{-1}\) as a local operation at \(q\) and leave the spectator list \(\ell \) untouched. This gives an actual composition \(D^f_{\ell }:(a\ell )^f\to (b\ell )^f\).
The complete operation satisfies
If \(\| D\| \leq 1\), this composition is allowed and contains no pair sources. For \(f=g\) and \(q=*\), this retains an exterior gate as its original aggregate contraction, without expanding its branch sum.
Apply the concatenation identity for the canonical owner isometries. The two isometries conjugating \(D\) cancel with the corresponding input and output identifications, giving (1059). Conjugation by isometries preserves the contraction bound. A local operation creates no pair sources.
For a finite family of compositions \(F_\xi :a\to b\) and arbitrary coefficients \(c_\xi \in \mathbb {C}\),
Sum the exact identity \(F_\xi ^fR_{f,a}=R_{f,b}F_\xi \) with coefficients \(c_\xi \). Composition is linear on both sides.
For common finite source data and any map \(f:P\to Q\), define the grouped aggregate by \(G^f=R_{f,b}GR_{f,a}^{-1}\).
The exact branch expansion is \(G^f=\sum _\xi c_\xi F_\xi ^f\), and \(\| G^f\| \leq 1\). This bound comes from the original aggregate contraction and does not involve \(\sum _\xi |c_\xi |\).
Compose (1060) with \(R_{f,a}^{-1}\). Both canonical owner maps are isometries, so conjugating the aggregate preserves its norm bound.
27.148.3 Preparing only the fixed sources
Let \(E\) be an ordered list of pair positions, with halfspaces \(U_e,V_e\), and let \(E_{\rm free}\subseteq E\) be selected. The selected source list retains these occurrences in their original order, with their original endpoint parties and the prescribed spaces and vectors. Its register layout is independent of those vectors. Denote its memory by \(\mathcal S_{\rm free}\) and the memory of all positions by \(\mathcal S\).
Fix vectors \(\eta _e\in U_e\otimes V_e\) only at positions outside \(E_{\rm free}\). Define \(\Phi _\eta :\mathcal S_{\rm free}\otimes \mathcal H_\ell \to \mathcal S\otimes \mathcal H_\ell \) by preparing the fixed sources and retaining all free registers in their original positions. This is one actual composition, independent of the vectors later supplied to the free registers.
For an arbitrary family \(\zeta _e\in U_e\otimes V_e\), write \(Q^{\rm free}_{\zeta ,\mathcal H_\ell }\) for preparation at just the free positions. The completed family \(\theta \) has \(\theta _e=\zeta _e\) at free positions and \(\theta _e=\eta _e\) elsewhere.
The source list of \(\Phi _\eta \) is exactly the list of fixed original positions, in their original order, with vectors \(\eta _e\). In particular, each prepared source has the original endpoint parties and halfspaces of its fixed position. If every fixed vector has norm one, then \(\Phi _\eta \) is allowed.
Induct on the source list. At a free position, retain the two registers and apply the construction to the remaining positions. At a fixed position, prepare its source after the remaining positions. The source inventory records precisely this fixed occurrence. Its preparation is allowed by normalization.
For every family of free vectors, without a normalization assumption,
Induct on the ordered positions. At a free head, preparation of its vector commutes with the operations on the remaining registers. At a fixed head, both sides prepare the same fixed vector. In each case the induction hypothesis gives the equality for the remaining positions. The empty source list gives the identity on the original input memory.
Let \(f:P\to \{ 0,1\} \) specify two sides. Suppose every fixed source has both endpoints on the same side and has norm one. Let \(T:\mathcal S\otimes \mathcal H_\ell \to \mathcal H_{\ell '}\) be an allowed composition containing no pair sources. There are two allowed compositions \(A_+,A_-\), containing no pair sources and each having norm at most one, such that
Here \(L\) is the free-source register list followed by \(\ell \), and \(J_{f,L}\) is its canonical partition isometry. The same two compositions work for every family \(\zeta \) of free vectors. Thus the contraction bounds apply on all free inputs.
The only sources in \(T\Phi _\eta \) are the fixed original positions. They are internal by hypothesis. Apply Theorem 27.146.10 to obtain (1062) and the two contraction bounds. Compose that identity with \(Q^{\rm free}_{\zeta ,\mathcal H_\ell }\) and use (1061) to obtain (1063). The factors were chosen before \(\zeta \).
27.148.4 Operations beside untouched registers
Given an actual composition \(W:a\to b\) and a spectator list \(\ell \), move \(\ell \) before \(a\), perform \(W\) while leaving those registers unchanged, and move them back after \(b\). Denote the resulting composition \(a\ell \to b\ell \) by \(W_{\ell }\).
The spectator extension has exactly the same ordered source list as \(W\), is allowed whenever \(W\) is allowed, and satisfies
The operator identity holds without norm or normalization assumptions.
On an elementary input \(x\otimes y\), the first exchange gives \(y\otimes x\), the framed composition gives \(y\otimes Wx\), and the last exchange gives \(Wx\otimes y\). Linearity gives (1064). Exchanges create no sources, and leaving a register unchanged preserves the source list and allowedness of the operations on the other registers.
Let \(g:P\to Q\) be any map of party labels, let \(W:a\to b\) be an arbitrary chronological composition, and let \(t\) be an ordered spectator list. Write \(W^g\) and \(t^g\) for their relabellings. Under the canonical concatenation identifications \((at)^g=a^g t^g\) and \((bt)^g=b^g t^g\), the constructed compositions satisfy \((W_t)^g=(W^g)_{t^g}\). The map \(g\) need not be injective. Pair sources are permitted, including those whose endpoints acquire the same owner; the two constructions use the same relabelled core \(W^g\). No contraction or normalization assumption is imposed. This is a compatibility identity for the register operations in [ 248 , proof of Theorem 5.2 ] .
Relabelling preserves the spaces, order and concatenation of register lists. For a register \(r\), a block \(A\) and an adjacent exchange, each frame and exchange acquires exactly its relabelled owners,
so induction on a prefix shows that moving a register through it commutes with relabelling. Induction on the second block then gives the block exchange identity \((\mathrm{Exch}_{A,B})^g=\mathrm{Exch}_{A^g,B^g}\), and induction on \(t\) gives \((W_{[t]})^g=(W^g)_{[t^g]}\) for the framing of \(W\) by \(t\). Apply these identities to the initial exchange, the framed core and the final exchange defining \(W_t\) and compose,
under the canonical concatenation identifications.
Identify equal ordered input, intermediate and output register lists. If chronological compositions \(W\) and \(W'\) agree under these identifications, and so do \(V\) and \(V'\), then \(V\circ W=V'\circ W'\). Framing the same register \(r\) preserves this equality: \(\mathrm{frame}_r W=\mathrm{frame}_r W'\). Identity compositions on equal lists agree, and exchanging the same two registers beside equal spectator lists gives the same composition. Identifying equal input and output lists preserves the original composition itself. If two constructed words agree under equalities of their ordered input and output lists, their evaluated operators also agree under the induced canonical memory identifications. These assertions concern the constructed compositions and permit pair sources and arbitrary local maps. They are compatibility identities for the register operations in [ 248 , proof of Theorem 5.2 ] .
Replace each equal register list by its identified list, and each equal composition by the same composition. Both sides then have identical chronological operations, including the identity, frame and exchange operations. Evaluating the same constructed word gives the same operator on the identified input and output memories.
Let \(f:P\to Q\) be an arbitrary owner map and \(W:a\to b\) an arbitrary chronological composition. Given equalities \(h_a:a=a'\) and \(h_b:b=b'\), denote by \(W[h_a,h_b]\) the same composition with its input and output lists identified accordingly. Write \(h_a^f:a^f=(a')^f\) and \(h_b^f:b^f=(b')^f\) for the induced equalities of relabelled lists. Then \((W[h_a,h_b])^f=W^f[h_a^f,h_b^f]\). This identity permits pair sources and does not require \(f\) to be injective. It is an auxiliary compatibility identity for the register operations in [ 248 , proof of Theorem 5.2 ] .
After identifying \(a'\) with \(a\) and \(b'\) with \(b\), both sides are the same relabelled composition \(W^f\).
Let \(f:P\to Q\) be any map of party labels, let \(B\subseteq Q\) be any selection of new owners, and let \(W:a\to b\) be a chronological composition containing no pair-source preparations. Under the canonical list identifications \((a^f)_B=(a_{f^{-1}(B)})^f\) and \((b^f)_B=(b_{f^{-1}(B)})^f\), the constructed compositions satisfy
No injectivity, finiteness or norm assumption is imposed. This is an auxiliary identity for the party restrictions in [ 248 , proof of Theorem 5.2 ] .
Induct on the chronological operations of \(W\). For a local map \(A:\mathcal H_a\to \mathcal H_b\) at \(p\), relabelling gives the same map \(R_{f,b}AR_{f,a}^{-1}\) in both constructions. Both restrictions retain it precisely when \(f(p)\in B\); otherwise both give the identity on the selected spectator registers. For exchanges, the same condition selects each exchanged register on both sides. Framing retains the framed register on both sides or removes it on both sides. The induction hypotheses combine under composition using the equal register lists, and the identity operation has the same selected list on both sides. Identifying equal lists commutes with relabelling throughout.
Let \(f:P\to Q\) be any map of party labels and let \(W:a\to b\) be an arbitrary chronological composition. The relabelled lists \(a^f\) and \(b^f\) have the same ordered register spaces as \(a\) and \(b\), respectively. Under these identifications of the memory spaces, \(W^f\) and \(W\) have the same continuous linear operator. If the two owners of a pair source merge, its original vector is retained as a local preparation on the same ordered spaces. No injectivity, finiteness, contraction or normalization assumption is imposed. This is an auxiliary identity for the relabellings in [ 248 , proof of Theorem 5.2 ] .
The canonical isometries \(R_{f,a}\) and \(R_{f,b}\) preserve each tensor factor and act as the identity after identifying the ordered memory spaces. The intertwining relation \(W^fR_{f,a}=R_{f,b}W\) therefore gives \(W^f x=W x\) for every input \(x\). Equality on all inputs identifies the two continuous linear operators.
Let \(f:P\to Q\) be any map of party labels, let \(W:a\to b\) be an arbitrary chronological word, and let \(s\) be an ordered list of spectator registers. Write \(F_s(W):sa\to sb\) for framing \(W\) by \(s\), leaving these registers unchanged. Under the canonical concatenation identifications, the constructed words satisfy \((F_s(W))^f=F_{s^f}(W^f)\). Pair sources are permitted, including those whose owners merge, and \(f\) need not be injective. This is an auxiliary identity for the register operations in [ 248 , proof of Theorem 5.2 ] .
Induct on \(s\). For the empty list, both constructions are \(W^f\). For a first register \(r\), relabelling changes its owner and preserves its space. Frame the inductive identity by this relabelled register, identifying the input and output concatenations by \((sa)^f=s^f a^f\) and \((sb)^f=s^f b^f\). The relabelled core is the same word \(W^f\) in both constructions.
Let \(f:P\to Q\) be an arbitrary owner map, and let \(p\in P\), \(q\in Q\) satisfy \(f^{-1}(\{ q\} )=\{ p\} \). Let \(S\) be any ordered pair-source list, with arbitrary source vectors, and write \(\ell _S\) for its ordered register list. Let \(S_f\) be the surviving source list of Definition 27.146.3, which removes precisely the sources whose two owners merge under \(f\). Then \((\ell _{S_f})_q=((\ell _S)_p)^f\) as ordered register lists. Thus selecting the source registers at \(q\) after grouping owners retains exactly the relabelled original registers at \(p\), with their original spaces and order. Owners away from this fibre may merge arbitrarily. The empty source list is included, and no normalization assumption is imposed. This is an auxiliary register identity for [ 248 , proof of Theorem 5.2 ] .
For any register list, the fibre condition identifies selection at \(q\) after relabelling with relabelling the selection at \(p\). Now induct on \(S\). The empty list gives the identity directly. If the first source has distinct owners \(x,y\) whose images coincide, their common image cannot be \(q\): otherwise the fibre condition would imply \(x=y=p\). Neither half-register is selected, and removing that source leaves the inductive identity. If the two images differ, the source survives with its original spaces and vector. Apply the register-list selection identity to its two half-registers, followed by the induction hypothesis for the tail. Concatenation preserves their order.
Let \(W:a\to b\) be a chronological composition and let \(\mathcal A\subseteq P\) be a selection of original parties. Enumerate its retained source occurrences in their original order as \(e_0,\ldots ,e_{r-1}\). At position \(i\), use the endpoint owners after grouping by \(\mathcal A\) and the prescribed original endpoint spaces \(\mathbb {C}^{u_{e_i}}\) and \(\mathbb {C}^{v_{e_i}}\). The concatenation of these two-register lists is exactly the ordered register list of the fixed partial source inventory. This equality preserves every dimension and register position and requires no branch choice or source vector. It is an auxiliary coordinate identity for [ 248 , proof of Theorem 5.2 ] .
The fixed-slot specification identifies each endpoint space with its prescribed original dimension. Concatenating these two-register lists in the fixed occurrence order gives the register list of the original partial inventory. The vectors recorded at the slots do not enter this equality.
Fix \(p\in P\). Let \(g_p\) group the exterior of \(\{ p\} \) at one owner, and let \(g\) retain all original parties. Denote their selected copies of \(p\) by \(q_p\) and \(g(p)\), respectively. Let \(S_p\) be the coordinate source list for the singleton selection, and let \(S_P\) be the all-party source list. The owner map \(\lambda _p\) sends the singleton exterior owner to the all-party exterior owner and sends each retained original owner to its all-party copy. Then \((S_p|_{q_p})^{\lambda _p}=S_P|_{g(p)}\) as ordered register lists. In particular, the selected endpoint spaces and their original order agree without a branch choice. This is an auxiliary register identity for [ 248 , proof of Theorem 5.2 ] .
Replace the coordinate lists by their fixed partial inventories. Traverse the original source occurrences in order. A half-register is selected in either grouping precisely when its original endpoint is \(p\). Its space is unchanged, and \(\lambda _p\) sends its singleton owner to \(g(p)\). The other half-registers are erased in both selections. Concatenating the retained halves gives the stated equality.
Let \(W:a\to b\), fix an original party \(p\in P\), and supply its incident gate labels \(\eta \) and the recorded product input \(\psi \in \mathcal H_a\). The labels \(\eta \) determine a singleton partial choice \(\chi _\eta \in \Xi _{\{ p\} }\). Let \(V_{p,\chi _\eta }:S_p a^{g_p}\to b^{g_p}\) be its actual source-free chronological residual, and let \(P_\psi :\mathbb {C}\to \mathcal H_a\) prepare the recorded input. Form the composition \(U_{p,\eta }=V_{p,\chi _\eta }\circ F_{S_p}(P_\psi ^{g_p})\), where \(F_{S_p}\) frames the preparation by the source registers. Define \(K_{p,\eta }=(U_{p,\eta }|_{q_p})^{\lambda _p}\), with the canonical input and output list identifications. Thus \(K_{p,\eta }:S_P|_{g(p)}\to (b^g)|_{g(p)}\) is an actual chronological word determined by \(W,p,\psi ,\eta \) alone. Its input uses precisely the selected original source endpoint spaces. Identifying its operator with the restriction of a supplied all-party branch requires a separate locality identity. This construction is auxiliary to the local contraction in [ 248 , proof of Theorem 5.2 ] .
For every supplied incident label assignment \(\eta \) and recorded product input \(\psi \), the word \(K_{p,\eta }\) contains no pair-source preparations. Its selected source-register input is free. This is an auxiliary property of the source-to-output composition in [ 248 , proof of Theorem 5.2 ] .
The original product preparation and the canonical residual contain no pair preparations. Framing and composing them preserves this property, as do restriction, owner relabelling and the canonical register-list identifications. Hence the constructed incident-label word has empty source inventory.
Fix an original party \(p\in P\) and an arbitrary ordered register list \(\ell \) on \(P\). Let \(g\) retain every original party, and select precisely those original owners whose image under \(g\) is \(g(p)\). This selection is \(\{ p\} \); let \(g_p\) group its exterior at one owner and denote its retained copy of \(p\) by \(q_p\). Let \(\lambda _p\) send the singleton exterior owner to the all-party exterior owner and each retained original owner to its all-party copy. Then
The equality preserves the original spaces and register order, including the empty list. No source inventory, branch choice or prepared vector is required. It is an auxiliary register identity for [ 248 , proof of Theorem 5.2 ] .
Induct on \(\ell \). If the first register belongs to \(p\), both selections retain its original space, and \(\lambda _p\) maps its singleton owner to \(g(p)\). Prefix this same register to the identity for the tail. If its owner differs from \(p\), the singleton selection and the all-party selection both erase it, leaving the identity for the tail.
Let \(A,A',B,B',C,C',D,D'\) be complex inner product spaces, with fixed identifications \(A=A'\), \(B=B'\), \(C=C'\) and \(D=D'\). Let \(T:A\to B\), \(T':A'\to B'\), \(U:C\to D\) and \(U':C'\to D'\) be continuous linear maps. Suppose \(T\) agrees with \(T'\) under the first two identifications, and \(U\) agrees with \(U'\) under the last two. Then \(T\otimes U\) and \(T'\otimes U'\) agree under the induced identifications of \(A\otimes C\) with \(A'\otimes C'\) and \(B\otimes D\) with \(B'\otimes D'\). The spaces and maps are arbitrary, including zero maps and zero-dimensional spaces. This is an auxiliary congruence identity for the tensor maps in [ 248 , proof of Theorem 5.2 ] .
Identify the four pairs of spaces using the given equalities. The corresponding maps then satisfy \(T=T'\) and \(U=U'\). Substituting these equalities into their tensor products gives the asserted identity.
27.149 Chronological partial source expansions
We follow the chronological expansion in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 342–434. Each gate has its own participating parties and the common source data constructed above. The following statements establish the exact partial expansion and its scalar cost. The estimate for corrected source operators requires the subsequent separation and second-moment arguments.
Let \(P\) be any set of parties. A chronological composition from register list \(a\) to register list \(b\) is formed from identities, consecutive composition, private operations, exchanges of adjacent registers, and nonprivate gates. A gate has a finite participant set \(C\), an injection \(C\hookrightarrow P\), and common finite source data on \(C\). It acts on its prescribed input registers and as the identity on all remaining registers. Any composition may also be extended by an idle register. A composition is allowed when each private operation is a contraction. Its complete operator \(W:\mathcal H_a\to \mathcal H_b\) uses the aggregate operator of each nonprivate gate.
Every allowed chronological composition satisfies \(\| W\| \leq 1\).
Each gate is a contraction. Consecutive composition, extension by an identity, and unitary identifications of the register order preserve this bound. Induction on the composition proves the claim.
Let \(\mathcal G\) be the finite set of occurrences of nonprivate gates. Repeated uses of the same gate give distinct elements of \(\mathcal G\). For \(g\in \mathcal G\), let \(C_g\subseteq P\) be its participant set, \(E_g\) its ordered source slots, \(\Xi _g\) its original branch set, and \(c_{g,\xi }\) its original coefficients. The source positions are \(\mathcal E=\{ (g,e):g\in \mathcal G,\ e\in E_g\} \). Each position has the two original endpoints of its slot, transported along the participant injection. No monomial label enters the definition of \(\mathcal E\).
The two endpoints of \((g,e)\) are distinct and both belong to \(C_g\).
Each original slot has distinct endpoints in its gate’s participant set. The participant injection preserves distinctness. Consecutive composition retains the occurrence of the gate containing the slot.
