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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.
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 27.130.8 hold for the network contraction.
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.
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.
In the setting of Theorem 27.139.11, 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 \(\delta \).
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
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 \(1-m\varepsilon \).
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 \(\varepsilon _s=1\).
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 \(\mathbb Z^2\).
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.
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 \(F\subset \mathbb Z^2\) 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.
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.
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.
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 ] .
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.
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.
![\begin{tenkz}[size=s, metrics=compact, rows={wire,wire,wire,wire,wire,wire,wire,wire,wire}, cols=9, bonds=none]
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\tn[at=(1,5), name=fann, ports={0:virtual, 180:virtual, 270:virtual}]{}
\tn[at=(1,9), name=fanne, ports={180:virtual, 225:virtual, 270:virtual}]{}
\tn[at=(5,9), name=fane, ports={90:virtual, 180:virtual, 270:virtual}]{}
\tn[at=(9,9), name=fanse, ports={90:virtual, 135:virtual, 180:virtual}]{}
\tn[at=(9,5), name=fans, ports={0:virtual, 180:virtual}]{}
\tn[at=(9,1), name=fansw, ports={0:virtual, 45:virtual, 90:virtual}]{}
\tn[at=(5,1), name=fanw, ports={90:virtual, 270:virtual}]{}
\tnwire[name=fansidenw]{fannw.0}{fann.180}
\tnwire[name=fansiden]{fann.0}{fanne.180}
\tnwire[name=fansidene]{fanne.270}{fane.90}
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\tnwire[name=fansidese]{fanse.180}{fans.0}
\tnwire[name=fansides]{fans.180}{fansw.0}
\tnwire[name=fansidesw]{fansw.90}{fanw.270}
\tnwire[name=fansidew]{fanw.90}{fannw.270}
\tnwire[name=fanraynw]{fanc.135}{fannw.315}
\tnwire[name=fanrayn]{fanc.90}{fann.270}
\tnwire[name=fanrayne]{fanc.45}{fanne.225}
\tnwire[name=fanraye]{fanc.0}{fane.180}
\tnwire[name=fanrayse]{fanc.315}{fanse.135}
\tnwire[name=fanraysw]{fanc.225}{fansw.45}
\tnmark[form=label, label pos=s]{(5,5)}{$c$}
\tnmark[form=label, label pos=n]{(1,1)}{$a_e$}
\tnmark[form=label, label pos=n]{(1,5)}{$b_e$}
\tnmark[form=label]{(3,4)}{$P_e$}
\tnmark[form=label]{(8,5)}{$P_{e'}$}
\tnmark[form=label, label pos=se]{(9,9)}{$Q$}
\end{tenkz}](tenkz_svg/tenkz-b4108728f3932ff1.svg)
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 \(u\ne 0\).
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 \(4\).
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 ] .
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 \(\Lambda \subset \mathbb Z^2\) 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.
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 \(H_{\ell ,z,s,1}=[(a+b)/2,b]\).
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 ] .
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]
\tndeclare{species}{endpoint}{hue=source:red}
\tn[at=(5,5), skin=dot, species=endpoint]{}
\tnmark[form=enclosure, species=secondary, tint]{(2,2) .. (7,7) - (4,4) .. (6,6)}{}
\tnmark[form=enclosure, species=selected, tint]{(4,4) .. (6,6)}{}
\tnmark[form=enclosure, species=complement]{(2,2) .. (3,3)}{}
\tnmark[form=enclosure, species=complement]{(2,4) .. (3,5)}{}
\tnmark[form=enclosure, species=complement]{(2,6) .. (3,7)}{}
\tnmark[form=enclosure, species=complement]{(4,2) .. (5,3)}{}
\tnmark[form=enclosure, species=complement]{(4,4) .. (5,5)}{}
\tnmark[form=enclosure, species=complement]{(4,6) .. (5,7)}{}
\tnmark[form=enclosure, species=complement]{(6,2) .. (7,3)}{}
\tnmark[form=enclosure, species=complement]{(6,4) .. (7,5)}{}
\tnmark[form=enclosure, species=complement]{(6,6) .. (7,7)}{}
\tnmark[form=label, label pos=e]{midway (2,2) and (3,2)}{$D_k$}
\tnmark[form=label, label pos=e]{midway (4,4) and (5,4)}{$N_k$}
\tnmark[form=bracket, label pos=e]{(2,2) .. (7,7)}{$N_{k+1}$}
\end{tenkz}](tenkz_svg/tenkz-ee87101e9648f362.svg)
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
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 \(K_i\).
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 ] .
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.
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 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.
- Matrix.IsGappedGroundState
- Matrix.isGappedGroundState_iff
- TNLean.PEPS.AreaLaw.IsGappedGroundState
- TNLean.PEPS.AreaLaw.reducedState
- TNLean.PEPS.AreaLaw.reducedState_isHermitian
- TNLean.PEPS.AreaLaw.regionalEntropy
- TNLean.PEPS.AreaLaw.reducedState_eq_regionState
- TNLean.PEPS.AreaLaw.regionalEntropy_eq_regionEntropy
- TNLean.PEPS.AreaLaw.UniformAreaLaw
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.
- TNLean.PEPS.AreaLaw.IsAdmissibleSupport
- TNLean.PEPS.AreaLaw.AdmissibleSupport
- TNLean.PEPS.AreaLaw.LocalHamiltonian
- TNLean.PEPS.AreaLaw.LocalHamiltonian.operator
- TNLean.PEPS.AreaLaw.LocalHamiltonian.operator_isHermitian
- TNLean.PEPS.AreaLaw.LocalHamiltonian.zero
- TNLean.PEPS.AreaLaw.localLift
- TNLean.PEPS.AreaLaw.localLift_mem_supportedOperators
- TNLean.PEPS.AreaLaw.localLift_eq_entropyLocalLift
- TNLean.PEPS.AreaLaw.edgeBoundary
- TNLean.PEPS.AreaLaw.edgeBoundary_empty
- TNLean.PEPS.AreaLaw.edgeBoundary_univ
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
Assign \(\lambda (i)\in \mathbb Z/2\mathbb Z\) by
Run-color constancy makes the last value well defined.
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.
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 \(C(A)=\{ i\in I:S_i\cap A\ne \varnothing ,\ S_i\setminus A\ne \varnothing \} \).
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.
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
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)}{$U_j$}
\tnmark[form=enclosure, species=selected, label pos=n]{(1,6) .. (3,8)}{$U_{j'}$}
\tnmark[form=label, label pos=e]{midway (1,3) and (2,3)}{$R_j$}
\tnmark[form=label, label pos=s]{midway (3,3) and (3,4)}{$R_j$}
\tnmark[form=label, label pos=e]{midway (1,6) and (2,6)}{$R_{j'}$}
\end{tenkz}](tenkz_svg/tenkz-9b403b75bf529388.svg)
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.
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 ] .
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.
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
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 \(A\cap Q\).
- TNLean.PEPS.AreaLaw.IntRect
- TNLean.PEPS.AreaLaw.IntRect.size
- TNLean.PEPS.AreaLaw.IntRect.dilate
- TNLean.PEPS.AreaLaw.IsSafe
- TNLean.PEPS.AreaLaw.rectRegion
- TNLean.PEPS.AreaLaw.supDist
- TNLean.PEPS.AreaLaw.IntRect.toFinset
- TNLean.PEPS.AreaLaw.IntRect.mem_toFinset
- TNLean.PEPS.AreaLaw.IntRect.one_le_size
- TNLean.PEPS.AreaLaw.IntRect.width_le_size
- TNLean.PEPS.AreaLaw.IntRect.height_le_size
- TNLean.PEPS.AreaLaw.IntRect.mem_dilate
- TNLean.PEPS.AreaLaw.IntRect.toFinset_dilate
- TNLean.PEPS.AreaLaw.IntRect.dilate_zero
- TNLean.PEPS.AreaLaw.IntRect.dilate_dilate
- TNLean.PEPS.AreaLaw.IntRect.toFinset_dilate_mono
- TNLean.PEPS.AreaLaw.IntRect.mem_dilate_of_supDist_le
- TNLean.PEPS.AreaLaw.mem_rectRegion
- TNLean.PEPS.AreaLaw.rectRegion_dilate_mono
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.
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.
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 ] .
- TNLean.PEPS.AreaLaw.Geometry.IsAllowedSlope
- TNLean.PEPS.AreaLaw.Geometry.integerPoint
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon.region
- TNLean.PEPS.AreaLaw.Geometry.Template
- TNLean.PEPS.AreaLaw.Geometry.Template.IsSeparated
- TNLean.PEPS.AreaLaw.Geometry.Template.core
- TNLean.PEPS.AreaLaw.Geometry.Template.collar
- TNLean.PEPS.AreaLaw.ambientDilation
- TNLean.PEPS.AreaLaw.ambientDilation_zero
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 ] .
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.
For a finite induced domain \(\Lambda \subset \mathbb Z^2\) and \(A\subseteq \Lambda \), define
These are the conventions of [ 247 , Corollary 1.2 ] .
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 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 \(S\).
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 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.
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.
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 \(D_\ell \).
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.
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 \(g\in G\) 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 \(\ell \).
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 \(\kappa _p\).
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 \(I_p\).
- TNLean.PEPS.Approximation.DistributedOperatorContraction.LinkAt
- TNLean.PEPS.Approximation.DistributedOperatorContraction.starLinkAt
- TNLean.PEPS.Approximation.DistributedOperatorContraction.sampleLinkAt
- TNLean.PEPS.Approximation.DistributedOperatorContraction.starGateAt
- TNLean.PEPS.Approximation.DistributedOperatorContraction.rootGateAt
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.
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.
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 \(s\).
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.
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 \(s\in S_p\).
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.
Given \((\kappa ,j)\), define \(\eta (\kappa ,j)_{(g,\mathrm{star},q)}=\kappa _g\) and \(\eta (\kappa ,j)_{(g,\mathrm{sample},s)}=j_s\). Thus each star carries the complete common gate label and each sample edge carries its position’s sample value.
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 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.
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 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 edge of the cycle graph on \(n \ge 3\) vertices joins a site to its cyclic successor, so sites and bonds are in bijection, and every site has exactly two incident bonds: the bond shared with its cyclic predecessor and the bond shared with its cyclic successor. The cycle tensor of a tensor \(A = (A^i)_{i=0}^{d-1}\) of \(D \times D\) matrices places one copy of \(A\) at every site of the closed chain of \(n\) sites, every bond of dimension \(D\), with the row index of \(A^i\) read from the bond shared with the predecessor and the column index from the bond shared with the successor.
The map \(\Gamma _L\) in (1) is a \(\mathbb {C}\)-linear map from \(M_{D}(\mathbb {C})\) to \((\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\). In tensor-network notation,
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring]{X} &
\tn[label pos=135, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=135, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$\sigma_L$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L$}
\end{tenkz}](tenkz_svg/tenkz-850da2de312b2b1e.svg)
The chain denotes the word tensor \(A^\sigma \) with open physical indices \(\sigma _1,\ldots ,\sigma _L\), and the red capsule denotes the boundary matrix \(X\) inserted on the closing virtual bond before taking the trace. The brace records the chain length \(L\).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 \(W_{\ell }\).
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.
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 27.141.2.5.
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.
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 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 \((ab)_f=a_fb_f\).
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.
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 \(d_p=d\).
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 ] .
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 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.
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 \(V_{\mathcal A,\xi }\).
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 \(r\).
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.
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.
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.
Equip \(\mathbb {C}^I\) with \(\langle x,y\rangle =\sum _i\overline{x_i}y_i\). For a subspace \(S\subseteq \mathbb {C}^I\), let \(\Pi _S\) be its orthogonal projector in the coordinate basis. Then \(\Pi _S^2=\Pi _S=\Pi _S^*\), \(\operatorname {ran}\Pi _S=S\), and \(\Pi _Sx=x\) exactly when \(x\in S\).
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 \(C=\bigotimes _vC_v\).
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.
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 \(m=d\).
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 \(|G|^{-1}\).
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.
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.
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.
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 \(Q_\xi \colon \ell ^2(I)\to \mathcal H_C\).
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.
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.
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 \(P_m=|\eta ^{\otimes m}\rangle \langle \eta |\).
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 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, \(A_v^0(\eta ,s)=|G|^{-1}\sum _gW_v(g)_{s,\eta }\).
- TNLean.PEPS.DependentBondNetwork.representationAveragingSite
- TNLean.PEPS.DependentBondNetwork.representationAveragingSite_apply
- TNLean.PEPS.DependentBondNetwork.localSiteMap_representationAveragingSite
- TNLean.PEPS.DependentBondNetwork.averagingSite
- TNLean.PEPS.DependentBondNetwork.localSiteMap_averagingSite
- TNLean.PEPS.DependentBondNetwork.averagingSite_apply
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.
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 \(b_p=b_{(U_e(p_e))_e}\).
- TNLean.PEPS.DependentBondNetwork.siteBondEquiv
- TNLean.PEPS.DependentBondNetwork.bondRegrouping
- TNLean.PEPS.DependentBondNetwork.bondRegrouping_apply
- TNLean.PEPS.DependentBondNetwork.matrixBondProduct
- TNLean.PEPS.DependentBondNetwork.bondRegrouping_matrixBondProduct
- TNLean.PEPS.DependentBondNetwork.representationBondProduct
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 \(B_e(q)=U_e(q_{h(e)})B_eU_e(q_{t(e)}^{-1})\).
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.
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.
- TNLean.PEPS.DependentBondNetwork.CutEndpoint
- TNLean.PEPS.DependentBondNetwork.CutConfig
- TNLean.PEPS.DependentBondNetwork.cutRestriction
- TNLean.PEPS.DependentBondNetwork.cutInteriorWeight
- TNLean.PEPS.DependentBondNetwork.cutCoeff
- TNLean.PEPS.DependentBondNetwork.cutMap
- TNLean.PEPS.DependentBondNetwork.cutMap_apply
- TNLean.PEPS.DependentBondNetwork.cutSpace
- TNLean.PEPS.DependentBondNetwork.mem_cutSpace_iff
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 \(S\beta =(\beta _{e,+},\beta _{e,-})_{e\in \mathcal E}\).
- TNLean.PEPS.DependentBondNetwork.Endpoint
- TNLean.PEPS.DependentBondNetwork.endpointVertex
- TNLean.PEPS.DependentBondNetwork.IncidentEndpoint
- TNLean.PEPS.DependentBondNetwork.EndpointConfig
- TNLean.PEPS.DependentBondNetwork.LocalConfig
- TNLean.PEPS.DependentBondNetwork.endpointSiteEquiv
- TNLean.PEPS.DependentBondNetwork.endpointSiteEquiv_apply
- TNLean.PEPS.DependentBondNetwork.endpointSiteEquiv_symm_apply
- TNLean.PEPS.DependentBondNetwork.endpointPairEquiv
- TNLean.PEPS.DependentBondNetwork.endpointPairEquiv_apply
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.
- TNLean.PEPS.DependentBondNetwork.incidentMatrix
- TNLean.PEPS.DependentBondNetwork.incidentMatrix_one
- TNLean.PEPS.DependentBondNetwork.incidentMatrix_mul
- TNLean.PEPS.DependentBondNetwork.incidentRepresentation
- TNLean.PEPS.DependentBondNetwork.incidentRepresentation_apply
- TNLean.PEPS.DependentBondNetwork.localSiteMap
- TNLean.PEPS.DependentBondNetwork.localSiteMap_apply
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 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
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.
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.
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 \(n{\gt}0\) 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)
- TNLean.PEPS.Approximation.blockIndex
- TNLean.PEPS.Approximation.blockGuide
- TNLean.PEPS.Approximation.blockCorner
- TNLean.PEPS.Approximation.SquareEdge.nbrBlock
- TNLean.PEPS.Approximation.blockBaseline
- TNLean.PEPS.Approximation.blockIndex_of_mem_openSquare
- TNLean.PEPS.Approximation.blockIndex_of_mem_edgeNeighbor
- TNLean.PEPS.Approximation.blockIndex_ne_of_notMem_closedSquare
- TNLean.PEPS.Approximation.shiftGuide
- TNLean.PEPS.Approximation.shiftGuide_blockBaseline_guide
- TNLean.PEPS.Approximation.InRoot
- TNLean.PEPS.Approximation.levelLabels
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.
- TNLean.PEPS.Approximation.IsChamberSampling
- TNLean.PEPS.Approximation.IsChamberSampling.isDisplacedReading
- TNLean.PEPS.Approximation.IsChamberSampling.unique
- TNLean.PEPS.Approximation.normalLine
- TNLean.PEPS.Approximation.IsLocallyInLines
- TNLean.PEPS.Approximation.HasLineWalls
- TNLean.PEPS.Approximation.wallThresholds
- TNLean.PEPS.Approximation.wallNormals
- TNLean.PEPS.Approximation.IsGenericDisplacement
- TNLean.PEPS.Approximation.isGenericDisplacement_one_one
- TNLean.PEPS.Approximation.IsGenericDisplacement.smul
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 \(S\).
- TNLean.PEPS.Approximation.SquareEdge
- TNLean.PEPS.Approximation.bandWidth
- TNLean.PEPS.Approximation.edgeBand
- TNLean.PEPS.Approximation.closedEdgeBand
- TNLean.PEPS.Approximation.bandCoord
- TNLean.PEPS.Approximation.edgeLens
- TNLean.PEPS.Approximation.edgeNeighbor
- TNLean.PEPS.Approximation.edgeMarkDist
- TNLean.PEPS.Approximation.cornerMarkDist
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 \(S\).
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 27.132.1.
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 27.132.4 maps the endpoint \(Q\).
![\begin{tenkz}[rows={ket,ket,ket,ket,ket,ket,ket,ket}, cols=8,
bonds=none]
\tndeclare{species}{scale-one}{hue=source:blue}
\tndeclare{species}{route}{hue=source:green!50!black}
\tn[at=(5,5), skin=box]{}
\tn[at=(3,3), skin=ring, name=P, label pos=nw]{P}
\tn[at=(3,7), skin=ring]{}
\tn[at=(7,3), skin=ring]{}
\tn[at=(7,7), skin=ring]{}
\tn[at=(2,2), species=scale-one]{}
\tn[at=(2,4), species=scale-one]{}
\tn[at=(2,6), species=scale-one]{}
\tn[at=(2,8), species=scale-one]{}
\tn[at=(4,2), species=scale-one]{}
\tn[at=(4,4), species=scale-one]{}
\tn[at=(4,6), species=scale-one]{}
\tn[at=(4,8), species=scale-one]{}
\tn[at=(6,2), species=scale-one]{}
\tn[at=(6,4), species=scale-one]{}
\tn[at=(6,6), species=scale-one]{}
\tn[at=(6,8), species=scale-one, name=Q, label pos=e]{Q}
\tn[at=(8,2), species=scale-one]{}
\tn[at=(8,4), species=scale-one]{}
\tn[at=(8,6), species=scale-one]{}
\tn[at=(8,8), species=scale-one]{}
\tnwire[kind=string, species=route, route=orth, dir=to,
name=link, via={(3,8)}]{P}{Q}
\tnmark[form=label, label pos=s]{(4,2)}{$f(Q)$}
\tnmark[form=enclosure, species=selected, tint, label pos=nw]
{(1,1) .. (5,5)}{$L\times L$}
\end{tenkz}](tenkz_svg/tenkz-2cf239bfc4a1b21f.svg)
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.
Let a repainting of \(S=[0,n]^2\) start from the guide \(g\) with final label \(B\). Its local labels are \(B\) and the values of \(g\) within sup distance \(n\) of the closed square. A guide is local over a base guide, the starting guide \(g\) of the main sheet or the uniform guide \(C_e\) of an auxiliary sheet, if it takes only local labels within sup distance \(n\) of the closed square and agrees with the base guide outside the open neighbourhood of radius \(n/2\) of the closed square. For the repainting of the square with indices \((r,s)\) and side \(2^{j'}\) during the pass to level \(2^{j'}\), the starting guide is the block guide of the level at a transition time, moved to the origin, with final label the new party of the square. Patch data is that of a gate of this large-scale repainting when its old and new guides are local guides over one base, moved back to the square, and its footprint lies within sup distance \(2^{j'}\) of the closed square.
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.
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]
\tndeclare{species}{bandA}{hue=source:orange}
\tndeclare{species}{bandB}{hue=source:cyan}
\tndeclare{species}{bandC}{hue=source:gray}
\tndeclare{species}{lens}{hue=source:red}
\tn[at=(1,1), name=bandTick0, skin=none]{}
\tnmark[form=label]{0.5 w of bandTick0}{$-8$}
\tn[at=(1,4), name=bandTick2, skin=none]{}
\tnmark[form=label]{0.5 e of bandTick2}{$-6$}
\tn[at=(1,8), name=bandTick4, skin=none]{}
\tnmark[form=label]{0.5 e of bandTick4}{$-4$}
\tn[at=(1,12), name=bandTick6, skin=none]{}
\tnmark[form=label]{0.5 e of bandTick6}{$-2$}
\tn[at=(1,16), name=bandTick8, skin=none]{}
\tnmark[form=label]{0.5 e of bandTick8}{$0$}
\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
\tn[at=(3,1), skin=none]{}
\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]{}
\tn[at=(3,7), skin=none]{}
\tn[at=(3,8), skin=none]{}
\tn[at=(3,9), skin=none]{}
\tn[at=(3,10), skin=none]{}
\tn[at=(3,11), skin=none]{}
\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)
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 27.127.3. 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 \(e\).
- TNLean.PEPS.Approximation.RepaintingBaseline
- TNLean.PEPS.Approximation.openSquare
- TNLean.PEPS.Approximation.closedSquare
- TNLean.PEPS.Approximation.bandUpdate
- TNLean.PEPS.Approximation.RepaintingBaseline.centralGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.mainGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.auxGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.edgeInterface
- TNLean.PEPS.Approximation.auxSteps
- TNLean.PEPS.Approximation.mainSteps
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.
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.
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.
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.
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.
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 \(k\).
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 ] .
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 \(S_{f,\tau }\psi (s)=(\mathcal R\psi )(\tau [f\mapsto s])\).
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 \(|\mathcal E|=|V|^8\).
- TNLean.PEPS.TorusCutBoundaryConfig
- TNLean.PEPS.torusCutBoundaryRestriction
- TNLean.PEPS.torusCutInteriorWeight
- TNLean.PEPS.torusCutCoeff
- TNLean.PEPS.torusCutMap
- TNLean.PEPS.torusCutMap_apply
- TNLean.PEPS.torusCutSpace
- TNLean.PEPS.mem_torusCutSpace_iff
- TNLean.PEPS.fourTorusCutSpace
- TNLean.PEPS.mem_fourTorusCutSpace_iff
- TNLean.PEPS.card_torusCutBoundaryConfig_two
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.
- TNLean.PEPS.sitewiseTorusGClosure
- TNLean.PEPS.sitewiseTorusGClosure_const
- TNLean.PEPS.sitewiseCommutingClosureSpan
- TNLean.PEPS.torusMatchedHorizontalClosureAt
- TNLean.PEPS.torusMatchedVerticalClosureAt
- TNLean.PEPS.matchedTorusGClosure
- TNLean.PEPS.matchedTorusGClosure_const
- TNLean.PEPS.matchedCommutingClosureSpan
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 \(K_{s,c'}\).
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.
- TNLean.PEPS.torusMatchedLegMatrix
- TNLean.PEPS.torusMatchedLegRep
- TNLean.PEPS.torusMatchedLegRep_apply
- TNLean.PEPS.torusMatchedLegRep_const
- TNLean.PEPS.IsUnitaryTorusBondFamily
- TNLean.PEPS.IsSemiRegularTorusBondFamily
- TNLean.PEPS.representationAveragingSite
- TNLean.PEPS.siteMap_representationAveragingSite
- TNLean.PEPS.representationAveragingSite_torusLegRep
- TNLean.PEPS.torusMatchedAveragingSites
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 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 \(M^{E_\eta }_{c,r}(\theta )=\delta _{\eta ,\theta }\).
For fixed exterior crossing configurations \(\gamma _X,\gamma _Y\), define
The exterior endpoint has the complementary orientation to the retained regional endpoint.
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 \(Q_{\partial R}=\bigotimes _{f\in \partial E_R}Q_f\).
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.
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.
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 \(V^\dagger \).
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).
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).
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 \(T\cup E\).
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.
- TNLean.PEPS.EncodedFrame.TwoSheetExchange
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.frame₁
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.frame₂
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.newFrame₁
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.newFrame₂
- TNLean.PEPS.EncodedFrame.pairwiseDisjointOuter_append
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.renameEquiv
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.rename
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.newFrame₁_owner_of_mem
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.newFrame₂_owner_of_mem
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.ofFrames
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.mem_footprint_perm
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.mem_ofFrames_exchangeEnv
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.
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exchangeEnv
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.tSet
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.eSet
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.aSet
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.mem_exchangeEnv
- TNLean.PEPS.EncodedFrame.liftTags
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exchangeOp
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 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.
- TNLean.PEPS.EncodedFrame.siteRegs
- TNLean.PEPS.EncodedFrame.tagRegs
- TNLean.PEPS.EncodedFrame.sites
- TNLean.PEPS.EncodedFrame.nodup_sites
- TNLean.PEPS.EncodedFrame.mem_sites
- TNLean.PEPS.EncodedFrame.sitesEquiv
- TNLean.PEPS.EncodedFrame.owner_siteRegs
- TNLean.PEPS.EncodedFrame.layoutRegs
- TNLean.PEPS.EncodedFrame.Frame.regs
- TNLean.PEPS.EncodedFrame.Frame.map_owner_regs
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).
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 27.133.25. 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 27.133.16 also covers arbitrary tag orderings.
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.oldFrame
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.newFrame
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.Conditions
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.AvoidsUntouched
- TNLean.PEPS.EncodedFrame.Frame.innerHoles
- TNLean.PEPS.EncodedFrame.Frame.outerHoles
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 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.
- TNLean.PEPS.EncodedFrame.squareSample
- TNLean.PEPS.EncodedFrame.SquarePatch
- TNLean.PEPS.EncodedFrame.SquarePatch.inner
- TNLean.PEPS.EncodedFrame.SquarePatch.outer
- TNLean.PEPS.EncodedFrame.SquarePatch.Tag
- TNLean.PEPS.EncodedFrame.SquarePatch.cyl
- TNLean.PEPS.EncodedFrame.SquarePatch.proj
- TNLean.PEPS.EncodedFrame.SquarePatch.proj_of_outer_eq_empty
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 27.145.5.
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 \(\eta _J\).
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 \(\mathcal X_v\).
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.
- TNLean.PEPS.DependentTorus.Vertex
- TNLean.PEPS.DependentTorus.Bond
- TNLean.PEPS.DependentTorus.LocalConfig
- TNLean.PEPS.DependentTorus.cutBonds
- TNLean.PEPS.DependentTorus.fourCutSpace
- TNLean.PEPS.DependentTorus.mem_fourCutSpace_iff
- TNLean.PEPS.DependentTorus.cutBonds_cover
- TNLean.PEPS.DependentTorus.closure
- TNLean.PEPS.DependentTorus.commutingClosureSpan
Let \(g\mapsto U_g\) be a representation of a finite group \(G\) on the virtual system of a tensor \(A\), and let \(\Pi _U\) be the projector onto the \(U_g\)-invariant subspace. The tensor is \(G\)-injective if \(\mathcal{P}(A)U_g=\mathcal{P}(A)\) for every \(g\) and \(\mathcal{P}(A)\) is injective on the invariant subspace [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] . It is \(G\)-isometric if moreover \(\mathcal{P}(A)\) restricted to the invariant subspace is a multiple of an isometry, \(\langle \mathcal{P}(A)x,\mathcal{P}(A)y\rangle =c\, \langle x,y\rangle \) for invariant \(x,y\) and one constant \(c{\gt}0\) [ 235 , Section “Isometric PEPS” ] . When the virtual action factors as \(U_g=U_g^{\ell }\otimes U_g^r\otimes U_g^t\otimes U_g^b\), the invariance of a four-leg tensor reads
![\begin{tenkz}[rows={wire,wire,wire}, cols=3, bonds=none]
\tn[at=(2,2), name=symmetricSite, label pos=225,
ports={180:virtual, 0:virtual, 90:virtual, 270:virtual,
45:physical:$i$}]{A}
\tn[at=(2,1), name=symmetryLeft, skin=ring, label pos=270,
ports={180:virtual, 0:virtual}]{U_g^{\ell}}
\tn[at=(2,3), name=symmetryRight, skin=ring, label pos=270,
ports={180:virtual, 0:virtual}]{U_g^r}
\tn[at=(1,2), name=symmetryTop, skin=ring, label pos=180,
ports={90:virtual, 270:virtual}]{U_g^t}
\tn[at=(3,2), name=symmetryBottom, skin=ring, label pos=0,
ports={90:virtual, 270:virtual}]{U_g^b}
\tnwire{symmetryLeft.0}{symmetricSite.180}
\tnwire{symmetryRight.180}{symmetricSite.0}
\tnwire{symmetryTop.270}{symmetricSite.90}
\tnwire{symmetryBottom.90}{symmetricSite.270}
\end{tenkz}](tenkz_svg/tenkz-cc525d71ae61d9ec.svg)
\(=\) ![\begin{tenkz}[rows={wire}, cols=1, bonds=none]
\tn[label pos=225,
ports={180:virtual, 0:virtual, 90:virtual, 270:virtual,
45:physical:$i$}]{A}
\end{tenkz}](tenkz_svg/tenkz-9b5d07c99cc8ba21.svg)
The four factors include the contragredient actions required by the orientations of the virtual bonds.