Fix affected parties \(\mathcal A\subseteq P\). A gate occurrence is touched when \(C_g\cap \mathcal A\neq \varnothing \). Choose one label \(\xi _g\in \Xi _g\) at each touched occurrence and no label at the other occurrences. Let \(\Xi _{\mathcal A}\) be this finite set of choices. In the resulting composition \(W_\xi \), touched gates are replaced by their chosen prepared branches, while untouched gates retain their complete operators. Identify all exterior owners as in Definition 27.148.2.1. The coefficient of the partial branch is \(c_\xi =\prod _{g:\, C_g\cap \mathcal A\neq \varnothing }c_{g,\xi _g}\), and its weighted sum is \(\sum _{\xi \in \Xi _{\mathcal A}}c_\xi W_\xi \). An untouched gate has one partial choice with coefficient one, even if its original branch set is empty.
If the original composition is allowed, each partial branch is allowed. Its source list is obtained from the chosen sources of the touched gates by retaining exactly those with at least one endpoint in \(\mathcal A\), and then grouping the exterior owners. Each retained source has its original endpoint spaces and vector. The order and multiplicity of the source occurrences are preserved.
A chosen prepared branch is allowed. An untouched gate is a single contraction on exterior memory. The owner identification turns a pair with two exterior endpoints into a local normalized preparation. Induction on the chronological composition gives the asserted source list and allowedness.
Let \(I_a,I_b\) be the canonical isometries identifying the input and output memories with their grouped-owner memories. Then
The private operations need not be contractions for this identity.
At a touched gate, sum its original branch expansion and extend the sum by the spectator identity. At an untouched gate, the single partial branch is its aggregate operator. The resulting identity is compatible with consecutive composition, since the two independent finite sums distribute and their coefficients multiply. It is also compatible with adjoining idle registers. Induction gives (1067).
A touched gate contributes \(\sum _{\xi \in \Xi _g}|c_{g,\xi }|\) to the sum of absolute partial coefficients. An untouched gate, identity, private operation, or register exchange contributes one. Consecutive compositions multiply these sums. Adjoining an idle register leaves the sum unchanged.
Use the defining choice set at each elementary operation. For consecutive compositions, apply \(|cd|=|c|\, |d|\) and distribute the sum over the Cartesian product of the two choice sets.
Let \(\mathcal G_{\mathcal A}\) denote the touched gate occurrences and put \(t=|\mathcal G_{\mathcal A}|\). Then
and the sum of \(|c_\xi \overline{c_\zeta }|\) over all independent ket and bra choices is the square of this product. If each touched gate’s coefficient sum is at most \(B\), these two costs are bounded by \(B^t\) and \(B^{2t}\), respectively. Every occurrence contributes separately, including repeated uses of one gate.
Induction on the chronological composition gives the product formula from the local contributions. Bound each nonnegative factor by \(B\). The ket and bra sums are independent, and complex conjugation preserves absolute value, so their total cost is the square of the first sum.
Choose finite orthonormal bases of the grouped input and output memories. Let \(M\) be the matrix of \(I_bWI_a^{-1}\) and \(K_\xi \) the matrix of the partial branch \(W_\xi \). For every input matrix \(\rho \),
In each summand, the untouched gates retain their complete operators. The matrix \(\rho \) need not be positive or normalized.
Compose (1067) with \(I_a^{-1}\) and take matrices. Distribute the finite sums in \(M\rho M^\dagger \). Taking the adjoint conjugates each bra coefficient.
The positions \(\mathcal E\) admit a fixed exhaustive ordered list without repetitions, in the preparation order of the actual composition. Consecutive compositions list the later composition’s positions first; the original slot order within a gate is retained. Each position \(e\) has fixed halfspaces \(\mathbb {C}^{u_e}\) and \(\mathbb {C}^{v_e}\).
For every partial choice \(\xi \), its source inventory is obtained from this fixed list by retaining exactly those positions with an endpoint in \(\mathcal A\). A retained position has the grouped original endpoints, the fixed halfspaces, and its normalized chosen branch vector. Consequently the ordered layout of all surviving source registers is independent of \(\xi \).
Induct on the chronological composition. The two occurrence sets of consecutive compositions are disjoint, so concatenating their lists preserves uniqueness. At a gate, list its prescribed slots. The participant injection preserves their distinct endpoints, and grouping owners removes a slot exactly when both endpoints are exterior. The remaining fields are its prescribed spaces and vector. Removing vectors from these records leaves a layout determined solely by the original positions and \(\mathcal A\).
27.150 Fixed sources for the chronological partial expansion
We now place the sources of every partial branch in the same ordered register spaces. The order is inherited from the original gate occurrences, before any branch labels are chosen. This supplies the common free inputs used in the two separated contractions in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 342–434.
Let \(\mathcal E\) be the original ordered source positions and let \(\mathcal A\) be the affected parties. Retain the ordered sublist
Write \(r=|\mathcal E_{\mathcal A}|\). Reading this sublist in order gives a bijection from \(\{ 0,\ldots ,r-1\} \) to its positions. The reference source list assigns to the corresponding position \(e\) its grouped endpoints, its fixed halfspaces \(\mathbb {C}^{u_e}\) and \(\mathbb {C}^{v_e}\), and the zero vector. Thus the reference list specifies common register spaces. Its vectors will be replaced by the normalized vectors of each partial branch.
The source at reference index \(i\) has the grouped original endpoints of its enumerated position \(e_i\) and the halfspaces \(\mathbb {C}^{u_{e_i}}\), \(\mathbb {C}^{v_{e_i}}\). The endpoints remain distinct. Every partial branch has precisely this ordered source-register layout, independently of its labels. Membership in the reference list is equivalent to having an affected endpoint.
Filter the fixed exhaustive list of original positions by endpoint incidence. Its lack of repetitions gives the stated bijection. At least one endpoint of every retained position is affected, so grouping exterior owners cannot identify its two endpoints. The ordered inventory formula of Theorem 27.149.11 gives the same endpoint spaces for every branch.
Let the chronological composition be allowed. For every partial choice \(\xi \) there are unit vectors
and an allowed composition \(V_\xi \) containing no pair sources, such that the actual ordered source list of the partial branch consists exactly of these vectors at the reference positions. If \(P_{\eta _\xi }\) prepares them and leaves the original input registers unchanged, then
The input register list of \(V_\xi \) is the common source layout followed by the grouped original inputs. Its output register list is the grouped original output list. Both lists are independent of \(\xi \). The assertion includes an empty source list and an empty set of partial choices.
Equal ordered layouts identify the two halfspaces at each source position. Transport the actual normalized source vectors along these identifications. The resulting reference list is exactly the branch’s source inventory. The source-preparation theorem moves this inventory to the initial registers and gives an allowed remaining composition containing no sources. Its exact recovery identity is the displayed equality.
For a touched gate occurrence \(g\), a partial choice \(\xi \) determines its original label \(\xi _g\in \Xi _g\). Write \(\nu _{e,\alpha }\) for the original vector of source position \(e=(g,j)\) at local label \(\alpha \in \Xi _g\). At the retained reference index \(i\), with original position \(e_i=(g_i,j_i)\), define
Both sides belong to the fixed coordinate space \(\mathbb {C}^{u_{e_i}}\otimes \mathbb {C}^{v_{e_i}}\).
Every retained source belongs to a touched gate. All vectors \(\nu _{e,\alpha }\) have norm one. The actual source record at a retained position is its original endpoint pair, grouped by affected ownership, together with \(\eta ^{\mathrm{orig}}_{\xi ,i}\) in the fixed halfspaces.
Consequently any exact representation of the ordered source inventory in the reference positions uses precisely these vectors. In Theorem 27.150.3 one may take \(\eta _{\xi ,i}=\eta ^{\mathrm{orig}}_{\xi ,i}\). In particular, this vector depends only on the local label at its original owning gate, independently of every other gate label.
A retained endpoint belongs to its owning gate, so that gate is touched. Induction on the chronological composition recovers its chosen local label and original vector. At a gate the vector is the given normalized source vector; composition and idle registers preserve its occurrence. Compare the two ordered inventory lists at reference index \(i\). Equality of their source records gives equality of the vectors in their common halfspaces. Substitution in the fixed preparation gives the final assertion.
Let \(W\) be an allowed composition and divide its parties into two sides. Choose a set of free positions in its actual ordered source list, containing every source that crosses the division. Let \(P_{\mathrm{fix}}\) prepare the original source vectors at the remaining positions. There are an allowed composition \(V\) containing no pair sources, and two allowed compositions \(A_+,A_-\) containing no pair sources, such that
Here \(J_{\mathrm{free}}\) and \(J_{\mathrm{out}}\) group the input and output registers by side. The equality holds on the entire free input space. The maps \(V,A_+,A_-\) can be chosen before any vectors are supplied at the free source positions.
More precisely, for every family \(\zeta \) of vectors at those positions, let \(P_\zeta ^{\mathrm{free}}\) prepare that family and let \(P_{\mathrm{fill}(\zeta )}\) prepare the whole source list, retaining the original vectors at fixed positions. Then
Taking the original source vectors gives the same equality with \(VP_{\mathrm{fill}(\zeta )}=W\).
Enumerate the actual source list in its original halfspaces. The source-preparation theorem gives \(W=VP_\eta \) with \(V\) allowed and containing no sources. Every fixed source is internal to one side and its original vector has norm one. Apply selective factorization to \(V\) and these fixed preparations. Its equality holds before evaluating any free source vectors, and hence specializes both to arbitrary \(\zeta \) and to the original vectors.
For affected parties \(\mathcal A\), the map \(\xi \mapsto (\xi _g)_{g:\, C_g\cap \mathcal A\neq \varnothing }\) is a bijection from \(\Xi _{\mathcal A}\) onto \(\prod _{g:\, C_g\cap \mathcal A\neq \varnothing }\Xi _g\), also when some \(\Xi _g\) is empty. For \(\mathcal A=P\) the touched occurrences are those of \(\mathcal G^+=\{ g:C_g\neq \varnothing \} \), so \(\Xi _P\cong \prod _{g\in \mathcal G^+}\Xi _g\). Restricting \(\xi \in \Xi _P\) to the occurrences touched by \(\mathcal A\) gives the projection \(\pi _{\mathcal A}\colon \Xi _P\to \Xi _{\mathcal A}\).
For \(\xi \in \Xi _P\) and an occurrence \(g\) touched by \(\mathcal A\), \((\pi _{\mathcal A}\xi )_g=\xi _g\). An occurrence is touched by \(P\) exactly when \(C_g\neq \varnothing \).
Induct on the chronological composition. At a touched gate both choices are the same label transported along the same type identification; at an untouched gate the choice set has one element.
Let \(p\in P\) and take \(\mathcal A=\{ p\} \) after grouping owners for all parties. The occurrences touched by \(\{ p\} \) are exactly the occurrences \(g\in \mathcal G^+\) with \(p\in C_g\). Hence a choice in \(\Xi _{\{ p\} }\) is the same as a local assignment \(\prod _{g\in \mathcal G^+,\, p\in C_g}\Xi _g\) on the gates containing \(p\), and for \(\xi \in \Xi _P\) the local assignment \(\xi _p\) recovers \(\pi _{\{ p\} }\xi \). No label at a gate not containing \(p\) is used.
A grouped owner selects the original party \(p\) exactly. Compare the two partial choices label by label using Theorem 27.150.8.
For \(\xi \in \Xi _{\mathcal A}\), \(c_\xi =\prod _g c_{g,\xi _g}\) over the occurrences touched by \(\mathcal A\), with factor one at the other occurrences. For \(\xi \in \Xi _P\) this is \(c_\xi =\prod _{g\in \mathcal G^+}c_{g,\xi _g}\), so each factor depends only on the label \(\xi _g\) at its own occurrence, as in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 351–381 and 565–573.
Induct on the chronological composition; composition multiplies the coefficients and the products over the two disjoint occurrence sets. Restrict the product over all occurrences to \(\mathcal G^+\), where the touched condition holds.
Let \(w\) be a chronological source composition on an arbitrary party set \(P\), and let \(\xi ,\zeta \in \Xi _P\) be two all-party choices. Write \(\mathcal G^+=\{ g:C_g\neq \varnothing \} \) for the original nonempty participant occurrences and \(c_{g,\alpha }\) for their branch coefficients. Then
The product over an empty occurrence set is one. Empty-participant occurrences remain in the residual composition and contribute no factor to this coefficient product. This is an auxiliary identity for the expansion in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 351–381 and 565–573.
Apply Theorem 27.150.10 to each choice. Complex conjugation preserves finite products, so \(\overline{c_\zeta }=\prod _{g\in \mathcal G^+}\overline{c_{g,\zeta _g}}\). Multiply the two products term by term.
With the same chronological composition, let \(L=\prod _{g\in \mathcal G^+}\Xi _g\) and let \(e\colon \Xi _P\to L\) be the bijection that records each original gate label. For arbitrary index sets \(I,J\) and any matrix family \(K\colon \Xi _P\times \Xi _P\to \mathbb {C}^{I\times J}\),
The identity includes empty party sets and empty label sets. An empty label-function set gives an empty sum; an empty occurrence set gives a singleton label-function set and coefficient one. This is an auxiliary reindexing of the expansion in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 351–381 and 565–573.
Replace each mixed coefficient by Theorem 27.150.11. Reindex the outer sum by \(\eta =e\xi \) and the inner sum by \(\theta =e\zeta \). The inverse identities give \(K(\xi ,\zeta )=K(e^{-1}\eta ,e^{-1}\theta )\).
27.151 Source substitutions and corrected density terms
We follow [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 279–383. Let \(W:\mathcal H_a\to \mathcal H_b\) be a chronological composition. Its original source positions form the finite set \(\mathcal E\). For \(e=(g,j)\), write \(\Xi _g\) for the local labels of its owning gate, \(c_{g,\alpha }\) for the gate coefficients, and \(\nu _{e,\alpha }\in \mathbb {C}^{u_e}\otimes \mathbb {C}^{v_e}\) for its original source vector. A replacement at \(e\) may depend on \(\alpha \), but not on labels of other gates. All coordinate matrices below use fixed finite orthonormal bases on the original input and output memories. The input matrix \(\rho \) is arbitrary. The finite-dimensional convention is that of the manuscript, lines 5–19; no bound on private dimensions is imposed.
Given vectors \(\eta _{e,\alpha }\) in the original source halfspaces, replace the preparations in branch \(\alpha \) of gate \(g\) by these vectors, retain its remaining operations, and sum with its original coefficient \(c_{g,\alpha }\). Compose these substituted gate operators in the original order; denote the result by \(W(\eta )\).
For affected parties \(\mathcal A\), denote by \(W_{\mathcal A,\xi }(\eta )\) the partial branch obtained by the same substitution at touched gates, retaining each untouched gate as its original aggregate operator. For \(S\subseteq \mathcal E\), let \(\nu [S\leftarrow \eta ]\) equal \(\eta \) on \(S\) and \(\nu \) elsewhere.
Original vectors recover \(W(\nu )=W\) and the original partial branches. Suppose \(\eta _{e,\alpha }=\nu _{e,\alpha }\) whenever the owning gate of \(e\) does not touch \(\mathcal A\). If \(J_a,J_b\) are the canonical owner-identification isometries, then
In particular, (1079) holds for \(\nu [S\leftarrow \eta ]\) if every position in \(S\) has an endpoint in \(\mathcal A\). The substituted vectors need not be normalized.
Induct on the chronological composition. At a touched gate, the sum is its defining branch sum. At an untouched gate all source vectors are original, and every participant has exterior owner; its branch sum is therefore its original aggregate operator. Composition distributes through the finite sums, while idle registers tensor each term with an identity. For a composition \(W=W''W'\) with intermediate identification \(J_c\), the two induction hypotheses give
The pairs \((\zeta ',\zeta )\) are the branches \(\xi \) of \(W\), with \(c_{\mathcal A,\xi }=c_{\mathcal A,\zeta }c_{\mathcal A,\zeta '}\) and \(W_{\mathcal A,\xi }(\eta )=W''_{\mathcal A,\zeta }(\eta )W'_{\mathcal A,\zeta '}(\eta )\), which is (1079). Every endpoint of a selected source belongs to its owning gate, which proves the final specialization.
Moving the actual sources of a word to its initial registers leaves a specified composition of its original remaining operations. Identify its source-register list with any equal prescribed list. Denote the resulting composition by \(V\). For the partial branch \(W_{\mathcal A,\xi }\), use the fixed retained positions \(\mathcal E_{\mathcal A}\) and write \(V_{\mathcal A,\xi }\).
The composition \(V\) has no pair sources. It is allowed if the original word is allowed. Whenever a preparation in the prescribed positions has exactly the original ordered source inventory, \(VP_\eta \) is the original word operator. Consequently the allowed partial branches satisfy
Transport the source-preparation identity along the equality of ordered register lists. The inventory at index \(i\) is the original vector selected by \(\xi _{g_i}\), which gives (1081).
For arbitrary \(\eta \), the ordered pair-source inventory of \(W_{\mathcal A,\xi }(\eta )\) has the same retained positions and has vector \(\eta _{e_i,\xi _{g_i}}\) at position \(i\). If every source with both endpoints outside \(\mathcal A\) retains its original vector, the canonical remaining composition is literally \(V_{\mathcal A,\xi }\), independently of the retained vectors. Thus
Induction preserves the complete ordered source records. Surviving pair preparations change only their vectors. A source whose endpoints are both exterior becomes a local preparation; the stated condition keeps that local operation unchanged. All other remaining operations agree. Hence \(W_{\mathcal A,\xi }(\eta )\) has canonical remaining composition \(V_{\mathcal A,\xi }\) and exact ordered inventory \(\eta _\xi \). Theorem 27.151.4, applied to this word, replaces \(\nu _\xi \) by \(\eta _\xi \) in (1081) and gives (1082).
For finite coefficient families \(c_\alpha ,d_\beta \), rectangular matrices \(K_\alpha ,L_\beta \), and a compatible rectangular matrix \(R\),
Distribute the products over the finite sums and conjugate the coefficients in the adjoint.
Fix \(S\subseteq \mathcal E\). A choice \(u\) specifies at each \(e=(g,j)\in S\) a local label \(\alpha _e\in \Xi _g\) and an endpoint basis index \(a_e\). In \(W\), replace the vector at \((e,\gamma )\) by the basis tensor indexed by \(a_e\) if \(\gamma =\alpha _e\), and by zero otherwise; retain original vectors outside \(S\). Let \(M_{S,u}\) be the resulting circuit matrix. For arbitrary source matrices \(E_{e,\alpha ,\beta }\), define
This definition uses the original positions and local labels, before any affected set is chosen.
If \(M\) is the original circuit matrix, then \(F_\varnothing (E;\rho )=M\rho M^\dagger \).
There is one assignment on the empty set, its product is one, and every source remains original.
Let \(M_{\mathcal A,\xi ;S,u}\) be the matrix of the corresponding substituted partial branch, transported back to the original input and output coordinates by \(J_a,J_b\). A coordinate choice \(a\) on \(S\) specifies only the endpoint basis indices. Write \(K_{\mathcal A,\xi ;S,a}\) for the same partial matrix with these basis tensors at selected positions, without imposing additional local-label conditions.
Suppose every \(e\in S\) has an endpoint in \(\mathcal A\). Then \(M_{S,u}=\sum _\xi c_{\mathcal A,\xi }M_{\mathcal A,\xi ;S,u}\). A summand \(M_{\mathcal A,\xi ;S,u}\) equals \(K_{\mathcal A,\xi ;S,a}\) if \(\alpha _e=\xi _{g_e}\) at every selected position, and is zero otherwise. Consequently
Each \(K\) is the matrix of the canonical remaining composition applied to the indicated basis preparations, with the original locally selected vectors at every unselected retained position.
Apply (1079) to each basis substitution and distribute the ket and bra sums using (1083). If a selected label disagrees with the branch label, its source vector is zero. Multilinearity of preparation makes the corresponding matrix zero. Otherwise the substitution is exactly the prescribed basis preparation, by (1082). Summing the local-label conditions leaves (1085). Selected positions at the same gate impose the same label; the coefficient of that gate still occurs only once in \(c_{\mathcal A,\xi }\).
Let \(i\mapsto e_i\) enumerate the retained positions. Set \(S_{\mathcal A}=\{ i:e_i\in S\} \). If every selected position has an affected endpoint, \(i\mapsto e_i\) restricts to a bijection \(S_{\mathcal A}\to S\). Precomposition with this bijection identifies the two families of endpoint-coordinate assignments, point by point.
Under this bijection, \(K_{\mathcal A,\xi ;S,a}\) is exactly the prepared matrix of \(V_{\mathcal A,\xi }\) with the corresponding basis vectors at \(S_{\mathcal A}\) and original source vectors elsewhere.
Compare the vector at each retained index \(i\). Membership in \(S_{\mathcal A}\) is precisely \(e_i\in S\), and both expressions select the same coordinate in that case.
For a fixed ordered source list, let \(K_V(a)\) be the matrix obtained from a remaining composition \(V\) by preparing its endpoint vectors indexed by \(a\). For ket and bra endpoint families, put \(H_{V,V'}(a,b)=K_V(a)\rho K_{V'}(b)^\dagger \) and write
The endpoint families on the two sides may have different finite index sets.
Suppose \(\eta _i=\sum _a r_i(a)u_i(a_1)\otimes v_i(a_2)\) and \(\zeta _i=\sum _b s_i(b)u'_i(b_1)\otimes v'_i(b_2)\). With \(X_i=r_i s_i^\dagger \), the actual prepared matrices satisfy
No spanning or normalization condition is needed for these finite expansions. In particular the identity holds for the coordinate vectors of arbitrary sources in independently chosen endpoint orthonormal bases.
Pair the ket and bra assignments at each position. Expanding their rank-one matrix entries gives the product of the ket coefficients and conjugated bra coefficients. The finite preparation expansion then gives (1087).
For actual remaining compositions \(V_\xi \) and weights \(c_\xi \), define
Source matrices may depend on both branch labels; ket and bra endpoint-coordinate sets may vary with their respective branch labels. The matrices need not be positive or Hermitian.
If \(X\) is any family of source matrices, then
For exact rank-one coordinate matrices of sources \(\eta _\xi \), \(\mathcal R(X)=M\rho M^\dagger \), where \(M=\sum _\xi c_\xi K_{V_\xi }(\eta _\xi )\).
Let \(X_i=|\eta _i\rangle \! \langle \zeta _i|\) in the chosen ket and bra endpoint bases. For any set \(S\) of positions,
Here \(a,b\) range only over endpoint-coordinate assignments on \(S\), and the notation on the right replaces each selected vector by its endpoint basis tensor. Ket and bra bases may differ.
Suppose every \(e\in S\) has an endpoint in \(\mathcal A\). In the common retained positions, set
The same globally defined term then satisfies
Taking every party to be affected retains every original source position. Its ordered enumeration is a bijection \(\{ 0,\ldots ,r-1\} \to \mathcal E\). No finiteness assumption on the set of parties is needed.
This bijection identifies sums over nonempty subsets of the retained indices with sums over nonempty subsets of \(\mathcal E\), for every function valued in a commutative additive monoid.
The induced bijection on finite subsets preserves emptiness.
Put \(X_{e,\alpha ,\beta }=|\nu _{e,\alpha }\rangle \! \langle \nu _{e,\beta }|\). To define \(\widehat\rho (Y)\), take all parties to be affected in (1088), use the actual canonical remaining compositions, and assign at position \(e_i\) the matrix \(Y_{e_i,\xi _{g_i},\zeta _{g_i}}\). Thus \(\widehat\rho (Y)\) is defined directly by local source-operator replacement, independently of any corrected-subset formula.
The exact sources recover \(\widehat\rho (X)=M\rho M^\dagger \). Replacing precisely the original positions in \(S\) gives \(\widehat\rho (E\text{ on }S,X\text{ off }S)=F_S(E;\rho )\). Consequently
Suppose the final ordered register list is the physical list followed by the discarded list. Choose finite orthonormal bases of both memories and their tensor-product basis on the full output, transported through the canonical append isometry. Define
Exact sources recover \(\widehat\rho _{\mathrm{phys}}(X)=\rho _{\mathrm{phys}}\). The physical replacement error satisfies
Apply the linear partial trace to (1095), then use the triangle inequality for trace norm. No discarded-memory dimension factor appears in either assertion.
27.151.1 Preparing the original private input
Let \(W\) be a distributed composition with input memory \(\mathcal H_a\), let \(\psi \in \mathcal H_a\) be its recorded private product input, and let \(\xi \) choose a monomial at every gate that meets a party. Write \(\mathcal H_S\) for its fixed source-register memory and \(V_\xi \) for its canonical remaining operations when every party is selected. Relabel private owners by the canonical owner identification.
Extend the recorded preparation of \(\psi \) by the identity on \(\mathcal H_S\), obtaining \(P_\psi :\mathcal H_S\otimes \mathbb {C}\to \mathcal H_S\otimes \mathcal H_a\). The actual source-to-output composition is \(T_\xi =V_\xi P_\psi \). Both compositions contain no pair-source preparations. This uses the private input and fixed source spaces of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 137–139 and 246–267.
With the canonical append and owner identifications understood, every \(y\in \mathcal H_S\) satisfies
The vector \(y\) may be zero or correlated between different source registers.
A composition extended by untouched registers acts as the identity on their tensor factor. Apply this to the recorded preparation after relabelling owners; by (1054) its value at \(1\) is the relabelled vector \(\psi \). Then compose with \(V_\xi \). The source lists of the recorded preparation and \(V_\xi \) are empty; extending by untouched registers, relabelling owners, and composing preserve that assertion.
27.152 Schmidt coordinates and one Gaussian source law
We use the original source positions of a chronological composition, with the notation of Definition 27.151.1. Write \(\mathcal E\) for these positions, \(\Xi _g\) for the labels of gate \(g\), and \(\nu _{e,\alpha }\in \mathbb {C}^{u_e}\otimes \mathbb {C}^{v_e}\) for the normalized source vector at position \(e=(g,j)\) in local branch \(\alpha \). All choices below are made by original position and local label. They precede any choice of affected parties or partial branch. The construction follows [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 279–434.
Let \(z_{e,\alpha }\in \mathbb {C}^{u_ev_e}\) be the coordinates of \(\nu _{e,\alpha }\) in the tensor product of the standard orthonormal endpoint bases. Put \(r_e=\min (u_e,v_e)\).
For every original position \(e=(g,j)\) and local label \(\alpha \in \Xi _g\), the vector \(z_{e,\alpha }\) has norm one. There are weights \(\lambda _{e,\alpha ,t}\geq 0\), summing to one, and matrices \(E_{e,\alpha }:\mathbb {C}^{r_e}\to \mathbb {C}^{u_e}\) and \(F_{e,\alpha }:\mathbb {C}^{r_e}\to \mathbb {C}^{v_e}\) such that
The same choices represent every retained source in every partial expansion: one reads the label of its original owning gate. For two partial branches, the mixed source matrix is the outer product of these two represented vectors. Zero Schmidt weights are permitted; neither endpoint frame is required to span its ambient space.
Coordinates in an orthonormal basis preserve the norm. Apply the finite Schmidt decomposition to each \(z_{e,\alpha }\) independently. The actual retained vector at a partial slot is its original source vector at the selected local label, so both the vector and outer-product assertions follow by substitution.
Let \(U,V\) be complex inner-product spaces and let \(z\) in the algebraic tensor product \(U\otimes V\) have norm one (cf. [ 244 , Proposition 1.1 ] ). There exist a natural number \(r\), nonnegative weights \(\lambda _t\) with \(\sum _t\lambda _t=1\), and linear isometries \(f:\mathbb {C}^r\to U\), \(g:\mathbb {C}^r\to V\) such that
No completeness or finite-dimensionality of \(U\) or \(V\) is assumed.
An algebraic tensor belongs to the tensor product of two finite-dimensional subspaces. Choose orthonormal coordinates on these subspaces, apply the finite Schmidt decomposition, and compose the Schmidt frames with the coordinate inclusions. Isometry preserves both the vector and its norm.
Fix a sample count \(k\geq 0\). For every \(e=(g,j)\), every ordered pair \((\alpha ,\beta )\in \Xi _g^2\), every \(0\leq s{\lt}k\), and every \(p,q\in \{ 1,\ldots ,r_e\} \), choose a standard circular complex Gaussian coordinate \(\omega _{e,\alpha ,\beta ,s,p,q}\). Give all these coordinates their product probability law, denoted by \(\mathbb P_k\). The ordered pair of local labels is part of the sample index.
Selecting a fixed original position and a fixed ordered pair of local labels gives the standard Gaussian law on its sample coordinates. More generally, for any \(S\subseteq \mathcal E\) and fixed labels \(\alpha _e,\beta _e\) for \(e\in S\), the selected families are independent with their prescribed Gaussian laws. The labels in this assertion are chosen independently of the random sample.
Coordinate projections preserve product measures. Project first to the selected original positions, then to the fixed ordered local labels at those positions.
For \(k{\gt}0\), let \(\widehat X_{e,\alpha \beta }\) be the average of the Gaussian product operators in Equation (5.6) of the manuscript, transported by \(E_{e,\alpha },F_{e,\alpha }\) on the ket side and \(E_{e,\beta },F_{e,\beta }\) on the bra side. Thus
Here \(U_s\) uses the Gaussian coordinates and \(V_s\) their complex conjugates, with the quarter-power weights of that equation. The averaged operator need not have rank one.
For every \(k{\gt}0\), the Schmidt weights and frames may be chosen so that each \(\widehat X_{e,\alpha \beta }\) is integrable and
The choices precede all corrected subsets and partial branches. More generally, these assertions hold for any nonnegative weights and endpoint matrices representing the actual vectors as in (1104); their normalization and isometry identities are not needed for the expectation calculation. The ambient correction is exactly the four-frame transport of the centered correction in Schmidt coordinates.
Use the single-source expectation formula and the projection from Theorem 27.152.5. The exact source representation identifies the mean with \(X_{e,\alpha \beta }\). Subtraction gives the centered identity; linearity of the four-frame transport gives the final assertion. The required representatives are supplied by Theorem 27.152.2.
Fix \(k{\gt}0\), \(S\subseteq \mathcal E\), deterministic local labels \(\alpha _e,\beta _e\), and nonnegative weights. Let \(\delta _e\) denote the centered Schmidt-coordinate source matrix at these labels, and set
Both \(u_e\) and \(v_e\) are independent pairs of Schmidt indices. Every conjugate pair product is integrable, and
If \(S\) is nonempty, then \(\mathbb E_k Z_{u,v}=0\). These identities require no normalization of the weight lists.
The product is the established selected Gaussian coefficient with precisely the same original positions and ordered local labels. Substitute its diagonal covariance formula. Independence multiplies the factors \(k^{-1}\), one for each original corrected position.
Let \(T\) be a complex multilinear map, let \(S\) be a finite set of its argument positions, and fix the arguments \(x_i\) outside \(S\). Replacing argument \(i\in S\) by a finite sum \(\sum _j c_{ij}v_{ij}\) gives
Restrict the multilinear map to the selected arguments and expand the finite sums. Multilinearity brings out one scalar per selected position. The empty selected set gives a single summand.
Let \(V,V'\) be words acting after the same ordered pair-source layout, with finite orthonormal coordinates on the input and output memories and on each source halfspace. Ket and bra endpoint bases may differ. Fix arbitrary source vectors \(\eta _i,\zeta _i\) and a selected set \(S\). For \(i\in S\), let \(M_i\) have independent row indices \(K_i^2\) and column indices \((K'_i)^2\), and transport it by arbitrary endpoint families \(u_i,v_i\) and \(u'_i,v'_i\). Contracting these transported matrices against the actual prepared density coefficient, with \(|\eta _i\rangle \langle \zeta _i|\) elsewhere, equals
Here \(\eta _i^a=u_i((a_i)_1)\otimes v_i((a_i)_2)\) on \(S\) and \(\eta _i\) elsewhere; the bra definition is analogous. The matrix \(K_V\) is the actual preparation followed by \(V\). No spanning, normalization, or contraction hypothesis is imposed on the endpoint families or words. The input matrix \(\rho \) is arbitrary.
Expand each transported matrix as the sum of its entries times the corresponding rank-one endpoint matrices. Apply Theorem 27.152.9, and identify each rank-one source contraction with the actual prepared ket and bra matrices. Separate the independent ket and bra assignments.
Let \(S\subseteq \mathcal E\) and suppose every position in \(S\) has an endpoint among the affected parties \(\mathcal A\). The actual ordered retained slots identify the selected partial slots bijectively with \(S\). Hence their Schmidt assignments are identified with \(\prod _{e\in S}\{ 1,\ldots ,r_e\} ^2\) by reading the same original position. For a partial branch \(\xi \), replace a selected source by the two endpoint frame columns at its original local label, indexed independently by this assignment, and keep all other sources original.
Under the preceding incidence condition, suppose the fixed weights and frames represent every original source vector. For every sample \(\omega \) and every input matrix \(\rho \), the globally defined corrected term satisfies
Here \(Z^{\xi ,\zeta }\) reads exactly the original positions and the local labels selected by \(\xi ,\zeta \). The matrices \(K_{\xi ,u}\) are the canonical residual words with the source vectors of Definition 27.152.11 prepared at their inputs. Each gate coefficient occurs once on each side.
Express the actual corrected term as its partial-branch source contraction. The ambient correction is the transport of its Schmidt correction. Apply Theorem 27.152.10 at the selected retained slots, and use their original-position identification. This is an operator identity; it precedes the expectation and norm estimates.
Suppose the final memory is the ordered tensor product of physical and discarded registers. Choose finite orthonormal bases on the input and both output factors. Let \(\widehat\rho _{\rm phys}(\omega )\) be the operator obtained from the independently defined local sampled source replacements, followed by the trace over discarded registers. Then, for every sample and every input matrix,
The finite bases follow the manuscript’s finite-space convention; no bound on private dimensions enters these identities.
Substitute the sampled operators into the established nonempty-subset expansion of local source replacements, then apply the triangle inequality after the actual final partial trace.
The statements above construct the source samples and derive their exact covariance and density identities. Bounding each physical corrected term requires the further identification of its original registers with the separated affected and exterior contractions.
27.153 Physical contractions and Gaussian source estimates
We retain the original source positions, local branch labels, and common Gaussian law of Definition 27.152.4. The estimates below concern actual allowed words and specified isometric input frames. They allow different ket and bra input spaces. Finite orthonormal coordinates are chosen whenever matrices or partial traces are used; their dimensions are unrestricted. This follows the finite-space convention in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 5–19 and 383–558. The intrinsic component and Gram maps also make sense for arbitrary inner-product discard spaces.
Let \(V:H\to K\) be the evaluation of a word and let \(J:K\simeq \mathbb {C}^X\otimes D\) be a linear isometric equivalence. For \(x\in X\), put
After choosing finite orthonormal bases of \(H\) and \(D\), let \(A_{x,V}\) be its matrix. For an output layout consisting of physical registers followed by discarded registers, the canonical choice of \(J\) is the concatenation isometry followed by the physical coordinate isometry. It sends a concatenated elementary tensor \(s\otimes t\) to \([s]_{\rm phys}\otimes t\).
If \(V\) is allowed, then \(\| C_{x,V}\| \leq 1\) and \(\| A_{x,V}\| \leq 1\). The first bound requires neither completeness nor finite dimension of \(D\).
Since \(C_{x,V}=(\langle x|\otimes I_D)JV\), the coordinate functional, the isometry \(J\) and the allowed word give \(\| C_{x,V}\| \leq \| \langle x|\otimes I_D\| \, \| J\| \, \| V\| \leq 1\). Orthonormal coordinates preserve this bound.
Let \(W:H'\to K\) be another word with the same output identification \(J\). Assume that \(H,H'\) have finite orthonormal bases. Define \(O:H\to H'\) by
Its matrix has bra rows and ket columns:
Only the finite-dimensional domain of the Riesz representation is needed; the discard space \(D\) may be incomplete.
For allowed \(V,W\), both the mixed Gram operator and its matrix satisfy \(\| O\| \leq 1\).
By Cauchy–Schwarz and Theorem 27.153.2, \(|\langle t,Os\rangle | =|\langle C_{y,W}t,C_{x,V}s\rangle | \leq \| C_{y,W}t\| \, \| C_{x,V}s\| \leq \| t\| \, \| s\| \). Passing to orthonormal coordinates preserves this estimate.
Choose a finite orthonormal basis of \(D\). Write \(K_V\) for the matrix of \(JV\) in the resulting product output basis and the chosen input basis. Its rows are indexed by a physical coordinate and a discard coordinate.
For an arbitrary rectangular matrix \(\rho :H'\to H\),
If \(V,W\) are allowed, then \(\| K_V\| ,\| K_W\| \leq 1\) and
No positivity or self-adjointness is required of \(\rho \).
Expand the partial trace in the discard basis and use the Gram entry formula. The output matrices are contractions in orthonormal coordinates. The rectangular trace-norm contraction estimate for their mixed partial trace gives the last assertion.
Let \(F:\mathbb {C}^n\to H\) and \(F':\mathbb {C}^{n'}\to H'\) be linear isometries. Replace the orthonormal input coordinate maps of Definitions 27.153.1 and 27.153.3 by \(F,F'\), respectively. Thus
These frames need not span either input space.
For allowed words, the three matrices in Definition 27.153.7 have operator norm at most one. For every rectangular \(Z\),
Compose the actual word with the given isometric input frame. The component and mixed Gram estimates, followed by the rectangular partial-trace inequality, apply without a spanning condition.
Let \(F:\mathbb {C}^{I\times T}\to H\) be an isometry and let \(\tau _t\in \mathbb {R}\). For fixed physical coordinate \(x\), set
In the product basis its \((t,d)\) coordinate is \(\sqrt{\tau _t}(A_{x,V,F})_{d,(i,t)}\). A bra word, independent index sets \(L,U\), frame \(F'\), and weights \(\tau '\) define vectors \(\phi _l\) in the same manner.
For arbitrary complex coefficients \(z_{il}\), tracing the common discard factor in \(\sum _{i,l}z_{il}|\psi _i\rangle \langle \phi _l|\) gives the rectangular matrix \(W:T\leftarrow U\) with entries
when both weight families are nonnegative. For arbitrary real weights, the factor is \(\sqrt{\tau _t}\sqrt{\tau '_u}\). The coefficient \(z_{il}\) is not conjugated. The bra coordinates in the outer product supply the conjugation in \(O\).
Expand both actual vectors in their product bases and sum over the common discard coordinate. This gives precisely the weighted source matrix with the ordinary transpose of the bra-row, ket-column Gram matrix.
Let \(V':H_E\to K_E\) be an exterior word, let \(J':K_E\simeq \mathbb {C}^Y\otimes D'\) be its output identification, and let \(G:\mathbb {C}^T\to H_E\) be an isometry. The joint output at coordinate \(i\) is
A separate bra word and its own frame and weights define \(\Phi _l\). Reorder the four coordinate indices so that physical coordinates precede discarded coordinates, and put
At fixed affected physical coordinates \(x,y\), the vectors used in this formula are those of Definition 27.153.9 tensored with the corresponding exterior outputs.