Before multiplying by the outer gate coefficients, define
Use finite orthonormal output coordinates for this matrix.
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.
- TNLean.PEPS.ThreeBlockDependent.Vertex
- TNLean.PEPS.ThreeBlockDependent.Bond
- TNLean.PEPS.ThreeBlockDependent.tail
- TNLean.PEPS.ThreeBlockDependent.head
- TNLean.PEPS.ThreeBlockDependent.exteriorBonds
- TNLean.PEPS.ThreeBlockDependent.leftCut
- TNLean.PEPS.ThreeBlockDependent.rightCut
- TNLean.PEPS.ThreeBlockDependent.coreVertices
- TNLean.PEPS.ThreeBlockDependent.leftVertices
- TNLean.PEPS.ThreeBlockDependent.rightVertices
- TNLean.PEPS.ThreeBlockDependent.Core
- TNLean.PEPS.ThreeBlockDependent.CoreVertex
- TNLean.PEPS.ThreeBlockDependent.LocalConfig
- TNLean.PEPS.ThreeBlockDependent.core_degree
- TNLean.PEPS.ThreeBlockDependent.exteriorBonds_card
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 matrix of the full product physical map is \((F_V)_{\tau \sigma }=\prod _{v\in V}(F_v)_{\tau _v\sigma _v}\).
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.
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 ] .
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 ] .
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.
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 \(D^f_{\ell }:(a\ell )^f\to (b\ell )^f\).
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)
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.
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 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 ] .
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 ] .
A PEPS tensor \(A\) is injective if it is vertex-injective and every virtual bond space is nonzero, that is, \(D_e{\gt}0\) for every edge \(e\). The first condition is the injectivity of [ 220 , Section 3 ] . The second is the standing convention of that section, which picks reference virtual configurations and full matrix algebras on the bonds. Without it a zero-dimensional bond makes every state coefficient vanish, while vertex injectivity can hold vacuously at its endpoints. The convention is recorded in [ 67 ] .
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 \(q\in Q\) 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 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.
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 \(A_v\) 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,
physical=updown]
\tnmark[form=label, label pos=e]{(2,3)}{$A_v$}
\end{tenkz}](tenkz_svg/tenkz-9591bb68f19a8578.svg)
\(\quad \longmapsto \quad \) ![\begin{tenkz}[rows={op,op,op}, cols=3,
frame={plane, basis={op at (0,0), ket at (1,-2)}}, bonds=none]
\tndeclare{species}{input}{hue=source:blue}
\tn[at=(1,1,2), skin=dot, species=input, label pos=e]{}
\tnwire{(1,1,1)}{(1,1,2)}
\tnwire{(1,1,1)}{open up}
\tnwire{(1,1,1)}{(1,2,1)}
\tnwire{(1,1,1)}{(2,1,1)}
\tn[at=(1,2,2), skin=dot, species=input, label pos=e]{}
\tnwire{(1,2,1)}{(1,2,2)}
\tnwire{(1,2,1)}{open up}
\tnwire{(1,2,1)}{(1,3,1)}
\tnwire{(1,2,1)}{(2,2,1)}
\tn[at=(1,3,2), skin=dot, species=input, label pos=e]{}
\tnwire{(1,3,1)}{(1,3,2)}
\tnwire{(1,3,1)}{open up}
\tnwire{(1,3,1)}{(2,3,1)}
\tn[at=(2,1,2), skin=dot, species=input, label pos=e]{}
\tnwire{(2,1,1)}{(2,1,2)}
\tnwire{(2,1,1)}{open up}
\tnwire{(2,1,1)}{(2,2,1)}
\tnwire{(2,1,1)}{(3,1,1)}
\tn[at=(2,2,2), skin=dot, species=input, label pos=e]{}
\tnwire{(2,2,1)}{(2,2,2)}
\tnwire{(2,2,1)}{open up}
\tnwire{(2,2,1)}{(2,3,1)}
\tnwire{(2,2,1)}{(3,2,1)}
\tn[at=(2,3,2), skin=dot, species=input, label pos=e]{z_v}
\tnwire{(2,3,1)}{(2,3,2)}
\tnwire{(2,3,1)}{open up}
\tnwire{(2,3,1)}{(3,3,1)}
\tn[at=(3,1,2), skin=dot, species=input, label pos=e]{}
\tnwire{(3,1,1)}{(3,1,2)}
\tnwire{(3,1,1)}{open up}
\tnwire{(3,1,1)}{(3,2,1)}
\tn[at=(3,2,2), skin=dot, species=input, label pos=e]{}
\tnwire{(3,2,1)}{(3,2,2)}
\tnwire{(3,2,1)}{open up}
\tnwire{(3,2,1)}{(3,3,1)}
\tn[at=(3,3,2), skin=dot, species=input, label pos=e]{}
\tnwire{(3,3,1)}{(3,3,2)}
\tnwire{(3,3,1)}{open up}
\end{tenkz}](tenkz_svg/tenkz-db08096545db02bb.svg)
- TNLean.PEPS.Approximation.Pinned.operatorMatrix
- TNLean.PEPS.Approximation.Pinned.fixInput
- TNLean.PEPS.Approximation.Pinned.contractPEPS_fixInput_apply
- TNLean.PEPS.Approximation.Pinned.contractPEPS_fixInput
- TNLean.PEPS.Approximation.Vector.Tensor.operatorMatrix
- TNLean.PEPS.Approximation.Vector.Tensor.fixInput
- TNLean.PEPS.Approximation.Vector.Tensor.bondDim_fixInput
- TNLean.PEPS.Approximation.Vector.Tensor.maxBondDim_fixInput
- TNLean.PEPS.Approximation.Vector.Tensor.contract_fixInput_apply
- TNLean.PEPS.Approximation.Vector.Tensor.contract_fixInput
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 \(w,\ell \ge 0\) 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.
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 \(\kappa _{c,r}(g,h)\).
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.
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.
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 \(M\).
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 \(b\).
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 ] .
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.
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.
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 \(t\).
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 \(\mathcal E(M)\).
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 \(V=\mathbb Z_w\times \mathbb Z_h\) 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 \(y+1=r\).
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 \(\mathcal A x(s)=\sum _{\xi \in X^4}a(\xi ,s)x(\xi )\).
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.
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 \(e\in C\).
- TNLean.PEPS.graphEdgeTail
- TNLean.PEPS.graphEdgeHead
- TNLean.PEPS.graphBondAlphabet
- TNLean.PEPS.graphIncidentEndpoint
- TNLean.PEPS.graphDependentTensor
- TNLean.PEPS.regionExteriorCut
- TNLean.PEPS.regionCutBoundaryLabel
- TNLean.PEPS.graphDependentTensor_diagonal
- TNLean.PEPS.network_graphDependentTensor_one
- TNLean.PEPS.mem_regionExteriorCut_of_boundary
- TNLean.PEPS.mem_regionExteriorCut_of_outside
- TNLean.PEPS.regionCutBoundaryLabel_diagonal
- TNLean.PEPS.graphEndpointIncident
- TNLean.PEPS.graphIncidentEndpoint_endpointIncident
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.
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 \(u\ge 0\), put \(m=\lfloor u/2\rfloor +1\). For each integer \(0\le j{\lt}m\), define
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.
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 \(P=\bigotimes _vP_v\).
For a coefficient matrix \(M\) of a pure state on \(\mathcal H_R\otimes \mathcal H_S\), define
The inverse is taken to be zero when \(M=0\). For nonzero \(M\), \(\widehat M\) represents a normalized pure state.
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 \(\rho _R\).
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.
For a region \(R\) in a finite PEPS graph, let \(T_R(\mu ;\sigma )\) be the contraction of its site tensors with physical configuration \(\sigma \), fixing the crossing-bond labels \(\mu \) and summing only the internal-bond labels. It defines the linear map from the crossing virtual indices to the physical system of \(R\). No edge wholly outside \(R\) enters this contraction. This is the finite-region tensor used when blocking and cutting the PEPS in [ 235 , Section 6, proof of Theorem 6.9 ] .
- TNLean.PEPS.RegionIncidentConfig
- TNLean.PEPS.regionIncidentBoundaryLabel
- TNLean.PEPS.isRegionIncidentEdge_of_regionVertex
- TNLean.PEPS.regionIncidentWeight
- TNLean.PEPS.openRegionWeight
- TNLean.PEPS.openRegionMap
- TNLean.PEPS.openRegionMap_apply
- TNLean.PEPS.openRegionMap_single
- TNLean.PEPS.openRegionCoefficientMatrix
- TNLean.PEPS.openRegionMap_apply_eq_mulVec
- TNLean.PEPS.toMatrix_openRegionMap
- TNLean.PEPS.regionExteriorMultiplicity
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 \(X\) 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.
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.
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.
- TNLean.PEPS.PairEffect.SourceInventory.sourcePositionsLayout
- TNLean.PEPS.PairEffect.SourceInventory.prepareSourcePositions
- TNLean.PEPS.PairEffect.SourceInventory.slotReference
- TNLean.PEPS.PairEffect.SourceInventory.sourcePositionsLayout_eq_layout
- TNLean.PEPS.PairEffect.SourceInventory.sourcePositionsLayout_append
- TNLean.PEPS.PairEffect.SourceInventory.selectedSources_eq_map
- TNLean.PEPS.PairEffect.SourceInventory.sourcePositionsLayout_filter
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 ] .
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 \(e\in E_{R^c}\).
For fixed crossing exterior labels \(\gamma _Y,\gamma _X\), put
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 \(M_o(Q)=M((T_e(Q))_e)\).
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 \(E_v^{-1}\rho _v^o(g)E_v\).
- TNLean.PEPS.graphOrientedIncidentMatrix
- TNLean.PEPS.graphOrientedIncidentMatrix_one
- TNLean.PEPS.graphOrientedIncidentMatrix_mul
- TNLean.PEPS.graphOrientedIncidentRepresentation
- TNLean.PEPS.graphOrientedIncidentRepresentation_apply
- TNLean.PEPS.graphOrientedNumberedIncidentRepresentation
- TNLean.PEPS.graphOrientedNumberedIncidentRepresentation_apply
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 \(o_e=1\).
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.
- TNLean.PEPS.graphOrientedOpenBoundaryFactor
- TNLean.PEPS.graphOrientedOpenInternalFactor
- TNLean.PEPS.graphOrientedOpenBondCoordinates
- TNLean.PEPS.graphOrientedOpenBondSpace
- TNLean.PEPS.graphOrientedOpenBoundaryFactor_false
- TNLean.PEPS.graphOrientedOpenInternalFactor_false
- TNLean.PEPS.graphOrientedOpenBondCoordinates_false
- TNLean.PEPS.graphOrientedOpenBondSpace_false
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 \(L_n^o(L_m^o)^{-1}\).
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.
- TNLean.PEPS.PairEffect.EffectCircuit.gateLocations
- TNLean.PEPS.PairEffect.EffectCircuit.participants
- TNLean.PEPS.PairEffect.OriginalCircuit.isAllowed_produce
- TNLean.PEPS.PairEffect.OriginalCircuit.isExpansionBounded_produce_iff
- TNLean.PEPS.PairEffect.OriginalCircuit.nonprivateLocations
- TNLean.PEPS.PairEffect.OriginalCircuit.participants
- TNLean.PEPS.PairEffect.OriginalCircuit.participants_produce
- TNLean.PEPS.PairEffect.OriginalCircuit.produce
- TNLean.PEPS.PairEffect.OriginalCircuit.produceLocationsEquiv
- TNLean.PEPS.PairEffect.OriginalCircuit
- TNLean.PEPS.PairEffect.OriginalCircuit.eval
- TNLean.PEPS.PairEffect.OriginalCircuit.nonprivateCount
- TNLean.PEPS.PairEffect.OriginalCircuit.IsExpansionBounded
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.
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.
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.
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 ] .
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 ] .
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.
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 \(o_v(r_C\beta )=(R\beta )_v\).
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.
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.
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.
- TNLean.PEPS.PairEffect.Layout.mapOwner
- TNLean.PEPS.PairEffect.Layout.mapOwner_nil
- TNLean.PEPS.PairEffect.Layout.mapOwner_cons
- TNLean.PEPS.PairEffect.Layout.mapOwner_append
- TNLean.PEPS.PairEffect.Layout.mapOwnerIso
- TNLean.PEPS.PairEffect.Layout.mapOwnerIso_nil_apply
- TNLean.PEPS.PairEffect.Layout.mapOwnerIso_cons_tmul
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.
Two pairs are equivalent if \((g,h)=(xg'x^{-1},xh'x^{-1})\) for some \(x\in G\). Their equivalence classes are the pair-conjugacy classes of [ 235 , Definition 5.8 ] . Commutation is constant on each class, so the classes of commuting pairs are well-defined.
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 \(h\in R\).
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.
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 \(\mathcal H_R(U,W)=\operatorname {span}_{\mathbb {C}} \{ H_R^{U,W}(\theta ):\theta \in X^{\partial E_R}\} \).
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 \(m_i{\gt}0\).
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
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 \((\mathcal M(F,K)\psi )(\tau ,\beta ) =\sum _{\sigma ,\alpha }\mathcal M(F,K)_{(\tau ,\beta ),(\sigma ,\alpha )} \psi (\sigma ,\alpha )\).
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
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.
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.
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.
- TNLean.PEPS.DependentBondNetwork.partialAveragingSite
- TNLean.PEPS.DependentBondNetwork.localSiteMap_partialAveragingSite
- TNLean.PEPS.DependentBondNetwork.partialAveragingSite_of_eq_id
- TNLean.PEPS.DependentBondNetwork.localSiteMap_partialAveragingSite_of_eq_one
- TNLean.PEPS.DependentBondNetwork.partialAveragingSite_of_eq_one
- TNLean.PEPS.DependentBondNetwork.partialAveragingSite_apply
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.
- TNLean.PEPS.PairEffect.SourceCircuit.partialSourceBasisMatrix
- TNLean.PEPS.PairEffect.SourceCircuit.SelectedSourceCoordinates
- TNLean.PEPS.PairEffect.SourceCircuit.selectedSourceCoordinatesFintype
- TNLean.PEPS.PairEffect.SourceCircuit.coordinateBasisVectors
- TNLean.PEPS.PairEffect.SourceCircuit.partialCoordinateBasisMatrix
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.
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.
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 \(\mathbb {C}^{u_{e_i}}\otimes \mathbb {C}^{v_{e_i}}\).
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.
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 27.153.12.
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 27.149.2.1. 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.
- TNLean.PEPS.PairEffect.SourceCircuit.IsTouched
- TNLean.PEPS.PairEffect.SourceCircuit.isTouched_gate_iff
- TNLean.PEPS.PairEffect.SourceCircuit.Choices
- TNLean.PEPS.PairEffect.SourceCircuit.choicesFintype
- TNLean.PEPS.PairEffect.SourceCircuit.partialWord
- TNLean.PEPS.PairEffect.SourceCircuit.coefficient
- TNLean.PEPS.PairEffect.SourceCircuit.expandedEval
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.
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.
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 \((a\mathbin {+\! +}c)_p=a_p\mathbin {+\! +}c_p\).
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 27.139.7, whose pieces between effects are source-only. This is the notion of allowed monomial of [ 248 , Section 5.1 ] , lines 32–35.
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.
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.
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 \(U_\ell :\mathcal H_\ell \to \bigotimes _{j=1}^s\mathcal H_{\ell ,p_j}\).
- TNLean.PEPS.PairEffect.Layout.restrict
- TNLean.PEPS.PairEffect.Layout.restrict_nil
- TNLean.PEPS.PairEffect.Layout.restrict_cons
- TNLean.PEPS.PairEffect.Layout.restrict_append
- TNLean.PEPS.PairEffect.Layout.restrict_eq_self_of_owner
- TNLean.PEPS.PairEffect.Layout.restrict_eq_nil_of_owner
- TNLean.PEPS.PairEffect.Layout.memCongr
- TNLean.PEPS.PairEffect.Layout.memCongr_rfl
- TNLean.PEPS.PairEffect.Layout.partitionIso
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_true
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_false
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_true_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_false_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_append_true_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_append_false_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_cons_true_true_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_cons_true_false_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_cons_false_true_tmul
- TNLean.PEPS.PairEffect.Layout.partitionIso_cons_cons_false_false_tmul
- TNLean.PEPS.PairEffect.Layout.withoutParty
- TNLean.PEPS.PairEffect.Layout.atParty_withoutParty
- TNLean.PEPS.PairEffect.Layout.mem_withoutParty
- TNLean.PEPS.PairEffect.Layout.owners_withoutParty
- TNLean.PEPS.PairEffect.partyLayout_withoutParty
- TNLean.PEPS.PairEffect.groupByPartyIso
- TNLean.PEPS.PairEffect.groupByPartyIso_cons
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.
- TNLean.PEPS.PairEffect.Word.parties
- TNLean.PEPS.PairEffect.Word.owners_mem_parties
- TNLean.PEPS.PairEffect.Layout.atParty
- TNLean.PEPS.PairEffect.partyLayout
- TNLean.PEPS.PairEffect.tensorPartyMaps
- TNLean.PEPS.PairEffect.tensorPartyMaps_cons_tmul
- TNLean.PEPS.PairEffect.partyLayout_eq_of_atParty_eq
- TNLean.PEPS.PairEffect.tensorPartyMaps_heq
- TNLean.PEPS.PairEffect.Layout.conj_memCongr_heq
- TNLean.PEPS.PairEffect.Layout.norm_conj_memCongr
- TNLean.PEPS.PairEffect.Layout.eq_conj_memCongr_of_heq
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 \(1\).
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).
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.
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 \([s]_{\rm phys}\otimes t\).
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 \(\bigotimes _{v\in R}F_v^{-1}\).
If \(R\) is the constructed source-only replacement, define \(R_{\rm phys}=\tau R\). The exchange acts on pure tensors by
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 27.154.1 and 27.154.3 by \(F,F'\), respectively. Thus
These frames need not span either input space.
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.
Let \(F\) be a rectangular physical matrix. Its product over a finite site set \(\Lambda \) has coefficients
The transformed site tensor is \((FA)^q_{t,r,b,l}=\sum _s F_{q,s}A^s_{t,r,b,l}\). Input and output physical dimensions may differ.
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
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.
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 27.139.13, 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.
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 \(\widetilde V_\xi \).
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.
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.
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.
- TNLean.PEPS.PairEffect.ProductInput.mem_owners_wholePartyLayout
- TNLean.PEPS.PairEffect.ProductInput.nodup_owners_wholePartyLayout
- TNLean.PEPS.PairEffect.ProductInput.ofAllParties
- TNLean.PEPS.PairEffect.ProductInput.owners_wholePartyLayout
- TNLean.PEPS.PairEffect.ProductInput.wholePartyLayout
- TNLean.PEPS.PairEffect.ProductInput
- TNLean.PEPS.PairEffect.ProductInput.eval_prepare
- TNLean.PEPS.PairEffect.ProductInput.eval_prepare_smul
- TNLean.PEPS.PairEffect.ProductInput.isAllowed_prepare
- TNLean.PEPS.PairEffect.ProductInput.mapOwner
- TNLean.PEPS.PairEffect.ProductInput.mapOwnerIso_vector
- TNLean.PEPS.PairEffect.ProductInput.norm_vector
- TNLean.PEPS.PairEffect.ProductInput.ofParties
- TNLean.PEPS.PairEffect.ProductInput.partitionIso_symm_mapOwner_vector
- TNLean.PEPS.PairEffect.ProductInput.partitionIso_symm_vector
- TNLean.PEPS.PairEffect.ProductInput.partitionIso_vector
- TNLean.PEPS.PairEffect.ProductInput.prepare
- TNLean.PEPS.PairEffect.ProductInput.prepareIsometry
- TNLean.PEPS.PairEffect.ProductInput.restrict
- TNLean.PEPS.PairEffect.ProductInput.sources_prepare
- TNLean.PEPS.PairEffect.ProductInput.vector
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.
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.
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.
- TNLean.PEPS.EncodedFrame.RankOneBlock
- TNLean.PEPS.EncodedFrame.RankOneBlock.op
- TNLean.PEPS.EncodedFrame.RankOneBlock.ketAt
- TNLean.PEPS.EncodedFrame.RankOneBlock.braAt
- TNLean.PEPS.EncodedFrame.RankOneBlock.op_apply
- TNLean.PEPS.EncodedFrame.RankOneBlock.ketAt_congr
- TNLean.PEPS.EncodedFrame.RankOneBlock.braAt_congr
- TNLean.PEPS.EncodedFrame.zeroPair
- TNLean.PEPS.EncodedFrame.norm_zeroPair
- TNLean.PEPS.EncodedFrame.zeroPair_apply
- TNLean.PEPS.EncodedFrame.zeroBlock
- TNLean.PEPS.EncodedFrame.zeroBlock_ketAt
- TNLean.PEPS.EncodedFrame.zeroBlock_braAt
- TNLean.PEPS.EncodedFrame.emptyBlock
- TNLean.PEPS.EncodedFrame.emptyBlock_ketAt
- TNLean.PEPS.EncodedFrame.emptyBlock_braAt
- TNLean.PEPS.EncodedFrame.emptyBlock_op
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 \(L_v\).
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 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 ] .
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
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 \(S\subseteq \mathcal H_R\), 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, \(\mathcal C_R(0)=0\).
For arbitrary finite coordinate sets \(X_v\), put \(X_R=\prod _{v\in R}X_v\) and \(\mathcal H_R=\mathbb {C}^{X_R}\). Write a full configuration as \((\alpha ,\tau )\in X_R\times X_{V\setminus R}\). Extend \(K\) on \(\mathcal H_R\) by \(\widehat K_R=K\otimes I_{V\setminus R}\), so that \(\widehat K_R((\alpha ,\tau ),(\beta ,\rho ))=K(\alpha ,\beta )\delta _{\tau ,\rho }\). For \(R\subseteq U\), its extension inside \(U\) is \(K\otimes I_{U\setminus R}\).
For \(b=n+1\) regular boundary legs, let \(r\in G^n\) denote relative coordinates. Define the boundary embedding
For an open map \(T\), write \(C=TU\) and \(Q=C^\dagger C\). Its supported boundary Hamiltonian is \(H=-\log Q\), with the logarithm defined by spectral calculus and the value at a zero eigenvalue set to zero. For \(G\)-injective \(T\), the operator \(Q\) is positive definite and this is the usual spectral logarithm. The logarithm gives the inverse relation below when \(T\) is \(G\)-injective.
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
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.
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 positive density \(\rho \) and \(\alpha {\gt}0\), \(\alpha \ne 1\), define
where the real power is defined spectrally. Set \(S_1(\rho )=S(\rho )\) and \(S_0(\rho )=\log \operatorname {rank}\rho \).
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.
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 \(m^{-n}\).
- TNLean.PEPS.PairEffect.EffectChain.stackVector
- TNLean.PEPS.PairEffect.EffectChain.replace
- TNLean.PEPS.PairEffect.EffectChain.replaceTerm
- TNLean.PEPS.PairEffect.GateExpansion
- TNLean.PEPS.PairEffect.gate
- TNLean.PEPS.PairEffect.inventory
- TNLean.PEPS.PairEffect.inventoryVector
- TNLean.PEPS.PairEffect.replaceGate
- TNLean.PEPS.PairEffect.termList
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.
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 \(\widetilde T_g=T_g/(1+\delta )\).
Under the assumptions of Theorem 27.158.1, 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 \(S\) 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 ] .
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.
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 \(i\).
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 \(i\).
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.
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 27.141.1.12. 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.
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, \(\mathbf q=(*)\).
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.
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).
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.
- TNLean.PEPS.PairEffect.SourceCircuit.selectedPartialSlots
- TNLean.PEPS.PairEffect.SourceCircuit.mem_selectedPartialSlots
- TNLean.PEPS.PairEffect.SourceCircuit.selectedPartialSlotEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.selectedSourceCoordinateEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.selectedSourceCoordinateEquiv_apply
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.
A representation of \(G\) is semi-regular if it contains every irreducible representation of \(G\), that is, if every irreducible representation admits a nonzero intertwining map into it [ 235 , Section “MPS: the \(G\)-injective case” ] . The left-regular representation is semi-regular.
For \(a,b\in \{ 0,\ldots ,d_i-1\} \) and \(g\in G\), define
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 \(E_v:\mathcal V_v\cong \mathbb {C}^p\).
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 \(A_{\mathrm{reg}}=A(L,e^G)\).
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 27.154.9 tensored with the corresponding exterior outputs.
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 \(P\) 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.
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 \(\mathcal E\).
- TNLean.PEPS.PairEffect.SourceCircuit.gateLocations
- TNLean.PEPS.PairEffect.SourceCircuit.gateLocationsFintype
- TNLean.PEPS.PairEffect.SourceCircuit.slotCount
- TNLean.PEPS.PairEffect.SourceCircuit.sourceLocations
- TNLean.PEPS.PairEffect.SourceCircuit.sourceLocationsFintype
- TNLean.PEPS.PairEffect.SourceCircuit.participants
- TNLean.PEPS.PairEffect.SourceCircuit.endpoints
- TNLean.PEPS.PairEffect.SourceCircuit.branchLabels
- TNLean.PEPS.PairEffect.SourceCircuit.branchLabelsFintype
- TNLean.PEPS.PairEffect.SourceCircuit.branchCoefficient
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.
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.
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.
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.
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.
- TNLean.PEPS.PairEffect.PairSource
- TNLean.PEPS.PairEffect.PairSource.layout
- TNLean.PEPS.PairEffect.SourceInventory
- TNLean.PEPS.PairEffect.SourceInventory.layout
- TNLean.PEPS.PairEffect.SourceInventory.IsNormalized
- TNLean.PEPS.PairEffect.SourceInventory.prepare
- TNLean.PEPS.PairEffect.SourceInventory.vector
- TNLean.PEPS.PairEffect.SourceInventory.layout_nil
- TNLean.PEPS.PairEffect.SourceInventory.layout_cons
- TNLean.PEPS.PairEffect.SourceInventory.layout_append
- TNLean.PEPS.PairEffect.SourceInventory.isAllowed_prepare_iff
- TNLean.PEPS.PairEffect.SourceInventory.isNormalized_cons
- TNLean.PEPS.PairEffect.SourceInventory.eval_prepare_eq_appendIso_symm
- TNLean.PEPS.PairEffect.SourceInventory.eval_prepare_append
- TNLean.PEPS.PairEffect.Word.sources
- TNLean.PEPS.PairEffect.Word.sources_prepare
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.
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 ] .
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.
- TNLean.PEPS.PairEffect.EffectCircuit
- TNLean.PEPS.PairEffect.EffectCircuit.IsAllowed
- TNLean.PEPS.PairEffect.EffectCircuit.eval
- TNLean.PEPS.PairEffect.EffectCircuit.norm_eval_le_one
- TNLean.PEPS.PairEffect.PartyChain.isPreparation_prepStack
- TNLean.PEPS.PairEffect.SourceCircuit.castLayouts
- TNLean.PEPS.PairEffect.SourceCircuit.eval_castLayouts
- TNLean.PEPS.PairEffect.SourceCircuit.eval_exchangeBlocks
- TNLean.PEPS.PairEffect.SourceCircuit.eval_frameList_tmul
- TNLean.PEPS.PairEffect.SourceCircuit.eval_ofLocalWord
- TNLean.PEPS.PairEffect.SourceCircuit.exchangeBlocks
- TNLean.PEPS.PairEffect.SourceCircuit.frameList
- TNLean.PEPS.PairEffect.SourceCircuit.isAllowed_castLayouts
- TNLean.PEPS.PairEffect.SourceCircuit.isAllowed_frameList
- TNLean.PEPS.PairEffect.SourceCircuit.isAllowed_ofLocalWord
- TNLean.PEPS.PairEffect.SourceCircuit.ofLocalWord
- TNLean.PEPS.PairEffect.SourceInventory.norm_vector
- TNLean.PEPS.PairEffect.Word.IsPreparation
- TNLean.PEPS.PairEffect.Word.eval_isPreparation_cast
- TNLean.PEPS.PairEffect.Word.eval_isPreparation_heq
- TNLean.PEPS.PairEffect.Word.eval_localPrepare
- TNLean.PEPS.PairEffect.Word.isAllowed_localPrepare
- TNLean.PEPS.PairEffect.Word.localPrepare
- TNLean.PEPS.PairEffect.Word.output_of_isPreparation
- TNLean.PEPS.PairEffect.exists_prepared_effectReplacement
- TNLean.PEPS.PairEffect.isPreparation_prepGate
- TNLean.PEPS.PairEffect.isPreparation_prepWord
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 ] .
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.
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.
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.
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.
- TNLean.PEPS.PairEffect.SourceInventory.ofSlots
- TNLean.PEPS.PairEffect.SourceInventory.ofSlots_nil
- TNLean.PEPS.PairEffect.SourceInventory.ofSlots_cons
- TNLean.PEPS.PairEffect.SourceInventory.isNormalized_ofSlots
- TNLean.PEPS.PairEffect.SourceInventory.layout_ofSlots_eq
- TNLean.PEPS.PairEffect.SourceInventory.partyPairs_ofSlots
- TNLean.PEPS.PairEffect.SourceInventory.slotLayout
- TNLean.PEPS.PairEffect.SourceInventory.prepareSlots
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.
A local tensor may be written as \(A_v(s_v,a|_v)\) or \(A_v(a|_v,s_v)\). In either case its contraction is the vector in \(\ell ^2(\Lambda \to \{ 0,\ldots ,q-1\} ;\mathbb {C})\) whose \(s\) coordinate is \(\sum _a\prod _v A_v(s_v,a|_v)\), with the arguments reversed in the second presentation. The second presentation includes the requirement \(D_e{\gt}0\). Its maximum bond dimension is \(\max (\{ D_e:e\in E\} \cup \{ 0\} )\).
- TNLean.PEPS.Approximation.Pinned.IncidentEdge
- TNLean.PEPS.Approximation.Pinned.State
- TNLean.PEPS.Approximation.Pinned.LocalTensor
- TNLean.PEPS.Approximation.Pinned.contractPEPS
- TNLean.PEPS.Approximation.Vector.IncidentEdge
- TNLean.PEPS.Approximation.Vector.State
- TNLean.PEPS.Approximation.Vector.Tensor
- TNLean.PEPS.Approximation.Vector.Tensor.contract
- TNLean.PEPS.Approximation.Vector.Tensor.maxBondDim
For a lattice point \(x\), set \(\rho (x)=\max \{ |x_1-c_1|,|x_2-c_2|\} \). Then \(K(c,r)=\{ x\in \Lambda :\rho (x)\le r\} \).
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 \(Q_{R,S,\tau }\).
A PEPS tensor on \(G\) with physical dimension \(d\) consists of a bond dimension \(D_e\) for every edge \(e\), together with a local tensor at each vertex \(v\) whose virtual indices are indexed by the edges incident to \(v\) and whose physical index lies in \(\{ 0,\ldots ,d-1\} \). The local tensor at \(v\) has one physical leg and one virtual leg for each incident edge:
![\begin{tenkz}[rows={wire}, cols=1, bonds=none]
\tn[label pos=w,
ports={210:virtual:$e_1$, 270:virtual:$e_k$,
330:virtual:$e_{\deg(v)}$,
90:physical:$\sigma(v)$}]{A_v}
\end{tenkz}](tenkz_svg/tenkz-afba3d26a5457d3a.svg)
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 \(m(j_{\xi _0}X)=X\).
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.
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 \(\eta |_{I_v}\).
- TNLean.PEPS.ThreeBlockDependent.CoreEndpoint
- TNLean.PEPS.ThreeBlockDependent.ExteriorEndpoint
- TNLean.PEPS.ThreeBlockDependent.OpenBoundaryConfig
- TNLean.PEPS.ThreeBlockDependent.CoreEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.ExteriorEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.tail_mem_core
- TNLean.PEPS.ThreeBlockDependent.head_mem_core_iff
- TNLean.PEPS.ThreeBlockDependent.exteriorEndpoint_mem_exterior
- TNLean.PEPS.ThreeBlockDependent.coreEndpoint_card
- TNLean.PEPS.ThreeBlockDependent.exteriorEndpoint_card
- TNLean.PEPS.ThreeBlockDependent.assembleCoreEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.openBoundaryRestriction
- TNLean.PEPS.ThreeBlockDependent.coreEndpointLocalConfig
- TNLean.PEPS.ThreeBlockDependent.assembleExteriorCutConfig
- TNLean.PEPS.ThreeBlockDependent.exteriorCutRestriction
- TNLean.PEPS.ThreeBlockDependent.cutRestriction_assembleCoreEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.exteriorCutRestriction_assemble
- TNLean.PEPS.ThreeBlockDependent.endpointSiteEquiv_assembleCoreEndpointConfig
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.
- TNLean.PEPS.ThreeBlockDependent.ExtendedPhys
- TNLean.PEPS.ThreeBlockDependent.extendedTensor
- TNLean.PEPS.ThreeBlockDependent.coreOutputEquiv
- TNLean.PEPS.ThreeBlockDependent.exteriorOutputEquiv
- TNLean.PEPS.ThreeBlockDependent.localSiteMap_extendedTensor_core
- TNLean.PEPS.ThreeBlockDependent.localSiteMap_extendedTensor_exterior
- TNLean.PEPS.ThreeBlockDependent.localSiteMap_extendedTensor_exterior_range
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 \(\mathbb {C}^{Y_0}\otimes \mathbb {C}^{Y_1}\otimes \mathbb {C}^{Y_2}\).
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.
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 \(\xi \).
- TNLean.PEPS.ThreeBlockDependent.RegionEndpoint
- TNLean.PEPS.ThreeBlockDependent.OutsideRegionEndpoint
- TNLean.PEPS.ThreeBlockDependent.RegionBoundaryEndpoint
- TNLean.PEPS.ThreeBlockDependent.RegionEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.OutsideRegionEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.RegionBoundaryConfig
- TNLean.PEPS.ThreeBlockDependent.left_regionBoundary_card
- TNLean.PEPS.ThreeBlockDependent.right_regionBoundary_card
- TNLean.PEPS.ThreeBlockDependent.assembleRegionEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.regionBoundaryRestriction
- TNLean.PEPS.ThreeBlockDependent.regionEndpointLocalConfig
- TNLean.PEPS.ThreeBlockDependent.assembleRegionalCutConfig
- TNLean.PEPS.ThreeBlockDependent.regionalCutBoundaryRestriction
- TNLean.PEPS.ThreeBlockDependent.outsideRegionalCutRestriction
- TNLean.PEPS.ThreeBlockDependent.cutRestriction_assembleRegionEndpointConfig
- TNLean.PEPS.ThreeBlockDependent.regionalCutBoundaryRestriction_assemble
- TNLean.PEPS.ThreeBlockDependent.outsideRegionalCutRestriction_assemble
- TNLean.PEPS.ThreeBlockDependent.endpointSiteEquiv_assembleRegionEndpointConfig
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.
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.
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 \(r_Ta_T(s,\tau )=s\).
- TNLean.PEPS.ThreeBlockDependent.CorePhysicalConfig
- TNLean.PEPS.ThreeBlockDependent.OutsideCorePhysicalConfig
- TNLean.PEPS.ThreeBlockDependent.corePhysicalRestriction
- TNLean.PEPS.ThreeBlockDependent.corePhysicalExtension
- TNLean.PEPS.ThreeBlockDependent.assembleCorePhysicalConfig
- TNLean.PEPS.ThreeBlockDependent.corePhysicalRestriction_assemble
Let every site \(v\) of the torus carry a tensor \(A_v\) with four virtual indices, ordered top, right, down, left, and let every bond carry a \(D\times D\) matrix \(O\): \(O^{\mathrm h}_v\) on the horizontal bond from \(v\) to \(v+(1,0)\) and \(O^{\mathrm v}_v\) on the vertical bond from \(v+(0,1)\) to \(v\). Split every bond into its two ends, and give a labelling of the ends the weight \(O_{ij}\) on each bond whose head reads \(i\) and whose tail reads \(j\). The network is the sum over all labellings of the ends of the product of these weights and of the entries of the site tensors, each reading the ends of its four legs. For a representation \(g\mapsto U_g\) on one bond, \(U_g\) acts on the virtual system of a site by \(U_g\) on the incoming legs (top and left) and \(U_g^{-1}\) on the outgoing legs (right and down), each along the orientation of its edge [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] .
On a \(w\times \ell \) torus let \(H_v=U_h\) when \(v+(1,0)\) crosses the horizontal seam and \(H_v=\mathbb 1\) elsewhere; let \(V_v=U_g\) when the vertical bond crosses the vertical seam and \(V_v=\mathbb 1\) elsewhere. Define the vector \(|\mathcal M(A\mid g,h)\rangle \) by contracting the site tensors with these bond operators. Its coefficient at \(\sigma \) is
This is [ 235 , Definition 5.6 ] , which uses a square lattice; the same contraction defines the vector on a rectangle. The four-site diagram suppresses the physical legs at fixed \(\sigma \). Crossings without a vertex do not denote contractions.
For \(n,m{\gt}1\), the discrete \(n\times m\) torus has vertex set \(\mathbb {Z}/n\mathbb {Z}\times \mathbb {Z}/m\mathbb {Z}\). Its graph joins \((x,y)\) to \((x+1,y)\) and to \((x,y+1)\), with the coordinates read cyclically; every edge is of one of these two forms, so every edge is horizontal or vertical. Translation by \((a,b)\) is the graph automorphism \(\tau _{a,b}(x,y)=(x+a,y+b)\). The induced action on edges sends an edge with unordered endpoint pair \(\{ u,v\} \) to the edge with unordered endpoint pair \(\{ \tau _{a,b}(u),\tau _{a,b}(v)\} \); in particular it preserves the horizontal and vertical orientation classes. For a general graph isomorphism \(\phi :G\simeq G'\), the same endpoint action gives a bijection on edges and a bijection from the edges incident to \(v\) to the edges incident to \(\phi (v)\).
- TNLean.PEPS.TorusVertex
- TNLean.PEPS.torusGraph
- TNLean.PEPS.torusGraph_adj_right
- TNLean.PEPS.torusGraph_adj_up
- TNLean.PEPS.Edge.map
- TNLean.PEPS.Edge.equiv
- TNLean.PEPS.IncidentEdge.equiv
- TNLean.PEPS.translateEquiv
- TNLean.PEPS.translate
- TNLean.PEPS.translateEdge
- TNLean.PEPS.translateEdge_isHorizontal
- TNLean.PEPS.translateEdge_isVertical
- TNLean.PEPS.translateIncidentEdge
- TNLean.PEPS.torusEdge_horizontal_or_vertical
Let \(X\) be the vertex set of a torus with positive periods and let \(\mathcal B=X\times \{ \mathrm h,\mathrm v\} \) be its horizontal and vertical bond positions. Write \(\sigma _v=(t_v,r_v,b_v,l_v)\) for the four original physical coordinates at \(v\). Their bond regrouping is
Each pair is ordered head, tail: horizontal bonds point right and vertical bonds point down. The inverse reads
Thus regrouping is a bijection of the actual physical basis configurations. Its induced coefficient map is \((\mathcal R\psi )(\beta )=\psi (\sigma (\beta ))\), and is a linear isometric equivalence of the finite-dimensional physical Hilbert spaces. This is the grouping into the two physical bond endpoints used in [ 235 , Section 7 ] .
- TNLean.PEPS.torusSiteBondEndpointEquiv
- TNLean.PEPS.torusSiteBondEndpointEquiv_apply_horizontal
- TNLean.PEPS.torusSiteBondEndpointEquiv_apply_vertical
- TNLean.PEPS.torusSiteBondEndpointEquiv_symm_apply
- TNLean.PEPS.torusBondRegrouping
- TNLean.PEPS.torusBondRegrouping_apply
- TNLean.PEPS.torusBondRegroupingIsometry
- TNLean.PEPS.torusBondRegroupingIsometry_apply
On the torus vertices, retain only consecutive coordinate values without wraparound. The result is the ordinary rectangular grid, a subgraph of the periodic square lattice.
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 \(\pi _{\mathcal A}\colon \Xi _P\to \Xi _{\mathcal A}\).
- TNLean.PEPS.PairEffect.SourceCircuit.projectAllChoices
- TNLean.PEPS.PairEffect.SourceCircuit.TouchedLabels
- TNLean.PEPS.PairEffect.SourceCircuit.choicesOfTouchedLabels
- TNLean.PEPS.PairEffect.SourceCircuit.choicesTouchedLabelsEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.nonemptyGateLocations
- TNLean.PEPS.PairEffect.SourceCircuit.allChoicesNonemptyLabelsEquiv
The uniform corrected-input identity asserts the existence of one allowed, source-free \(U\) such that the evaluation equality in Theorem 27.156.12 holds for every partial branch and every free-source family, using the actual residual and fixed vectors of that branch.
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.
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 ] .
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.
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 \(\operatorname {span}\{ \omega _e\} \).
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.
- TNLean.PEPS.Approximation.Configuration
- TNLean.PEPS.Approximation.StateSpace
- TNLean.PEPS.Approximation.SquareHamiltonian
- TNLean.PEPS.Approximation.SquareHamiltonian.operator
- TNLean.PEPS.Approximation.SquareHamiltonian.IsGappedGroundState
- TNLean.PEPS.Approximation.pepsVector
- TNLean.PEPS.Approximation.HasPEPSApproximation
- TNLean.PEPS.Approximation.PolynomialPEPSApproximation
A local circuit of depth \(T\) on a chain of \(N\) sites is a product of \(T\) layers, each a tensor product of unitaries acting on disjoint pairs of neighbouring sites. A state is prepared in depth \(T\) if it equals a local circuit of depth \(T\) applied to a product state.
More generally, a bond geometry on a finite set of sites \(I\) is a family \((B_b)_{b\in \beta }\) of subsets of \(I\), the bonds, and a layer is a product of unitaries acting on pairwise disjoint bonds. The model of [ 227 ] on a lattice takes for bonds the pairs of nearest-neighbour sites, that is the edges of a graph. Local circuits of depth \(T\) and states prepared in depth \(T\) are defined in the same way for every bond geometry.
The chain is a ring with sites \(1,\dots ,N\): the neighbouring pairs are \(\{ k,k+1\} \) with indices modulo \(N\); for \(N=1\) the only pair is \(\{ 1\} \), and for \(N=2\) both pairs are \(\{ 1,2\} \). The open chain has the pairs \(\{ k,k+1\} \) with \(k{\lt}N\). An operator acts on a set of sites \(X\) if it lies in \(M_{d}(\mathbb {C})^{\otimes X}\otimes \mathbb 1\), the span of the products \(\bigotimes _i m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\).
Acting on a set of sites needs no geometry: for any finite set of sites \(I\) and \(X\subseteq I\), an operator on \((\mathbb {C}^d)^{\otimes I}\) acts on \(X\) if it lies in the span of the products \(\bigotimes _{i\in I}m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\). The chain of \(N\) sites is the case \(I=\{ 1,\dots ,N\} \).
- QuantumCircuit.IsLocalCircuitOfDepth
- QuantumCircuit.IsPreparedInDepth
- QuantumCircuit.Layer
- QuantumCircuit.circuitOp
- QuantumCircuit.BondLayer
- QuantumCircuit.IsBondCircuitOfDepth
- QuantumCircuit.IsBondPreparedInDepth
- QuantumCircuit.ringBond
- QuantumCircuit.supportedOperators
- QuantumCircuit.rectKronecker_mem_supportedOperators
The map \(E_K\) on the operators of \((\mathbb {C}^q)^{\otimes I}\) defined above.
Let \(I\) be a finite set of sites and let \(d,e\geq 0\) be integers. At each site identify \((a,b)\in \{ 0,\dots ,d-1\} \times \{ 0,\dots ,e-1\} \) with \(ea+b\in \{ 0,\dots ,de-1\} \). Applied at every site, this gives a bijection \(p\) from pairs of configurations to configurations of local dimension \(de\).
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.
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.
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 ] .
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 \(A\).
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.
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.
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.
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 \(I(X:J)\).
Fix exponents \(0{\lt}\ell {\lt}1\), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \), \(\nu {\gt}0\) and \(0{\lt}e{\lt}1\) with \(\kappa \le e\), and put \(L=\lfloor n^{1-\ell }\rfloor \), \(m=\lfloor n^\mu \rfloor \), \(K=\lfloor L/(8m)\rfloor \), \(D=\lceil n^\kappa \rceil \) and \(\epsilon =n^{-\nu }\), the scales of [ 247 , Section 9 ] . There is a constant \(C_1\ge 1\), depending only on the exponents, such that for all sufficiently large integers \(n\): \(n\ge 2\), \(K\ge 1\), \(m\ge 1\), \(D\ge 1\), \(L\le n\), \(0{\lt}\epsilon {\lt}1\), and
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.
- TNLean.PEPS.AreaLaw.mem_latticeNeighbors_iff
- TNLean.PEPS.AreaLaw.card_latticeNeighbors_le
- TNLean.PEPS.AreaLaw.neighbor_mem_ambientDilation_succ
- TNLean.PEPS.AreaLaw.card_ambientBoundary
- TNLean.PEPS.AreaLaw.mem_ambientBoundary_iff
- TNLean.PEPS.AreaLaw.ambientBoundary_orientation_unique
- TNLean.PEPS.AreaLaw.ambientBoundary_empty
For finite lattice sets \(S,U\),
The neighbor relation is symmetric. If a finite set \(R\) meets every edge of \(\partial S\), then \(|\partial S|\le 4|R|\). In particular,
If \(C_{\rm tpl}\ge 24\) and \(0\le j\le s_0\), then
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 ] .
- TNLean.PEPS.AreaLaw.Geometry.latticeDyadicCell_eq_Icc_product
- TNLean.PEPS.AreaLaw.Geometry.latticeDyadicCell_interval_length
- TNLean.PEPS.AreaLaw.Geometry.Template.IsSeparated.lt_dist_of_mem_ambientDilation
- TNLean.PEPS.AreaLaw.Geometry.Template.IsSeparated.side_lt_dist_of_mem_ambientDilation
- TNLean.PEPS.AreaLaw.Geometry.Template.IsSeparated.dyadicCell_clearance
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.
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.
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 27.127.28.
- TNLean.PEPS.Approximation.blockRepainting_final
- TNLean.PEPS.Approximation.levelLabels_insert_of_ne
- TNLean.PEPS.Approximation.blockGuide_levelLabels_start
- TNLean.PEPS.Approximation.levelLabels_mem
- TNLean.PEPS.Approximation.levelLabels_empty_mem
- TNLean.PEPS.Approximation.levelLabels_mem_of_subset
- TNLean.PEPS.Approximation.blockRepainting_labels
- TNLean.PEPS.Approximation.edgeWords
- TNLean.PEPS.Approximation.edgeWords_mem
- TNLean.PEPS.Approximation.bandWord_mem
- TNLean.PEPS.Approximation.levelLabels_mem_owners
- TNLean.PEPS.Approximation.SquareEdge.nbrBlock_adjacent
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_mem
- TNLean.PEPS.Approximation.RepaintingBaseline.mainGuide_mem
- TNLean.PEPS.Approximation.RepaintingBaseline.mainGuide_of_notMem
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 27.127.14, 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.
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 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 27.127.6, have diameter at most \(n\).
- TNLean.PEPS.Approximation.disjoint_edgeBand
- TNLean.PEPS.Approximation.edgeBand_neg_subset_edgeNeighbor
- TNLean.PEPS.Approximation.dist_le_of_mem_closedEdgeBand
- TNLean.PEPS.Approximation.BandOperation.changed_subset_closedEdgeBand
- TNLean.PEPS.Approximation.BandOperation.dist_le_of_mem_closure_changed
- TNLean.PEPS.Approximation.dist_le_of_mem_closure_centralRegion
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.
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 \(v\).
- TNLean.PEPS.Approximation.IsLocallyInLines.eventually_notMem
- TNLean.PEPS.Approximation.HasLineWalls.isChamberSampling_iff
- TNLean.PEPS.Approximation.HasLineWalls.exists_isChamberSampling
- TNLean.PEPS.Approximation.hasLineWalls_lt
- TNLean.PEPS.Approximation.hasLineWalls_floor
- TNLean.PEPS.Approximation.isLocallyInLines_bandWidth_eq
- TNLean.PEPS.Approximation.isLocallyInLines_div_eq_int
- TNLean.PEPS.Approximation.hasLineWalls_mem_edgeBand
- TNLean.PEPS.Approximation.hasLineWalls_mem_centralRegion
- TNLean.PEPS.Approximation.hasLineWalls_mem_ball
- TNLean.PEPS.Approximation.hasLineWalls_bandCoord_lt
- TNLean.PEPS.Approximation.hasLineWalls_bandWord
- TNLean.PEPS.Approximation.word_subset_wallThresholds
- TNLean.PEPS.Approximation.hasLineWalls_blockGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide.isRayConstant
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide.hasLineWalls
- TNLean.PEPS.Approximation.hasLineWalls_schedule
- TNLean.PEPS.Approximation.isChamberSampling_schedule
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.
- TNLean.PEPS.Approximation.le_dist_blockCorner_of_ne
- TNLean.PEPS.Approximation.disjoint_ball_blockCorner
- TNLean.PEPS.Approximation.bandWidth_of_le_half
- TNLean.PEPS.Approximation.bandWidth_of_half_le
- TNLean.PEPS.Approximation.mem_edgeBand_iff_of_mem_ball
- TNLean.PEPS.Approximation.mem_edgeBand_iff_of_mem_ball_end
- TNLean.PEPS.Approximation.eq_point_of_mem_rim
- TNLean.PEPS.Approximation.eq_point_of_mem_rim_end
- TNLean.PEPS.Approximation.rim_separation
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 \(\{ \alpha '{\lt}x{\lt}\beta '\} \).
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.
- TNLean.PEPS.Approximation.IsTrueVertex.eq_blockCorner
- TNLean.PEPS.Approximation.IsBlockCornerOf
- TNLean.PEPS.Approximation.isBlockCornerOf_of_mem_closedSquare
- TNLean.PEPS.Approximation.le_dist_blockCorner
- TNLean.PEPS.Approximation.isTrueVertex_blockGuide_update
- TNLean.PEPS.Approximation.levelLabels_insert
- TNLean.PEPS.Approximation.isTrueVertex_levelLabels_insert
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 \(a_1=a_0\).
- TNLean.PEPS.Approximation.holeRadius
- TNLean.PEPS.Approximation.holeRadius_of_le
- TNLean.PEPS.Approximation.homogenize
- TNLean.PEPS.Approximation.edgeTreated
- TNLean.PEPS.Approximation.cornerTreated
- TNLean.PEPS.Approximation.homogenize_changed_subset
- TNLean.PEPS.Approximation.homogenize_ne_subset
- TNLean.PEPS.Approximation.homogenize_noncommon_subset
- TNLean.PEPS.Approximation.floor_of_subset
- TNLean.PEPS.Approximation.le_edgeMarkDist_of_notMem
- TNLean.PEPS.Approximation.le_cornerMarkDist_of_notMem
- TNLean.PEPS.Approximation.BandOperation.floor_bandUpdate
- TNLean.PEPS.Approximation.RepaintingBaseline.floor_central
- TNLean.PEPS.Approximation.RepaintingBaseline.floor_exchange
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 \(x\in \{ u,\, 2L-3-u\} \).
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.
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.
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.mono
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.relabel
- TNLean.PEPS.Approximation.similarity
- TNLean.PEPS.Approximation.dist_similarity
- TNLean.PEPS.Approximation.similarity_inv
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.image_similarity
- TNLean.PEPS.Approximation.isCompact_footprintJointSet
- TNLean.PEPS.Approximation.exists_hasTwoOwnerFootprints_Icc
- TNLean.PEPS.Approximation.exists_footprintRatio
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.of_template
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.of_le_radii
- TNLean.PEPS.Approximation.norm_smul_inv_sub_le_iff
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.
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.
- TNLean.PEPS.Approximation.isTrueVertex_congr
- TNLean.PEPS.Approximation.not_isTrueVertex_of_subset_pair
- TNLean.PEPS.Approximation.incidentLabels_congr
- TNLean.PEPS.Approximation.incidentLabels_subset
- TNLean.PEPS.Approximation.mem_range_of_mem_incidentLabels
- TNLean.PEPS.Approximation.incidentLabels_eq_or_subset_pair
- TNLean.PEPS.Approximation.isTrueVertex_iff_of_two_labels
- TNLean.PEPS.Approximation.incidentLabels_piecewise
- TNLean.PEPS.Approximation.isTrueVertex_piecewise_iff
- TNLean.PEPS.Approximation.incidentLabels_homogenize_subset
- TNLean.PEPS.Approximation.IsTrueVertex.of_homogenize
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 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.
- TNLean.PEPS.Approximation.subset_or_subset_compl_of_disjoint_frontier
- TNLean.PEPS.Approximation.closedBall_subset_or_subset_compl
- TNLean.PEPS.Approximation.exists_mem_frontier_dist_le
- TNLean.PEPS.Approximation.le_dist_of_frontier
- TNLean.PEPS.Approximation.RepaintingBaseline.outerHole_subset_or_subset_compl
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.
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 \(\varepsilon {\gt}0\).
- TNLean.PEPS.Approximation.IsRayConstant
- TNLean.PEPS.Approximation.IsRayAffine
- TNLean.PEPS.Approximation.IsRayConstant.exists_isDisplacedReading
- TNLean.PEPS.Approximation.IsDisplacedReading.unique
- TNLean.PEPS.Approximation.eventually_affine_trichotomy
- TNLean.PEPS.Approximation.IsRayAffine.isRayConstant_lt
- TNLean.PEPS.Approximation.IsRayAffine.isRayConstant_div_lt
- TNLean.PEPS.Approximation.IsRayAffine.isRayConstant_floor
- TNLean.PEPS.Approximation.isRayConstant_mem_edgeBand
- TNLean.PEPS.Approximation.isRayConstant_mem_centralRegion
- TNLean.PEPS.Approximation.isRayConstant_mem_ball
- TNLean.PEPS.Approximation.isRayConstant_bandWord
- TNLean.PEPS.Approximation.isRayConstant_blockGuide
- TNLean.PEPS.Approximation.IsRayConstant.homogenize
- TNLean.PEPS.Approximation.exists_eq_bandWord_of_mem_edgeWords
- TNLean.PEPS.Approximation.isRayConstant_blockSchedule
- TNLean.PEPS.Approximation.exists_readings_exchange
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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- TNLean.PEPS.Approximation.bandWord_eventually_eq
- TNLean.PEPS.Approximation.edgeCurves
- TNLean.PEPS.Approximation.mainWords_eventually_eq
- TNLean.PEPS.Approximation.auxWords_eventually_eq
- TNLean.PEPS.Approximation.eventually_bandUpdate_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.eventually_centralGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.eventually_completedGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide.eventually_eq
- TNLean.PEPS.Approximation.rimPattern
- TNLean.PEPS.Approximation.rimPattern_finite
- TNLean.PEPS.Approximation.dist_of_mem_rimPattern
- TNLean.PEPS.Approximation.le_dist_of_mem_rimPattern
- TNLean.PEPS.Approximation.mem_rimPattern_of_grid
- TNLean.PEPS.Approximation.mem_rimPattern_of_curve
- TNLean.PEPS.Approximation.mem_rimPattern_of_incidentLabels
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 27.127.13 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.