For each pair \(x,y\) of affected physical coordinates, the corresponding \(Y\times Y\) block of \(M(z)\) equals
This equality requires no allowedness assumption on the words and no normalization of the real weight families.
Assume all four words are allowed. Let \(p_i,p'_l,\tau _t,\tau '_u\) be nonnegative probability weights and let \(\kappa \geq 0\). On a probability space, suppose every \(z_{il}\overline{z_{i'l'}}\) is integrable and
Then, for each \(x,y\), the exterior block satisfies
The input frames are only required to be isometries. The Gram contraction in this estimate follows from the actual allowed words.
Use the Gram norm bound and the weighted-source second-moment estimate. The exterior partial trace is a trace-norm contraction. The pair-product integrability supplies the integrability needed to integrate this pointwise inequality.
For \(S\subseteq \mathcal E\), write \(\mathcal I_S=\prod _{e\in S}\{ 0,\ldots ,r_e-1\} ^2\). Each original position contributes two independent endpoint indices. The two local labels at each position are fixed before sampling.
Use the actual Gaussian coefficient product \(Z_{u,v}\) at original positions \(S\) and fixed local ket and bra labels. Assume \(k{\gt}0\) and that the local Schmidt weights and the two crossing weight families are probability weights. Let the specified input frames be isometries and all four separated words be allowed. Then
The full trace norm is integrable. Neither estimate depends on the exterior physical dimension or either discard dimension.
The original-source law gives diagonal covariance with \(\kappa =k^{-|S|}\) and product probabilities of total mass one. Apply Theorem 27.153.13 to each block. The trace norm of the full matrix is at most the sum of the block trace norms; sum over the \(|X|^2\) affected coordinate pairs. Integrability follows from the finite Gaussian coefficient expansion.
For the actual preparation followed by a word, let \(T\) denote its linear map and \(K_T\) its matrix in finite orthonormal bases. Then
These identities hold for arbitrary prepared source vectors, including the canonical source-basis substitutions in the original circuit. They also hold with preparation written explicitly on the input side of the residual word. No normalization or allowedness is required.
Apply the defining matrix action to each input vector. Matrix multiplication of a mixed rank-one operator gives the two resulting output vectors, with complex conjugation on the bra side.
For a mixed pure input \(|\psi \rangle \langle \phi |\), the trace norm of the actual physical corrected term is unchanged by replacing either the physical or the discard orthonormal basis. The input basis, original corrected positions, and arbitrary local correction matrices are fixed.
Expand the actual corrected term in source-basis substitutions. Theorem 27.153.16 expresses every summand as the outer product of its actual output vectors. Independence of the finite partial trace and trace norm from orthonormal output coordinates then applies to their finite sum.
For a partial branch \(\xi \) and selected endpoint assignment \(u\), let \(\Psi _{\xi ,u}(\psi )\) be the vector obtained by preparing the selected Schmidt frame columns and the original unselected sources, adjoining the owner-relabeled input \(\psi \), and applying the canonical partial residual. It belongs to the original output memory after the inverse owner identification. These are the actual prepared vectors of Theorem 27.152.12.
Suppose the affected set meets every corrected source and the weights and frames represent every original source vector. For arbitrary \(\psi ,\phi \),
The two endpoint assignments are independent, and each partial gate coefficient occurs exactly once on each side.
Apply the actual corrected-term matrix expansion and then the mixed-vector identity of Theorem 27.153.16.
For \(k{\gt}0\) and nonnegative local weights, every original-position Gaussian coefficient product is square integrable. The same holds after reading the local labels from partial branches, provided every selected source meets the affected set. Consequently, the trace norm of any finite linear combination of these products with fixed rectangular matrix coefficients is integrable. Normalization of the weights is unnecessary.
Use the diagonal second moment of the original coefficient products. Reindex the retained positions by the original positions. Square integrability on the probability space, followed by the finite matrix-sum integrability theorem, gives the conclusion.
Let \(k{\gt}0\), let the weights be nonnegative, and suppose the fixed weights and endpoint matrices represent the original source vectors. For any fixed complex-linear map \(L\) between finite matrix spaces and any input matrix \(\rho \), the function
is integrable whenever the affected set meets \(S\). In particular, the actual physical corrected term obtained by tracing the final discarded registers has integrable trace norm.
The corrected operator is a finite sum of actual coefficient products times fixed matrices. Apply Theorem 27.153.20; a fixed linear map changes only the matrix coefficients.
Before multiplying by the outer gate coefficients, define
Use finite orthonormal output coordinates for this matrix.
If the affected set meets \(S\), the two sums defining \(B_{\xi \zeta }\) may be indexed by \(\mathcal I_S\) using the actual retained-slot bijection. At \(e\in S\) the coefficient reads exactly the original local labels selected by \(\xi ,\zeta \); the output vectors are transported by the same coordinate bijection. For \(k{\gt}0\) and nonnegative local weights, every \(\| L(B_{\xi \zeta })\| _1\) is integrable. If the weights and frames also represent the original vectors, then
Reindex both finite sums and each source product by the retained original positions. Their coefficient products are square integrable. Finally use the actual mixed-vector expansion, the trace-norm triangle inequality, and linearity of the integral for the resulting finite sum.
Fix \(k{\gt}0\), nonnegative representing source weights and matrices, finite orthonormal input and physical/discard output bases, and an arbitrary input matrix. There exists one sample \(\omega \) such that
Here the sampled operator is defined by the actual local source replacements, independently of the error expansion.
All summands are nonnegative and integrable. On the probability space, their finite sum takes a value no larger than its integral. At that sample, apply the established pointwise physical error bound.
For \(S\subseteq \mathcal E\), let \(\mathcal A_S\) be the set of all endpoints of sources in \(S\). A retained source is free precisely when its original position lies in \(S\) or its endpoints lie on opposite sides of \(\mathcal A_S\). Every position in \(S\) has its corresponding free retained slot. The other retained sources are prepared with their original vectors.
A gate occurrence is touched exactly when one of its participating parties belongs to \(\mathcal A_S\). In particular, the gate containing each source in \(S\) is touched. Suppose every party in \(\mathcal A_S\) participates in at most \(b\) gate occurrences. Then the number of touched occurrences is at most \(2b|S|\). If each touched gate has coefficient sum at most \(B\geq 1\), then
More precisely, this sum times \(\prod _{p\in \mathcal A_S}d_p^2\) is the corrected-position choice cost for the actual occurrences and their branch coefficients. If the prescribed numbers \(d_p\) satisfy \(d_p\leq d\) on \(\mathcal A_S\), with \(d\geq 1\), then
The numbers \(d_p\) are supplied physical dimensions; no bound on private memory is inferred from this assertion.
Every source endpoint participates in its gate. Count touched occurrences by summing the lifetime bound over the at most \(2|S|\) affected parties. The exact product formula for partial coefficient norms gives the ket and bra factors. Apply the existing corrected-position cost estimate, retaining repeated gate occurrences separately.
A word admits a tensor partition for a Boolean owner map when there are two allowed source-free words \(C,D\), on the respective restricted input and output layouts, with \(\| C\| ,\| D\| \leq 1\), such that
The maps \(P\) are the canonical owner relabelings followed by the ordered register partitions. The equality holds on every input vector.
For every allowed chronological composition, every corrected set \(S\), and every partial branch for \(\mathcal A_S\), prepare precisely the nonfree sources of Definition 27.153.25 and then apply the canonical partial residual. This actual word admits a tensor partition into the affected and exterior sides. Its free input spaces are fixed across branches. The two factors are chosen before any vectors in these free spaces are supplied.
The nonfree source vectors are normalized and both endpoints lie on one side. The canonical residual is allowed and source-free. The selective-preparation factorization therefore applies on the entire free-source memory, with the exterior owners kept together.
For \(N\geq 0\), real \(Q\), and \(\epsilon {\gt}0\), put
Then \(k{\gt}0\) and \(k{\lt}(8\max (1,N)Q/\epsilon )^2+2\). The ceiling is a nonnegative integer ceiling.
For a finite set \(\mathcal E\) of cardinality \(N\),
The second formula uses \(k{\gt}0\). If \(Q\geq 0\) and \(\epsilon {\gt}0\), the count in Definition 27.153.29 satisfies
For \(0{\lt}\epsilon \leq 1\), its finite subset sum is therefore at most \(\epsilon /4\). These are numerical inequalities; applying them to the actual corrected physical terms also requires the corresponding individual term estimates.
Expand the product \(\prod _{e\in \mathcal E}(1+q)\) and remove the empty subset. Substitute \(q=Q/\sqrt k\). The ceiling bounds its defining square, giving the inverse-square-root estimate. Finally, \((1+q)^N\leq \exp (Nq)\) and \(Nq\leq \epsilon /8\) imply the stated bound. The empty source set gives zero throughout.
The estimates for specified separated words and the identities for actual corrected circuit terms have been stated separately. To obtain the full compression estimate, one must identify the actual free-source input and physical/discard output registers with the separated words and frames, then combine the branch, lifetime, and finite-subset estimates.
27.154 Source-only reduction of distributed densities
We prove the source-only reduction from [ 248 , proof of Theorem 5.2 ] . All registers are retained until the final partial trace. The Hilbert spaces have arbitrary finite dimensions.
For an ordered register list with spaces \(H_1,\ldots ,H_n\), choose unit vectors \(x_j\in H_j\) and put \(x=x_1\otimes \cdots \otimes x_n\). The empty tensor is the scalar \(1\). Registers may instead be grouped into one entire private memory for each party, so no factorization inside a party’s memory is required. Local preparation is the isometry \(z\mapsto zx\) from \(\mathbb {C}\) to the input memory.
Restricting to a set of owners selects the corresponding factors. The canonical owner partition sends \(x\) to \(x_A\otimes x_{A^c}\); each factor has norm one. Changing owner names leaves these vectors unchanged under the canonical memory identification.
A source-only operation is composed from local contractions, register permutations, and normalized pair-source preparations. A local vector \(x\) is prepared by \(y\mapsto x\otimes y\). Preparing an ordered inventory of pair vectors gives the tensor product of those vectors followed by the unchanged input. Its norm is one when every pair vector has norm one.
For an operation \(T\) and an additional register memory \(R\), retaining \(R\) means applying \(I_R\otimes T\). Exchanging two register blocks sends \(r\otimes s\otimes y\) to \(s\otimes r\otimes y\). Equal register lists are identified by the canonical memory isometry. These operations preserve the indicated vectors and introduce no additional nonprivate gate. An allowed original monomial expansion can consequently be replaced by its actual prepared-source expansion.
An original circuit is an ordered composition of contractions on owned registers. A private operation acts at one party. Every nonprivate operation has a specified participating set and an expansion \(G_j=\sum _\xi c_{j,\xi }M_{j,\xi }\) into allowed monomials. Registers outside the participating set are spectators. Let \(N\) denote the number of nonprivate occurrences. Fix \(r\) and \(S\) so that each original monomial has at most \(r\) pair effects and \(\sum _\xi |c_{j,\xi }|\le S\) at every nonprivate occurrence.
Let \(T\) be the operator of an original circuit and let \(\delta {\gt}0\). Choose \(m=\lceil (rS/\delta )^2\rceil +1\), using the original expansions. Replacing each nonprivate operation by its rescaled source-only expansion gives a contraction \(\widetilde T\) and a normalized auxiliary vector \(\Gamma \) such that
Each previously added auxiliary register is a spectator for all subsequent original operations.
Induct on the composition of the original operations. The ideal replacement of each nonprivate gate is its original operator tensored with that gate’s fixed normalized auxiliary vector. Their tensor product is \(\Gamma \). Private contractions and permutations contribute no error. In a composition, insert the ideal intermediate operator and use contractivity to obtain \(e(T_2T_1)\le e(T_1)+e(T_2)\). Each replaced nonprivate occurrence contributes at most \(2\delta \), proving (1146).
The nonprivate occurrences of the original circuit correspond bijectively to the expanded gates of the replacement. Corresponding occurrences have exactly the same participating parties. Auxiliary preparations and register permutations introduce no additional expanded gate occurrences.
Induct on the circuit. An original nonprivate occurrence contributes one replacement gate with the original participating set. Composition joins the two occurrence sets, while spectator registers leave them unchanged. Private operations and permutations contribute empty sets.
Fix a decomposition of the original output memory into physical and discarded factors. For an input \(x\) with \(\| x\| \le 1\), let \(\rho \) be the original physical density and let \(\widetilde\rho \) be the replacement density after also discarding its auxiliary registers. Then
If \(N\le M\) and \(\varepsilon {\gt}0\), choose \(\delta =\varepsilon /(8\max \{ 1,M\} )\). The density error is at most \(\varepsilon /2\). For \(M{\gt}0\) this is precisely \(\delta =\varepsilon /(8M)\). The estimate applies in particular to a tensor product of normalized vectors, one in each party’s entire private memory.
Put \(u=\widetilde T x\) and \(v=\Gamma \otimes Tx\), expressed in the common physical and discarded coordinates. Both norms are at most one, and (1146) gives \(\| u-v\| \le 2\delta N\). For the final partial trace \(\operatorname {Tr}_E\),
Discarding the normalized factor \(\Gamma \) leaves the original physical density unchanged. This proves (1147); substituting the chosen budget and \(N\le M\) gives the final bound. When \(M=0\), one has \(N=0\) and the error is zero.
The actual physical memory selected by the original party set \(A\) has dimension
whenever \(d_p\le D\) on \(A\). The regional readout splits the same original configuration into its restrictions to \(A\) and its complement, and discards the actual private memory. With this readout, the original whole-party product input and the gate budget of Theorem 27.154.6 give replacement density error at most \(\varepsilon /2\).
Selecting owned physical registers filters the original party ordering. Its tensor basis is indexed by one choice of \(d_p\) values at each \(p\in A\), so its cardinality is \(\prod _{p\in A}d_p\). Multiplying the local bounds gives (1149). Reindexing the full tensor basis by the two restrictions of the original configuration gives the regional isometry. Apply Theorem 27.154.6 to this actual partial trace.
Suppose \(d_p=d\) at every party and the actual discarded private memory is finite-dimensional. For the original whole-party product input, contraction circuit and positive error budget of Theorem 27.154.6, the regional readout retains the original physical coordinates in \(A\) and discards the exterior physical and private coordinates. If the original gate count is bounded by \(M\), the resulting source-only replacement density has trace-norm error at most \(\varepsilon /2\).
Specialize the physical dimension family to \(d_p=d\), using the same complete party ordering and its coverage. Apply the density-error estimate to the resulting regional isometry and the original normalized whole-party product vector.
Suppose each original nonprivate gate has at most \(K\) monomials, absolute coefficient sum at most \(S\), and at most \(r\) pair effects per monomial. Suppose also that each gate involves at most \(b\) parties and each party participates in at most \(b\) nonprivate occurrences over the entire circuit.
With the stack length chosen above, each replacement gate has at most \(Km^r\) monomials and absolute coefficient sum at most \(S\). Moreover,
After combining sources on equal party pairs, a gate with \(q\) participating parties has exactly \(\binom q2\) common source slots, including one-dimensional padding. Thus the entire replacement has at most \(N\binom b2\) source positions. Each party participates in exactly as many expanded gate occurrences as originally, so its bound \(b\) over the entire circuit is preserved. At each actual chronological occurrence, the branch cardinality and coefficient sum satisfy the same bounds. The participation count is the number of incident nonprivate occurrences, with repeated uses counted separately.
Expanding the permutation averages preserves the absolute sum of the original coefficients. The common rescaling has modulus \((1+\delta )^{-1}\le 1\), so it cannot increase that sum. The monomial count is at most \(Km^r\) by the gate expansion. The ceiling estimate gives (1151). Inducting on the original chronology carries both bounds through composition and retained spectators.
Each prepared gate has one common slot for every unordered pair of distinct participating parties. Summing \(\binom q2\le \binom b2\) over its \(N\) occurrences bounds all source positions. The occurrence correspondence of Theorem 27.154.5 restricts to a bijection on the occurrences involving any fixed party, proving exact preservation of its count.
27.155 Corrected source inputs and actual local outputs
We construct the local input isometries and output expansion used in [ 248 , proof of Theorem 5.2 ] .
Source occurrences retain their chronological positions, even when their endpoint parties coincide with those of another occurrence. A register list \(L\) has memory \(\mathcal H_L\); concatenation is identified with tensor product by the canonical isometry \(\alpha _{L,M}\colon \mathcal H_{L+M}\to \mathcal H_L\otimes \mathcal H_M\). For a division of the parties into two classes, write \(\Pi _L\) for the isometry gathering the two classes in their inherited orders. Relabelling parties changes neither the order nor the Hilbert spaces.
Identifying two equal register lists leaves their vectors and the evaluation of a composition unchanged. Such identifications commute with source preparation and with the concatenation isometries \(\alpha _{L,M}\).
Identify the equal lists. Each assertion is then an equality of the same vector or linear map.
For a family of pair sources \(r_i\) and a finite ordered list \(J\) of positions, let \(L_J\) be the concatenation of their two-register lists, and let \(P_J(\eta )\) prepare the vectors \(\eta _i\) in that order. Then \(L_{J+K}=L_J+L_K\). Selecting positions by a predicate retains their inherited order and their literal endpoint parties and spaces. The same layout is obtained by assigning zero vectors to these fixed spaces.
Preparation indexed by positions equals preparation of the corresponding ordered source records. Under the concatenation identifications, \(P_{J+K}(\eta )=P_J(\eta )P_K(\eta )\). Restricting \(J\) to a selected set of positions agrees with preparing precisely those selected sources.
Induct on the list of positions. The induction step prepends the actual source vector and identifies the unchanged spectator lists.
If \(J\) and \(J'\) are permutations of the same list of source positions, fix a spectator register list. There is an allowed composition \(U\) containing no sources such that, for every source-vector family,
In particular, one may place any selected subset of free positions before its complement, preserving the order within each subset. The same \(U\) works for every vector family; no normalization is required.
Induct on a decomposition of the permutation into adjacent exchanges. Use the identity for the empty list, retain a common first pair as spectators of the inductively constructed composition, exchange adjacent two-register blocks, and compose successive permutations. Apply this result to the selected list followed by its complement.
For selected positions \(J\), let \(\eta _J\in \mathcal H_{L_J}\) be their ordered tensor product, with the scalar \(1\) for the empty list. Its memory is fixed by the endpoint spaces, independently of the vectors. Each register is exactly one of the two registers at a selected original position. For a party division, define the crossing vector by relabelling the selected registers by their side and applying \(\Pi _{L_J}\) to \(\eta _J\).
For every spectator vector \(x\),
The joint vector depends only on the selected vectors. If each selected vector has norm one, both \(\eta _J\) and its crossing vector have norm one, including when \(J\) is empty.
Preparation gives the ordered tensor product by induction. The norm of an elementary tensor is the product of the norms; the register identifications and the party partition are isometries. Equality on the selected positions gives equality of the ordered source records.
Let \(S\) be a set of original source occurrences of a circuit \(W\), and let \(A(S)\) consist of all their endpoint parties. Among the surviving partial source positions, call a position corrected if its original occurrence is in \(S\), and crossing if its two endpoints lie on opposite sides of \(A(S)\). Write \(C\) and \(T\) for the corrected and crossing register lists, respectively, with all lists in the inherited original order. Their Boolean-labelled versions record only the side of each register.
Both endpoints of every corrected position lie in \(A(S)\). Thus corrected and crossing positions are disjoint, and the free positions are exactly their union. All registers in \(C\) lie on the affected side. Consequently, the two restrictions of \(C+T\) are \(C+T_A\) and \(T_E\), where \(T_A,T_E\) are the affected and exterior restrictions of \(T\). Every occurrence in \(S\) meets \(A(S)\).
Membership in \(A(S)\) is defined by being an endpoint of an occurrence in \(S\). Apply this to each corrected position. The two Boolean masks are then disjoint, and their union is precisely the defining free mask.
There is an allowed composition \(U\), containing no sources, from the corrected block followed by the crossing block and original input to the original free-register order. It satisfies (1152) for every free-source vector family. In particular, \(U\) is chosen before the partial branch and before any corrected-source coordinate.
Apply the uniform permutation theorem to the free positions, selecting the corrected positions. The complementary list is the crossing list by the disjointness of the two masks.
Preparing an inventory on \(1\in \mathbb C\) gives its ordered joint vector. Concatenating two inventories gives the vector \(\alpha _{L,M}^{-1}(\eta _L\otimes \eta _M)\). If a scalar-input composition \(p_0\) acts with these registers as spectators, it acts as the identity on their joint vector and prepares \(p_0(1)\) in the original input memory.
Use the preparation identity (1153) and the scalar tensor identity. Tensoring with the identity commutes with preparing the unchanged source registers. Apply this once to an inventory and then to its two concatenated parts.
Given a scalar-input composition \(p_0\), a register permutation \(U\), fixed source vectors, and a residual composition \(V\), form \(\widetilde V\) by retaining the two free blocks as spectators of \(p_0\), applying \(U\), preparing the fixed sources, and applying \(V\). Its input is just \(C+T\). The corresponding input vector is \(\alpha _{C,T}^{-1}(\eta _C\otimes \eta _T)\). For a partial circuit branch \(\xi \), use its actual residual and its original fixed-source vectors; denote the resulting composition by \(\widetilde V_\xi \).
Suppose \(U\) has the uniform preparation property (1152). Then \(\widetilde V\) evaluated on \(\alpha _{C,T}^{-1}(\eta _C\otimes \eta _T)\) equals the original residual applied after full source preparation on \(p_0(1)\), with the prescribed fixed vectors outside the free positions. The same identity holds for every actual partial branch and every free-vector family.
First move the input preparation past the two free-source preparations. Apply the defining equality for \(U\), then compose free preparation with fixed preparation. Their composition is exactly full source preparation.
The uniform corrected-input identity asserts the existence of one allowed, source-free \(U\) such that the evaluation equality in Theorem 27.155.12 holds for every partial branch and every free-source family, using the actual residual and fixed vectors of that branch.
Every circuit and scalar-input composition satisfy the uniform corrected-input identity. No normalization or contraction assumption is needed for this equality.
Suppose the circuit is allowed. After the common rearrangement, its selectively prepared residual factors across the affected/exterior division into two actual allowed compositions with no sources and operator norms at most one. If \(p_0\) is also allowed and contains no sources, the same assertion holds after absorbing \(p_0\); the free input is then exactly \(C+T\). The factorization holds on every free input vector.
Every source remaining in the selectively prepared composition is internal to the division. Relabel by the side, so these preparations become local operations, and apply the source-free binary factorization. Retaining spectators and composing the allowed source-free \(p_0\) preserves these properties.
A matrix \(E\) with \(E^*E=1\) defines the linear isometry whose columns are \(E\). Tensor the two endpoint isometries in the order of a list of source positions. Restricting this construction to a selected list gives an isometry from the Euclidean space indexed by the selected endpoint coordinates into its actual register memory. Repeated endpoint parties remain distinct tensor factors.
The column at a supplied endpoint assignment is the ordered tensor of its literal endpoint columns. With an arbitrary spectator vector appended, this column gives exactly preparation of those elementary pair vectors. For a selected list, coordinates outside the selected positions make no contribution.
Evaluate each matrix isometry on a coordinate basis vector. Induct on the list using the canonical tensor-product basis and source preparation. Then reindex the selected coordinate assignments by the filtered list.
Fix a partial branch \(\xi \) and the original endpoint matrices \(E_e,F_e\), each with orthonormal columns. For \(I=\prod _{e\in S}(J_e\times J_e)\), tensor their columns at the actual local labels of \(\xi \), in the original occurrence order. Relabelling these registers by the affected side gives an isometry \(Q_\xi \colon \ell ^2(I)\to \mathcal H_C\).
For every corrected coordinate \(i\in I\), \(Q_\xi e_i\) is the actual joint corrected-source vector formed from the original \(E_e,F_e\) columns at \(i\). The coordinate correspondence is the bijection between selected partial positions and original occurrences in \(S\).
Apply the selected-column identity. The existence of the branch and the normalization of its original source vectors make each needed endpoint coordinate set nonempty. Extend a selected assignment outside the selected positions, then remove those choices using dependence only on the selected source vectors.
For branch \(\xi \), let \(\beta _\xi \in \mathcal H_{T_A}\otimes \mathcal H_{T_E}\) be the actual ordered tensor of its crossing sources, relabelled and partitioned into affected and exterior registers. Its two ambient memories depend on the fixed register lists, not on the branch vectors.
For every partial branch, \(\| \beta _\xi \| =1\). There are a finite index set \(J\), probabilities \(\tau _j\geq 0\) with \(\sum _j\tau _j=1\), and linear isometries \(L\colon \ell ^2(J)\to \mathcal H_{T_A}\) and \(R\colon \ell ^2(J)\to \mathcal H_{T_E}\) such that
The isometries need not span either ambient memory.
Every original source vector is normalized by the prepared-gate construction, so the crossing vector has norm one. Place this algebraic tensor in finite dimensional subspaces of its two memories, apply the finite dimensional Schmidt decomposition, and include the resulting orthonormal families back into the original memories.
Fix the corrected isometry \(Q\) and the crossing vector \(\beta \). Thus \(Q\colon \ell ^2(I)\to \mathcal H_C\). Choose probabilities \(\tau \) and a linear isometry \(A\colon \ell ^2(I\times J)\to \mathcal H_{(C+T)_A}\). Choose also \(B\colon \ell ^2(J)\to \mathcal H_{(C+T)_E}\). A Schmidt input representation requires the identity
The probabilities and isometries are chosen before \(i\).
For each actual partial branch and original endpoint matrices with orthonormal columns, the corrected isometry \(Q_\xi \) and exact crossing vector \(\beta _\xi \) admit the representation (1155).
Tensor \(Q_\xi \) with the affected isometry in (1154), and use the exterior isometry unchanged. The corrected registers all lie on the affected side, so canonical concatenation identifies their targets with the two actual restrictions of \(C+T\). Distribute the tensor product over the Schmidt sum.
A family \(z_i\) in the affected/exterior output tensor product has a local-word Schmidt representation if there are actual allowed source-free compositions \(D_A,D_E\), each of operator norm at most one, probabilities \(\tau \), and linear input isometries \(A,B\) such that
All these choices precede the coordinate \(i\).
Transport the input isometries along the canonical equal-list identifications, apply the exact tensor-product factorization to (1155), and use linearity. The norm bounds come from allowedness of the two local words.
The corrected-column input identity asserts that one allowed source-free rearrangement, chosen before every branch and corrected coordinate, identifies the original partially prepared output with the absorbed composition evaluated on the actual corrected-column vector tensor the original crossing vector.
Every circuit, endpoint matrix family and scalar-input composition satisfy the corrected-column input identity. No orthonormality or contraction hypothesis is needed for this equality.
In the uniform input identity, substitute the original endpoint columns at corrected positions and the original source vectors elsewhere. Fixed positions are not corrected; crossing positions are not corrected either. Dependence only on selected positions therefore preserves the literal crossing vector and every fixed source.
Let \(W\) be an allowed circuit, let all original endpoint matrices have orthonormal columns, and let \(p_0\) be an allowed scalar-input composition containing no sources. For each partial branch \(\xi \), replace precisely the corrected source vectors by their original endpoint columns at \(i\in I\), and use \(p_0(1)\) as the original input. After the canonical owner relabelling and affected/exterior output partition, this actual output has the representation (1156). Thus the two local words, probabilities and linear input isometries are constructed from the circuit and are chosen before \(i\).
Choose the common rearrangement from the corrected-column input identity. Factor the actual absorbed composition into its two local words. Construct its input isometries from the corrected-source isometry and exact crossing vector, then apply Theorem 27.155.25. Finally substitute the original partial-output identity. This keeps the original occurrence-coordinate bijection and both owner relabellings.
27.156 Physical coordinates of the two local outputs
We use the actual physical and discarded register lists in the regional decomposition of [ 248 , proof of Theorem 5.2 ] . All coordinate bases in this section are finite orthonormal bases. The private dimensions are unrestricted.
Write the output as a physical register list followed by a discarded register list. Partition each list into the affected and exterior owners. Choose orthonormal bases \((a_x)\) and \((e_y)\) of the two physical memories, and \((d_i)\) and \((f_j)\) of the two discarded memories. Transport their tensor products by the canonical register permutations. This gives a basis with coordinates \(((x,y),(i,j))\) in physical/discard order, and a basis with coordinates \(((x,i),(y,j))\) in affected/exterior order.
On the affected memory, the output isometry sends \(a_x\otimes d_i\) to \(e_x\otimes d_i\), where \((e_x)\) is the standard coordinate basis. The exterior isometry is defined in the same way. If a register list is identified with its physical/discard concatenation, these maps include the canonical identification of equal lists.
For every output vector \(z\), its coordinate at \(((x,y),(i,j))\) in physical/discard order is its coordinate at \(((x,i),(y,j))\) after the affected/exterior partition. The affected output isometry preserves the pair \((x,i)\) of coordinates, and similarly on the exterior. For \(v\in \mathbb {C}^X\otimes D\), taking the partial inner product with \(e_x\) and then the \(d_i\) coordinate gives exactly the \((x,i)\) coordinate of \(v\).
On an elementary basis tensor the two orders are \((a_x\otimes e_y)\otimes (d_i\otimes f_j)\) and \((a_x\otimes d_i)\otimes (e_y\otimes f_j)\). The canonical partition sends the first to the second. Taking inner products with an arbitrary \(z\) gives the coordinate identity because the partition is isometric. Partial inner products give the stated coordinate projection first on elementary tensors and then by linearity.
Relabel owners by any map, or successively by two maps. Transport the physical and discarded bases by the corresponding memory isometries. Every physical/discard coordinate of the original vector equals the same coordinate of the relabelled vector. In particular, partitioning after the two relabellings gives the coordinate order \(((x,i),(y,j))\) described above. No injectivity of the owner maps is needed.
Owner relabelling preserves the ordered register spaces and commutes with concatenation. Its isometry therefore sends each transported physical/discard basis tensor to the original tensor. Preserve inner products, apply this twice, and then apply the partition identity.
Let \(V,W\) be the two actual local output maps, let \(F,G\) be input isometries, and suppose that the affected/exterior partition of an output vector \(z_k\) is
In the partition-adapted physical/discard bases, the coordinate of \(z_k\) at \(((x,y),(i,j))\) is
where \(C_{x,V}\) and \(C_{y,W}\) are the physical-coordinate projections of the two local outputs. The same identity holds for the original output before either of the two owner relabellings, in the transported bases. This identity does not require contractivity or normalization in addition to the displayed vector equality.
Substitute (1157) into the partition coordinate identity. Linearity gives the sum over \(t\), and the tensor inner product gives the product of the two discarded coordinates. The partial-inner-product identity identifies these coordinates with \(C_{x,V}\) and \(C_{y,W}\). Transport the bases back through the two owner maps.
Let the original parties form a finite list without repetitions, containing every party. Party \(p\) has physical space \(\mathbb {C}^{d_p}\). Relabel affected parties individually, identify all exterior parties, and then map owners to the two sides. The selected physical memory is canonically isometric to the original memory on that side. For an affected set \(A\), its actual orthonormal coordinates are indexed by assignments \(s_p\in \{ 0,\ldots ,d_p-1\} \) for \(p\in A\), and its dimension is \(\prod _{p\in A}d_p\).
Selecting after relabelling is the same as relabelling the registers selected by the composite owner test. For the two stated owner maps, that test is exactly membership in \(A\). Transport the original regional tensor basis through these isometries and count its assignments.
27.157 Sampling the actual source circuit
We complete the Gaussian sampling estimate for an actual source-only circuit in [ 248 , proof of Theorem 5.2 ] . The source occurrences, local branch labels, physical registers, and discarded registers are those of the original circuit. The sampled physical quantity is an operator; positivity is not asserted. The tensor-network representation of that operator is a separate statement.
Let \(R\) be a finite ordered list of registers, each with a finite-dimensional complex Hilbert space. Its tensor-product memory is finite dimensional. The same is true after relabelling owners twice and selecting either side of any division of the resulting owners. Zero-dimensional register spaces are allowed.
Induct on the register list, using finite dimensionality of tensor products. Relabelling changes no register space, and selection retains a sublist.
Under the assumptions of Theorem 27.157.1, choose an orthonormal basis of the selected memory indexed by the integers from zero to one less than its dimension. The registerwise finite-dimensional assumption supplies this basis; no dimension bound is required.
Let \(P\) be the finite set of parties, listed without repetition, and let party \(p\) have physical dimension \(d_p\). For \(A\subseteq P\), the tensor product of the original coordinate bases outside \(A\) is indexed by \(\prod _{p\notin A}\{ 0,\ldots ,d_p-1\} \). Transport this basis through the canonical owner identifications to the actual exterior physical memory. This is an orthonormal basis of that memory.
The dimension of the actual exterior physical memory after the two owner identifications is \(\prod _{p\notin A}d_p\).
Count the coordinate indices of the exterior orthonormal basis.
Let \(W\) be an allowed circuit with output memory consisting of physical registers followed by discarded registers. Fix a set \(S\) of original source occurrences and let \(A(S)\) be their endpoint parties. Let \(p_0\) be an allowed scalar-input composition containing no sources, and fix two actual partial branches \(\xi ,\zeta \) for \(A(S)\).
At every original occurrence and local branch label, let the two endpoint matrices have orthonormal columns, and let the source weights be nonnegative with sum one. Choose orthonormal bases for the affected and exterior physical memories, indexed by \(X,Y\), and for the affected and exterior discarded memories, indexed by \(D,E\). These four bases, transported through the two owner relabellings and canonical register partitions, determine bases of the original physical and discarded memories.
For \(k{\gt}0\), let \(M_{\xi ,\zeta }(\omega )\) be the actual Gaussian partial branch operator on the input \(p_0(1)\), traced over the discarded memory in these bases. Then its trace norm is integrable for the original source Gaussian law and
Construct the actual local-word Schmidt output representation independently for \(\xi \) and \(\zeta \). Denote their separated ket and bra outputs by \(\Psi _u\) and \(\Phi _v\), as in Definition 27.153.11. At each original occurrence \(s\in S\), let \(\xi _s,\zeta _s\) be the selected local labels and let \(\delta _s^{\xi _s,\zeta _s}(\omega )\) be the centered correction in Schmidt coordinates. For assignments \(u,v\) of pairs of Schmidt indices at these occurrences, put
The canonical output-coordinate identities identify each original prepared output vector with the separated coordinates of those local words. Reindex both corrected-coordinate sums by the original occurrences in \(S\). Tracing both discarded memories gives
Here \(M(z)\) is the operator in Definition 27.153.11, and \(\operatorname {tr}_{\mathrm{disc}}\) traces both discarded memories. Apply Theorem 27.153.15 to (1161) with the coefficients in (1160) to obtain integrability and (1159).
Let \(W\) be an allowed circuit, with output registers consisting of physical registers followed by discarded registers. Let \(p_0\) be an allowed scalar-input composition containing no sources, and use \(|p_0(1)\rangle \langle p_0(1)|\) as the input. Fix a set \(S\) of original source occurrences and actual Schmidt decompositions of every source vector, with nonnegative weights of sum one and orthonormal endpoint columns.
Let \(A(S)\) be the endpoint parties of \(S\). Choose orthonormal bases of the affected and exterior physical memories and of the affected and exterior discarded memories after the canonical owner identifications. Write \(X\) for the affected physical basis index. For \(k{\gt}0\), the actual physical corrected term \(F_S\), in any original physical and discarded orthonormal bases, satisfies
Here \(\xi ,\zeta \) are the actual partial choices at gate occurrences touching \(A(S)\), and \(c_\xi ,c_\zeta \) are their original coefficients.
Express the actual corrected term in the bases adapted to \(A(S)\). Trace norm is unchanged by these physical and discarded basis changes. Expand the term into its actual partial branches, apply the expectation triangle inequality with the factors \(|c_\xi \overline{c_\zeta }|\), and use Theorem 27.157.5 for every pair. The integrability needed to pass through the finite sums follows from the original Gaussian coefficient products. Factor the common bound out of the sum.
Let the finite set of parties be listed without repetition, with one physical register of dimension \(d_p\) at each party. Assume every discarded register is finite dimensional; no upper bound on these dimensions is imposed. Let \(W\) be an allowed circuit and let \(p_0\) be an allowed source-free scalar-input preparation.
Suppose each party participates in at most \(b\) gate occurrences over the whole circuit. At every gate occurrence, suppose the sum of the absolute values of its branch coefficients is at most \(B\), and suppose \(d_p\leq D\) at every party, where \(B,D\geq 1\). Fix actual normalized Schmidt decompositions of all source vectors. Then for every set \(S\) of original source occurrences and every \(k{\gt}0\), in arbitrary original input, physical output and discarded orthonormal coordinates,
The input is \(|p_0(1)\rangle \langle p_0(1)|\) and the expectation is with respect to the original source Gaussian law.
Use the original affected and complementary physical tensor bases. Finite dimensionality of each discarded register supplies the two regional discarded bases. Their dimensions do not enter (1162). The squared affected physical dimension is \(\prod _{p\in A(S)}d_p^2\). The whole-lifetime incidence estimate bounds the product of this quantity with the actual ket–bra coefficient sum by \(\bigl(B^{4b}D^4\bigr)^{|S|}\). Multiply by the nonnegative factor \(k^{-|S|/2}\).
Fix a source-only circuit with finite input, physical-output and discarded-output orthonormal coordinates, and an arbitrary input matrix. Fix endpoint data reproducing every original source vector, with nonnegative source weights. Let \(N\) be the number of original source occurrences, let \(Q\geq 0\), and let \(0{\lt}\varepsilon \leq 1\). Put
Suppose that for every nonempty set \(S\) of original source occurrences,
Then there is one actual global Gaussian sample for which the sampled physical operator differs from the original physical operator in trace norm by at most \(\varepsilon /4\).
Integrability follows from the finite Gaussian expressions for the actual source corrections. The exact nonempty-subset expansion bounds the mean error by the sum of the displayed expectations. Choose a sample with error at most that sum, and apply the scalar sample-count estimate. This implication assumes the corrected-term estimates; the preceding circuit estimate supplies them in the application below.
Let \(W\) be an allowed circuit on a finite set of parties, listed without repetition, with physical output \(\bigotimes _p\mathbb C^{d_p}\) followed by discarded registers. Every discarded register is finite dimensional, with no bound imposed on its dimension. Let \(p_0\) be an allowed scalar-input composition containing no sources, and use \(|p_0(1)\rangle \langle p_0(1)|\) as input.
Suppose every party participates in at most \(b\) gate occurrences over the whole circuit, every gate has sum of absolute branch coefficients at most \(B\), and \(d_p\leq D\), where \(B,D\geq 1\). Let \(N\) be the number of original source occurrences. For \(0{\lt}\varepsilon \leq 1\), put
There are normalized Schmidt data for every original source occurrence and local branch label, and one global Gaussian sample \(\omega \) using \(k\) samples for each original occurrence and ordered local label pair, such that
Here \(\rho \) is the original physical output operator, and \(\widehat\rho (\omega )\) is obtained by replacing each mixed source operator by its averaged Gaussian operator and tracing the original discarded registers. The conclusion holds in arbitrary original input, physical and discarded orthonormal coordinates.