- TNLean.PEPS.Approximation.IsDisplacedReading
- TNLean.PEPS.Approximation.isDisplacedReading_const
- TNLean.PEPS.Approximation.tendsto_add_smul_nhdsGT
- TNLean.PEPS.Approximation.IsDisplacedReading.eq_of_eventuallyEq
- TNLean.PEPS.Approximation.IsDisplacedReading.ne_subset_closure
- TNLean.PEPS.Approximation.IsDisplacedReading.closure_ne_subset
- TNLean.PEPS.Approximation.IsDisplacedReading.closure_ne_const_subset
- TNLean.PEPS.Approximation.IsDisplacedReading.closure_noncommon_subset
- TNLean.PEPS.Approximation.IsDisplacedReading.closure_noncommon_inter_subset
- TNLean.PEPS.Approximation.IsDisplacedReading.clearance
- TNLean.PEPS.Approximation.IsDisplacedReading.dist_le
- TNLean.PEPS.Approximation.blockRepainting_central_of_reading
- TNLean.PEPS.Approximation.blockRepainting_auxStep_of_reading
- TNLean.PEPS.Approximation.blockRepainting_mainStep_of_reading
- TNLean.PEPS.Approximation.blockRepainting_exchange_of_reading
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.
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_of_mem_edgeBand
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_eq_edgeStartWord
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_eq_mainGuide_start
- TNLean.PEPS.Approximation.RepaintingBaseline.lensExchange_mainGuide
- TNLean.PEPS.Approximation.auxSteps_chain
- TNLean.PEPS.Approximation.mainSteps_chain
- TNLean.PEPS.Approximation.RepaintingBaseline.mainGuide_mainWordFive
- TNLean.PEPS.Approximation.RepaintingBaseline.auxGuide_uniform
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_of_notMem_closedSquare
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_of_mem_openSquare
- TNLean.PEPS.Approximation.exists_edgeBand_of_notMem_centralRegion
- TNLean.PEPS.Approximation.mem_centralRegion_of_forall_notMem
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.
- TNLean.PEPS.Approximation.IsCellCenter
- TNLean.PEPS.Approximation.IsCellCenter.sub_blockCorner
- TNLean.PEPS.Approximation.IsCellCenter.mem_openSquare_or_notMem_closedSquare
- TNLean.PEPS.Approximation.IsCellCenter.notMem_edgeInterface
- TNLean.PEPS.Approximation.blockRepainting_start_site
- TNLean.PEPS.Approximation.blockRepainting_final_site
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.
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.
- TNLean.PEPS.Approximation.IsTrueVertex.exists_isTagHolder
- TNLean.PEPS.Approximation.isTagHolder_congr
- TNLean.PEPS.Approximation.IsTagHolder.mem_of_forall
- TNLean.PEPS.Approximation.isTagHolder_levelStart
- TNLean.PEPS.Approximation.isTagHolder_level
- TNLean.PEPS.Approximation.isTagHolder_blockRepainting
- TNLean.PEPS.Approximation.isTagHolder_levelEnd
- TNLean.PEPS.Approximation.RepaintingBaseline.nbrLabel_mem_range_mainGuide
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.
- TNLean.PEPS.Approximation.clearance_shift
- TNLean.PEPS.Approximation.dist_le_shift
- TNLean.PEPS.Approximation.le_dist_of_mem_closure
- TNLean.PEPS.Approximation.cornerMarkDist_sub
- TNLean.PEPS.Approximation.edgeMarkDist_sub
- TNLean.PEPS.Approximation.sub_mem_closure
- TNLean.PEPS.Approximation.sub_mem_frontier
- TNLean.PEPS.Approximation.mem_closure_shift
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.
- TNLean.PEPS.Approximation.HasTwoLabelsNear
- TNLean.PEPS.Approximation.HasTwoLabelsNear.not_isTrueVertex
- TNLean.PEPS.Approximation.HasTwoLabelsNear.congr
- TNLean.PEPS.Approximation.hasTwoLabelsNear_of_eventuallyEq_const
- TNLean.PEPS.Approximation.hasTwoLabelsNear_blockGuide
- TNLean.PEPS.Approximation.HasTwoLabelsNear.comp
- TNLean.PEPS.Approximation.bandWord_eventually_step
- TNLean.PEPS.Approximation.bandWord_eventually_mem
- TNLean.PEPS.Approximation.continuousAt_bandCoord
- TNLean.PEPS.Approximation.eq_point_zero_of_mem_closedEdgeBand
- TNLean.PEPS.Approximation.eq_point_end_of_mem_closedEdgeBand
- TNLean.PEPS.Approximation.hasTwoLabelsNear_bandUpdate
- TNLean.PEPS.Approximation.IsSquareCorner
- TNLean.PEPS.Approximation.isSquareCorner_point_zero
- TNLean.PEPS.Approximation.isSquareCorner_point_end
- TNLean.PEPS.Approximation.isOpen_openSquare
- TNLean.PEPS.Approximation.isOpen_centralRegion
- TNLean.PEPS.Approximation.exists_edge_of_mem_closedSquare
- TNLean.PEPS.Approximation.mem_openSquare_of_mem_closure_centralRegion
- TNLean.PEPS.Approximation.mem_centralRegion_of_nor_eq_two_mul
- TNLean.PEPS.Approximation.notMem_closedEdgeBand_of_nor_eq_two_mul
- TNLean.PEPS.Approximation.mainWords
- TNLean.PEPS.Approximation.auxWordExchanged
- TNLean.PEPS.Approximation.auxWords
- TNLean.PEPS.Approximation.mainWords_props
- TNLean.PEPS.Approximation.auxWords_props
- TNLean.PEPS.Approximation.exists_eq_bandWord_of_mem_mainWords
- TNLean.PEPS.Approximation.exists_eq_bandWord_of_mem_auxWords
- TNLean.PEPS.Approximation.mem_mainWords_of_mem_mainSteps
- TNLean.PEPS.Approximation.mem_auxWords_of_mem_auxSteps
- TNLean.PEPS.Approximation.RepaintingBaseline.hasTwoLabelsNear_centralGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.eventually_completedGuide_nbrLabel
- TNLean.PEPS.Approximation.RepaintingBaseline.eventually_completedGuide_finalLabel
- TNLean.PEPS.Approximation.RepaintingBaseline.hasTwoLabelsNear_completedGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.hasTwoLabelsNear_mainGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.hasTwoLabelsNear_auxGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide.hasTwoLabelsNear
- TNLean.PEPS.Approximation.IsTrueVertex.eq_blockCorner_of_isUnmodifiedGuide
- TNLean.PEPS.Approximation.eq_blockCorner_of_isTrueVertex_of_mem_closedBall
- TNLean.PEPS.Approximation.exists_eventuallyEq_bandUpdate_of_mem_closedEdgeBand
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 \(W\).
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).
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.sitesY
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.sitesN
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.renameWord
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.eval_renameWord
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.twoLayoutIso_renameWord
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.isReordering_renameWord
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).
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 \(\psi \)
- TNLean.PEPS.EncodedFrame.pairSource_conjTranspose_mul_self
- TNLean.PEPS.EncodedFrame.pairSource_mulVec
- TNLean.PEPS.EncodedFrame.birthOp_isHermitian
- TNLean.PEPS.EncodedFrame.norm_birthOp_le_one
- TNLean.PEPS.EncodedFrame.norm_act_birthOp_sub_le
- TNLean.PEPS.EncodedFrame.birthEffect_eq_submatrix
- TNLean.PEPS.EncodedFrame.birthOp_eq_submatrix
- TNLean.PEPS.EncodedFrame.commute_birthOp
- TNLean.PEPS.EncodedFrame.sheetBirthOp_isHermitian
- TNLean.PEPS.EncodedFrame.norm_sheetBirthOp_le_one
- TNLean.PEPS.EncodedFrame.norm_act_sheetBirthOp_sub_le
- TNLean.PEPS.EncodedFrame.commute_sheetBirthOp
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 \(B=\operatorname{id}_E\otimes K^\dagger K\).
- TNLean.PEPS.EncodedFrame.pairHeadIso_birthWord_eval
- TNLean.PEPS.EncodedFrame.pairHeadIso_deathChain_eval
- TNLean.PEPS.EncodedFrame.pairHeadIso_birthPrivateWord_eval
- TNLean.PEPS.EncodedFrame.sourceThenAdjoint
- TNLean.PEPS.EncodedFrame.sourceThenAdjoint_pairVec
- TNLean.PEPS.EncodedFrame.pairHeadIso_frame_localMap_source
- TNLean.PEPS.EncodedFrame.effectNext
- TNLean.PEPS.EncodedFrame.effectNext_apply
- TNLean.PEPS.EncodedFrame.effectNext_act
- TNLean.PEPS.EncodedFrame.norm_birthKernel_le_one
- Matrix.IsIsometry.l2_opNorm_le_one
- TNLean.PEPS.EncodedFrame.norm_toLp_eq_one_of_star_dotProduct
- TNLean.PEPS.EncodedFrame.isAllowed_birthWord
- TNLean.PEPS.EncodedFrame.isAllowed_birthPrivateWord
- TNLean.PEPS.EncodedFrame.isAllowed_deathChain
- TNLean.PEPS.EncodedFrame.usesOnly_birthWord
- TNLean.PEPS.EncodedFrame.sourceCount_birthWord
- TNLean.PEPS.EncodedFrame.usesOnly_birthPrivateWord
- TNLean.PEPS.EncodedFrame.sourceCount_birthPrivateWord
- TNLean.PEPS.EncodedFrame.usesOnly_deathChain
- TNLean.PEPS.EncodedFrame.sourceCount_deathChain
- TNLean.PEPS.EncodedFrame.effectCount_deathChain
- TNLean.PEPS.EncodedFrame.sheetBirthOp_eq_submatrix
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).
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).
- TNLean.PEPS.EncodedFrame.SquarePatch.tagVec
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.branchSteps
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.branchRaw_eq_chainOp
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.patchVertices
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.card_patchVertices_le
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.mem_patchVertices_of_isOpen
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_owners_of_isOpen_inl
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_owners_of_isOpen_inr
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).
- TNLean.PEPS.EncodedFrame.PatchTags
- TNLean.PEPS.EncodedFrame.cylProd
- TNLean.PEPS.EncodedFrame.projProd_eq_sum_cylProd
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.Branch
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.branchRaw
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.branchOp
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.affectedRewrite_eq_sum_branchOp
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.rewrite_eq_sum_branch
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.card_branch_le
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).
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).
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 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.
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).
- TNLean.PEPS.EncodedFrame.partSwap
- TNLean.PEPS.EncodedFrame.correctionMatrix
- TNLean.PEPS.EncodedFrame.norm_correctionMatrix_le_one
- TNLean.PEPS.EncodedFrame.sheetSwapOp_eq_submatrix_of_subset
- TNLean.PEPS.EncodedFrame.sheetBufferCorrection_mul_sheetSwapOp_eq
- TNLean.PEPS.EncodedFrame.correctionWord
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exchange_correction_monomial
For a unit vector \(\omega \) and \(\delta =\lVert \mathcal V\omega -s\otimes s'\rVert \),
- EuclideanSpace.vecKron
- TNLean.PEPS.EncodedFrame.act_kronecker_vecKron
- EuclideanSpace.norm_vecKron
- EuclideanSpace.norm_vecKron_self_sub_le
- TNLean.PEPS.EncodedFrame.act_tbSwap_vecKron_tensorPurification
- TNLean.PEPS.EncodedFrame.norm_toLp_tensorPurification
- TNLean.PEPS.EncodedFrame.norm_bufferCorrection_le_one
- TNLean.PEPS.EncodedFrame.norm_act_bufferCorrection_mul_tSwap_sub_le
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).
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.rename_mul_encoder
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exists_out_form
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exists_in_form
- TNLean.PEPS.EncodedFrame.sheetSplit_sheetSwap_fst
- TNLean.PEPS.EncodedFrame.sheetSplit_sheetSwap_snd
- TNLean.PEPS.EncodedFrame.mul_submatrix_sheetSplit
With \(\mathcal V=\operatorname{id}_{TE}\otimes V\),
(05-frames.tex, equation (exchange-net-map), lines 516–526).
- TNLean.PEPS.EncodedFrame.tSwapW_toMatrix
- TNLean.PEPS.EncodedFrame.tbSwap_toMatrix
- Matrix.kronecker_kronecker_eq_submatrix
- Matrix.toMatrix_toPEquiv_prodCongr
- Matrix.toMatrix_toPEquiv_submatrix
- TNLean.PEPS.EncodedFrame.bufferCorrection_mul_tSwap
- TNLean.PEPS.EncodedFrame.sheetSwapOp_eq_submatrix
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.sheetBufferCorrection_mul_sheetSwapOp
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).
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 \(\lVert K_{\mathcal F}\rVert \le 1\).
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 \(\lVert (\operatorname{id}\otimes B)\Omega _{\mathcal F}-\Omega _{\mathcal F}\rVert \le \lVert B\Omega -\Omega \rVert \).
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 \(|c\rangle _\ell =|c'\rangle _\ell \).
- TNLean.PEPS.EncodedFrame.siteVec
- TNLean.PEPS.EncodedFrame.siteVec_congr
- TNLean.PEPS.EncodedFrame.siteIsoList
- TNLean.PEPS.EncodedFrame.siteIsoList_siteVec
- TNLean.PEPS.EncodedFrame.groupIso
- TNLean.PEPS.EncodedFrame.groupIso_siteVec
- TNLean.PEPS.EncodedFrame.tagVec
- TNLean.PEPS.EncodedFrame.tagIso
- TNLean.PEPS.EncodedFrame.tagIso_tagVec
- TNLean.PEPS.EncodedFrame.layoutVec
- TNLean.PEPS.EncodedFrame.layoutIso
- TNLean.PEPS.EncodedFrame.layoutIso_layoutVec
- TNLean.PEPS.EncodedFrame.layoutIso_symm_single
- TNLean.PEPS.EncodedFrame.Frame.regIso
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).
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).
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 \(\operatorname{id}_T\otimes A'\otimes \operatorname{id}_U\).
- TNLean.PEPS.EncodedFrame.sheetSplit
- TNLean.PEPS.EncodedFrame.threeSplit
- TNLean.PEPS.EncodedFrame.threeSplit_fst
- TNLean.PEPS.EncodedFrame.threeSplit_snd
- TNLean.PEPS.EncodedFrame.rectKronecker_submatrix_sheetSplit
- TNLean.PEPS.EncodedFrame.rectKronecker_submatrix_threeSplit
- TNLean.PEPS.EncodedFrame.exists_kronecker_one_of_mem_supportedOperators
- TNLean.PEPS.EncodedFrame.exists_one_kronecker_of_mem_supportedOperators
- TNLean.PEPS.EncodedFrame.exists_threeSplit_eq_kronecker_one
- TNLean.PEPS.EncodedFrame.exists_threeSplit_eq_one_kronecker_kronecker_one
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.
- TNLean.PEPS.EncodedFrame.SplittingData
- TNLean.PEPS.EncodedFrame.SplittingData.error
- TNLean.PEPS.EncodedFrame.splitVec
- TNLean.PEPS.EncodedFrame.partialTraceRight_splitVec
- TNLean.PEPS.EncodedFrame.quantumRelativeEntropy_splitVec_eq_mutualInformation
- TNLean.PEPS.EncodedFrame.exists_sheetSplitting
- TNLean.PEPS.EncodedFrame.exists_sheetSplitting_zpow
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 \(\operatorname{id}\otimes Y\).
- TNLean.PEPS.EncodedFrame.layoutIso_eq_act_of_basis
- TNLean.PEPS.EncodedFrame.regBra
- TNLean.PEPS.EncodedFrame.regKet
- TNLean.PEPS.EncodedFrame.norm_regBra_le
- TNLean.PEPS.EncodedFrame.norm_regKet_le
- TNLean.PEPS.EncodedFrame.tagBraWord
- TNLean.PEPS.EncodedFrame.tagKetWord
- TNLean.PEPS.EncodedFrame.eval_tagBraWord
- TNLean.PEPS.EncodedFrame.eval_tagKetWord
- TNLean.PEPS.EncodedFrame.tagBraWord_props
- TNLean.PEPS.EncodedFrame.tagKetWord_props
- TNLean.PEPS.EncodedFrame.tagChangeWord
- TNLean.PEPS.EncodedFrame.eval_tagChangeWord
- TNLean.PEPS.EncodedFrame.tagChangeWord_props
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch_apply
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch_mul
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch_sum_smul
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch_smul
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.layoutIso_tagChangeWord
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).
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).
- TNLean.PEPS.EncodedFrame.SquarePatch.tagSample
- TNLean.PEPS.EncodedFrame.patchSquares
- TNLean.PEPS.EncodedFrame.holeSquares
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.oldSample_subset_patchSquares
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.newSample_subset_patchSquares
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_old_owners_of_patch_input
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_new_owners_of_patch_output
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.ownerOld_eq_ownerNew_of_direct
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.owners_mem_parties_of_mem_sample
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 \(j\).
- TNLean.PEPS.EncodedFrame.act
- TNLean.PEPS.EncodedFrame.act_apply
- TNLean.PEPS.EncodedFrame.act_apply_apply
- TNLean.PEPS.EncodedFrame.ofLp_act
- TNLean.PEPS.EncodedFrame.norm_act
- TNLean.PEPS.EncodedFrame.act_add
- TNLean.PEPS.EncodedFrame.act_mul
- TNLean.PEPS.PairEffect.oneIso
- TNLean.PEPS.PairEffect.twoIso
- TNLean.PEPS.PairEffect.pairHeadIso_eq
- TNLean.PEPS.PairEffect.appendIso_one_symm
- TNLean.PEPS.PairEffect.pairHeadIso_appendIso_symm
- TNLean.PEPS.PairEffect.matLocal₂₁
- TNLean.PEPS.PairEffect.matLocal₁₂
- TNLean.PEPS.PairEffect.matLocal₂₂
- TNLean.PEPS.PairEffect.eval_localMap
- TNLean.PEPS.PairEffect.eval_localMap₂₁
- TNLean.PEPS.PairEffect.eval_localMap₁₂
- TNLean.PEPS.PairEffect.pairHeadIso_eval_localMap₂₂
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 \(MQ=M\).
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.
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 \(\operatorname{id}_{\rm tags}\otimes (\operatorname{id}_E\otimes M)\).
- TNLean.PEPS.EncodedFrame.groupWord
- TNLean.PEPS.EncodedFrame.ungroupWord
- TNLean.PEPS.EncodedFrame.eval_groupWord
- TNLean.PEPS.EncodedFrame.eval_ungroupWord
- TNLean.PEPS.EncodedFrame.groupWord_props
- TNLean.PEPS.EncodedFrame.ungroupWord_props
- TNLean.PEPS.EncodedFrame.relabelRest
- TNLean.PEPS.EncodedFrame.isReordering_relabelRest
- TNLean.PEPS.EncodedFrame.eval_place
- TNLean.PEPS.EncodedFrame.sheetPlace
- TNLean.PEPS.EncodedFrame.sum_single_merge
- TNLean.PEPS.EncodedFrame.layoutIso_place
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.
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.
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.
Let \(V=\bigoplus _{i\in I}V_i\) be a finite orthogonal direct sum. Any injective relabelling of orthonormal endpoint coordinates defines an isometric inclusion. In particular, the natural inclusion
is an isometry. If a bond vector is obtained by arranging the entries of matrices \(X_i\) in their respective summands, its image under \(E_V\) is the vector of entries of \(\bigoplus _i X_i\).
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 ] .
- TNLean.PEPS.torusIncidentCoordinatesEquiv
- TNLean.PEPS.torusIncidentCoordinatesEquiv_apply
- TNLean.PEPS.torusIncidentCoordinatesEquiv_smul
- TNLean.PEPS.regularSiteMap_torusIncidentSite_eq_siteMap
- TNLean.PEPS.IsGInjective.isGInjective_torusIncidentSite
- TNLean.PEPS.torusGClosure_one_eq_stateCoeff_torusIncidentSite
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 ] .
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.
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.
- TNLean.PEPS.SiteChain.eq_of_not_earlierTouch
- TNLean.PEPS.SiteChain.eq_of_not_laterTouch
- TNLean.PEPS.SiteChain.eq_of_isOpen_inl
- TNLean.PEPS.SiteChain.eq_of_isOpen_inr
- TNLean.PEPS.SiteChain.not_untouched_of_isOpen
- TNLean.PEPS.SiteChain.fixedIn_iff
- TNLean.PEPS.SiteChain.fixedOut_iff
- TNLean.PEPS.SiteChain.prod_braBlocks_apply
- TNLean.PEPS.SiteChain.prod_ketBlocks_apply
- TNLean.PEPS.SiteChain.prod_zeroBlocks_apply
- TNLean.PEPS.SiteChain.termOp_eq_smul_prod
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
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.
- TNLean.PEPS.EncodedFrame.exists_rankOneBlock_chain
- TNLean.PEPS.EncodedFrame.exists_pairVec
- TNLean.PEPS.EncodedFrame.exists_groupBlock
- TNLean.PEPS.EncodedFrame.zeroGroupVec
- TNLean.PEPS.EncodedFrame.norm_zeroGroupVec
- TNLean.PEPS.EncodedFrame.zeroGroupVec_apply
- TNLean.PEPS.EncodedFrame.exists_braVertexBlock
- TNLean.PEPS.EncodedFrame.exists_ketVertexBlock
- TNLean.PEPS.EncodedFrame.prod_map_one_kronecker
- TNLean.PEPS.EncodedFrame.exists_chain_listProd
- TNLean.PEPS.EncodedFrame.exists_zeroChain
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 \(c\).
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 \(p_A=p_B\).
- TNLean.PEPS.PairEffect.PartyChain.append
- TNLean.PEPS.PairEffect.PartyChain.eval_append
- TNLean.PEPS.PairEffect.PartyChain.append_props
- TNLean.PEPS.PairEffect.braLocal
- TNLean.PEPS.PairEffect.ketLocal
- TNLean.PEPS.PairEffect.norm_braLocal_le
- TNLean.PEPS.PairEffect.norm_ketLocal_le
- TNLean.PEPS.PairEffect.owner_of_mem_pair
- TNLean.PEPS.PairEffect.owner_of_mem_nil
- TNLean.PEPS.PairEffect.braChain
- TNLean.PEPS.PairEffect.ketWord
- TNLean.PEPS.PairEffect.rankOneChain
- TNLean.PEPS.PairEffect.eval_braChain
- TNLean.PEPS.PairEffect.pairHeadIso_ketWord
- TNLean.PEPS.PairEffect.act_vecMulVec_star
- TNLean.PEPS.PairEffect.pairHeadIso_rankOneChain
- TNLean.PEPS.PairEffect.rankOneChain_props
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 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 \(R\).
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.
- TNLean.PEPS.EncodedFrame.relabelHead
- TNLean.PEPS.EncodedFrame.relabelSites
- TNLean.PEPS.EncodedFrame.eval_relabelSites
- TNLean.PEPS.EncodedFrame.partSites
- TNLean.PEPS.EncodedFrame.partSites_eq
- TNLean.PEPS.EncodedFrame.partWord
- TNLean.PEPS.EncodedFrame.unpartWord
- TNLean.PEPS.EncodedFrame.eval_partWord
- TNLean.PEPS.EncodedFrame.eval_unpartWord
- TNLean.PEPS.PairEffect.Word.eval_frameList_appendIso_symm
- TNLean.PEPS.PairEffect.Word.appendNil
- TNLean.PEPS.PairEffect.Word.dropNil
- TNLean.PEPS.PairEffect.Word.eval_appendNil
- TNLean.PEPS.PairEffect.Word.eval_dropNil
- TNLean.PEPS.PairEffect.Word.assocWord
- TNLean.PEPS.PairEffect.Word.unassocWord
- TNLean.PEPS.PairEffect.Word.eval_assocWord_appendIso_symm
- TNLean.PEPS.PairEffect.Word.eval_unassocWord_appendIso_symm
- TNLean.PEPS.PairEffect.Word.swapPairs
- TNLean.PEPS.PairEffect.Word.eval_swapPairs
- TNLean.PEPS.EncodedFrame.partWordApp
- TNLean.PEPS.EncodedFrame.unpartWordApp
- TNLean.PEPS.EncodedFrame.eval_partWordApp
- TNLean.PEPS.EncodedFrame.eval_unpartWordApp
- TNLean.PEPS.EncodedFrame.relabelSitesApp
- TNLean.PEPS.EncodedFrame.eval_relabelSitesApp
- TNLean.PEPS.EncodedFrame.tagSplit
- TNLean.PEPS.EncodedFrame.tagMerge
- TNLean.PEPS.EncodedFrame.eval_tagSplit
- TNLean.PEPS.EncodedFrame.eval_tagMerge
- TNLean.PEPS.EncodedFrame.exists_tagWord_of_perm
- TNLean.PEPS.PairEffect.Word.isAllowed_frameList_iff
- TNLean.PEPS.PairEffect.Word.usesOnly_frameList_iff
- TNLean.PEPS.PairEffect.Word.sourceCount_frameList
- TNLean.PEPS.EncodedFrame.eval_relabelHead
- TNLean.PEPS.EncodedFrame.mem_partSites_fst
- TNLean.PEPS.EncodedFrame.mem_partSites_snd
- TNLean.PEPS.EncodedFrame.nodup_partSites_fst
- TNLean.PEPS.EncodedFrame.nodup_partSites_snd
- TNLean.PEPS.EncodedFrame.partStepApp
- TNLean.PEPS.EncodedFrame.unpartStepApp
- TNLean.PEPS.EncodedFrame.eval_partStepApp
- TNLean.PEPS.EncodedFrame.eval_unpartStepApp
- TNLean.PEPS.PairEffect.Word.IsReordering
- TNLean.PEPS.PairEffect.Word.IsReordering.isAllowed
- TNLean.PEPS.PairEffect.Word.IsReordering.usesOnly
- TNLean.PEPS.PairEffect.Word.IsReordering.sourceCount_eq
- TNLean.PEPS.PairEffect.Word.IsReordering.comp
- TNLean.PEPS.PairEffect.Word.IsReordering.frameList
- TNLean.PEPS.PairEffect.Word.isReordering_id
- TNLean.PEPS.PairEffect.Word.isReordering_swap
- TNLean.PEPS.PairEffect.Word.isReordering_comp_iff
- TNLean.PEPS.PairEffect.Word.isReordering_frame_iff
- TNLean.PEPS.PairEffect.Word.isReordering_frameList_iff
- TNLean.PEPS.PairEffect.Word.isReordering_moveHead
- TNLean.PEPS.PairEffect.Word.isReordering_exchangeBlocks
- TNLean.PEPS.PairEffect.Word.isReordering_appendNil
- TNLean.PEPS.PairEffect.Word.isReordering_dropNil
- TNLean.PEPS.PairEffect.Word.isReordering_assocWord
- TNLean.PEPS.PairEffect.Word.isReordering_unassocWord
- TNLean.PEPS.PairEffect.Word.isReordering_swapPairs
- TNLean.PEPS.EncodedFrame.isReordering_relabelHead
- TNLean.PEPS.EncodedFrame.isReordering_partStepApp
- TNLean.PEPS.EncodedFrame.isReordering_unpartStepApp
- TNLean.PEPS.EncodedFrame.isReordering_partWordApp
- TNLean.PEPS.EncodedFrame.isReordering_unpartWordApp
- TNLean.PEPS.EncodedFrame.isReordering_relabelSitesApp
- TNLean.PEPS.EncodedFrame.isReordering_relabelSites
- TNLean.PEPS.EncodedFrame.isReordering_partWord
- TNLean.PEPS.EncodedFrame.isReordering_unpartWord
- TNLean.PEPS.EncodedFrame.isReordering_tagSplit
- TNLean.PEPS.EncodedFrame.isReordering_tagMerge
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 ] .
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 ] .
- TNLean.PEPS.regularRegionHalfEdgeRightMul
- TNLean.PEPS.regularProjectorTwistedRegionMatrix_halfEdgeRightMul
- TNLean.PEPS.regularProjectorOpenRegionMatrix_halfEdgeRightMul
- TNLean.PEPS.regularProjectorOpenRegionMatrix_internalEdgeRightMul
- TNLean.PEPS.regionPhysicalMap_halfEdgeRightMul_of_mem_regularGroundSpace
- TNLean.PEPS.regionPhysicalMap_slice_halfEdgeRightMul_of_parent_annihilates
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 ] .