Choose the Schmidt data of all original source occurrences and local labels once. For every nonempty corrected set \(S\), the actual lifetime estimate gives \(\mathbb E\| F_S\| _1\leq Q^{|S|}k^{-|S|/2}\). The subset expansion and integrability of the Gaussian corrections give a single sample whose error is at most the sum of these expectations. The choice (1164) bounds that sum by \(\varepsilon /4\). The same sample supplies all source replacements in (1165).
Let \(L\geq 1\), \(Q\geq 0\), \(\varepsilon {\gt}0\), and suppose
where \(n,q,p\) are nonnegative integers, \(C_N\geq 1\), and \(C_Q\geq 0\). For the sample count above,
At \(\varepsilon =L^{-p}\) one may take \(C_\varepsilon =1\). In particular, if \(B,D\geq 0\), \(B\leq C_BL^m\), \(D\leq C_DL^r\), and \(C_B\geq 0\), then \(Q=B^{4b}D^4\) gives
These inequalities include the case of no source occurrences.
The source bound also bounds \(\max (1,N)\) because \(C_NL^n\geq 1\). Substitute the three power bounds into the defining square and use \(k{\lt}(8\max (1,N)Q/\varepsilon )^2+2\). Since the resulting power of \(L\) is at least one, the additive constant is absorbed by the displayed coefficient. For \(Q=B^{4b}D^4\), multiply the two power estimates.
Under the circuit and input assumptions of Theorem 27.157.9, let \(L\geq 1\) and suppose
where \(n,m,r\) are nonnegative integers, \(C_N\geq 1\), and \(C_B\geq 0\). Fix any real accuracy exponent \(p\), and let \(q=\max (0,\lceil p\rceil )\). Choose exactly the sample count (1164) at accuracy \(L^{-q}\). This positive integer satisfies
There are actual global Schmidt data and a sample at this same \(k\) for which \(\| \widehat\rho -\rho \| _1\leq L^{-p}/4\). In particular this covers every real \(p{\gt}0\) prescribed in the source.
Let \(r,N\) be nonnegative integers, let \(S\) be real, and let \(\varepsilon {\gt}0\). Use the per-gate error budget \(\delta =\varepsilon /(8\max (1,N))\). The chosen stack length is exactly
Suppose \(L\geq 1\), \(N\leq C_NL^n\) and \(0\leq S\leq C_SL^s\), where \(C_N\geq 1\), \(C_S\geq 0\), and \(n,s,p\) are nonnegative integers. At \(\varepsilon =L^{-p}\), put \(H=64C_N^2(rC_S)^2+2\). Then \(m{\lt}H L^{2(n+s+p)}\).
Substitute the per-gate budget into the stack-length formula. It is the same ceiling expression as the source sample count, with cost \(rS\). Apply the inverse-power sample-count estimate to \(rS\leq rC_SL^s\).
Under the preceding power bounds, let \(K\) be a nonnegative integer and suppose \(K\leq C_KL^a\), with \(C_K\geq 0\) and \(a\) a nonnegative integer. Then
If every original nonprivate gate has at most \(K\) monomials, at most \(r\) pair effects in each monomial, and absolute coefficient sum at most \(S\), every gate occurrence of the actual source-only replacement has at most the number in (1169) of branch labels. Original occurrences remain distinct. The bound also includes empty gate expansions and \(r=0\).
Raise the nonnegative stack bound to the fixed integer power \(r\) and multiply by the bound for \(K\). The replacement construction has at most \(Km^r\) monomials at each original gate occurrence; inserted private operations and register permutations introduce no expanded gates. Apply this bound at the chosen occurrence.
Let \(\psi \) be the recorded original product input. Use the original physical basis, with dimensions \(d_p\), and any finite orthonormal reference basis of the original private output memory. Choose arbitrary finite orthonormal local source-input bases and private output bases, and construct the local physical/private bases as in Definition 27.140.1.15. Let \(c_\xi \) be the actual chronological choice coefficients and let \(\lambda _W\) be the original empty-party gate scalar. For the actual sampled source matrices, write \(\widehat\rho _{\mathrm{phys},\psi }(\widehat X)\) for the source-replaced physical operator. If \(n\) is the number of original source occurrences, then
where \(M_{\xi ,\zeta ,j,p}\) is the actual local mixed physical trace of Theorem 27.164.30 at the selected owner of \(p\). Each \(j\) is one complete assignment on the original source occurrences, identified with their chronological slots; the same assignment is used at both endpoints and in every party factor. The ket and bra choices \(\xi ,\zeta \) are independent. The scalar may vanish, and no division or normalization is performed. Empty source inventories, empty party sets, zero-dimensional memories and \(k=0\) are included, with the finite-sum and reciprocal conventions already used for the sampled source matrices. This is an identity for the original physical operator in [ 248 , proof of Theorem 5.2 ] .
Expand the source-replaced operator into its independent ket and bra choice sums by Theorem 27.164.23. For each mixed term, Theorem 27.164.24 changes the original reference private basis to the constructed grouped basis without changing the physical partial trace. Separating retained and discarded coordinates and identifying the retained configurations with the original physical configurations commute with that trace. Apply Theorem 27.164.30 termwise, then distribute the common scalar over the two finite choice sums. This gives (1170).
27.158 Approximate original gate expansions
The final paragraph of the proof of [ 248 , Theorem 5.2 ] permits approximate monomial expansions. We retain the original gate occurrences and rescale only their scalar coefficients. The resulting operator and physical trace-norm estimates do not involve private dimensions.
Fix a chronological circuit whose gate occurrence \(g\) carries an ordered expansion \(T_g=\sum _j c_{g,j}M_{g,j}\). Assign an original operator \(G_g\) on the same local input and output memories to every occurrence. Let \(W_G\) be their chronological composition, with the prescribed private operations and spectator identities. For \(\delta \ge 0\), require all private operators and all \(G_g\) to be contractions, every listed monomial to be allowed, and \(\| T_g-G_g\| \le \delta \). Retain the convention that a nonprivate occurrence has a participating set of cardinality different from one; the empty participating set is permitted.
Under these local hypotheses, \(\| W_G\| \le 1\). The same assertion holds for every original contraction circuit.
Induct on the chronology. Composition multiplies operator norms, and tensoring with a spectator identity does not increase the norm. The prescribed owner and register identifications are isometries.
Replace each coefficient \(c_{g,j}\) by \(c_{g,j}/(1+\delta )\). Write \(\widetilde W_\delta \) for the chronological circuit with these new expansions and the original private operations. Thus its local operator at \(g\) is \(\widetilde T_g=T_g/(1+\delta )\).
The ordered monomial list is unchanged. Linearity gives the displayed local operator, and \(\| T_g\| \le \| G_g\| +\delta \le 1+\delta \) proves its contractivity. Each monomial remains allowed, while
Match every gate of \(\widetilde W_\delta \) to the same original occurrence, and let \(k_g\) be the length of its prescribed monomial list. Repeated occurrences remain distinct in this correspondence.
Rescaling preserves the number \(N\) of nonprivate occurrences, their participating sets, and each party’s number of participations. If each monomial has at most \(r\) pair effects, each gate has absolute coefficient sum at most \(S\), and \(k_g\le K\), the same three bounds hold for \(\widetilde W_\delta \).
The occurrence correspondence preserves each participating set. The monomials and their pair effects are identical, and (1171) bounds the new coefficient sums.
Let \(A,C:E\to F\) and \(B,D:F\to G\) be bounded linear maps between complex inner-product spaces. If \(\| B\| ,\| C\| \le 1\), then \(\| BA-DC\| \le \| A-C\| +\| B-D\| \).
Use \(BA-DC=B(A-C)+(B-D)C\), the triangle inequality, and submultiplicativity of the operator norm.
If the local approximation error is at most \(\delta \ge 0\) and the original chronology has \(N\) nonprivate occurrences, then \(\| \widetilde W_\delta -W_G\| \le 2\delta N\).
The local error satisfies
Apply the two-factor estimate at every composition. Private maps and register permutations add no error; spectator identities preserve the local bound. Induction adds one contribution for each gate.
Choose an isometric identification \(K\) of the final memory with finite physical and discarded coordinate spaces. For an input \(x\), set \(\rho _G(x)=\operatorname {Tr}_{\rm discard}|KW_Gx\rangle \langle KW_Gx|\). Define \(\widetilde\rho _\delta (x)\) by the same formula with \(\widetilde W_\delta \). These operators are not normalized by their trace.
If \(\| x\| \le 1\), then \(\| \widetilde\rho _\delta (x)-\rho _G(x)\| _1\le 4\delta N\). In particular, for \(\varepsilon {\gt}0\) and \(N\le M\), taking \(\delta =\varepsilon /(8\max \{ 1,M\} )\) gives error at most \(\varepsilon /2\). Replacing this budget parameter by \(\varepsilon /2\) reserves \(\varepsilon /4\) for this step.
Put \(u=K\widetilde W_\delta x\) and \(v=KW_Gx\). Contractivity and the isometry \(K\) give \(\| u\| ,\| v\| \le 1\). Therefore
Substitute the prescribed scalar budget into (1174).
Let \(N\le CL^n\), where \(n,p\) are nonnegative integers and \(L\ge \max \{ 1,16C\} \). Local accuracy \(\delta =L^{-(n+p+1)}\) gives \(\| \widetilde\rho _\delta (x)-\rho _G(x)\| _1\le L^{-p}/4\) for \(\| x\| \le 1\).
The scalar estimate is \(4L^{-(n+p+1)}N\le (16C/L)L^{-p}/4\le L^{-p}/4\).
Let \(L\ge 2\), let \(N,n\) be nonnegative integers, and suppose \(N\le C L^n\). For a real target exponent \(p\), set
Then
In particular, \(q\) depends on the fixed polynomial bound and on \(p\), but not on \(L\).
Write \(s=\lceil 16C\rceil _+\). The elementary inequalities \(16C\le s\le 2^s\le L^s\) imply
The argument includes \(N=0\) and requires no further size threshold.
Suppose a chronological contraction circuit has \(N\le C L^n\) nonprivate gate occurrences, where \(L\ge 2\). Fix a real exponent \(p\), and choose \(q\) as in Theorem 27.158.11. At every original gate, let an allowed monomial expansion approximate that gate in operator norm to accuracy \(L^{-q}\). Divide all coefficients of this expansion by \(1+L^{-q}\). For an input vector of norm at most one, the physical output of the resulting contraction circuit differs from the original physical density by at most \(L^{-p}/4\) in trace norm. The output density is not normalized; all private registers are traced out.
The chronological physical-density estimate is \(4L^{-q}N\). Apply Theorem 27.158.11. The comparison uses the actual original gate operators, the same private operations and the same physical readout.
An original contraction circuit has termwise bounds \((r,C)\) if, at every nonprivate gate occurrence, each monomial in its prescribed expansion contains at most \(r\) pair effects and has scalar coefficient of modulus at most \(C\). These conditions concern the actual ordered expansions. They do not impose a bound on the sum of the moduli. The constituent local maps already satisfy the contraction conditions of the original circuit.
Write a prescribed gate expansion as \(\sum _{j=1}^{k}c_jM_j\). If \(|c_j|\leq C\) for every listed term, then \(\sum _{j=1}^{k}|c_j|\leq kC\). In particular, if \(L\geq 0\), \(C_C\geq 0\), \(u,v\) are nonnegative integers, \(k\leq C_KL^u\), and \(|c_j|\leq C_CL^v\) for every listed term, then
Both statements include the empty expansion. This is the coefficient estimate used in Theorem 5.2, assumption 3, and the opening reduction in its proof.
Sum the individual inequalities over the ordered list. For the polynomial estimate, multiply its length bound by the nonnegative number \(C_CL^v\) and combine the powers of \(L\).
Suppose an original circuit has at most \(K\) monomials at each nonprivate gate occurrence and has termwise bounds \((r,C)\), where \(C\geq 0\). Then every such gate has absolute coefficient sum at most \(KC\), or at most any \(S\geq KC\), and each monomial still has at most \(r\) pair effects. Consequently, if \(K\leq C_KL^u\) and the termwise bounds are \((r,C_CL^v)\), with \(L,C_C\geq 0\), the uniform coefficient-sum bound is \(C_KC_CL^{u+v}\). No nonemptiness of the expansion or participating set is required.
Apply the preceding list estimate at each original gate occurrence. Composition and addition of spectator registers retain these same local conditions; private operations and register permutations add no nonprivate gate expansions. The effect count is unchanged.
27.159 Physical output after exchanging auxiliary registers
The source replacement of [ 248 , proof of Theorem 5.2 ] has output memory \(\mathcal A\otimes (\mathcal H\otimes \mathcal E)\), where \(\mathcal A\) contains the retained auxiliary registers, \(\mathcal H\) contains the original physical registers, and \(\mathcal E\) contains the original discarded private registers. Write \(\tau \) for the canonical isometry from this memory to \(\mathcal H\otimes (\mathcal A\otimes \mathcal E)\).
If \(R\) is the constructed source-only replacement, define \(R_{\rm phys}=\tau R\). The exchange acts on pure tensors by
Appending \(\tau \) preserves the gate occurrences, source occurrences, participating sets, branch-label types, branch coefficients, endpoint spaces and source vectors of \(R\). It preserves source order, each party’s total number of participating occurrences, and the bounds on monomial counts and coefficient sums.
The exchange consists only of permutations of register order and introduces no gate or source occurrence. Include each occurrence of \(R\) into the first part of the composition. The defining data at these occurrences are unchanged, and the second part contributes empty occurrence sets. The gate and source cardinalities are therefore unchanged, and allowedness is equivalent before and after the exchange. Formula (1179) extends by linearity to the full operator identity.
Every auxiliary register constructed by the source replacement is finite-dimensional. If every original discarded private register is finite-dimensional, the same holds for every register of \(\mathcal A\otimes \mathcal E\) and for their tensor product. No numerical bound on private dimensions is required.
A prepared pair stack consists of finitely many finite-dimensional Euclidean registers. Induction on the stack length, on the monomial, on the gate expansion, and on the original chronology proves the assertion for each auxiliary register. Changing an owner leaves the register space unchanged, and concatenation preserves this assertion. Finally, a finite tensor product of finite-dimensional spaces is finite-dimensional.
Choose orthonormal coordinates indexed by \(U\), \(X\) and \(V\) on \(\mathcal A\), \(\mathcal H\) and \(\mathcal E\), respectively. For \(z\in \mathcal A\otimes (\mathcal H\otimes \mathcal E)\), let \(z_{i,(k,j)}\) denote its coordinates when the physical coordinate precedes the private and auxiliary coordinates. The coordinates of \(\tau z\) are \((\tau z)_{i,(j,k)}=z_{i,(k,j)}\). Consequently,
This identity holds for every vector, without a normalization premise.
Formula (1179) gives the coordinate identity on pure tensors. Tensor-product induction extends it to arbitrary vectors. Their outer-product matrices are related by the simultaneous reindexing \((i,(j,k))\mapsto (i,(k,j))\). Reindexing the traced coordinate by the bijection \(U\times V\to V\times U\) preserves the partial trace. The physical-first coordinate isometry uses the tensor product of the physical basis and the auxiliary–private basis. Evaluation of a pure input has density equal to the outer product of its evaluated vector in these coordinates. Applying this to the output vector of the composed circuit proves the final assertion.
Let \(W\) be an original circuit with at most \(M\) nonprivate gates, satisfying the original expansion bounds. Set \(\delta =\varepsilon /(8\max \{ 1,M\} )\) for \(\varepsilon {\gt}0\), and construct its replacement \(R\) using this budget. For a pure input \(\psi \) with \(\| \psi \| \le 1\), let \(\rho _W\) be the original physical output state. In the source-replacement expression for \(R_{\rm phys}\), insert the exact operator of each original pair source. The resulting physical operator is the replacement state \(\rho _R\), and \(\| \rho _R-\rho _W\| _1\le \varepsilon /2\). The physical coordinates may be the labelled tensor coordinates with an arbitrary original local dimension \(d_p\) at each party \(p\).
27.160 Exact elimination of finite-dimensional messages
The next identities justify the preliminary message expansion in [ 248 , Theorem 5.2 ] . They act on the original transmitted register and arbitrary spectator memory. In particular, no bound on the spectator dimension occurs.
For a vector \(u\in H\) at a party \(a\), the local bra is \(\langle u|\otimes I\), acting on \(H\otimes K\) and discarding the \(H\) register. For parties \(a,b\) and an orthonormal basis \((e_i)_{i\in I}\) of \(H\), the \(i\)th message summand first applies the local bra \(\langle e_i|\) at \(a\), then prepares \(e_i\) at \(b\).
A local bra defined by a normalized vector has norm at most one. Every message summand is an allowed product of local contractions, containing no pair source. On \(x\otimes y\) its action is
where the first output register is assigned to the receiver.
The bra has norm at most \(\| e_i\| =1\), and preparation of \(e_i\) is an isometry. Their composition has the action (1181). Both operations are local at their prescribed parties.
Summing the message summands over an orthonormal basis gives exactly the identity transfer from sender to receiver, tensored with the identity on the spectator memory:
This includes zero-dimensional transmitted spaces.
The orthonormal-basis expansion gives \(\sum _i\langle e_i,x\rangle e_i=x\). The asserted equality follows first on pure tensors and then by linearity.
Prepend a word to the initial word of a monomial. To compose two monomials with matching intermediate registers, retain the first monomial’s effects in their original order and append the second monomial at its final word.
An allowed monomial may be preceded by an allowed word containing no pair effects, without changing its number of pair effects. More generally, two monomials with matching intermediate registers have a chronological composition. Its operator is the composition of their operators, its pair-effect count is the sum of their counts, and it is allowed whenever both original monomials are allowed.
Prepend the word to the initial word of the monomial. For composition, retain the first monomial’s successive effects in their original order and append the second monomial at its final word. Induction on the first monomial proves the operator identity, allowedness and the exact count.
The finite expansion consisting of the message summands, each with coefficient one, has exactly \(\dim H\) terms and absolute coefficient sum \(\dim H\). Every term is allowed and has no pair effect. The operator of this actual expansion is exactly the transmitted identity operator. For \(\dim H=0\) the expansion is empty and represents the zero operator on the zero space.
The expansion is indexed once by each basis vector. Its operator is the sum of the weighted monomial operators, so (1182) applies directly. Counting the unit coefficients proves both assertions about size.
Let \(L=((c_j,M_j))_j\) and \(K=((d_k,N_k))_k\) be ordered gate expansions with matching intermediate memories. Their chronological composition lists \((c_jd_k,N_kM_j)\) in lexicographic order on \((j,k)\). Each composed occurrence comes from one occurrence in each original list.
The composed list evaluates to the second gate operator after the first. Its length is \(|L||K|\), and its absolute coefficient sum is \((\sum _j|c_j|)(\sum _k|d_k|)\). If the original monomials are allowed, so are their compositions. Bounds \(r,s\) on their pair-effect counts give the bound \(r+s\) on every composed occurrence.
Distribute the two finite operator sums and use the exact operator identity for monomial composition. Counting the ordered pairs gives the length formula. Since \(|c_jd_k|=|c_j||d_k|\), the coefficient sums multiply. The local contractions and pair effects remain in their original order, so allowedness is preserved and effect counts add.
A monomial with messages is a chronological composition of the existing source-and-effect monomials and finite-dimensional register transfers. A transfer is the identity on a register’s Hilbert space, changing its owner from the sender to the receiver and leaving all spectators unchanged. The monomial is allowed when all its original source-and-effect segments are allowed. Record its message dimensions \(d_1,\ldots ,d_s\) in chronological order and let \(r_M\) count its original pair effects. Choose an orthonormal basis of each message space and expand its identity as the sum of sender bras followed by receiver kets. Compose these finite lists in the original order to obtain an expansion \(\mathcal E(M)\).