The set \(\mathcal D\) is compact and nonempty. Each \(L_j\) and the objective \(F\) are continuous on \(\mathcal D\).
Let \(G\) be a finite group and assign to every site \(v\) of the torus two elements \(a_v,b_v\in G\). Call \(f\) from the sites to \(G\) a parallel section if \(f_{v+(1,0)}=a_vf_v\) and \(f_{v+(0,1)}=b_vf_v\) for every \(v\). Parallel sections exist if and only if \(b_{v+(1,0)}a_v=a_{v+(0,1)}b_v\) for every \(v\) and the ordered products \(a_{(w-1,0)}\cdots a_{(0,0)}\) and \(b_{(0,h-1)}\cdots b_{(0,0)}\) are the identity. If they exist, there are exactly \(|G|\) of them, and any two differ by right multiplication with one group element.
- TNLean.PEPS.zmodTransport
- TNLean.PEPS.zmodTransport_zero
- TNLean.PEPS.zmodTransport_succ
- TNLean.PEPS.apply_natCast_eq_zmodTransport_mul
- TNLean.PEPS.zmodTransport_self_eq_one_of_apply_add_one
- TNLean.PEPS.zmodTransport_val_add_one
- TNLean.PEPS.IsTorusParallelSection
- TNLean.PEPS.IsTorusFlat
- TNLean.PEPS.IsTorusParallelSection.mul_right
- TNLean.PEPS.IsTorusParallelSection.isTorusFlat
- TNLean.PEPS.IsTorusParallelSection.zmodTransport_row
- TNLean.PEPS.IsTorusParallelSection.zmodTransport_column
- TNLean.PEPS.IsTorusFlat.zmodTransport_column_mul
- TNLean.PEPS.IsTorusParallelSection.eq_mul_right
- TNLean.PEPS.IsTorusFlat.zmodTransport_column_eq_one
- TNLean.PEPS.exists_isTorusParallelSection_iff
- TNLean.PEPS.card_isTorusParallelSection
Every pair of physical coefficient vectors satisfies
For any complex bond factors \(F_e\), the literal product identity is
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 ] .
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.
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 ] .
- TNLean.PEPS.torusIncidentLeg
- TNLean.PEPS.torusContiguousRectangle_one_eq_singleton
- TNLean.PEPS.torusGraph_compl_singleton_connected
- TNLean.PEPS.torusGraph_singleton_connected
- TNLean.PEPS.torusIncidentLeg_surjective
- TNLean.PEPS.torusIncidentLeg_injective
- TNLean.PEPS.torusIncidentLegEquiv
- TNLean.PEPS.card_regionBoundaryEdge_torus_singleton
- TNLean.PEPS.torusGraph_connected
The isotypic projectors \(P_i\) commute with every \(U_g\), are linearly independent, and
If \(U\) is semi-regular, then the operators \(U_g\) are linearly independent, and so are the twirls \(\sigma (U_g)\) of one representative \(g\) of each conjugacy class [ 235 , Section “Parent Hamiltonians” ] .
- Representation.rep_mul_charProjector
- Representation.apply_conj_of_mem_irreducibleCharacters
- Representation.sum_rep_conj_apply_of_mem
- Representation.averageMap_linHom_rep
- Representation.linearIndependent_charProjector
- Representation.linearIndependent_of_isSemiRegular
- Representation.linearIndependent_averageMap_conjClasses
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.
Let \(\sigma \) be a permutation of the configurations of a finite set of sites, such as the ring, that changes only the sites of a set \(X\), the new values at these sites depending only on the old values there. Its permutation matrix \(P_\sigma \), \((P_\sigma v)(x)=v(\sigma x)\), is a unitary acting on \(X\). Examples are the shift \(x\mapsto x+f(x)e_t\) at a site \(t\in X\), with values in \(\mathbb Z_d\) and \(f\) reading only the sites of \(X\setminus \{ t\} \). If \(\sigma _k\), \(k\in K\), are such permutations for pairwise disjoint pairs \(\{ k,k+1\} \), the layer of the gates \(P_{\sigma _k}\) acts as \(v\mapsto v\circ \rho \), where \(\rho (x)\) agrees with \(\sigma _k(x)\) on the pair \(\{ k,k+1\} \) and with \(x\) off all the pairs; here the sites form the ring. For shifts \(\sigma _k\colon x\mapsto x+f_k(x)e_{t_k}\) this reads \(\rho (x)_{t_k}=x_{t_k}+f_k(x)\), and \(\rho (x)_i=x_i\) at every site \(i\) that is none of the \(t_k\). Finally, for permutations \(\tau _i\) of the labels of the sites \(i\) of the ring, the product \(\bigotimes _iP_{\tau _i}\) acts as \(v\mapsto v\circ \tau \), where \(\tau (x)_i=\tau _i(x_i)\).
- QuantumCircuit.IsLocalPerm
- QuantumCircuit.IsLocalPerm.permMatrix_mem_supportedOperators
- QuantumCircuit.shiftPerm
- QuantumCircuit.shiftPerm_apply
- QuantumCircuit.isLocalPerm_shiftPerm
- QuantumCircuit.permLayer
- QuantumCircuit.exists_permLayer_op_mulVec
- QuantumCircuit.exists_shiftLayer_op_mulVec
- Matrix.finKronecker_permMatrix_mulVec
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 27.116.4, 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.
For positive \(d\), the ambient depth row is \(T_d\setminus T_{d-1}\); its physical part has no larger cardinality. Graph distance bounds depth variation: \(|d(x)-d(y)|\le d_\Lambda (x,y)\). Let \(Z\) be the actual endpoints of edges crossing \(A\). If \(\operatorname {dist}_\infty (T,Z){\gt}2L+10r_0\), every graph \(r_0\)-ball meeting \(A\cap T_L\) is wholly contained in \(A\).
- TNLean.PEPS.AreaLaw.Scan.ambientSupDistance
- TNLean.PEPS.AreaLaw.Scan.ambientDepth
- TNLean.PEPS.AreaLaw.Scan.ambientDepth_nonneg
- TNLean.PEPS.AreaLaw.Scan.abs_ambientDepth_sub_le
- TNLean.PEPS.AreaLaw.Scan.ambientDepth_le_iff_mem_dilation
- TNLean.PEPS.AreaLaw.Scan.ambientDepth_eq_iff_mem_dilation_sdiff
- TNLean.PEPS.AreaLaw.Scan.abs_ambientDepth_sub_le_walk_length
- TNLean.PEPS.AreaLaw.Scan.abs_ambientDepth_sub_le_of_edist_le
- TNLean.PEPS.AreaLaw.Scan.ambientSupDistance_le_one_of_domainGraph_adj
- TNLean.PEPS.AreaLaw.Scan.abs_ambientDepth_sub_le_domainGraph
- TNLean.PEPS.AreaLaw.Scan.CollarScan.abs_depth_sub_anchor_le_domainGraph_of_mem_ball
- TNLean.PEPS.AreaLaw.Scan.depthRow_ambientDepth_length_le
- TNLean.PEPS.AreaLaw.Scan.depthRow_ambientDepth_length_le_of_row_bound
- TNLean.PEPS.AreaLaw.Scan.domainGraph_ball_subset_of_clearance
If every positive ambient layer through \(L\) has at most \(n\) sites, then
Let \(V\) be a finite graph embedded injectively in \(\mathbb {Z}^2\), with edge displacement at most one. Let \(\mathcal F\) be a finite family of distinct finite supports such that any two sites in a support admit a walk in the full graph of length at most the nonnegative integer \(R\). Put \(v_R=1+2R(R+1)\) and \(\mu _R=2^{v_R-1}\). Each original support has at most \(v_R\) sites, and at most \(\mu _R\) original supports contain any specified site. For \(J\ge 0\) and real weights \(w_X\le J\),
If also \(w_X\ge 0\) for each \(X\in \mathcal F\), then
The continuation bound holds for each original support \(X\) and permits repetitions. Both bounds apply to \(w_X=\lVert h_X\rVert \) for elements \(h_X\) of any seminormed additive group with \(\lVert h_X\rVert \le J\). These are the interaction budgets used in [ 247 , Sections 2 and 4 ] .
- TNLean.PEPS.AreaLaw.card_support_le_diamond
- TNLean.PEPS.AreaLaw.card_supports_containing_le_diamond
- TNLean.PEPS.AreaLaw.sum_supportWeights_containing_le_diamond
- TNLean.PEPS.AreaLaw.sum_supportNorms_containing_le_diamond
- TNLean.PEPS.AreaLaw.interactionChainWeightSum_le_diamond_pow
- TNLean.PEPS.AreaLaw.interactionChainWeightSum_norm_le_diamond_pow
The diamond \(D_R(a)\) has cardinality \(v_R=1+2R(R+1)\). Let \(\iota :V\to \mathbb {Z}^2\) be injective. A finite set \(S\subseteq V\) with \(\ell (\iota (a),\iota (x))\le R\) for each \(x\in S\) has at most \(v_R\) elements. If every graph edge has displacement at most one, the same bound holds when every \(x\in S\) is reached from \(a\) by a graph walk of length at most \(R\), or when \(d_G(a,x)\le R\). This is the local counting bound of [ 247 , Section 2 ] .
Fix a common origin \(o\in \mathbb R^2\), finite endpoint set \(Z\), integer width \(C_0\ge 2\) and natural indices \(k_0,k,h\) 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.
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 \(i\).
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 \((a_s+b_s)/2\).
The fixed fine exponents \(\ell _k=\lfloor \zeta k\rfloor \) are nondecreasing. For every nonnegative integer \(k\),
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 \(t/8\).
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 ] .
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.
- TNLean.PEPS.AreaLaw.Geometry.Template.exists_sample_four_strip_bounds
- TNLean.PEPS.AreaLaw.Geometry.Template.exists_latticeRow_profile
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_nonempty_between
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_min_step
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_max_step
- TNLean.PEPS.AreaLaw.Geometry.Template.exists_sample_in_window_between
- TNLean.PEPS.AreaLaw.Geometry.Template.exists_nearby_sample_in_row
Let \(D\), \(D_{\rm ret}\) and \(L'\) be square matrices on one finite-dimensional Hilbert space, let \(\zeta \) and \(g\) be vectors, and let \(M_0,\varepsilon \) be real. Suppose that \(Dg=g\), \(\| g\| \le 1\), \(\| \zeta \| \le M_0\),
Then \(\| L'\zeta \| \le 1+3\varepsilon \). If moreover \(M_0\ge 1\) and \(D_{\rm ret}^\dagger w=w\) for a vector with \(\| w\| \le 1\), then \(\| (L')^\dagger w-w\| \le \varepsilon \). With \(g\) the ground component of the restoring vector, these are the two estimates that precede the polar decomposition in the proof of Proposition 10.2 of [ 247 ] .
Let \(\Lambda \) be a finite induced domain, let \(A\subseteq \Lambda \), let \(T\subset \mathbb Z^2\) be finite, and let \(Z\) be the set of endpoints of the cut edges of \(A\). Put \(L=\lfloor n^{1-\varepsilon }\rfloor \) and \(S_0=A\cap T_L\). For every \(C\) there is \(N\) such that for all \(n\ge N\), every integer \(r\le Cn^{(1-\varepsilon )/\alpha }\) and every \(\Lambda \), \(A\), \(T\) with \(\operatorname {dist}_\infty (T,Z){\gt}L+n^\beta \) satisfy \(N_{2r}^\Lambda (S_0)\subseteq A\). This is the last assertion of Proposition 10.2 of [ 247 ] , given its radius bound.
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.
Fix \(q,R,J,\Delta \). Suppose that, under the hypotheses of the amplification bound, for every \(C_0{\gt}0\) and every integer \(M\ge 1\) there are \(C_M\) and \(N\) such that at every \(n\ge \max (2,N)\) some integer \(r\le C_Mn^{(1-\varepsilon )/\alpha }\) satisfies \(I_\Omega (X':J)\le C_Mn^{-M}\) for every \(X'\subseteq X\) and every \(J\subseteq \Lambda \setminus N_{2r}^\Lambda (S_0)\). Then the amplification bound holds. This reduces Proposition 10.2 of [ 247 ] to its radius and output bounds; it does not prove the proposition.
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
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 \(X,Y,Z\le \varepsilon \).
Let \(R'\) be a square matrix on a finite-dimensional Hilbert space \(\mathcal H\), and let \(\phi ,\eta \) be unit vectors in \(\mathcal H\otimes \mathcal K\) for a finite-dimensional space \(\mathcal K\). If \(\| (R'\otimes I)^\dagger \eta -\phi \| \le a\) and \(\| (R'\otimes I)\phi \| \le 1+b\), then
and there is a unitary \(U\) on \(\mathcal H\) with \(\| (U\otimes I)\phi -\eta \| \le a+\sqrt{2a+2b+b^2}\). No invertibility of \(R'\) is assumed. These are the direct and unitary errors in the proof of Proposition 10.2 of [ 247 ] .
Let \(R_0\) and \(L'\) be square matrices on a finite-dimensional space \(\mathcal H\), and let \(D\), \(D_{\rm ret}\) act on \(\mathcal H\otimes \mathcal K\). Let \(\phi ,\eta \) be unit vectors, put \(\zeta =(R_0\otimes I)\phi \), and suppose that \(\| \zeta \| \le M_0\), that \(D\) fixes a vector \(g\) with \(\| g\| \le 1\), that
and that \(\| ((L'R_0)\otimes I)^\dagger \eta -\phi \| \le a\). Then some unitary \(U\) on \(\mathcal H\) satisfies \(\| (U\otimes I)\phi -\eta \| \le a+\sqrt{2a+6\varepsilon +9\varepsilon ^2}\).
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.
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.
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.
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 \(\mathcal M(o,t/4)\).
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.
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.
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 \(T\).
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.
Fix the scanner exponents described in Section 28.9, 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.
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 \(D_0\).
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 ] .
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 \(n\).
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 \(g_0-\ell -\nu /4{\lt}1\).
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 ] .
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.
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
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 \(Q\).
- TNLean.PEPS.AreaLaw.IntRect.child
- TNLean.PEPS.AreaLaw.IntRect.exists_mem_child
- TNLean.PEPS.AreaLaw.IntRect.dist_le_of_mem_child_dilate
- TNLean.PEPS.AreaLaw.IntRect.child_dilate_subset
- TNLean.PEPS.AreaLaw.IntRect.width
- TNLean.PEPS.AreaLaw.IntRect.height
- TNLean.PEPS.AreaLaw.IntRect.numChunksX
- TNLean.PEPS.AreaLaw.IntRect.numChunksY
For \(q\ge 1\) the function \(F_{\mathrm{box}}\) is nonnegative, nondecreasing in \(r\), and satisfies \(F_{\mathrm{box}}(r)\le r^2\log q\).
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 \(F_{\mathrm{box},D_1}(r)\le F_{\mathrm{box},D_0}(r)\).
With \(C_{\mathrm{pad}},C_{\mathrm{buf}}\) from Theorem 27.120.3, \(P=2C_{\mathrm{pad}}+1\) and \(p=1/(2P^2)\), for every \(M\) and \(r\ge 1\),
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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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 ] .
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 \(K=0\).
The squares selected by \(\mathcal P_K(S)\) partition \(S\). In particular,
The empty set has an empty partition, and \(\mathcal P_0(S)=\{ (0,x):x\in S\} \).
- TNLean.PEPS.AreaLaw.Geometry.mem_cappedDyadicPartition
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- TNLean.PEPS.AreaLaw.Geometry.sum_pow_cappedDyadicPartition
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- TNLean.PEPS.AreaLaw.Geometry.cappedDyadicPartition_empty
- TNLean.PEPS.AreaLaw.Geometry.cappedDyadicPartition_zero
If \((k,z)\in \mathcal P_K(S)\) and \(k{\lt}K\), then \(\lfloor z/2\rfloor \in M_{k+1}(S)\). Consequently,
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 \(f\).
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.
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 ] .
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
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.
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 \(\{ c\} \).
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 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 \(c\in \mathbb R^2\) 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 ] .
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 ] .
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.
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 \(|S_0|\le 10n^2\).
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 ] .
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.
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.
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.
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 \(x=b\).
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 \(q\).
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 \(q\).
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.
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 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.
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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.
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- TNLean.PEPS.AreaLaw.rectangleConfigurationEquiv
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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 \(x=v\).
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 \(x=b_{\ell _k,z,s}\).
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.
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
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 ] .
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 ] .
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 \(P\).
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 ] .
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.
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.
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.
For every finite \(P\), depth \(d\), and index \(u\in \mathbb Z^2\),
For every pair of scales \(\ell \le k\) and every \(x\in \mathbb R^2\),
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.
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.
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.
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 \([u,v]\subseteq [a,b]\).
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
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 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.
- TNLean.PEPS.AreaLaw.Exponents.boxError
- TNLean.PEPS.AreaLaw.Exponents.scannerMu
- TNLean.PEPS.AreaLaw.Exponents.scannerNu
- TNLean.PEPS.AreaLaw.Exponents.scannerEll
- TNLean.PEPS.AreaLaw.Exponents.scannerOmega
- TNLean.PEPS.AreaLaw.Exponents.scannerKappa
- TNLean.PEPS.AreaLaw.Exponents.amplificationEpsilon
- TNLean.PEPS.AreaLaw.Exponents.beta
- TNLean.PEPS.AreaLaw.Exponents.amplificationPower
- TNLean.PEPS.AreaLaw.Exponents.alpha
- TNLean.PEPS.AreaLaw.Exponents.geometryDelta
- TNLean.PEPS.AreaLaw.Exponents.zeta
- TNLean.PEPS.AreaLaw.Exponents.rho
- TNLean.PEPS.AreaLaw.Exponents.geometryEll
- TNLean.PEPS.AreaLaw.Exponents.alpha_gt
- TNLean.PEPS.AreaLaw.Exponents.geometry_gaps
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.
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.
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.
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.
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.
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.
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.
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.
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
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 \(T\).
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.
For every scale \(k\),
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
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.
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 ] .
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.
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 ] .
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 \(r\ge 1\).
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.
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 \(H\) 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 ] .
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
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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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 \(S_v/128\).
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.
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 ] .
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 ] .
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 ] .
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 ] .
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 ] .
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 \(v\).
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.
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 ] .
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.
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.
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.
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.
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.
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.
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, \(C_{k,z}\subseteq C_{k+1,\lfloor z/2\rfloor }\).
- TNLean.PEPS.AreaLaw.Geometry.dyadicAncestor_eq_cellIndex
- TNLean.PEPS.AreaLaw.Geometry.mem_latticeDyadicCell
- TNLean.PEPS.AreaLaw.Geometry.mem_latticeDyadicCell_iff_mem_cell
- TNLean.PEPS.AreaLaw.Geometry.card_latticeDyadicCell
- TNLean.PEPS.AreaLaw.Geometry.latticeDyadicCell_zero
- TNLean.PEPS.AreaLaw.Geometry.latticeDyadicCell_subset_of_common_mem
- TNLean.PEPS.AreaLaw.Geometry.latticeDyadicCell_subset_parent
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 \(U=Y\).
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 ] .
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.
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 \(w\)
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.
- TNLean.PEPS.AreaLaw.Geometry.mem_latticeDyadicCell_iff_bounds
- TNLean.PEPS.AreaLaw.Geometry.mixedDyadicIndices_empty
- TNLean.PEPS.AreaLaw.Geometry.mixedDyadicIndices_biUnion_subset
- TNLean.PEPS.AreaLaw.Geometry.card_mixedDyadicIndices_biUnion_le
- TNLean.PEPS.AreaLaw.Geometry.mixedDyadicIndices_sdiff_subset
- TNLean.PEPS.AreaLaw.Geometry.card_mixedDyadicIndices_sdiff_le
Let \(P,x,Y,F\) partition \(\Lambda \) and let \(\theta \) be a unit vector. Then
where \(S_\theta (x|R)=S_\theta (R\cup x)-S_\theta (R)\).
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.
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 ] .
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.
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, \(|\partial _\Lambda (A\cap U)|\le |\partial U|\).
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,
- TNLean.PEPS.AreaLaw.Geometry.Template.mem_latticeRow_dilation_of_between
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_dilation_eq_Icc
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_dilation_succ_bounds
- TNLean.PEPS.AreaLaw.Geometry.Template.card_latticeRow_layer_le_four
- TNLean.PEPS.AreaLaw.Geometry.Template.card_latticeRow_dilation_le
- TNLean.PEPS.AreaLaw.Geometry.Template.exists_dilation_row_domain
- TNLean.PEPS.AreaLaw.Geometry.Template.card_dilation_layer_le
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.
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.
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.
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.
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
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 \(p=\ell \).
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 \(x,y\in F_{j,z}\).
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.
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.
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.
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 ] .
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.
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 \(K=0\).
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 ] .
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 ] .
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
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 \(\Lambda \subset \mathbb Z^2\) 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.
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.
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 ] .
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 \(d^{j}(S)\ge S_*\),
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 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.
- TNLean.PEPS.AreaLaw.Geometry.rho_pos_lt_one
- TNLean.PEPS.AreaLaw.Geometry.repairHalfSide_le_rpow
- TNLean.PEPS.AreaLaw.Geometry.half_rpow_le_repairHalfSide
- TNLean.PEPS.AreaLaw.Geometry.repairHalfSide_lt_self
- TNLean.PEPS.AreaLaw.Geometry.eventually_repairHalfSide_le
- TNLean.PEPS.AreaLaw.Geometry.eventually_repairHalfSide_margins
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
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.
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 ] .
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
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.
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 ] .
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:
- TNLean.PEPS.AreaLaw.Geometry.latticeRow
- TNLean.PEPS.AreaLaw.Geometry.mem_latticeRow
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon.convex_region
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon.isCompact_region
- TNLean.PEPS.AreaLaw.Geometry.horizontal_mem_of_mem_of_le
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon.isCompact_horizontalSection
- TNLean.PEPS.AreaLaw.Geometry.TemplatePolygon.horizontalSection_eq_Icc
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_eq_Icc_ceil_floor
- TNLean.PEPS.AreaLaw.Geometry.Template.mem_sample_of_row_between
- TNLean.PEPS.AreaLaw.Geometry.Template.latticeRow_eq_Icc
- TNLean.PEPS.AreaLaw.Geometry.ceil_affine_row
- TNLean.PEPS.AreaLaw.Geometry.floor_affine_row
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.
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 ] .
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.
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 ] .
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.
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.
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 \(S\).
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 \(R\).
If \(C_{\rm tpl}\ge 1\), a template satisfies \(|T|\le 9ns_0\).
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.
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.
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 ] .
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 ] .
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.
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.
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 \(q\).
Under the hypotheses and notation of Theorem 27.118.3, 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 ] .
Under the hypotheses and notation of Theorem 27.118.3, let \(q\ge 1\), let \(n\ge 0\) be an integer and \(M\ge 0\), and put
with \(C\) as in Theorem 27.118.7. 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 \(M{\gt}0\).
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 \(\eta \).
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 \(b,e,e_0,\eta ,C_2\in \mathbb R\) satisfy \(0{\lt}b\le e\le e_0\), \(\eta \ge 0\), and \(C_2\ge 0\). Then
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.
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.
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 \(s(v)\).
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 \(h\ge k_0\).
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.
Both \(\partial _{\rm in}A\) and \(Z_A\) are empty when \(A=\varnothing \) or \(A=\Lambda \).
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 \(K\ge 0\) 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.
Fix integers \(q\geq 1\) and \(R\geq 0\), and a real number \(J\). There are positive constants \(C,v,c\) with the following property. Let \(\Lambda \subseteq \mathbb {Z}^2\) be a finite domain, let \(H'\) be a Hamiltonian of range \(R\) and term-norm bound \(J\) on \(\Lambda \), and let \(H\) be any Hermitian operator on the same space. Suppose that \(W\) acts on \(Q\subseteq \Lambda \) and that \(D=H'-H\) acts on \(Z\subseteq \Lambda \). For each integer \(l\geq 0\), put \(B=N_l(Q\cup Z)\), where distance is measured in the induced graph. There is a differentiable path \(U\) such that
For every real \(t\), the operator \(U(t)\) is unitary and acts on \(B\). Choose this path before the Gaussian variance and time cutoff. For every \(\eta \geq 0\) and \(T\geq 0\), let \(\gamma _\eta \) be the centered Gaussian measure of variance \(\eta \), and define
Then \(M_{\eta ,T}\) acts on \(B\), its norm is at most \(\| W\| \), and
The integrals retain their original Gaussian mass. Zero variance, empty domains and empty regions are included. This is the supported Gaussian approximation used in [ 248 ] .
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 \(A\cap C_{k,z}\).
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 ] .
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 ] .
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 \(C_e\).
- TNLean.PEPS.AreaLaw.Geometry.Template.regionalEntropy_core_le_of_safe_box
- TNLean.PEPS.AreaLaw.Geometry.Template.regionalEntropy_shell_le_of_safe_box
- TNLean.PEPS.AreaLaw.Geometry.Template.regionalEntropy_prefix_le_of_safe_box
- TNLean.PEPS.AreaLaw.Geometry.exists_template_entropy_bounds_of_arbitrary_safe_box
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 ] .
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 \(2^K a_K\le 2n\).
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.
If \(r\in P\) and \(|P|=n\), then \(|T(P,r)|=n-1+\binom n2\). If \(n\le b\), this is at most \(b-1+\binom b2\).
The constructed operator has the exact entries
The identity permits empty branch alphabets, empty link families, singleton gates and \(k=0\).
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.
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.
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 \(g\in G\).
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 \(|P|-1\).
For every gate occurrence \(g\) and party \(p\), \(|\mathcal L_g|=|T(P_g,r_g)|\) and \(|\mathcal L_g(p)|=|T_p(P_g,r_g)|\).
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.
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 ] .
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 ] .
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 ] .
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.
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 \(i\).
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.
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.
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 27.145.5, 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.
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
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.
- TNLean.PEPS.PairEffect.SourceCircuit.sourceCorrection_product_eq_selected
- TNLean.PEPS.PairEffect.SourceCircuit.integrable_sourceCorrection_product_mul_conj
- TNLean.PEPS.PairEffect.SourceCircuit.integral_sourceCorrection_product_mul_conj
- TNLean.PEPS.PairEffect.SourceCircuit.integral_sourceCorrection_product_eq_zero
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.
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.
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.
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.
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.
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.
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 27.165.22, 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 ] .
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 27.141.1.15. 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 27.165.30 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 ] .
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.
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 ] .
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.
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 ] .
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 27.153.4) at each source \(e\); write \(U_{\xi ,\zeta ,j,\omega }\) for the actual sampled source-input word of Definition 27.165.17 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 ] .
The sampled endpoint word and its transport \(U_{\xi ,\zeta ,j}\) to the actual source input contain no source preparation.
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 ] .
In the notation of Theorem 27.165.20, 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.
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.
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 ] .
In the notation of Theorem 27.165.28, 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 ] .
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.
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.
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 ] .
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.
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.
- TNLean.PEPS.PairEffect.SourceCircuit.isTouched_correctedMask_iff
- TNLean.PEPS.PairEffect.SourceCircuit.corrected_slot_isTouched
- TNLean.PEPS.PairEffect.SourceCircuit.card_touched_correctedMask_le
- TNLean.PEPS.PairEffect.SourceCircuit.coefficient_physical_cost_eq
- TNLean.PEPS.PairEffect.SourceCircuit.coefficient_physical_cost_le
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_mul_conj_le_lifetime
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.
Under the circuit and input assumptions of Theorem 27.158.9, 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 \(W\) 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.
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 \(VP_{\mathrm{fill}(\zeta )}=W\).
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.
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 \(\prod _{p\in A}d_p\).
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.
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.
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 ] .
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 27.150.5, 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 ] .
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 \(s\).
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.
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 27.149.4.1. 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 ] .
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 \(L\).
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 \(\| x\| \le 1\).
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 27.159.11. 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.
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.
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.
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.
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 ] .
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, \(J_{a,\varnothing }^{-1}(x\otimes 1)=E_{a\varnothing =a}^{-1}x\).
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.
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.
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 \(x'+y'\).
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 \(M(ab)=M(a'b')\).
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.
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 \(x\in M(a)\).
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 \(U_{g\circ q}^a x\).
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 27.141.2.13 for these four lists,
With the bases of Definition 27.141.2.15, 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 \(U_f^c\).
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.
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.
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.
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 \(\mathbb {C}\).
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.
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:
Under the hypotheses of Definition 27.141.2.23, 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 \(\| R_A\| _{\rm op}\le 1\).
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.
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.
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.
For every \(q\), the assignment \(q\cdot p\) is flat if and only if \(p\) is flat.
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 orthogonal projector onto \(\mathcal G_R^\perp \) is a regional parent interaction.
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
- TNLean.PEPS.PairEffect.Word.sources_residualInSlots
- TNLean.PEPS.PairEffect.Word.isAllowed_residualInSlots
- TNLean.PEPS.PairEffect.Word.eval_residualInSlots_prepare
- TNLean.PEPS.PairEffect.SourceCircuit.slotLayout_partialSlots_eq
- TNLean.PEPS.PairEffect.SourceCircuit.sources_partialResidual
- TNLean.PEPS.PairEffect.SourceCircuit.isAllowed_partialResidual
- TNLean.PEPS.PairEffect.SourceCircuit.eval_partialResidual_prepare
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
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.
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.
- TNLean.PEPS.PairEffect.EffectCircuit.referenceEval_apply
- TNLean.PEPS.PairEffect.EffectCircuit.norm_auxiliaryVector
- TNLean.PEPS.PairEffect.EffectCircuit.norm_replacement_sub_reference_le
- TNLean.PEPS.PairEffect.EffectCircuit.isAllowed_replacement
- TNLean.PEPS.PairEffect.OriginalCircuit.eval_produce
- TNLean.PEPS.PairEffect.OriginalCircuit.expandedGateCount_produce
If \(r\ge 0\), then \(|K(c,r)|\le (2r+1)^2\). If also \(r\le 4\), then \(|K(c,r)|\le 81\).
Let \(d_i{\gt}0\) and let \(M(a)=\bigoplus _i\sqrt{d_i}\, D_i(a)\) be a family of matrices. Every finite coherent sum of products of these matrices belongs to the product of matching-sector bond spaces. The multiplicity-restoring map sends it to the same coherent sum with bond matrices \(\bigoplus _i D_i(a)\otimes I_{d_i}\). If an isometric coordinate change identifies these repeated blocks with another matrix family, one fixed full-domain product isometry gives the corresponding identity for every choice of coefficients and bond labels, and preserves all inner products. These are auxiliary matrix statements for the construction in Section 7 of [ 235 ] ; an actual contraction formula is still required when applying them to a tensor-network state.
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.
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.
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 \((S_\xi )_{\xi \in \Xi }\) 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.
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.
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 \(T\).
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.
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.
- TNLean.PEPS.PairEffect.PartyChain.eval_prependWord
- TNLean.PEPS.PairEffect.PartyChain.isAllowed_prependWord
- TNLean.PEPS.PairEffect.PartyChain.effectCount_prependWord
- TNLean.PEPS.PairEffect.PartyChain.eval_comp
- TNLean.PEPS.PairEffect.PartyChain.isAllowed_comp
- TNLean.PEPS.PairEffect.PartyChain.effectCount_comp
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.
Let \(R\) be a nonempty connected finite region whose site tensors are \(G\)-injective for the simultaneous regular action on their incident legs. Then the actual open tensor of \(R\) is \(G\)-injective for the simultaneous regular action on its crossing legs. Internal cycles and an empty boundary are allowed. This is the regular-representation connected-region consequence of [ 235 , Lemma 5.2 and the following observation ] ; the argument does not assume local isometry.
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 27.155.6, 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 \(\varepsilon /2\).
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.
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 \(m=d\).
- TNLean.PEPS.graphRegionCopyMatrix_maps_openBondSpace
- TNLean.PEPS.graphRegionCopyMatrix_adjoint_maps_openBondSpace
- TNLean.PEPS.numberedRegionCopyMatrix_maps_regionGroundSpace
- TNLean.PEPS.numberedRegionCopyMatrix_adjoint_maps_regionGroundSpace
- TNLean.PEPS.graphRegionMultiplicityMatrix_maps_openBondSpace
- TNLean.PEPS.graphRegionMultiplicityMatrix_adjoint_maps_openBondSpace
- TNLean.PEPS.numberedRegionMultiplicityMatrix_maps_regionGroundSpace
- TNLean.PEPS.numberedRegionMultiplicityMatrix_adjoint_maps_regionGroundSpace
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.
- TNLean.PEPS.graphOpenInternalFactor_copy
- TNLean.PEPS.graphOpenInternalFactor_copy_adjoint
- TNLean.PEPS.graphOpenBoundaryFactor_copy
- TNLean.PEPS.graphOpenBoundaryFactor_copy_adjoint
- TNLean.PEPS.graphOpenInternalFactor_multiplicity
- TNLean.PEPS.graphOpenInternalFactor_multiplicity_adjoint
- TNLean.PEPS.graphOpenBoundaryFactor_multiplicity
- TNLean.PEPS.graphOpenBoundaryFactor_multiplicity_adjoint
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 \(m=d\).
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.
- TNLean.PEPS.map_regionParentGroundSpace_copy_eq
- TNLean.PEPS.canonicalCopyParentEquiv
- TNLean.PEPS.finrank_regionParentGroundSpace_copy_eq
- TNLean.PEPS.map_ker_regionParentHamiltonian_copy_eq
- TNLean.PEPS.map_regionParentGroundSpace_multiplicity_eq
- TNLean.PEPS.canonicalMultiplicityParentEquiv
- TNLean.PEPS.finrank_regionParentGroundSpace_multiplicity_eq
- TNLean.PEPS.map_ker_regionParentHamiltonian_multiplicity_eq
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.
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 \(m=d\).
- TNLean.PEPS.graphNumberedCopyMatrix_leftInverse_parent
- TNLean.PEPS.graphNumberedCopyMatrix_rightInverse_parent
- TNLean.PEPS.graphNumberedCopyMatrix_injOn_parent
- TNLean.PEPS.graphNumberedMultiplicityMatrix_leftInverse_parent
- TNLean.PEPS.graphNumberedMultiplicityMatrix_rightInverse_parent
- TNLean.PEPS.graphNumberedMultiplicityMatrix_injOn_parent
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,
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, \(T_m(\ker H)=\ker K\).
For every \(g\in G\),
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 \(A(S)\).
- TNLean.PEPS.PairEffect.SourceCircuit.correctedSlotMask_endpoints
- TNLean.PEPS.PairEffect.SourceCircuit.correctedFreeSlots_eq_or
- TNLean.PEPS.PairEffect.SourceCircuit.crossingSlotMask_eq_false_of_corrected
- TNLean.PEPS.PairEffect.SourceCircuit.correctedFreeSlots_masks
- TNLean.PEPS.PairEffect.SourceCircuit.correctedSourceLayout_owners
- TNLean.PEPS.PairEffect.SourceCircuit.restrict_correctedSourceLayout_append
- TNLean.PEPS.PairEffect.SourceCircuit.correctedMask_meets
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 27.153.11 prepared at their inputs. Each gate coefficient occurs once on each side.
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.
- TNLean.PEPS.PairEffect.SourceCircuit.sourceBasisMatrix_eq_sum_partial
- TNLean.PEPS.PairEffect.SourceCircuit.correctedSourceTerm_eq_sum_partial
- TNLean.PEPS.PairEffect.SourceCircuit.partialSourceBasisMatrix_eq_ite
- TNLean.PEPS.PairEffect.SourceCircuit.correctedSourceTerm_eq_sum_coordinates
- TNLean.PEPS.PairEffect.SourceCircuit.partialCoordinateBasisMatrix_eq_preparedMatrix
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.
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 \(S\).
Suppose every \(e\in S\) has an endpoint in \(\mathcal A\). In the common retained positions, set
The same globally defined term then satisfies
For every allowed chronological composition, every corrected set \(S\), and every partial branch for \(\mathcal A_S\), prepare precisely the nonfree sources of Definition 27.154.25 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.
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.
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.
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.
The map \(T_m\) is a contraction, and \(T_m\eta =\eta ^{\otimes m}\).
For a finite family of subspaces \(S_i\subseteq \mathcal H_R\), a linear operator \(A\) on \(\mathcal H_R\), and \(R\subseteq U\subseteq V\),
Let \(A\) act on \(\mathcal H_R\), and let \(P_S\) be the orthogonal projector onto \(S\subseteq \mathcal H_R\). Then
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.
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 \(\mathcal P\mathcal N_I(B)=\mathcal N_{A^0}(B)\).
Every \(A_v^0\) is \(G\)-injective for its representation \(\rho _v\). In particular this holds for independently chosen incident bond actions.
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 \(B\).
- TNLean.PEPS.DependentBondNetwork.dependentProduct_coordinateSlice_eq_smul
- TNLean.PEPS.DependentBondNetwork.bondRegrouping_network_averagingSite
- TNLean.PEPS.DependentBondNetwork.network_averagingSite_eq_sum_matrixBondProduct
- TNLean.PEPS.DependentBondNetwork.bondCoefficientExtraction_network_averagingSite
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.
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.
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.
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.
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.
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,
- TNLean.PEPS.DependentBondNetwork.exists_physicalMap_to_representationAveragingSite
- TNLean.PEPS.DependentBondNetwork.physicalMap_recover_representationAveragingSite
- TNLean.PEPS.DependentBondNetwork.exists_cutMap_localInverse
- TNLean.PEPS.DependentBondNetwork.cutMap_recover_representationAveragingSite
- TNLean.PEPS.DependentBondNetwork.iInf_cutSpace_eq_map_representationAveragingSite
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 \(\mathcal C_{A,C}(\delta _\eta )=\mathcal N_A(E^\eta )\).
- TNLean.PEPS.DependentBondNetwork.cutMap_eq_sum_single
- TNLean.PEPS.DependentBondNetwork.cutSpace_eq_span_single
- TNLean.PEPS.DependentBondNetwork.cutBondBoundary
- TNLean.PEPS.DependentBondNetwork.cutBondBoundary_restriction
- TNLean.PEPS.DependentBondNetwork.cutCoeff_bondBoundary
- TNLean.PEPS.DependentBondNetwork.cutBondUnits
- TNLean.PEPS.DependentBondNetwork.cutBondBoundary_units
- TNLean.PEPS.DependentBondNetwork.cutMap_single_eq_network
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
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 \(G\).
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,
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.
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 \(\mathcal F\mathcal N_A(B)=\mathcal N_{FA}(B)\).
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.
- TNLean.PEPS.dependentPhysicalProductFamilyMatrix_mul
- TNLean.PEPS.dependentPhysicalProductFamilyMap_comp
- TNLean.PEPS.dependentPhysicalProductFamilyMatrix_one
- TNLean.PEPS.dependentPhysicalProductFamilyMap_one
- TNLean.PEPS.dependentPhysicalProductFamilyMap_oneCoordinate_apply
- TNLean.PEPS.dependentPhysicalProductFamilyMap_fixed_of_slices_fixed
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.
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 ] .
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,
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.
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 ] .
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.
- TNLean.PEPS.regionReplaceConfig
- TNLean.PEPS.regionLocalTerm_mulVec_apply
- TNLean.PEPS.regionReplaceConfig_apply_of_disjoint
- TNLean.PEPS.regionReplaceConfig_comm_of_disjoint
- TNLean.PEPS.regionLocalTerm_commute_of_disjoint
- TNLean.PEPS.regionLocalTermES
- TNLean.PEPS.regionLocalTermES_commute_of_disjoint
- TNLean.PEPS.regionParentHamiltonianES
- TNLean.PEPS.regionParentHamiltonianES_eq_sum
The actual physical density of a distributed construction is positive semidefinite and satisfies \(\operatorname {Tr}\rho \leq 1\).
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 27.144.12, 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.
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.
- TNLean.PEPS.Approximation.levelLabels_eq_party
- TNLean.PEPS.Approximation.OwnsLevelBlock.scale
- TNLean.PEPS.Approximation.levelLabels_empty_eq_party
- TNLean.PEPS.Approximation.levelLabels_eq_party_of_subset
- TNLean.PEPS.Approximation.repaintingUses_of_levelLabels
- TNLean.PEPS.Approximation.incidentLabels_blockGuide_corner
- TNLean.PEPS.Approximation.junctionUses_of_incidentLabels
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 27.127.20. 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.
- TNLean.PEPS.Approximation.blockRepainting_central
- TNLean.PEPS.Approximation.blockRepainting_auxStep
- TNLean.PEPS.Approximation.blockRepainting_mainStep
- TNLean.PEPS.Approximation.blockRepainting_exchange
- TNLean.PEPS.Approximation.RepaintingBaseline.clearance_central
- TNLean.PEPS.Approximation.RepaintingBaseline.dist_le_central
- TNLean.PEPS.Approximation.RepaintingBaseline.clearance_auxStep
- TNLean.PEPS.Approximation.RepaintingBaseline.clearance_mainStep
- TNLean.PEPS.Approximation.RepaintingBaseline.clearance_exchange
- TNLean.PEPS.Approximation.BandOperation.clearance_bandUpdate
- TNLean.PEPS.Approximation.BandOperation.dist_le_bandUpdate
- TNLean.PEPS.Approximation.dist_le_of_mem_closure_edgeLens
- TNLean.PEPS.Approximation.RepaintingBaseline.exchange_changed_subset
- TNLean.PEPS.Approximation.RepaintingBaseline.dist_le_exchange
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 27.127.13 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.
- TNLean.PEPS.Approximation.exists_isChamberSampling_blockRepainting
- TNLean.PEPS.Approximation.blockRepainting_central_of_chamberSampling
- TNLean.PEPS.Approximation.blockRepainting_auxStep_of_chamberSampling
- TNLean.PEPS.Approximation.blockRepainting_mainStep_of_chamberSampling
- TNLean.PEPS.Approximation.blockRepainting_exchange_of_chamberSampling
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.
- TNLean.PEPS.Approximation.BandOperation.clearance
- TNLean.PEPS.Approximation.auxBirthB
- TNLean.PEPS.Approximation.auxBirthA
- TNLean.PEPS.Approximation.auxBirthC
- TNLean.PEPS.Approximation.mainDeathC
- TNLean.PEPS.Approximation.mainDeathA
- TNLean.PEPS.Approximation.mainBirthC
- TNLean.PEPS.Approximation.mainDeathB
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.
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.
- TNLean.PEPS.Approximation.blockWindow
- TNLean.PEPS.Approximation.BlockSlot
- TNLean.PEPS.Approximation.blockSlot
- TNLean.PEPS.Approximation.blockSlotLabel
- TNLean.PEPS.Approximation.blockSlotLabel_blockSlot
- TNLean.PEPS.Approximation.floor_mem_Icc
- TNLean.PEPS.Approximation.blockGuide_eq_blockSlot
- TNLean.PEPS.Approximation.exists_blockGuide_footprintRatio
- TNLean.PEPS.Approximation.exists_directRepainting_footprintRatio
- TNLean.PEPS.Approximation.exists_resizing_footprintRatio
- TNLean.PEPS.Approximation.exists_resizing_footprintRatio_mixed
- TNLean.PEPS.Approximation.blockGuide_eq_classRep
- TNLean.PEPS.Approximation.norm_smul_inv_sub_lt_iff
- TNLean.PEPS.Approximation.mem_Icc_of_mem_holeRadius_pair
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
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\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]{}
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\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]{}
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\tn[at=(4,1), skin=none]{}
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\tn[at=(7,7), name=centralTipSE, skin=none]{}
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\tn[at=(7,9), skin=none]{}
\tn[at=(8,1), skin=none]{}
\tn[at=(8,2), name=centralCornerSW, skin=dot, species=contact]{}
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\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]{}
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\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}
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\tnmark[form=label]{(1,5)}{$C_e$}
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\end{tenkz}](tenkz_svg/tenkz-884ec5190d5dcc6e.svg)
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.
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.
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 patch data: for a repainting, the direct repainting of its square or a gate of its large-scale repainting; for a junction, the resize of its hole. 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, the conclusion of the small-patch rewrite lemma and of the birth and exchange lemmas for the parties of the patch data. This is a conditional form of the bounded lifetime participation of [ 248 , Proposition 7.1, item 2 ] .
Let \(n=2^{j'}\) and consider two local guides over one base of the repainting of a square of side \(n\). If \(0{\lt}t\), \(2t\le n\), \(\epsilon _0,\nu {\gt}0\), \(2\epsilon _0+3\nu {\lt}1\) and \(c\) is a point of the closed square, a homogenization or removal of a point treatment about \(c\) with affected holes centred within \(t\) of \(c\) and of outer radius \(2\epsilon _0t\) has patch data of a gate of the large-scale repainting whose patches form a square net of mesh \(\nu t\). There are at most \((2(1+2\epsilon _0+\nu )/\nu +1)^2\) patches, and their inner squares cover the treated square of radius \(t\) about \(c\) and the affected outer holes. If \(0{\lt}\nu \le 1/12\) and \(0{\lt}R\le n/4\), a bulk birth, death or exchange with affected holes of outer radius \(R\) centred within \(n/2\) of the square has patch data of a gate of the large-scale repainting whose patches form a square net of mesh \(\nu n\), at most \((2(5/4+\nu )/\nu +1)^2\) of them, covering the changed set and the affected outer holes. The point-treatment nets are those of [ 248 , Section 7 ] , lines 482–505. The bulk step is implemented by the birth or exchange lemma, not by its net; the net bounds its parties, the owners on the changed set “compactly located in diameter \(O(n)\)” and on the affected holes, lines 383–387 and 421–430.
Label the blocks by an injective assignment of parties, none of them the placeholder. In the repainting of the square with indices \((r,s)\) and side \(2^{j'}\), every local label that is a block party is used by the repainting, and so is every block party of the patch data of a gate of the large-scale repainting. This is “every active main owner is an \(n\)- or \(2n\)-block owner. On the affected footprints those blocks lie at distance \(O(n)\) from the repainted square” in [ 248 , Section 7 ] , lines 585–596.
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.
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.
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.
The labels \(A\), \(B\) and \(C_e\) are local labels, and so is every value of a normal word of an edge construction. The starting guide and the guide after the central birth, every completed guide, and every main guide of an edge construction with any of its normal words are local over the starting guide; the uniform guide \(C_e\) and every auxiliary guide of the construction along \(e\) with any of its normal words are local over the uniform guide \(C_e\). Homogenizing a local guide to a local label on a set inside the neighbourhood of radius \(n/2\), as the point treatment does on the treated squares of radius \(t\le n/2\), and replacing it on a set inside that neighbourhood by a local guide over another base give local guides over the same base. The lens lies in that neighbourhood, so after the lens exchange along \(e\), for \(t\le n/2\), the main sheet is local over the starting guide and the auxiliary sheet is local over the uniform guide \(C_e\). Two local guides over one base differ only inside the neighbourhood of radius \(n/2\). This is the locality of the large-scale repaintings in [ 248 , Section 7 ] , lines 585–596: “An auxiliary uses only the labels of the local edge and point diagrams; its uniform initial owner \(C\) is the actual neighboring-block owner.”
- TNLean.PEPS.Approximation.SquareEdge.point_mem_closedSquare
- TNLean.PEPS.Approximation.edgeBand_subset_thickening
- TNLean.PEPS.Approximation.edgeTreated_subset_thickening
- TNLean.PEPS.Approximation.cornerTreated_subset_thickening
- TNLean.PEPS.Approximation.edgeLens_subset_thickening
- TNLean.PEPS.Approximation.RepaintingBaseline.finalLabel_mem_localLabels
- TNLean.PEPS.Approximation.RepaintingBaseline.guide_mem_localLabels
- TNLean.PEPS.Approximation.RepaintingBaseline.oldLabel_mem_localLabels
- TNLean.PEPS.Approximation.RepaintingBaseline.nbrLabel_mem_localLabels
- TNLean.PEPS.Approximation.RepaintingBaseline.edgeWords_mem_localLabels
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_guide
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_const
- TNLean.PEPS.Approximation.RepaintingBaseline.IsLocalGuide.update
- TNLean.PEPS.Approximation.RepaintingBaseline.IsLocalGuide.homogenize
- TNLean.PEPS.Approximation.RepaintingBaseline.IsLocalGuide.piecewise
- TNLean.PEPS.Approximation.RepaintingBaseline.IsLocalGuide.bandUpdate
- TNLean.PEPS.Approximation.RepaintingBaseline.IsLocalGuide.changed_subset
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_centralGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_completedGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_mainGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_auxGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_treatedStartMain
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_treatedAux
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_exchangedMain
- TNLean.PEPS.Approximation.RepaintingBaseline.isLocalGuide_exchangedAux
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.
- TNLean.PEPS.Approximation.BandOperation.InsertsOne
- TNLean.PEPS.Approximation.edgeOperations_insertsOne
- TNLean.PEPS.Approximation.angularConstant_mul_le_mul_max
- TNLean.PEPS.Approximation.BandOperation.treated_surrounding_eq
- TNLean.PEPS.Approximation.disjoint_closedBall_of_forall_le_dist
- TNLean.PEPS.Approximation.BandOperation.isTrueVertex_treated
- TNLean.PEPS.Approximation.RepaintingBaseline.isTrueVertex_central_treated
- TNLean.PEPS.Approximation.RepaintingBaseline.treatedStartMain
- TNLean.PEPS.Approximation.RepaintingBaseline.treatedAux
- TNLean.PEPS.Approximation.RepaintingBaseline.exchange_corridor
- TNLean.PEPS.Approximation.RepaintingBaseline.isTrueVertex_exchange_treated
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 \(2(2c+1)B\Delta \).
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.
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.
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.
- TNLean.PEPS.Approximation.GuidePatchRewrite.parties_subset
- TNLean.PEPS.Approximation.junctionUses_of_blockGuide_eq
- TNLean.PEPS.Approximation.repaintingUses_of_blockGuide_eq
- TNLean.PEPS.Approximation.GuidePatchRewrite.IsJunctionResize.uses
- TNLean.PEPS.Approximation.GuidePatchRewrite.IsDirectRepainting.uses
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 27.127.40).
- TNLean.PEPS.Approximation.SquareEdge.par_smul
- TNLean.PEPS.Approximation.SquareEdge.nor_inv_smul
- TNLean.PEPS.Approximation.bandWidth_inv_mul
- TNLean.PEPS.Approximation.mem_edgeBand_inv_smul
- TNLean.PEPS.Approximation.bandCoord_inv_smul
- TNLean.PEPS.Approximation.mem_centralRegion_inv_smul
- TNLean.PEPS.Approximation.bandUpdate_inv_smul
- TNLean.PEPS.Approximation.comp_bandWord
- TNLean.PEPS.Approximation.bandUpdate_comp
- TNLean.PEPS.Approximation.comp_mainWords
- TNLean.PEPS.Approximation.comp_auxWords
- TNLean.PEPS.Approximation.exists_mainWords_comp
- TNLean.PEPS.Approximation.exists_auxWords_comp
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale.centralGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale.completedGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale.mainGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale.auxGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRescale.isUnmodifiedGuide
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRelabel
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRelabel.centralGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRelabel.completedGuide_eq
- TNLean.PEPS.Approximation.RepaintingBaseline.IsRelabel.isUnmodifiedGuide
- TNLean.PEPS.Approximation.IsConicalAt
- TNLean.PEPS.Approximation.IsSquareCorner.trichotomy
- TNLean.PEPS.Approximation.SquareEdge.par_add_smul
- TNLean.PEPS.Approximation.SquareEdge.nor_add_smul
- TNLean.PEPS.Approximation.mem_edgeBand_conical
- TNLean.PEPS.Approximation.bandCoord_conical
- TNLean.PEPS.Approximation.mem_centralRegion_conical
- TNLean.PEPS.Approximation.isConicalAt_blockGuide
- TNLean.PEPS.Approximation.IsConicalAt.bandUpdate
- TNLean.PEPS.Approximation.RepaintingBaseline.IsUnmodifiedGuide.isConicalAt
- TNLean.PEPS.Approximation.isTrueVertex_comp_iff
- TNLean.PEPS.Approximation.isTrueVertex_comp_homeomorph
- TNLean.PEPS.Approximation.similarityInv
- TNLean.PEPS.Approximation.similarityInv_apply
- TNLean.PEPS.Approximation.image_similarity_closedBall
- TNLean.PEPS.Approximation.HasTwoOwnerFootprints.congr
- TNLean.PEPS.Approximation.referenceRepainting
- TNLean.PEPS.Approximation.nbrBlock_zero_mem_blockWindow
- TNLean.PEPS.Approximation.isRelabel_referenceRepainting
- TNLean.PEPS.Approximation.isRescale_blockBaseline
- TNLean.PEPS.Approximation.RepaintingBaseline.completedGuide_congr
- TNLean.PEPS.Approximation.RepaintingBaseline.finite_isUnmodifiedGuide
- TNLean.PEPS.Approximation.pointTreated
- TNLean.PEPS.Approximation.classRep
- TNLean.PEPS.Approximation.classRep_spec
- TNLean.PEPS.Approximation.injOn_range_classRep
- TNLean.PEPS.Approximation.TemplateIndex
- TNLean.PEPS.Approximation.footprintTemplate
- TNLean.PEPS.Approximation.cornerOffset
- TNLean.PEPS.Approximation.isSquareCorner_cornerOffset
- TNLean.PEPS.Approximation.blockCorner_sub_blockCorner
- TNLean.PEPS.Approximation.blockIndex_mem_blockWindow
- TNLean.PEPS.Approximation.exists_reference_eq
- TNLean.PEPS.Approximation.exists_template_eq
- TNLean.PEPS.Approximation.exists_pointTreatment_footprintRatio
- TNLean.PEPS.Approximation.homogenize_iUnion_eq_of_mem_ball
- TNLean.PEPS.Approximation.exists_pointTreatment_footprintRatio_iUnion
- TNLean.PEPS.Approximation.exists_template_injOn
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.
There is a ratio \(\nu {\gt}0\), depending only on the fixed constants, such that for every generic displacement \(v\) the conclusions of Theorems 27.127.38 and 27.127.39 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 27.127.19; 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.
- TNLean.PEPS.Approximation.IsDisplacedReading.range_subset
- TNLean.PEPS.Approximation.IsDisplacedReading.eq_on_ball
- TNLean.PEPS.Approximation.IsRayConstant.comp_affine
- TNLean.PEPS.Approximation.IsRayConstant.pointTreated
- TNLean.PEPS.Approximation.HasLineWalls.pointTreated
- TNLean.PEPS.Approximation.isChamberSampling_pointTreated
- TNLean.PEPS.Approximation.isRayConstant_footprintTemplate
- TNLean.PEPS.Approximation.IsLocallyInLines.preimage_affine
- TNLean.PEPS.Approximation.HasLineWalls.comp_affine
- TNLean.PEPS.Approximation.hasLineWalls_footprintTemplate
- TNLean.PEPS.Approximation.HasLineWalls.eventually_eq_of_sameSide
- TNLean.PEPS.Approximation.IsDisplacedReading.of_sameSide
- TNLean.PEPS.Approximation.finite_wallNormals
- TNLean.PEPS.Approximation.HasLineWalls.finite_genericReadings
- TNLean.PEPS.Approximation.exists_template_footprintRatio_reading
- TNLean.PEPS.Approximation.exists_pointTreatment_footprintRatio_reading
- TNLean.PEPS.Approximation.exists_pointTreatment_footprintRatio_iUnion_reading
- TNLean.PEPS.Approximation.exists_blockGuide_footprintRatio_reading
- TNLean.PEPS.Approximation.exists_directRepainting_footprintRatio_reading
- TNLean.PEPS.Approximation.exists_resizing_footprintRatio_reading
- TNLean.PEPS.Approximation.exists_resizing_footprintRatio_mixed_reading
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 27.127.22.
- TNLean.PEPS.Approximation.isTrueVertex_homogenize_cases
- TNLean.PEPS.Approximation.le_dist_of_isTrueVertex_homogenize
- TNLean.PEPS.Approximation.homogenize_outerHoles_disjoint
- TNLean.PEPS.Approximation.IsDisplacedReading.preimage_subset_closure
- TNLean.PEPS.Approximation.IsDisplacedReading.isTrueVertex
- TNLean.PEPS.Approximation.homogenize_reading_outerHoles_disjoint
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 \(r=0\).