The expansion \(\mathcal E(M)\) evaluates exactly to the original monomial. If \(M\) is allowed, every term is an allowed existing monomial with exactly \(r_M\) pair effects, and
These statements include zero-dimensional transfers and the empty list of message occurrences.
The basis expansion of a transfer has one unit-coefficient local monomial for each basis vector and introduces no pair effect. Composition multiplies the list lengths and absolute coefficient sums, adds the original effect counts, and preserves the evaluated operator. Induction over the actual monomial gives the assertions.
Suppose \(M\) contains at most \(m\) messages, each of dimension at most \(D\geq 1\). Both the number of expanded monomials and their absolute coefficient sum are at most \(D^m\). In particular, if \(d_j\leq C_D L^v\), where \(C_D,L\geq 1\) and \(v\) is a nonnegative integer, both quantities are at most \(C_D^mL^{vm}\). Thus boundedly many polynomial-dimensional messages change a gate expansion by only a polynomial factor, as required in Theorem 5.2, assumption 3, and the preliminary reduction in its proof.
Bound each factor in (1183) by \(D\). Since \(D\geq 1\), replacing the actual message count by \(m\) can only increase this upper bound. Substitute \(D=C_D L^v\) for the final assertion.
27.161 Gates containing bounded messages
Let \(H_a,H_b\) be the input and output memories of a fixed finite set of participating parties. We use the monomials that permit local contractions, normalized pair sources and effects, and finite-dimensional transfers, as in [ 248 , Proof of Theorem 5.2 ] .
A weighted gate list is a finite ordered list \(L=((c_j,M_j))_{j=0}^{K-1}\), where each \(M_j:H_a\to H_b\) is such a monomial and \(c_j\in \mathbb {C}\). Its operator is \(G_L=\sum _j c_jM_j\). Occurrences are retained separately when two monomials agree.
Write the coordinate expansion of the messages in \(M_j\) as the ordered list \(((a_{j,t},N_{j,t}))_t\). Replace the occurrence \((c_j,M_j)\) by \(((c_ja_{j,t},N_{j,t}))_t\) and concatenate these lists in their original order. Thus every resulting occurrence has coefficient \(c_ja_{j,t}\) and monomial \(N_{j,t}\) for an original occurrence \(j\) and one of its coordinate choices \(t\).
The expanded list has operator \(G_L\). If every \(M_j\) is allowed, so is every resulting monomial. If every \(M_j\) is allowed and has at most \(r\) pair effects, the same bound holds for every resulting monomial.
Each coordinate expansion satisfies \(\sum _t a_{j,t}N_{j,t}=M_j\). Hence \(\sum _{j,t}c_ja_{j,t}N_{j,t}=\sum _jc_jM_j\). Its factors are the original allowed factors and the local coordinate maps of the transfers. The latter introduce no pair effects.
Let \(d_{j,1},\ldots ,d_{j,m_j}\) be the dimensions of the messages in \(M_j\), and put \(q_j=\prod _{t=1}^{m_j}d_{j,t}\). The number of resulting occurrences and their absolute coefficient sum are respectively \(\sum _jq_j\) and \(\sum _j|c_j|q_j\). If \(|L|\le K\), \(m_j\le m\), and \(d_{j,t}\le D\) with \(D\ge 1\), then
In particular, \(|c_j|\le C\) with \(C\ge 0\) gives the bound \(KCD^m\).
A monomial with message dimensions \(d_{j,t}\) has exactly \(q_j\) coordinate choices, and the absolute coefficient sum of its expansion is \(q_j\). Multiplication by \(c_j\) scales this sum by \(|c_j|\). Summing proves the exact formulas. The inequalities follow from \(q_j\le D^{m_j}\le D^m\) and \(\sum _j|c_j|\le KC\).
Suppose the local participant set \(C\) has \(|C|\ne 1\), its embedding in the global parties is \(\iota :C\hookrightarrow P\), all monomials are allowed, and \(\| G_L\| \le 1\). Insert the expanded list as one nonprivate gate with the same participant embedding and the same spectator memory. Empty participant sets are included.
The constructed gate’s operator is the canonical relabelling of \(G_L\otimes I\). It has one nonprivate occurrence, with participant set \(\iota (C)\).
The expanded list has the original operator, so adjoining the spectator identity and relabelling the participant registers gives the stated operator. Its norm bound follows from equality of the expanded and original operators. The construction inserts exactly one nonprivate occurrence with the prescribed participant embedding.
Under the dimension and message-count bounds of Theorem 27.161.4, if the original coefficient sum is at most \(S\) and every original monomial has at most \(r\) pair effects, the constructed gate has the same effect bound and coefficient sum at most \(SD^m\). If the original list has at most \(K\) terms, it has at most \(\lceil KD^m\rceil \) terms.
Apply the effect-count preservation and the two size estimates to the actual expanded list used in the gate. The ceiling converts its real upper bound into an integer bound.
27.162 Original input and physical-output resources
The following bounds concern the actual source replacement and its prescribed star and sample alphabets. They do not assert an identification of the sampled physical operator with a contraction of local tensors.
Let \(P\) be a finite set of parties and let \(H_p\) be their original finite-dimensional private memories. The ordered memory \(\bigotimes _{p\in P}H_p\) is finite-dimensional and hence admits a finite orthonormal basis. There is no numerical bound on the dimensions \(\dim H_p\).
Each register in the original party list has space \(H_p\) for one \(p\in P\). Finite tensor products of finite-dimensional spaces are finite-dimensional. This includes the empty tensor product and zero-dimensional factors.
Let \(W\) be an original contraction circuit with \(N\) nonprivate gate occurrences, each involving at most \(a\) parties. Its physical-first source replacement has at most \(N\binom a2\) source positions. This count does not depend on the effect-elimination accuracy or on private dimensions.
Each replacement gate has one source slot per unordered pair of its original participants. The final physical output exchange introduces no new gate or source occurrence.
Let \(W,N,a\) be as in Theorem 27.162.2. Let \(L\ge 1\), let \(n,s,d,p\) be nonnegative integers, and suppose
Here \(B\) bounds the original gate coefficient sums, \(D\) is a nonnegative parameter, and \(\ell \) is a nonnegative integer. In applications they are chosen as a physical-dimension bound and a whole-lifetime participation bound. Put \(C_{\rm src}=\max \{ 1,C_N\binom a2\} \). For the actual sample count of the physical-first replacement, chosen at accuracy \(L^{-p}\) and cost \(B^{4\ell }D^4\), one has
Every coefficient and exponent in this bound depends only on the fixed original resource bounds and the prescribed accuracy.
The source count is at most \(C_{\rm src}L^n\). Apply the explicit sample-count estimate with this bound and (1186). The maximum with one covers the case of no sources.
Suppose \(L\ge 1\), \(C_N\ge 1\), \(C_K,C_S,S\ge 0\), and
Let every original gate contain at most \(K\) monomials with at most \(r\) pair effects each and coefficient sum at most \(S\). Use gate budget \(\delta =L^{-p}/(8\max \{ 1,N\} )\), where \(u,n,s,p\) are nonnegative integers. Every gate of the actual physical-first replacement then has at most
branches.
The replacement has at most \(K m^r\) branches, where \(m\) is the chosen stack length. Substitute its polynomial bound. The physical output exchange changes neither branch labels nor coefficient sums.
Let \(E\) be the bijection from original nonprivate gate occurrences to physical-first replacement gate occurrences obtained by composing the correspondences for effect elimination and the physical output exchange.
The bijection \(E\) preserves participating sets. Choose any collection of replacement gate occurrences and a root assignment on all replacement occurrences whose value participates at every retained occurrence. Every star or sample link in the resulting construction joins two parties that participate in a common gate of the original circuit.
The two gate correspondences preserve participating sets, so their composition does as well. A link’s gate label places both endpoints in its replacement gate’s participating set; the correspondence identifies this set with the original one.
Suppose every original gate involves at most \(b\) parties and each party participates in at most \(b\) original nonprivate occurrences over the whole construction. For the retained gate collection and participating roots above, each party is incident to at most \(2b(b-1)\) actual links, counting parallel links separately.
The gate correspondence preserves arity and lifetime participation. Discarding some gate occurrences can only reduce the latter count. Each participating gate contributes at most \(b-1\) star links and \(b-1\) sample links at a party. Sum over its at most \(b\) original occurrences.
Fix a set of gate occurrences of an actual source circuit, and the root assignment defining its star links. A star link at occurrence \(g\) carries the ordered ket–bra pair of its actual branch labels; each sample link carries one of \(k\) sample indices. Suppose \(L\geq 1\), \(C_M,C_k\geq 0\), and \(u,v\) are nonnegative integers. If each of these gate occurrences has at most \(C_ML^u\) branches and \(k\leq C_kL^v\), then every constructed link alphabet satisfies
No dimension of a private memory or source Schmidt space occurs in this bound. The statement concerns the actual alphabets; the identification of the sampled physical operator with a contraction of local tensors is a separate assertion.
At a star link the cardinality is the square of the actual branch count, hence at most \(C_M^2L^{2u}\). At a sample link it is exactly \(k\leq C_kL^v\). Since \(L\geq 1\), increase either exponent to \(\max (2u,v)\) and bound its nonnegative coefficient by \(C_M^2+C_k\). These are the star and sample alphabets in [ 248 , Proof of Theorem 5.2 ] .
27.163 Sampling an original construction
We now combine the local approximation, pair-effect replacement and source sampling estimates for the original gate maps. These are the approximation steps in [ 248 , proof of Theorem 5.2 ] . The sampled physical operator need not be positive. Its identification with a contraction of local tensors is a separate assertion.
A distributed construction has a finite nonempty set \(P\) of parties. For each \(p\in P\), it specifies a finite-dimensional initial private memory \(H_p\) and a unit vector \(x_p\in H_p\). It also specifies a physical dimension \(d_p\), an ordering of the parties without repetition, a finite ordered collection \(R\) of finite-dimensional private output registers, and an actual contraction circuit
The circuit retains the original gate expansions and participant labels. There is no numerical bound on private dimensions in this definition. Resource estimates and approximation conclusions are separate statements.
The nonempty-party convention permits scalar factors to be absorbed in an existing local tensor, as in the proof of Theorem 5.2. It does not exclude gates with no participating parties. The zero-party scalar boundary case is recorded in [ 251 ] .
The input of a distributed construction is the normalized product \(\psi =\bigotimes _{p\in P}x_p\). Its physical density is
expressed in the physical coordinates indexed by the original parties. The private trace uses a finite orthonormal basis of the actual retained registers. No normalization by \(\| W\psi \| \) is performed.
The actual physical density of a distributed construction is positive semidefinite and satisfies \(\operatorname {Tr}\rho \leq 1\).
The outer product of the original output vector is positive semidefinite, and partial trace preserves positivity. Partial trace also preserves the full trace, so \(\operatorname {Tr}\rho =\| W\psi \| ^2\). Each gate is a contraction and \(\| \psi \| =1\), giving the asserted bound.
Let \(L\ge 2\), \(C\ge 1\), \(M\le CL^n\), where \(M,n\) are nonnegative integers, and let \(p\) be real. Set \(q=n+\lceil p\rceil _++\lceil 16C\rceil _+\). Then
The exponent is fixed by the original count bound and the target accuracy, and works simultaneously at every size \(L\ge 2\).
The bound \(\max \{ 1,M\} \le CL^n\) follows from \(C,L\ge 1\). Apply Theorem 27.158.11 with this enlarged count, and divide by its positive denominator.
Increasing the permitted local approximation error preserves the original operators, approximate monomial lists and all contraction and allowedness conditions. Consequently, under the preceding count assumptions, gate expansions available at accuracy \(L^{-q}\) satisfy the local budget \(L^{-p}/(16\max \{ 1,M\} )\) required to reserve one quarter of the final physical error for this approximation step.
Only the inequality bounding each local operator difference changes. Apply Theorem 27.163.4 and transitivity at every original occurrence. All the other local hypotheses are unchanged.
Let \(L\geq 1\). Suppose every nonprivate gate of the original circuit has at most \(K\leq C_KL^u\) monomials, each with at most \(r\) pair effects and coefficient of absolute value at most \(C_CL^v\), where \(C_K,C_C\geq 0\) and \(u,v\) are nonnegative integers. Then there is a bound \(B\geq 1\) for the absolute coefficient sum of every gate, with the same effect bound \(r\), such that
Thus the lower bound \(B\geq 1\) required in the source-sampling estimate follows from the original polynomial assumptions.
Choose \(B=(1+C_KC_C)L^{u+v}\). This is at least one since \(L\geq 1\). Each coefficient sum is at most \(K C_CL^v\leq C_KC_CL^{u+v}\leq B\).
Every finite original circuit has an integer \(K\) bounding the lengths of all its gate expansions. Any larger integer is also such a bound. Suppose its monomials contain at most \(r\) pair effects and the absolute coefficient sum at every gate is at most \(B\). For any positive replacement accuracy, every gate of the actual source replacement, with the physical output registers placed first, has absolute branch coefficient sum at most \(B\).
Take a common bound on the finitely many original gate lists. The coefficient estimate of Theorem 27.159.2 then gives the bound. The final exchange of physical and auxiliary registers introduces no gate and changes no coefficient.
Let \(W\) be a circuit on a finite party set \(P\), with initial vector \(\psi =\bigotimes _{p\in P}\psi _p\), where each \(\psi _p\) is normalized in the whole initial private memory of party \(p\). The final physical register at \(p\) has dimension \(d_p\), and every discarded private register is finite-dimensional. Suppose that the original nonprivate gate count is at most \(M\), each party participates in at most \(\ell \) nonprivate occurrences, and the original expansion bounds hold with effect budget \(r\) and coefficient-sum bound \(B\ge 1\). Let \(D\ge 1\) satisfy \(d_p\le D\) for every \(p\).
Fix finite orthonormal coordinates on the whole initial memory. Fix \(0{\lt}\varepsilon \le 1\). Set \(\delta =\varepsilon /(8\max \{ 1,M\} )\), and form the physical-first replacement \(R\) using the corresponding prepared stack length. Let \(\mathcal E_R\) be its actual source occurrences and \(N_R=|\mathcal E_R|\). Choose
There exist Schmidt data for every actual source and branch, and one Gaussian sample array \(\omega \), such that the sampled physical operator \(\widehat\rho _R(\omega )\) satisfies
Here \(\rho _W\) is the physical partial trace of \(|W\psi \rangle \langle W\psi |\), in the original party-labelled coordinates. No numerical bound on private-memory dimensions is assumed. These are the density-approximation steps in [ 248 , proof of Theorem 5.2 ] .
Prepare each \(\psi _p\) by a local contraction from the scalar space. Their chronological composition prepares \(\psi \) and introduces no pair source. The replacement is a contraction, preserves the original participating occurrences, and retains coefficient-sum bound \(B\). A finite bound on its monomial counts follows from the finite original gate lists; it is not an additional hypothesis. Every auxiliary register is finite-dimensional by its Euclidean construction, so the individual private-register assumption suffices for all discarded registers of \(R\).
Apply Theorem 27.157.9 to this actual circuit, preparation, and discarded memory. It gives original-source Schmidt data and one sample \(\omega \) with \(\| \widehat\rho _R(\omega )-\rho _R\| _1\le \varepsilon /4\). Theorem 27.159.5 gives \(\| \rho _R-\rho _W\| _1\le \varepsilon /2\). The trace-norm triangle inequality yields (1196).
Let a finite distributed construction begin in a product of normalized party vectors. Suppose its original gates are contractions, its number of nonprivate gates is at most \(M\), and each party participates in at most \(\ell \) such gates. All private spaces are finite-dimensional, without numerical dimension bounds. Let the physical dimension at each party be at most \(D\ge 1\).
Fix \(0{\lt}\varepsilon \le 1\). Suppose the actual monomial list for each original nonprivate gate approximates that gate in operator norm within \(\delta _a=\varepsilon /(16\max (1,M))\). Its monomials are allowed, each has at most \(r\) pair effects, and its absolute coefficient sum is at most \(B\ge 1\). Rescale each such list by \((1+\delta _a)^{-1}\) to obtain a contraction circuit \(W\). Set
Apply the constructed pair-effect replacement of length \(m\) to \(W\), and let \(N\) be the number of sources of the resulting circuit. With
there are normalized Schmidt data for these actual sources and a Gaussian sample of size \(k\) whose sampled physical operator \(\widehat\rho \) satisfies \(\| \widehat\rho -\rho _G\| _1\le \varepsilon \). Here \(\rho _G\) is the physical density computed from the original gate operators, with all private registers traced out.
The input coordinates are obtained from finite-dimensionality of the original party memories. Rescaling preserves the original gate occurrences and participating sets, and does not increase the coefficient sums or pair-effect counts. The actual sampling theorem applied to \(W\) therefore gives \(\| \widehat\rho -\rho _W\| _1\le 3\varepsilon /4\) with the stated replacement and sample counts. The local approximation estimate gives \(\| \rho _W-\rho _G\| _1\le 4M\delta _a\le \varepsilon /4\). Thus
Let an original circuit have at most \(N\) nonprivate gate occurrences, each with at most \(b\) participants. Suppose each gate has at most \(K\) monomials, at most \(r\) pair effects in each monomial, and absolute coefficient sum at most \(B\). Let \(L\geq 1\), and assume
where \(C_K,C_B\geq 0\), \(C_N\geq 1\), and \(u,n,s,d,p\) are nonnegative integers. Use the per-gate budget \(\delta =L^{-p}/(8\max (1,N))\), and let \(R\) be the resulting source replacement, with the physical output registers ordered first. For any nonnegative integer \(\tau \), choose the actual sample count
Define constants and exponents by
For every retained set of gate occurrences, every root assignment, and every corresponding star or sample link \(e\), its actual alphabet \(\mathcal A_e\) satisfies
These constants depend only on the displayed uniform resource bounds. No private-memory dimension or source Schmidt rank occurs. This is a dimension estimate for the constructed alphabets; the error estimate and equality with a contraction of local tensors are separate assertions.
Theorem 27.162.4 gives at most \(C_{\mathrm{br}}L^{a_{\mathrm{br}}}\) actual branches per gate. There are at most \(N\binom b2\) actual source occurrences, so the estimate of Theorem 27.162.3 gives \(k{\lt}C_{\mathrm{sam}}L^{a_{\mathrm{sam}}}\). A star alphabet has cardinality equal to the square of its gate’s branch count; a sample alphabet has cardinality \(k\). Apply the bound of Theorem 27.162.8 to these two estimates. The use of \(C_{\mathrm{src}}\geq 1\) includes circuits without sources.
27.164 The exterior source memory when all parties are selected
Let \(P\) be the set of original parties and let \(W\) be a finite chronological composition. Each source occurrence \(e\) has distinct endpoints \(p_e,q_e\in P\) and prescribed complex halfspaces \(\mathbb {C}^{u_e}\) and \(\mathbb {C}^{v_e}\), whose dimensions may be zero. When every party is selected, write \(Q=P\sqcup \{ *\} \) for the owner set. An original endpoint \(p\) retains its own owner, and \(*\) denotes the exterior owner.
Let \(\mathcal E_W\) be the source occurrences in their original preparation order. Define \(R_W\) to be the concatenation, in that order, of the two-register lists \([(p_e,\mathbb {C}^{u_e}),(q_e,\mathbb {C}^{v_e})]\) for \(e\in \mathcal E_W\). Its owners and spaces depend only on the original composition, before a monomial or a prepared source vector is chosen.