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 \(m{\lt}H L^{2(n+s+p)}\).
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.
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.
In the joint source basis, the endpoint operator satisfies
In particular, constructing the endpoint maps from matrices \(L_i,H_i\) gives \([Z_R]_{x,y}=\prod _i (L_i)_{x_i^{(1)},y_i^{(1)}} (H_i)_{x_i^{(2)},y_i^{(2)}}\).
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 ] .
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.
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.
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.
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 \(\rho \).
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 \(1\).
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 27.155.6 give replacement density error at most \(\varepsilon /2\).
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 \(\kappa (1-m\varepsilon )\).
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.
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 \(aH_k\le H_h\le bH_k\).
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.
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.
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.
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.
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 \(\dim H_p\).
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 \(L\ge 2\).
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.
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.
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
- TNLean.PEPS.torusPhysicalBondSlice_mem_range_of_mem_torusCutSpace
- TNLean.PEPS.torusBondRegrouping_mem_product_range_of_mem_fourTorusCutSpace
- TNLean.PEPS.exists_bondCoefficients_of_mem_fourTorusCutSpace
- TNLean.PEPS.exists_labelCoefficients_of_mem_fourTorusCutSpace
- TNLean.PEPS.eq_sum_extracted_bondProducts_of_mem_fourTorusCutSpace
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.
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, \(\mathcal S^{A^0}=\mathcal Z_{A^0}\).
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.
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 \(p\).
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.
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, \(\mathcal Q_\rho B_p\in \mathcal Z_{A^0}\).
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
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 ] .
- TNLean.PEPS.physicalMapSite_representationAveragingSite
- TNLean.PEPS.exists_sitewisePhysicalMap_to_representationAveragingSite
- TNLean.PEPS.exists_torusCutMap_localInverse
- TNLean.PEPS.torusCutMap_recover_representationAveragingSite
- TNLean.PEPS.fourTorusCutSpace_eq_map_representationAveragingSite
- TNLean.PEPS.map_matchedCommutingClosureSpan_recover
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.
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.
For arbitrary inserted bond matrices, with \(O^q_f\) as in (233) and (234),
This identity requires no semi-regularity assumption.
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 \(E_p(\psi )=0\).
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.
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 \(\mathcal T_B\psi =\psi \).
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.
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 ] .
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 \(l\).
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.
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.
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 \(\mathcal N_A(O^{\mathrm h,q},O^{\mathrm v,q}) =\mathcal N_A(O^{\mathrm h},O^{\mathrm v})\).
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.
- TNLean.PEPS.graphNumberedBondMatrix_coordinates_rightInverse
- TNLean.PEPS.graphNumberedBondMatrix_coordinates_injective
- TNLean.PEPS.map_regionParentGroundSpace_coordinates_eq
- TNLean.PEPS.canonicalCoordinateParentEquiv
- TNLean.PEPS.finrank_regionParentGroundSpace_coordinates_eq
- TNLean.PEPS.map_ker_regionParentHamiltonian_coordinates_eq
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.
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 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 \(Q_R(Q^\dagger )_R=I\).
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.
The identity \(L(g)=QU(g)Q^\dagger \) alone implies \(QQ^\dagger =I\). If also \(Q^\dagger Q=I\), then \(U(g)=Q^\dagger L(g)Q\).
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, \(T(\ker H)=\ker K\).
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.
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.
In the setting of the canonical birth map, with its hypotheses, there are splitting data with the reference error of Theorem 27.134.2.6 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 27.134.2.17.
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 \(Y\).
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exchange
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.exchangeOp_mul_encoder
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.commute_correction
- TNLean.PEPS.EncodedFrame.TwoSheetExchange.norm_exchangeOp_le_one
- TNLean.PEPS.EncodedFrame.liftTags_mul_kronecker
- TNLean.PEPS.EncodedFrame.commute_sheetBufferCorrection
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
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.
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 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.
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.
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.norm_rewrite_le_one_and_refVec_sub_le
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.norm_rewrite_refVec_sub_le
- TNLean.PEPS.EncodedFrame.exists_tagEquiv_of_perm
- TNLean.PEPS.EncodedFrame.frameEncoder_submatrix_of_rawProd_eq
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_norm_le_one_and_refVec_sub_le_of_perm
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 27.133.25). The expansion is stated in the canonical coordinates of the two frames, the tags of the untouched holes first.
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 ] .
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 27.133.23 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 27.138.8, and the polynomial bound on the number of terms is Theorem 27.133.25.
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_branch_truncation
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranchLinearMap
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranchLinearMap_apply
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.liftBranch_sub
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.norm_liftBranch_le
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_contractive_approx
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).
- TNLean.PEPS.EncodedFrame.exists_of_mem_encSteps
- TNLean.PEPS.EncodedFrame.exists_of_mem_decSteps
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.owners_mem_parties_of_mem_branchSteps
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.owners_mem_parties_of_not_untouched
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.ownerOld_eq_ownerNew_of_untouched
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.card_open_vertices_le
- TNLean.PEPS.EncodedFrame.SmallPatchRewrite.exists_termChain
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.
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.
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 27.145.5, with independent endpoint bases on the two sides.
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.
If all eight bond representations are semi-regular, then \(\mathcal I_{A^0}\subseteq \mathcal Z_{A^0}\) and in fact \(\mathcal I_{A^0}=\mathcal Z_{A^0}\).
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 \(\mathcal P b_p\in \mathcal Z_{A^0}\).
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.
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 \(\mathcal T(\mathcal Z_{A^0})=\mathcal Z_A\).
Put \(V'=\bigoplus _i(V_i\otimes \mathbb C^{M_i})\) and define
Then \(F^\dagger F=E_VE_V^\dagger \), so \(F\) preserves inner products on the matching-sector subspace. More precisely, whenever \(E_VE_V^\dagger \phi =\phi \),
It sends the full endpoint vector of \(\bigoplus _i\sqrt{m_i}X_i\) to that of \(\bigoplus _i(X_i\otimes I_{M_i})\). For \(m_i=\dim V_i\), this is the local physical bond transformation used in [ 235 , Section 7 ] .
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.
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.
Let \(U\) be a semi-regular representation of a finite group \(G\), and let \(A\) be \(G\)-injective for its four-leg virtual representation. On a nonempty rectangular torus, let \(\mathcal V_A\) be the span of the vectors \(|\mathcal M(A\mid C)\rangle \) indexed by commuting simultaneous-conjugacy classes \(C\). Then
This is the dimension of the commuting closure span appearing in [ 235 , Theorem 5.9 ] .
- TNLean.PEPS.IsGInjective.finrank_span_torusGClosure_commuting_eq_sum_card_centralizer
- TNLean.PEPS.IsGInjective.finrank_span_torusGClosureClass_commuting_eq_sum_card_centralizer
- TNLean.PEPS.IsGInjective.finrank_span_torusGClosureClass_commuting_of_isSemiRegular
- TNLean.PEPS.range_torusGClosure_commuting_eq_range_torusGClosureClass
Let \(G\) be a finite group acting linearly on a complex virtual space \(\mathcal V\), and let \(P_G=|G|^{-1}\sum _{g\in G}U_g\) be the projector onto \(\mathcal V^G\). If \(T:\mathcal V\to \mathcal P\) is \(G\)-injective, then
In finite dimension, \(\dim \operatorname {im}T=\dim \mathcal V^G\). These are local consequences of [ 235 , Definition 5.1 ] . The virtual projector can be inserted before the physical map:
![\begin{tenkz}[rows={wire}, cols=2, bonds=none]
\tn[name=rangeProjector, skin=box,
ports={180:virtual, 0:virtual}]{P_G}
\tn[at=(1,2), name=rangeTensor,
ports={180:virtual, 0:physical}]{T}
\tnwire{rangeProjector.0}{rangeTensor.180}
\end{tenkz}](tenkz_svg/tenkz-0229e24c9fd184fb.svg)
\(=\) ![\begin{tenkz}[rows={wire}, cols=1, bonds=none]
\tn[ports={180:virtual, 0:physical}]{T}
\end{tenkz}](tenkz_svg/tenkz-10fc6a9dd2ea1a51.svg)
Let \(T:\mathcal V\to \mathcal P\) be \(G\)-injective and let \(F:\mathcal P\to \mathcal Q\) be linear. The composite \(FT\) is invariant under the original virtual representation. If \(F|_{\operatorname {im}T}\) is injective, then \(FT\) is \(G\)-injective. For a finite group \(G\), the converse holds: \(FT\) is \(G\)-injective if and only if \(\operatorname {im}T\cap \ker F=\{ 0\} \). In particular an invertible physical change of coordinates preserves \(G\)-injectivity in both directions. These are local consequences of [ 235 , Definition 5.1 ] .
- TNLean.PEPS.comp_invariant_of_invariant
- TNLean.PEPS.IsGInjective.comp_of_injOn_range
- TNLean.PEPS.IsGInjective.comp_of_injective
- TNLean.PEPS.IsGInjective.comp_equiv
- TNLean.PEPS.isGInjective_comp_equiv_iff
- TNLean.PEPS.IsGInjective.comp_iff_injOn_range
- TNLean.PEPS.IsGInjective.comp_iff_range_inf_ker_eq_bot
- TNLean.PEPS.IsGInjective.comp_iff_injective_rangeSubtype
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.
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
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 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.
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.
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.
![\begin{tenkz}[rows={wire}, cols=3, bonds=none]
\tn[at=1 e of (1,1), name=generalThreeA, skin=box,
ports={0:virtual,180:virtual,240:virtual,300:virtual,90:physical}]{A_0}
\tn[at=3 e of generalThreeA, name=generalThreeB, skin=box,
ports={180:virtual,0:virtual,240:virtual,300:virtual,90:physical}]{A_1}
\tn[at=3 e of generalThreeB, name=generalThreeC, skin=box,
ports={180:virtual,0:virtual,240:virtual,300:virtual,90:physical}]{A_2}
\tn[at=3 s of generalThreeB, name=generalThreeX, skin=ring,
ports={180:virtual,150:virtual,120:virtual,105:virtual,
75:virtual,60:virtual,30:virtual,0:virtual}]{X}
\tnwire[name=generalThreePhysicalA]{generalThreeA.90}{open n}
\tnmark[form=label, label pos=e]{on generalThreePhysicalA 0.6}{$i$}
\tnwire[name=generalThreePhysicalB]{generalThreeB.90}{open n}
\tnmark[form=label, label pos=e]{on generalThreePhysicalB 0.6}{$j$}
\tnwire[name=generalThreePhysicalC]{generalThreeC.90}{open n}
\tnmark[form=label, label pos=e]{on generalThreePhysicalC 0.6}{$k$}
\tnwire[name=generalThreeInternalA]{generalThreeA.0}{generalThreeB.180}
\tnmark[form=label, label pos=n]{on generalThreeInternalA 0.5}{$a$}
\tnwire[name=generalThreeInternalB]{generalThreeB.0}{generalThreeC.180}
\tnmark[form=label, label pos=n]{on generalThreeInternalB 0.5}{$b$}
\tnwire[name=generalThreeXtwo, route=arc]{generalThreeA.180}{generalThreeX.180}
\tnwire[name=generalThreeXthree]{generalThreeA.240}{generalThreeX.150}
\tnwire[name=generalThreeXfour]{generalThreeA.300}{generalThreeX.120}
\tnwire[name=generalThreeXfive]{generalThreeB.240}{generalThreeX.105}
\tnwire[name=generalThreeXsix]{generalThreeB.300}{generalThreeX.75}
\tnwire[name=generalThreeXseven]{generalThreeC.240}{generalThreeX.60}
\tnwire[name=generalThreeXeight]{generalThreeC.300}{generalThreeX.30}
\tnwire[name=generalThreeXnine, route=arc]{generalThreeC.0}{generalThreeX.0}
\end{tenkz}](tenkz_svg/tenkz-a8614674886584fa.svg)
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
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
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.
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 ] .
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.
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.
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 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.
For a finite family of compositions \(F_\xi :a\to b\) and arbitrary coefficients \(c_\xi \in \mathbb {C}\),
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.
- TNLean.PEPS.PairEffect.SourceInventory.exists_grouped
- TNLean.PEPS.PairEffect.SourceInventory.exists_combined
- TNLean.PEPS.PairEffect.Word.exists_grouped_source_preparation
- TNLean.PEPS.PairEffect.PairSource.combine
- TNLean.PEPS.PairEffect.PairSource.partyPair_combine
- TNLean.PEPS.PairEffect.PairSource.norm_combine_vector
- TNLean.PEPS.PairEffect.SourceInventory.Expands.combine
- TNLean.PEPS.PairEffect.expandCombinedPair
- TNLean.PEPS.PairEffect.isAllowed_expandCombinedPair
- TNLean.PEPS.PairEffect.eval_expandCombinedPair
For every \(\psi \in \mathcal H_V\),
Thus \(\| K\| \leq 1\).
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.
Identifying coordinate vectors with their Euclidean Hilbert space preserves the dimension of every inside subspace. The tag dimension satisfies
If \(n=0\) or every \(S_j=0\), then \(T\) is empty and \(K=0\).
The identity encoding satisfies \(K_{\mathrm{id}}^\dagger K_{\mathrm{id}}=I\). For every \(\psi \in \mathcal H_V\), \(\| K_{\mathrm{id}}\psi \| =\| \psi \| \).
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 ] .
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.
- TNLean.PEPS.PairEffect.SourceCircuit.partySelector_affectedOwner_all_apply
- TNLean.PEPS.PairEffect.SourceCircuit.isTouched_groupedParty_iff
- TNLean.PEPS.PairEffect.SourceCircuit.OriginalGateLabelsAt
- TNLean.PEPS.PairEffect.SourceCircuit.choicesOfOriginalGateLabelsAt
- TNLean.PEPS.PairEffect.SourceCircuit.originalGateLabelsAtOfAllChoices
- TNLean.PEPS.PairEffect.SourceCircuit.choicesOfOriginalGateLabelsAt_eq_projectAllChoices
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 ] .
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.
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 ] .
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.
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.
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.
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.barePhysicalInteraction
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.unusedPhysicalVector
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.barePhysicalInteraction_eq_zero
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.barePhysicalInteraction_posSemidef
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.twoSiteEmbedding_mulVec_unused
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.unusedPhysicalVector_not_mem_range
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.unusedPhysicalVector_mem_bareKernel
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.bareKernel_ne_productSiteRange
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.finrank_productSiteRange
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.finrank_bareKernel
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.
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.bondDimension
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.siteEmbedding
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.siteLeftInverse
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.twoSiteEmbedding
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.twoSiteLeftInverse
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.bellVector
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.virtualInteraction
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.bondDimension_pos
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.siteLeftInverse_mul_siteEmbedding
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.twoSiteLeftInverse_mul_twoSiteEmbedding
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.siteEmbedding_injective
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.twoSiteEmbedding_injective
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.bellVector_eq_one
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.virtualInteraction_posSemidef
- TNLean.PEPS.InjectiveParentReconstructionCounterexample.ker_virtualInteraction_eq_bellSpan
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 ] .
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.
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 ] .
- TNLean.PEPS.regionVertexLeftInverse_comp_regionVertexTensorMap
- TNLean.PEPS.regionVertexTensorMap_injective
- TNLean.PEPS.regionVertexTensorMap_single
- TNLean.PEPS.regionVertexTensorMap_regionVirtualBondWeight
- TNLean.PEPS.openRegionMap_eq_regionVertexTensorMap_comp_regionVirtualBondMap
- TNLean.PEPS.regionVertexLeftInverse_comp_openRegionMap
- TNLean.PEPS.regionGroundSpace_eq_map_regionVirtualBondMap_range
- TNLean.PEPS.comap_regionGroundSpace_regionVertexTensorMap
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 \(\eta ^{\otimes m}\).
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.
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.
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.
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.
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 \(\lVert v/\lVert v\rVert -e^{i\theta }\Omega \rVert \le 2\eta \).
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 \(\sigma \) 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.
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.
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.
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 ] .
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 \(E_{x,y}\).
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 \(\rho _v\).
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
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.
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.
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 \(M\).
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.
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.
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 ] .
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.
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.
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.
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 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 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.
- TNLean.PEPS.PairEffect.gate_toGate_eq_sum
- TNLean.PEPS.PairEffect.PartyGate.transfer
- TNLean.PEPS.PairEffect.PartyGate.length_transfer
- TNLean.PEPS.PairEffect.PartyGate.isAllowed_effectCount_transfer
- TNLean.PEPS.PairEffect.PartyGate.sum_norm_coeff_transfer
- TNLean.PEPS.PairEffect.PartyGate.eval_transfer
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.
Under the dimension and message-count bounds of Theorem 27.162.4, 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.
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 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.
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 \(KCD^m\).
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.
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.
Let \(U\) be a monoid homomorphism and let \(W\) commute with every \(U(a)\). Then, for \(n\geq 0\),
This is the endpoint-weight calculation used in [ 235 , Section 7 ] .
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 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 \(\rho _\square \).
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.
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.
- TNLean.PEPS.torusGClosure_slice_mem_plaquetteGroundSpace_of_invariant
- TNLean.PEPS.torusGClosure_mem_torusParentGroundSpace_of_invariant
- TNLean.PEPS.torusParent_annihilates_torusGClosure_of_invariant
- TNLean.PEPS.torusParentHamiltonian_annihilates_torusGClosure_of_invariant
- TNLean.PEPS.span_torusGClosure_commuting_le_torusParentKernel_of_invariant
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 \(E_v^{-1}\rho _v^oE_v\).
- TNLean.PEPS.graphOrientedIncidentMatrix_torus
- TNLean.PEPS.graphOrientedIncidentRepresentation_torus
- TNLean.PEPS.regularSiteMap_torusNativeIncidentSite_eq_siteMap
- TNLean.PEPS.IsGInjective.isGInjective_torusNativeIncidentSite
- TNLean.PEPS.localTensorMap_torusNativeIncidentSite
- TNLean.PEPS.IsGInjective.isGInjective_torusNativeIncidentTensor
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.
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.
- TNLean.PEPS.regionParentGroundSpace_le_iInf_cutSpace
- TNLean.PEPS.exists_cutBoundary_of_mem_regionParentKernel
- TNLean.PEPS.DependentBondNetwork.headTailConfigEquiv
- TNLean.PEPS.DependentBondNetwork.diagonalEndpointConfig
- TNLean.PEPS.DependentBondNetwork.sum_identityWeight_mul
- TNLean.PEPS.DependentBondNetwork.cutIdentityWeight
- TNLean.PEPS.DependentBondNetwork.cutIdentityWeight_mul_interior
- TNLean.PEPS.DependentBondNetwork.sum_cutIdentityWeight_mul
- TNLean.PEPS.sum_regionBoundaryLabel_mul_eq_openRegionMap
- TNLean.PEPS.outsidePhysicalToComplement
- TNLean.PEPS.graphRegionalCutBoundary
- TNLean.PEPS.liftedCutMap_graphRegionalCutBoundary
- TNLean.PEPS.mem_liftedCutSpace_of_region_slices
- TNLean.PEPS.regionParentGroundSpace_le_iInf_liftedCutSpace
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.
- TNLean.PEPS.torusPlaquetteParentGroundSpace_le_graphSeamCut
- TNLean.PEPS.exists_graphSeamBoundary_of_mem_torusPlaquetteParentKernel
- TNLean.PEPS.regularWalkHolonomy_eq_relative_of_region
- TNLean.PEPS.regularWalkHolonomy_eq_one_of_region
- TNLean.PEPS.IsTorusFlat.translate
- TNLean.PEPS.exists_vertexGauge_off_torusGraphSeamCut
- TNLean.PEPS.averagingProjector_bondProduct_mem_graphSeamCut
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 \(n=3\).
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.
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
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 \(v\).
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 ] .
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 ] .
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 \(\widetilde S_j=0\).
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 \(\sum _j\dim \widetilde S_j\leq \sum _j\dim S_j\).
- TNLean.PEPS.finrank_nestedCylinderInnovation_le
- TNLean.PEPS.nestedCylinderInnovation_sup
- TNLean.PEPS.nestedCylinderInnovation_orthogonalEarlier
- TNLean.PEPS.nestedCylinderInnovation_prefix_span
- TNLean.PEPS.nestedCylinderInnovation_span
- TNLean.PEPS.nestedCylinderInnovation_pairwise_orthogonal
- TNLean.PEPS.sum_finrank_nestedCylinderInnovation_le
- TNLean.PEPS.sum_int_finrank_nestedCylinderInnovation_le
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.
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
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.
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.
Let \(u\ge 4\), \(\kappa \in \mathbb R\), and assign each selected patch the same weight \(t_j=\kappa /m\). Then
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.
- TNLean.PEPS.AreaLaw.Geometry.nestedPatchCount_pos
- TNLean.PEPS.AreaLaw.Geometry.half_lt_nestedPatchCount
- TNLean.PEPS.AreaLaw.Geometry.le_nestedPatchRadius
- TNLean.PEPS.AreaLaw.Geometry.nestedPatchRadius_le_two_mul
- TNLean.PEPS.AreaLaw.Geometry.nestedPatchRadius_add_two_le
- TNLean.PEPS.AreaLaw.Geometry.nestedPatch_mono
- TNLean.PEPS.AreaLaw.Geometry.card_edgeBoundary_nestedPatch_le
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.
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.
- TNLean.PEPS.NativeTorus.closureLabels_conjugate
- TNLean.PEPS.NativeTorus.closureLabels_eq_iff
- TNLean.PEPS.NativeTorus.bondCoefficientExtraction_closure
- TNLean.PEPS.NativeTorus.bondCoefficientExtraction_closure_eq_zero_of_class_ne
- TNLean.PEPS.NativeTorus.bondCoefficientExtraction_closure_self
- TNLean.PEPS.NativeTorus.bondCoefficientExtraction_closure_self_ne_zero
- TNLean.PEPS.NativeTorus.closure_conjugate
- TNLean.PEPS.NativeTorus.closure_eq_of_pairConjugacyClass_eq
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.
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 ] .
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.
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 \(\Psi \).
- TNLean.PEPS.ker_nonuniformTorusPlaquetteParent_eq_map_nativeCommutingClosureSpan
- TNLean.PEPS.nonuniformNativeParentEquiv
- TNLean.PEPS.nonuniformNativeParentEquiv_apply
- TNLean.PEPS.nonuniformNativeParentEquiv_symm_apply
- TNLean.PEPS.existsUnique_nativeClosureRepresentative_of_mem_nonuniformParentKernel
- TNLean.PEPS.finrank_ker_nonuniformTorusPlaquetteParentHamiltonian
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.
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 \(\mathcal Z_A=(\bigotimes _vT_v)\mathcal Z_{A^0}\).
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.
- TNLean.PEPS.exists_nonuniform_seamBoundaries_of_mem_torusPlaquetteParentKernel
- TNLean.PEPS.physicalPaddingEmbedding
- TNLean.PEPS.graphPhysicalPaddingTensor
- TNLean.PEPS.physicalRestrictionProduct_padding
- TNLean.PEPS.padding_restriction_eq_self_of_mem_regionParentGroundSpace
- TNLean.PEPS.nonuniform_physicalRepresentative_unique
- TNLean.PEPS.physicalPaddingDimension
- TNLean.PEPS.physicalPaddingMatrix
- TNLean.PEPS.physicalRestrictionMatrix
- TNLean.PEPS.physicalRestrictionMatrix_mul_padding
- TNLean.PEPS.physicalPaddingMatrix_injective
- TNLean.PEPS.physicalPaddingProduct_injective
- TNLean.PEPS.graphDependentTensor_physicalPadding
- TNLean.PEPS.physicalMapSite_restrict_graphPhysicalPaddingTensor
- TNLean.PEPS.isGInjective_graphPhysicalPaddingTensor
- TNLean.PEPS.cutSpace_graphPhysicalPaddingTensor
- TNLean.PEPS.restriction_cutMap_graphPhysicalPaddingTensor
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.
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 \(\mathcal S\).
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.
If \(e\notin C\), then \(\operatorname {im}F_e\subseteq \operatorname {im}\widetilde F_e^C\). If \(e\in C\), then \(\operatorname {im}\widetilde F_e^C=\mathbb {C}^{X_e\times X_e}\).
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
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 \(\mathcal P^f b_{(Q_e^C(a_e))_e}\in \mathcal S_{A^f,C}\).
Suppose every bond space has positive dimension. The coefficient of the PEPS at physical configurations \(\sigma \) on \(R\) and \(\tau \) on its complement is exactly
In particular this holds for regular bonds of any finite group. If the region coefficient is instead defined by summing global edge assignments, it equals \(T_R\) multiplied by the product of the dimensions of all edges wholly outside \(R\).
This is an exact finite-region contraction identity. Identifying its cut maps with isometries on the regular invariant boundary remains a separate step in [ 235 , Theorem 6.9 ] .
- TNLean.PEPS.regionBlockedWeight_eq_exterior_mul_openRegionWeight
- TNLean.PEPS.regionExteriorMultiplicity_eq_prod
- TNLean.PEPS.regionInteriorBondProd_eq_exterior_mul_exterior
- TNLean.PEPS.regionExteriorMultiplicity_ne_zero
- TNLean.PEPS.stateCoeff_eq_openRegionComplement
- TNLean.PEPS.stateCoeff_eq_openRegionComplement_of_regularBonds
- TNLean.PEPS.stateCoeff_rectangle_eq_openRegionComplement_of_regularBonds
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.
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.
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 \(U,W,\theta ,\sigma \).
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.
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.
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
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 ] .
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.
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 \(U\).
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 \(M(F)M(L)=1\).
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.
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.
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.
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
- TNLean.PEPS.graphOrientedOpenBoundaryFactor_coordinates
- TNLean.PEPS.graphOrientedOpenInternalFactor_coordinates
- TNLean.PEPS.graphOrientedRegionCoordinateMatrix_open
- TNLean.PEPS.graphOrientedRegionCoordinateMatrix_open_superposition
- TNLean.PEPS.graphOrientedRegionCoordinateMatrix_maps_openBondSpace
- TNLean.PEPS.graphOrientedRegionCoordinateMatrix_map_openBondSpace
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 \(M_o(Q)\ker H_U=\ker H_L\).
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 \(\operatorname {vec}(M)(a,b)=M_{a,b}\).
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
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 \(B_m\).
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
- TNLean.PEPS.graphOrientedOpenInternalFactor_copy_adjoint
- TNLean.PEPS.graphOrientedOpenBoundaryFactor_copy_adjoint
- TNLean.PEPS.graphRegionCopyMatrix_adjoint_oriented_open
- TNLean.PEPS.graphOrientedOpenInternalFactor_copy
- TNLean.PEPS.graphOrientedOpenBoundaryFactor_copy
- TNLean.PEPS.graphRegionCopyMatrix_oriented_open
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, \(M_m\ker H_A=\ker H_B\).
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.
- TNLean.PEPS.graphNumberedCopyMatrix_maps_oriented_parent
- TNLean.PEPS.graphNumberedCopyMatrix_adjoint_maps_oriented_parent
- TNLean.PEPS.graphNumberedCopyMatrix_leftInverse_oriented_parent
- TNLean.PEPS.graphNumberedCopyMatrix_rightInverse_oriented_parent
- TNLean.PEPS.graphNumberedCopyMatrix_injOn_oriented_parent
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 \(m_i\).
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.
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.
- TNLean.PEPS.graphOrientedOpenBondCoordinates_eq_sum_prod
- TNLean.PEPS.graphOpenRegionNetwork_orientedAveragingSite_coherent
- TNLean.PEPS.graphOpenBondRegrouping_orientedAveragingSite
- TNLean.PEPS.openRegionWeight_numberedGraphOrientedAveragingTensor
- TNLean.PEPS.regionBondPhysicalEquiv_openRegionWeight_oriented
- TNLean.PEPS.map_regionGroundSpace_eq_graphOrientedOpenBondSpace
- TNLean.PEPS.regionBondPhysicalEquiv_mem_graphOrientedOpenBondSpace_iff
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 \(\operatorname {im}F^{\otimes E}\).
- TNLean.PEPS.graphOrientedOpenInternalFactor_mem_range_of_factors
- TNLean.PEPS.graphRegionInternalBondSlice_orientedAveragingSite
- TNLean.PEPS.graphRegionInternalBondSlice_mem_range_of_oriented_factors
- TNLean.PEPS.regionInternalBondSlice_mem_range_of_oriented_factors
- TNLean.PEPS.regionParentGroundSpace_mem_product_range_of_oriented_factors
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.