The sublist of \(R_W\) consisting of registers at owner \(*\) is empty. This holds for every chronological composition, including when \(P\) or the source inventory is empty.
A register in \(R_W\) belongs to one of the two halves of an actual source occurrence. The source-slot specification identifies its owner with the affected owner of the corresponding original endpoint. Under the all-party selection this is \(p\in P\), which is distinct from \(*\). No register is therefore retained by the exterior selection.
This identity concerns the fixed source registers in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267 and 342–434. The exterior memory of this empty register list is the scalar space \(\mathbb {C}\). The identity does not remove the scalar contributed by source-free operations on an empty memory.
For each source occurrence \(e\), take the standard bases of \(\mathbb {C}^{u_e}\) and \(\mathbb {C}^{v_e}\). Their ordered tensor product is an orthonormal basis \((b_x)_{x\in I_W}\) of \(\mathcal H_{R_W}\), where \(I_W=\prod _{e\in \mathcal E_W} (\{ 0,\ldots ,u_e-1\} \times \{ 0,\ldots ,v_e-1\} )\). Transport this basis through the canonical identification \(\mathcal H_{R_W}\simeq \mathcal H_{R_W\, \varnothing }\), obtaining \((b_x^+)_{x\in I_W}\) on the input of the actual private preparation. These bases depend on the original source positions and spaces, before a monomial choice or sampled source vector is specified. This is the fixed-coordinate convention of [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 253–267 and 279–355.
Let \(T_\xi \) be the actual source-to-output composition with recorded private input \(\psi \), and choose any finite orthonormal output basis \((\beta _j)_{j\in J}\). Its literal column matrix is
where \(J_{R_W,\varnothing }\) is the canonical append isometry. The ket and bra compositions may have independent monomial choices.
Every \(x\in I_W\) satisfies \(b_x^+=J_{R_W,\varnothing }^{-1}(b_x\otimes 1)\). Consequently \(M_\xi \) is precisely the matrix of \(T_\xi \) in the input basis \((b_x^+)\) and output basis \((\beta _j)\). Empty source inventories give the singleton scalar basis; a source halfspace of dimension zero gives an empty column-index set.
Appending an empty register list identifies its scalar unit with the original source vector. This gives the stated expression for \(b_x^+\). Substituting it into the defining matrix entries yields (1206).
At gate occurrence \(g\), let \(C_g\) be its participating party set, \(\Xi _g\) its monomial labels, and \(c_{g,\alpha }\) its recorded coefficients. Define
The empty product is \(1\), and an empty label set contributes \(0\). Concatenating compositions multiplies their scalar contributions; extending by spectator registers preserves them. Zero-party gates are included in the gate convention of [ 248 , Theorem 5.2 ] , 04-compression.tex, lines 24–28: a private gate has exactly one participating party.
A prepared gate with no participating parties acts on \(\mathbb {C}\) as \(z\mapsto (\sum _{\alpha \in \Xi _g}c_{g,\alpha })z\). Its contraction bound gives \(|\sum _{\alpha \in \Xi _g}c_{g,\alpha }|\le 1\). Consequently every distributed composition satisfies \(|\lambda _W|\le 1\), including compositions with vanishing scalar contributions.
A register list on an empty party set is empty, and a word with no participating parties acts as the identity on \(\mathbb {C}\) by the no-party clause of Theorem 27.140.5. Each monomial of the gate therefore has operator \(1\), so its weighted sum has operator \(\sum _{\alpha \in \Xi _g}c_{g,\alpha }\). Taking operator norms gives the coefficient-sum bound. Induction on the composition, using \(|\lambda _1\lambda _2|\le |\lambda _1||\lambda _2|\), proves the final bound.
Select all original parties. The restriction of the actual private preparation to the exterior owner is the identity on the empty memory \(\mathbb C\). This identity holds before any choice of gate monomials. It is the private-preparation part of the local contraction in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 137–139, 246–267 and 409–434.
Write the private preparation as \(F_\tau (V^f)\), where \(V\) prepares the recorded private input, \(V^f\) relabels each owner \(p\) by the selected original owner \(f(p)=p\in Q\), and \(F_\tau \) adjoins the source register list \(\tau \) unchanged. By Theorem 27.164.10, restriction to \(\{ *\} \) selects exactly the registers owned by \(*\). Every source register belongs to an original party, so \(\tau |_{\{ *\} }\) is empty by Theorem 27.164.2, and removing these spectators gives the first equality below. Since \(*\notin f(P)\), Theorem 27.147.3 gives the second:
Select all original parties, and consider an actual gate \(G\) with no participating parties and branch coefficients \((c_i)_{i\in I}\). For every monomial choice, the exterior restriction of the remaining operations of its partial expansion is
An empty branch set gives the zero map. No coefficient sum is cancelled. This is the empty-party case of the chronological contraction in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 24–28, 233–251 and 351–417.
The input and output register lists of \(G\) are empty. No participating party is selected, so the remaining operations of its partial expansion are the local map \(\Lambda _*(\widetilde G;\tau ^f)\) at the exterior owner, beside the relabelled spectator list \(\tau ^f\), where \(\widetilde G=UGU^{-1}\) for the canonical identifications \(U\) of the empty memories. By Theorem 27.164.7, \(G=(\sum _{i\in I}c_i)\operatorname {id}_{\mathbb C}\). Every original spectator belongs to a selected original party, so \(\tau ^f|_{\{ *\} }\) is empty. Hence
which is (1209).
In the owner set \(Q=P\sqcup \{ *\} \), the singleton selector at \(*\) is the indicator of the exterior owner.
The two selectors are \(1\) at \(*\) and \(0\) at every original party.
Select all original parties. If a gate has a participating party, then, for each choice of its monomial, the exterior restriction of its remaining operations is \(\mathbb {1}_{\mathbb {C}}\). The monomial vectors and spectator registers are arbitrary.
The gate is replaced by the chosen monomial. Relabelling its operations and spectator registers assigns every owner to a selected original party. The exterior owner is outside that image. Moving its pair preparations to the beginning preserves the participating parties, so the exterior restriction has no participating party and acts as \(\mathbb {1}_{\mathbb {C}}\).
Select all original parties of a composition \(W\). For every monomial choice, the exterior restriction of its remaining operations acts as \(\lambda _W\mathbb {1}_{\mathbb {C}}\). The same identity holds after identifying the source registers with their prescribed slots and after preparing the recorded private product input. It includes an empty original party set and a vanishing coefficient sum.
Induct on the composition, writing \(\lambda _u\) for the exterior scalar of a subword \(u\). A local operation or exchange at an original party, a gate with participating parties \(C\), and a chronological composition contribute
by Theorems 27.164.11, 27.164.9 and 27.140.3.5, and adjoining an excluded register preserves \(\lambda _u\). All exterior input, intermediate and source memories are empty. The prescribed slot identification adds no operation, and the private preparation contributes the identity, so \(\lambda _W\) is the product of the factors in (1213). No scalar is cancelled.
Let \(P\) be finite and set \(Q=P\sqcup \{ *\} \), where \(*\) denotes the exterior owner. Choose orthonormal bases \(B_q,C_q\) for the actual input and output register memories at each \(q\in Q\), and order all parties without repetition. Let \(B,C\) be the corresponding grouped bases. Write \(E\) for the literal source input basis. Define
where \(\widetilde W_\xi \) is the actual remaining word after preparing the recorded private input. Thus \(R\) is the matrix of the identity from \(E\) to \(B\) and is independent of the monomial choice.
The literal prepared source-column matrix, with output basis \(C\), satisfies
Every owner, including \(*\), remains in this product. The same bases and the same \(R\) may be used for independent ket and bra choices.
First express the actual word in the grouped input and output bases. By Theorem 27.140.2.27 its matrix is the product of its singleton-restriction matrices,
Composing with the identity from the literal source basis \(E\) to the grouped input basis \(B\), whose matrix is \(R\), gives
Let \(U\) be any source-free word from the actual source input memory to itself, and let \([U]_E\) be its matrix in the literal source input basis. Set \(U_q=[U|_q]_{B_q,B_q}\). For independent ket and bra choices,
Every owner, including the exterior owner, remains in the product. The identity applies to each word in a finite sampled sum separately.
Express the prepared words and \(U\) in orthonormal coordinates. The matrix of their mixed composition is independent of the chosen input basis: both expressions represent the same operator on the output memory. In the grouped input basis, the actual singleton restrictions give (1214).
At an original source occurrence \(e\), retain its ordered ket and bra branch labels \(\alpha ,\beta \), its Gaussian array, and its sample index \(s\). Denote the resulting left and right endpoint matrices by \(L_{e,\alpha \beta ,s}\) and \(H_{e,\alpha \beta ,s}\). The right factor uses the conjugate of the same Gaussian array. The constructed source matrix is
This algebraic formula also defines the zero-sample case using \(0^{-1}=0\) and an empty sum.
Fix independent global monomial choices \(\xi ,\zeta \) and one sample assignment \(j\) on all original source occurrences. At each source, form the endpoint word with its actual branch pair and the two endpoint matrices carrying \(j(e)\). Transport this word to the actual input of the recorded private preparation, including the terminal scalar memory, and denote it by \(U_{\xi ,\zeta ,j}\). The original occurrence, branch pair, Gaussian array and sample assignment are retained throughout the construction.
The sampled endpoint word and its transport \(U_{\xi ,\zeta ,j}\) to the actual source input contain no source preparation.
An endpoint word is a composition of endpoint operators and exchanges, none of which prepares a source, and the canonical transport to the actual input adds no operation.
Let \(n\) be the number of original source slots and let \(X_{\xi ,\zeta }\) be the product of their actual sampled source matrices in the literal source input basis \(E\). Let \(\sigma \) be the fixed equivalence from slot positions to original source occurrences. Then
The canonical identification with the terminal scalar memory preserves every matrix entry. The same assignment is used at both endpoints of each source. The identity includes \(k=0\), \(n=0\), empty coordinate sets and arbitrary sampled matrices.
Apply the finite endpoint-averaging identity to the original ordered source list. The equivalence between its positions and the original source occurrences transports one complete assignment \(j\). Conjugating the word by the canonical input identification leaves its coordinates unchanged, giving (1216) entry by entry.
In the same notation, let \(U_{\xi ,\zeta ,j,q}=[U_{\xi ,\zeta ,j\circ \sigma ^{-1}}|_q]_{B_q,B_q}\). The literal sampled source contraction satisfies
The sum is over complete assignments on the original ordered slots. Every owner, including the exterior owner, remains in the product. No output register is traced out in this identity. Empty sample sets, empty source lists, zero-dimensional memories and an empty set of original parties are included.
Insert the actual endpoint-word average between the prepared column matrices. Distribute multiplication over the finite sum and apply the individual-word mixed identity to each complete assignment.
Select all original parties and let \(*\) be the exterior owner. For any independent ket and bra choices \(\xi ,\zeta \), original sample assignment \(j\) and sample values \(\omega \), the actual sampled endpoint word \(U_{\xi ,\zeta ,j}\) restricts to \(\operatorname {id}_{\mathbb C}\) at \(*\). This identity holds both on the fixed source layout and after its canonical identification with the private preparation input. The endpoint matrices are arbitrary, and the source inventory may be empty. This is the exterior source factor in [ 248 , proof of Theorem 5.2 ] .
By Theorem 27.150.2, both owners of every source are original selected parties. Restriction to \(*\) excludes them both. Theorem 27.143.2 therefore gives \(U_{\xi ,\zeta ,j}|_*=\operatorname {id}_{\mathbb C}\). Identifying the source layout with the preparation input transports the restriction by the same canonical identification on its input and output; hence this scalar identity is preserved.
Let \(\psi \) be the recorded product input, let \(c_\xi \) be the original coefficient of the choice \(\xi \), and let \(D_\xi \) be its actual prepared source-column matrix in the specified finite orthonormal output basis. Let \(\widehat\rho _\psi (Y)\) be the full source-replaced output operator before tracing the private registers. For arbitrary replacement matrices \(Y_{e,\alpha ,\beta }\), put \(Y_{\xi ,\zeta }=\bigotimes _e Y_{e,\xi _e,\zeta _e}\). Then
The ket and bra choices are independent, and every source occurrence is the original occurrence of the circuit. Zero coefficients, empty source inventories and arbitrary complex replacement matrices are included. This is the prepared-input form of source replacement in [ 248 , proof of Theorem 5.2 ] .
Expand the definition of source replacement as a sum over independent choices \(\xi ,\zeta \). Transporting the input basis and the input vector through the same owner isometry preserves their coordinates. Apply Theorem 27.145.21 to each term, retaining its original scalar \(c_\xi \overline{c_\zeta }\).
Choose finite orthonormal bases of the original physical and private output memories, and use their tensor product through the canonical output identification. Write \(G\) for the original private output memory. With the notation of Theorem 27.164.22, the physical source-replaced operator \(\widehat\rho _{\mathrm{phys},\psi }(Y)\) on the recorded product input satisfies
The choices remain independent and the replacement matrices are arbitrary complex matrices. No positivity or normalization is required. This is the physical partial trace of the prepared-column expansion in [ 248 , proof of Theorem 5.2 ] .
Apply the partial trace over the original private output memory to (1218). The partial trace is complex-linear and therefore distributes over both finite sums and the coefficients \(c_\xi \overline{c_\zeta }\).
Let \(X\) be any matrix on the original joint source-coordinate space, and let \(\xi ,\zeta \) be independent chronological choices. Let \(D_\xi ,D_\zeta \) be their actual prepared-column matrices in the constructed local physical/private bases. Let \(D_\xi ^{\mathrm{ref}}\) and \(D_\zeta ^{\mathrm{ref}}\) instead use the original physical basis and any finite orthonormal reference basis of the original private output memory \(G\). Write \(\chi \) for the canonical separation of retained and discarded coordinates, followed by identification with the original physical configurations. Then
The matrix \(X\) may be correlated across source slots, and need not be positive or Hermitian. Empty source inventories, zero matrices and zero-dimensional physical memories are included. This identifies the local-coordinate trace with the actual physical partial trace in [ 248 , proof of Theorem 5.2 ] .
Expand \(X\) in matrix units. Writing \(e_s\) for the basis vectors of the original joint source-coordinate space,
and likewise for \(D_\xi ^{\mathrm{ref}}X(D_\zeta ^{\mathrm{ref}})^\dagger \). For each \((s,t)\) the summand is the rank-one operator formed from the actual \(s\)-th ket column and \(t\)-th bra column, and Theorem 27.143.12 applied to these actual output vectors gives
Complex linearity of the coordinate change \(\chi \) and of both partial traces gives the result for arbitrary \(X\).
In the notation of Theorem 27.164.20, choose each party output basis with index set \(H_q\times D_q\). Identify \(\prod _q(H_q\times D_q)\) with \((\prod _qH_q)\times (\prod _qD_q)\) by the canonical coordinate equivalence \(\chi \). Then
The same original global sample assignment \(j\) is retained in each summand. Every owner, including the exterior owner, remains in the product. This is an identity in the specified retained and discarded coordinates; identifying these bases with physical and private register bases is a separate assertion. Zero matrices and empty coordinate sets are included.
Reindexing and partial trace are complex-linear. Apply their composition to Theorem 27.164.20 and distribute it over the scalar multiple and finite sum. In each summand, Theorem 27.145.10 gives \(\operatorname {tr}_{\prod _qD_q}((\bigotimes _q M_q)^\chi ) =\bigotimes _q\operatorname {tr}_{D_q}(M_q)\). Substitute \(M_q=A_{\xi ,q}U_{\xi ,\zeta ,j,q}A_{\zeta ,q}^\dagger \) to obtain (1219).
Let \(S\) be a selection of parties, let \(u,t:a\to b\) and \(v:a\to a\) be chronological words containing no pair-source preparations, and choose finite orthonormal bases \(B\) and \(C\) of their selected input and output memories. Write the output basis index as \(H\times E\), with \(H\) the retained coordinate and \(E\) the discarded coordinate. Let \(A_u,A_t\) be the matrices of the restrictions of \(u,t\) in these bases, and let \(U_v\) be the matrix of the restriction of \(v\) in the input basis. Define the mixed physical matrix by
No contraction, positivity or normalization is required. Empty coordinate sets are permitted. This is the coordinate contraction in [ 248 , proof of Theorem 5.2 ] .
Fix independent chronological choices \(\xi ,\zeta \) and an original joint sample assignment \(j\). At a selected owner \(q\), let \(A_{\xi ,q}\) and \(A_{\zeta ,q}\) be the actual source-to-output matrices in the prescribed source bases and the constructed physical/private output basis. Let \(U_{\xi ,\zeta ,j,q}\) be the actual restricted sampled source-input matrix in that same source basis. Define
At an original party \(p\), this matrix has the original physical indices \(0,\ldots ,d_p-1\); the exterior owner has a singleton physical index. The private basis is arbitrary and finite. The same \(j\) is used at both endpoints of every original source. This definition includes zero matrices, empty sample sets and zero physical dimensions.
Use the actual local physical/private output bases, the prescribed selected-party order, and arbitrary compatible input bases. Let \(\chi \) separate the retained and discarded coordinate families and let \(\eta \) identify the retained configurations with the original configurations \(\prod _{p\in \mathcal P}\{ 0,\ldots ,d_p-1\} \). For one original global sample assignment \(j\), put \(M_{j,q}=\operatorname {tr}_{E_q} (A_{\xi ,q}U_{\xi ,\zeta ,j,q}A_{\zeta ,q}^\dagger )\) and \(\alpha _j=M_{j,*}(1,1)\), where \(1\) denotes the exterior singleton coordinate. Then
Each \(j\) is the same complete assignment on the original source slots in every factor of its summand, and \(\xi ,\zeta \) remain independent. The exterior scalar stays inside the sum and may vanish. Empty source inventories, empty original party sets and zero-dimensional memories are included. This is the local-coordinate contraction in [ 248 , proof of Theorem 5.2 ] .
Apply the retained configuration change to (1219). This change is complex-linear and therefore commutes with the scalar multiple and finite sum. Theorem 27.140.1.17 writes the reindexed product in each summand as its exterior singleton entry \(\alpha _j\) times the product over the original parties. No scalar is cancelled, giving (1222).
Let \(\lambda _W\) be the original empty-party gate scalar of \(W\). Choose any finite orthonormal basis of the exterior input memory and any finite orthonormal basis of the exterior output memory indexed by \(\{ 1\} \times J\). For independent ket and bra choices \(\xi ,\zeta \) and any original joint sample assignment \(j\), the actual local trace satisfies
In particular its sole entry is \(\lambda _W\overline{\lambda _W}\), independently of \(j,\xi ,\zeta \) and the chosen orthonormal coordinates. The scalar may vanish, and no physical register is introduced. This is the exterior contribution in [ 248 , proof of Theorem 5.2 ] .
The exterior input and output memories are both \(\mathbb {C}\). Their actual chronological restrictions are \(\lambda _W I\), and the sampled input restriction is \(I\). If \(V\) is the actual orthonormal coordinate change, then \(VV^\dagger =I\), so the mixed matrix is \(\lambda _W\overline{\lambda _W}\, I_{\{ 1\} \times J}\). The output basis is a basis of \(\mathbb {C}\), and hence \(|J|=1\). Tracing \(J\) gives (1223). No scalar is divided out.
In the notation of Theorem 27.164.28, let \(\lambda _W\) be the original empty-party gate scalar. Then
Each summand retains the same complete original sample assignment at every endpoint, and the ket and bra choices remain independent. The scalar may vanish; empty original party sets and source inventories are included. This is the common exterior contribution in [ 248 , proof of Theorem 5.2 ] .