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.
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.
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.
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.
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.
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 \(B\).
Under these local hypotheses, \(\| W_G\| \le 1\). The same assertion holds for every original contraction circuit.
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 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 \(F\).
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, \(R_{f,\ell m}C_{\ell ,m}^{-1} =C_{\ell ^f,m^f}^{-1}(R_{f,\ell }\otimes R_{f,m})\).
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 \(\mathbb {C}\).
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 ] .
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 ] .
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 ] .
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.
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 ] .
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 \(S\).
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 \(S\).
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 ] .
Let \(G\) be a group and choose a representative \(g_C\) of each conjugacy class \(C\). Write \(\operatorname {Cl}(H)\) for the set of conjugacy classes of a group \(H\). The simultaneous-conjugacy classes of commuting pairs are in bijection with \(\coprod _{C\in \operatorname {Cl}(G)}\operatorname {Cl}(Z_G(g_C))\). If \(G\) is finite, their number is therefore
This is the group-theoretic classification following [ 235 , Theorem 5.9 ] .
- TNLean.PEPS.card_commutingPairConjugacyClass_eq_sum_card_centralizer_of_representatives
- TNLean.PEPS.card_commutingPairConjugacyClass_eq_sum_card_centralizer
- TNLean.PEPS.commutingPairConjugacyClassEquivSigmaCentralizerOfRepresentatives
- TNLean.PEPS.commutingPairConjugacyClassEquivSigmaCentralizer
- TNLean.PEPS.centralizerConjClassEquivCommutingPairFiber
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.
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 27.139.2. The hypothesis that \(G\) is a contraction is not needed.
- TNLean.PEPS.PairEffect.pairEffectElimination
- TNLean.PEPS.PairEffect.norm_inventoryVector
- TNLean.PEPS.PairEffect.norm_replaceGate_sub_le
- TNLean.PEPS.PairEffect.replaceGate_eq_sum_termList
- TNLean.PEPS.PairEffect.length_termList_le
- TNLean.PEPS.PairEffect.sum_norm_coeff_termList
- TNLean.PEPS.PairEffect.norm_term_le_one
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.
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 \(\psi \ne 0\).
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.
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.
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.
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.
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 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 \(\mathcal G_{R_i}(A)\).
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
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 \(WD(g)=D(g)W=\bigoplus _i d_i^{1/4}D_i(g)\).
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: \((M_\lambda )_{x,r}=\lambda _{i(x)}M_{x,r} =\lambda _{i(r)}M_{x,r}\).
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
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
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.
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 \(s\),
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
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.
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 \(F\).
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.
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 \(\phi \in \operatorname {im}\mathcal F\),
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 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.
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.
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.
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
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 27.151.3 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.
- TNLean.PEPS.PairEffect.SourceCircuit.sourceVectorAt_norm
- TNLean.PEPS.PairEffect.SourceCircuit.isTouched_of_source_endpoint
- TNLean.PEPS.PairEffect.SourceCircuit.sourceAt_eq_mapOwner_sourceVectorAt
- TNLean.PEPS.PairEffect.SourceCircuit.eq_partialSlotVector_of_inventory_eq
- TNLean.PEPS.PairEffect.SourceCircuit.exists_partial_source_preparation_with_original_vectors
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 ] .
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.
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.
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 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.
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_gate_of_touches
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_gate_of_exterior
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_comp
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_frame
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_id
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_localMap
- TNLean.PEPS.PairEffect.SourceCircuit.sum_norm_coefficient_swap
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 \(\xi \).
- TNLean.PEPS.PairEffect.SourceCircuit.sourceOrder
- TNLean.PEPS.PairEffect.SourceCircuit.mem_sourceOrder
- TNLean.PEPS.PairEffect.SourceCircuit.nodup_sourceOrder
- TNLean.PEPS.PairEffect.SourceCircuit.sourceOrder_gate
- TNLean.PEPS.PairEffect.SourceCircuit.sourceDims
- TNLean.PEPS.PairEffect.SourceCircuit.sourceAt
- TNLean.PEPS.PairEffect.SourceCircuit.sourceAt_isSome_iff
- TNLean.PEPS.PairEffect.SourceCircuit.sourceAt_eq_some_spec
- TNLean.PEPS.PairEffect.SourceCircuit.filterMap_sourceAt_comp
- TNLean.PEPS.PairEffect.SourceCircuit.filterMap_sourceAt_gate
- TNLean.PEPS.PairEffect.SourceCircuit.sources_partialWord_eq_filterMap
- TNLean.PEPS.PairEffect.SourceCircuit.layout_sources_partialWord
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.
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 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 \(U_e(p_e)\).
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 \(v\).
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.
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.
Let \(L\ge 1\) and let an operator party network on dyadic anchors of the padded square satisfy the hypotheses of Theorem 27.132.11, 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 27.131.2.1 hold for this operator.
In the setting of Theorem 27.132.1.3, 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.
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 27.139.12.
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.
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.
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 \(q\).
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 \(\bigotimes _p X_p\).
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.
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.
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 \(A\).
- TNLean.PEPS.PairEffect.Word.castLayouts
- TNLean.PEPS.PairEffect.Word.sources_castLayouts
- TNLean.PEPS.PairEffect.Word.isAllowed_castLayouts
- TNLean.PEPS.PairEffect.Word.eval_castLayouts_heq
- TNLean.PEPS.PairEffect.Word.eval_castLayouts
- TNLean.PEPS.PairEffect.Word.restrict
- TNLean.PEPS.PairEffect.Word.sources_restrict
- TNLean.PEPS.PairEffect.Word.isAllowed_restrict
- TNLean.PEPS.PairEffect.Word.restrict_localMap_true_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_localMap_false_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_swap_true_true_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_swap_true_false_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_swap_false_true_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_swap_false_false_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_frame_true_eval_heq
- TNLean.PEPS.PairEffect.Word.restrict_frame_false_eval_heq
- TNLean.PEPS.PairEffect.Word.parties_castLayouts
- TNLean.PEPS.PairEffect.Word.parties_restrict
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 ] .
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 ] .
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.
- TNLean.PEPS.PairEffect.PartyChain.replaceTerm_toEffectChain
- TNLean.PEPS.PairEffect.termList_toGate
- TNLean.PEPS.PairEffect.isSourceOnly_termList
- TNLean.PEPS.PairEffect.gateIso_prepGate
- TNLean.PEPS.PairEffect.PartyChain.replaceWord
- TNLean.PEPS.PairEffect.wordList
- TNLean.PEPS.PairEffect.gateIso
- TNLean.PEPS.PairEffect.prepGate
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.
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
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 \(\mathcal H_V\).
![\begin{tenkz}[rows={wire}, cols=7, bonds=none]
\providecommand{\tnlFvOuter}{P_j^+}
\providecommand{\tnlFvInsert}{D_j}
\providecommand{\tnlFvInner}{P_j^-}
\tn[at=(1,1), name=coordinateouter, skin=box,
ports={180:physical, 0:physical}]{\tnlFvOuter}
\tn[at=(1,3), name=coordinateinsert, skin=box,
ports={180:physical, 0:physical}]{\tnlFvInsert}
\tn[at=(1,5), name=coordinateinner, skin=box,
ports={180:physical, 0:physical}]{\tnlFvInner}
\tn[at=(1,7), name=coordinatestate, skin=box,
ports={180:physical}]{\Omega}
\tnwire[name=coordinateoutput]{open w}{coordinateouter.180}
\tnwire[name=coordinatealpha]{coordinateouter.0}{coordinateinsert.180}
\tnwire[name=coordinatebeta]{coordinateinsert.0}{coordinateinner.180}
\tnwire[name=coordinategamma]{coordinateinner.0}{coordinatestate.180}
\tnmark[form=label, label pos=n]{on coordinateoutput 0.5}{$\sigma$}
\tnmark[form=label, label pos=n]{on coordinatealpha 0.5}{$\alpha$}
\tnmark[form=label, label pos=n]{on coordinatebeta 0.5}{$\beta$}
\tnmark[form=label, label pos=n]{on coordinategamma 0.5}{$\gamma$}
\end{tenkz}](tenkz_svg/tenkz-0ec27eb3139ed0e6.svg)
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 \(X_j\).
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.
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.
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 \(D\).
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.
Exact sources recover \(\widehat\rho _{\mathrm{phys}}(X)=\rho _{\mathrm{phys}}\). The physical replacement error satisfies
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.
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 \(p\).
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.
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.
- TNLean.PEPS.PairEffect.SourceCircuit.sourceOrder_physicalFirstOutput
- TNLean.PEPS.PairEffect.SourceCircuit.physicalFirstLocationsEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.physicalFirstSourcesEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.participants_physicalFirstLocationsEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.branchCoefficient_physicalFirstLocationsEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.endpoints_physicalFirstSourcesEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.sourceVectorAt_physicalFirstSourcesEquiv
- TNLean.PEPS.PairEffect.SourceCircuit.participationCount_physicalFirstOutput
- TNLean.PEPS.PairEffect.SourceCircuit.isExpansionBounded_physicalFirstOutput
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 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.
Let \(W,N,a\) be as in Theorem 27.163.2. 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.
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.
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.
For allowed \(V,W\), both the mixed Gram operator and its matrix satisfy \(\| O\| \leq 1\).
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 \(1\in \mathbb {C}\).
Let \(\mathcal B\) be a finite set, and let \(F:H\to H'\) and \(E:K\to H\) be linear maps between finite-dimensional complex Hilbert spaces. Suppose \(E^\dagger E=I\) and \(F^\dagger F=EE^\dagger \). Write \(\mathcal F=\bigotimes _{e\in \mathcal B}F\) and \(\mathcal E=\bigotimes _{e\in \mathcal B}E\). For every \(\psi \) and every \(\phi \in \operatorname {ran}\mathcal E\),
Products compose bond by bond: \((\bigotimes _e A)(\bigotimes _e B) =\bigotimes _e(AB)\). In particular, the product of any physical isometry is an isometry and preserves all physical overlaps.
Suppose the families in Definition 27.145.5 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.
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 \(\rho \).
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.
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 \(\sum _\xi |c_\xi |\).
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.
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.
For finite families as in Theorem 27.145.2,
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 \(Q_{\eta ,\mathcal H}\).
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 \(p\).
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.
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.
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.
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 \(M=\sum _\xi c_\xi K_{V_\xi }(\eta _\xi )\).
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.
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 27.131.12. The supplied estimates and the assumption of a network already on the square are distinguished from the protocol construction in [ 92 ] .
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 27.131.1.1, 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.
In the source bases of Definition 27.146.1, 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.
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.
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.
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.
- TNLean.PEPS.globalPhysicalMap_mem_parent_physicalDeform
- TNLean.PEPS.map_regionParentGroundSpace_physicalDeform_of_leftInverse
- TNLean.PEPS.map_regionParentGroundSpace_physicalDeform_of_injOn
- TNLean.PEPS.globalPhysicalMap_injOn_parent
- TNLean.PEPS.regionParentGroundSpaceRangeEquiv
- TNLean.PEPS.finrank_regionParentGroundSpace_physicalDeform_of_injOn
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.
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 \(\dim \ker K=\dim \ker H\).
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 \(F_{\mathrm{all}}(\ker H)=\ker K\).
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.
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 \(y\in \operatorname {im}B_v\).
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 \(\psi \in \mathcal P_{\mathcal R}(A)\).
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
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 ] .
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 \(u\in [q]^U\).
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 \(\alpha ,\beta \in \widetilde X_V\),
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 \(\kappa H_k\le H_h\le CH_k\).
If \(R\subseteq S\), then every \(\phi _S\in \mathcal G_S\) and every complementary physical configuration \(\tau \) satisfy \(Q_{R,S,\tau }\phi _S\in \mathcal G_R\).
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.
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.
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.
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 \(\ker H^F=F_V\ker H\).
- TNLean.PEPS.mem_regionParentGroundSpace_iff
- TNLean.PEPS.globalPhysicalMap_mem_regionParentGroundSpace
- TNLean.PEPS.globalPhysicalEquiv_mem_regionParentGroundSpace_iff
- TNLean.PEPS.map_regionParentGroundSpace_physicalDeform
- TNLean.PEPS.regionParentGroundSpacePhysicalEquiv
- TNLean.PEPS.finrank_regionParentGroundSpace_physicalDeform
- TNLean.PEPS.map_ker_regionParentHamiltonian_physicalDeform
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
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.
- TNLean.PEPS.openRegionWeight_physicalDeform
- TNLean.PEPS.openRegionMap_physicalDeform
- TNLean.PEPS.regionGroundSpace_physicalDeform
- TNLean.PEPS.regionGroundSpacePhysicalEquiv
- TNLean.PEPS.regionGroundSpacePhysicalEquiv_apply
- TNLean.PEPS.regionPhysicalEquiv_mem_regionGroundSpace_iff
- TNLean.PEPS.ker_openRegionMap_physicalDeform
- TNLean.PEPS.finrank_regionGroundSpace_physicalDeform
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.
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.
For a region \(R\), let \(F_v\) be a physical matrix at each vertex, allowing different input and output dimensions at different vertices. Their product matrix is
Then \((FL)_R=F_RL_R\), \((F^\dagger )_R=F_R^\dagger \), and identity local maps give the identity on the physical space of the region. In particular, local isometries give an isometry \(F_R\).
Applying \(F_R\) to the actual open-region tensor is the contraction of the transformed site coefficients:
This is the physical operation on a block used in [ 235 , §6.1, accessible virtual coordinates ] .
- TNLean.PEPS.regionPhysicalProductMatrix_mul
- TNLean.PEPS.regionPhysicalProductMatrix
- TNLean.PEPS.regionPhysicalProductMatrix_one
- TNLean.PEPS.regionPhysicalProductMatrix_conjTranspose
- TNLean.PEPS.regionPhysicalProductMatrix_isIsometry
- TNLean.PEPS.regionPhysicalMap
- TNLean.PEPS.regionPhysicalMap_apply
- TNLean.PEPS.regionPhysicalMap_comp
- TNLean.PEPS.regionPhysicalMap_openRegionWeight
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.
- TNLean.PEPS.regionPhysicalProductMatrix_isUnitaryBetween
- TNLean.PEPS.globalPhysicalMatrix_isUnitaryBetween
- TNLean.PEPS.toMatrix_regionPhysicalEquiv_symm
- TNLean.PEPS.deformedRegionInteraction_eq_unitary_conj
- TNLean.PEPS.regionParentHamiltonian_physicalDeform_unitary
- TNLean.PEPS.charpoly_regionParentHamiltonian_physicalDeform_unitary
- TNLean.PEPS.spectrum_regionParentHamiltonian_physicalDeform_unitary
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.
- TNLean.PEPS.canonicalRegionParentInteraction_isIdempotentElem
- TNLean.PEPS.canonicalRegionParentInteraction_apply_eq_self
- TNLean.PEPS.canonicalRegionParentInteraction_unit_norm_gap
- TNLean.PEPS.canonicalRegionParentInteraction_spectrum_subset
- TNLean.PEPS.regionLocalTerm_canonical_isIdempotentElem
- TNLean.PEPS.regionLocalTerm_canonical_isSymmetricProjection
- TNLean.PEPS.regionLocalTerm_canonical_unit_norm_gap
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.
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
- TNLean.PEPS.EncodedFrame.mem_listT
- TNLean.PEPS.EncodedFrame.mem_listR
- TNLean.PEPS.EncodedFrame.mem_listU
- TNLean.PEPS.EncodedFrame.mem_listE
- TNLean.PEPS.EncodedFrame.listT
- TNLean.PEPS.EncodedFrame.listR
- TNLean.PEPS.EncodedFrame.listU
- TNLean.PEPS.EncodedFrame.listE
- TNLean.PEPS.EncodedFrame.nodup_listT
- TNLean.PEPS.EncodedFrame.nodup_listU
- TNLean.PEPS.EncodedFrame.equivT
- TNLean.PEPS.EncodedFrame.equivU
- TNLean.PEPS.EncodedFrame.groupIso_listT_siteVec
- TNLean.PEPS.EncodedFrame.groupIso_listU_siteVec
For every \(\sigma ,\tau \in X_V\),
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 ] .
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.
- TNLean.PEPS.vertexVirtual_edgeLabelPerm_of_regionSlices
- TNLean.PEPS.vertexVirtual_eq_zero_of_regionSlices_off_diagonal
- TNLean.PEPS.fullRegionVertexConfigEquivEdgePair_edgeLabelPerm
- TNLean.PEPS.edgePair_eq_zero_of_regionSlices_off_diagonal
- TNLean.PEPS.edgePair_diagonal_update_eq_of_regionSlices
- TNLean.PEPS.virtualBondSlice_mem_span_of_regionVirtualSlices
- TNLean.PEPS.virtualBondSlice_mem_span_of_twoVertexRegionSlices
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.
- TNLean.PEPS.edgeLabelPerm
- TNLean.PEPS.regionVertexEdgeLabelPerm
- TNLean.PEPS.regionVirtualBondWeight_edgeLabelPerm
- TNLean.PEPS.regionVirtualBondWeight_eq_zero_of_internal_off_diagonal
- TNLean.PEPS.regionVirtualBondMap_range_edgeLabelPerm
- TNLean.PEPS.regionVirtualBondMap_range_eq_zero_of_internal_off_diagonal
Let \(V\) be any finite ordered site set with finite alphabets \(Y_v\), and choose bijections \(e_v:Y_v\to [n_v]\), where \([n]=\{ k\in \mathbb {N}:k{\lt}n\} \), so \([0]=\varnothing \). Write \(\mathcal H_X=\mathbb {C}^{\prod _{v\in X}Y_v}\). For \(X\subseteq V\), let \(E_X\) be the induced unitary coordinate relabelling, so \(E_X|(y_v)_{v\in X}\rangle = |(e_v(y_v))_{v\in X}\rangle \). For every complex matrix \(K\) on \(\mathcal H_X\) and every \(\xi \in \mathcal H_V\),
The extension on the right of (45) uses the relabelled bases. Equivalently, \(E_V(\widehat K_X\xi )=\widehat{E_XKE_X^*}_XE_V\xi \). No Hermiticity or normalization is required; the identities include empty regions, empty alphabets, and empty site sets.
Put \(C=V\setminus X\) and \(K'=E_XKE_X^*\). Write \(I'_C\) for the identity in the relabelled complementary basis. In product coordinates \(E_V=E_X\otimes E_C\), and (45) is the following equality.
![\begin{tenkz}[rows={wire,wire}, cols=5, bonds=none]
\tn[at=(1,1), name=regionbasis, skin=box,
ports={180:physical, 0:physical}]{E_X}
\tn[at=(1,3), name=regionmatrix, skin=box,
ports={180:physical, 0:physical}]{K}
\tn[at=(1,5), name=regioninverse, skin=box,
ports={180:physical, 0:physical}]{E_X^*}
\tn[at=(2,1), name=complementbasis, skin=box,
ports={180:physical, 0:physical}]{E_C}
\tn[at=(2,3), name=complementidentity, skin=box,
ports={180:physical, 0:physical}]{I_C}
\tn[at=(2,5), name=complementinverse, skin=box,
ports={180:physical, 0:physical}]{E_C^*}
\tnwire[name=regionout]{open w}{regionbasis.180}
\tnwire{regionbasis.0}{regionmatrix.180}
\tnwire{regionmatrix.0}{regioninverse.180}
\tnwire[name=regionin]{regioninverse.0}{open e}
\tnwire[name=complementout]{open w}{complementbasis.180}
\tnwire{complementbasis.0}{complementidentity.180}
\tnwire{complementidentity.0}{complementinverse.180}
\tnwire[name=complementin]{complementinverse.0}{open e}
\tnmark[form=label, label pos=n]{on regionout 0.5}{$i$}
\tnmark[form=label, label pos=n]{on regionin 0.5}{$j$}
\tnmark[form=label, label pos=s]{on complementout 0.5}{$\alpha$}
\tnmark[form=label, label pos=s]{on complementin 0.5}{$\beta$}
\end{tenkz}](tenkz_svg/tenkz-38a1e6b5ac8d441a.svg)
\(=\) ![\begin{tenkz}[rows={wire,wire}, cols=1, bonds=none]
\tn[at=(1,1), name=relabelledmatrix, skin=box,
ports={180:physical, 0:physical}]{K'}
\tn[at=(2,1), name=relabelledidentity, skin=box,
ports={180:physical, 0:physical}]{I'_C}
\tnwire[name=relabelledregionout]{open w}{relabelledmatrix.180}
\tnwire[name=relabelledregionin]{relabelledmatrix.0}{open e}
\tnwire[name=relabelledcomplementout]{open w}{relabelledidentity.180}
\tnwire[name=relabelledcomplementin]{relabelledidentity.0}{open e}
\tnmark[form=label, label pos=n]{on relabelledregionout 0.5}{$i$}
\tnmark[form=label, label pos=n]{on relabelledregionin 0.5}{$j$}
\tnmark[form=label, label pos=s]{on relabelledcomplementout 0.5}{$\alpha$}
\tnmark[form=label, label pos=s]{on relabelledcomplementin 0.5}{$\beta$}
\end{tenkz}](tenkz_svg/tenkz-5a6f491dd8571812.svg)
The upper and lower lines carry the region and its complement. Both panels represent the same operator after a change of coordinate labels.
For every region \(R\), the map \(K\mapsto \widehat K_R\) is continuous. For every operator \(K\) on \(\mathcal H_R\) and every \(\xi \in \mathcal H_V\), one has \(\| \widehat K_R\xi \| \le \| K\| \, \| \xi \| \).
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.
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.
- TNLean.PEPS.PairEffect.Word.eval_exchangeBlocks_appendIso_symm
- TNLean.PEPS.PairEffect.Word.exchangeBlocks
- TNLean.PEPS.PairEffect.Word.isAllowed_exchangeBlocks
- TNLean.PEPS.PairEffect.Word.moveHead
- TNLean.PEPS.PairEffect.Word.isAllowed_moveHead
- TNLean.PEPS.PairEffect.Word.eval_moveHead_appendIso_symm
- TNLean.PEPS.PairEffect.appendIso_symm_cons_tmul
- TNLean.PEPS.PairEffect.Word.sources_moveHead
- TNLean.PEPS.PairEffect.Word.sources_exchangeBlocks
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 \(\alpha _{L,M}\).
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 ] .
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 ] .
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.
Let \(b\geq 1\) and let \(G\) act on \(G^b\) by simultaneous left multiplication. Its invariant functions have dimension \(|G|^{b-1}\). Writing \(a=(a_0,\ldots ,a_{b-1})\), the relative coordinates \((a_0^{-1}a_i)_{i=1}^{b-1}\) label the orbits. The equal-weight orbit vectors form an orthogonal basis, each of squared norm \(|G|\). This is the invariant-boundary calculation used in the proof of [ 235 , Theorem 6.9 ] .
- TNLean.PEPS.regularBoundaryRepresentation_apply
- TNLean.PEPS.regularBoundaryRepresentation
- TNLean.PEPS.regularBoundaryRelativeEquiv
- TNLean.PEPS.regularBoundaryRelativeEquiv_smul
- TNLean.PEPS.regularBoundaryInvariantsEquiv
- TNLean.PEPS.finrank_regularBoundaryInvariants_succ
- TNLean.PEPS.finrank_regularBoundaryInvariants
- TNLean.PEPS.regularBoundaryOrbitBasis
- TNLean.PEPS.regularBoundaryOrbitBasis_apply
- TNLean.PEPS.regularBoundaryOrbitBasis_eq_sum
- TNLean.PEPS.regularBoundaryOrbitBasis_dotProduct
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.
If \(T\) is regular \(G\)-injective, then \(C=TU\) is injective, \(Q=C^\dagger C\) is positive definite, and the supported boundary Hamiltonian is Hermitian with
This establishes a finite Gibbs representation on the invariant support. It does not assert short-range interactions among the original virtual legs.
- TNLean.PEPS.IsGInjective.injective_regularBoundaryRestrictedMap
- TNLean.PEPS.IsGInjective.regularBoundarySupportedGram_posDef
- TNLean.PEPS.regularBoundaryGramHamiltonian_isHermitian
- TNLean.PEPS.IsGInjective.regularBoundarySupportedGram_eq_exp
- TNLean.PEPS.regularBoundarySupportedGram_eq_compression
- TNLean.PEPS.IsGInjective.regularBoundaryGram_eq_support_compression
- TNLean.PEPS.IsGInjective.regularBoundaryGram_eq_compressed_exp
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 ] .
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 ] .
- TNLean.PEPS.IsGInjective.commute_of_cornerParent_annihilates_torusGClosure
- TNLean.PEPS.IsGInjective.regionParent_annihilates_torusGClosure_of_isSimplyConnected
- TNLean.PEPS.IsGInjective.regionParentHamiltonian_annihilates_torusGClosure
- TNLean.PEPS.IsGInjective.span_torusGClosure_commuting_le_parentKernel
- TNLean.PEPS.IsGInjective.card_commutingPairConjugacyClass_le_finrank_parentKernel
Let \(G\) be a finite group and let \(T\) and \(S\) be \(G\)-injective maps from the regular virtual boundary with \(b\geq 1\) legs to the two physical spaces of a cut. The coefficient matrix \(C=TS^{\mathsf T}\) satisfies
Consequently, two such factorizations of the same coefficient matrix have the same first physical range, even when their virtual groups or numbers of boundary legs differ. This is a consequence of the regular boundary construction in [ 235 , Definition 5.1 and proof of Theorem 6.9 ] .
Let both torus circumferences be at least three, and let \(A\) be a \(G\)-injective site tensor with the regular virtual representation. Let \(R\) be a contiguous block: its induced nearest-neighbour graph is connected, and its actual closed-cell realization is simply connected. Let \(b\) be the number of bonds crossing its boundary. For any nonzero coherent superposition of actual closures with commuting pairs, let \(M\) be its physical coefficient matrix across the cut and put
The denominator is positive, \(\rho _R\) is a density matrix, and
Trees, roots, boundary numbering, and complementary paths are not supplied. This is the regular native-torus conclusion of [ 235 , Corollary 6.10 ] , under the stated contiguous closed-cell geometry. It makes no flat-spectrum assertion and does not identify the closure span with a parent-Hamiltonian ground space [ 96 ] .
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.
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.
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 \(\dim \operatorname {im}A_R=|G|^{b-1}\).
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.
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 ] .
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 ] .
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 ] .
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 ] .
- TNLean.PEPS.regionGroundSpace_regularTwisted_eq_of_internalGaugeFlat
- TNLean.PEPS.regionPhysicalMap_mem_regularProjectorTwistedRange
- TNLean.PEPS.regionPhysicalMap_leftInverse_on_regularGroundSpace
- TNLean.PEPS.regularProjectorOpenRegionRange_apply_eq_zero_of_cycle_ne_one
- TNLean.PEPS.regionPhysicalMap_apply_eq_zero_of_mem_regularGroundSpace_of_cycle_ne_one
- TNLean.PEPS.regionPhysicalMap_slice_apply_eq_zero_of_parent_annihilates
- TNLean.PEPS.disjoint_regularProjectorRegion_ranges_of_cycleResidual_ne_one
- TNLean.PEPS.disjoint_regionGroundSpace_regularTwisted_of_cycleResidual_ne_one
- TNLean.PEPS.regularWalkHolonomy_eq_one_of_mem_regionGroundSpace_of_mem_twisted
- TNLean.PEPS.regularWalkHolonomy_eq_one_of_regionParent_annihilates_twistedState
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 \(\operatorname {ran}T_R=\mathcal G_R\).
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.
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.
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 ] .
- TNLean.PEPS.IsGInjective.torusPlaquetteHolonomy_eq_one_of_parent_annihilates
- TNLean.PEPS.IsGInjective.regionParent_not_annihilates_torusPlaquetteFlux
- TNLean.PEPS.IsGInjective.torusPlaquetteHolonomy_eq_one_of_parentHamiltonian_annihilates
- TNLean.PEPS.torusPlaquetteHolonomy_eq_one_iff_squareTransport
- TNLean.PEPS.IsGInjective.isTorusFlat_of_parentHamiltonian_annihilates_twistedState
Let \(A\) be regular \(G\)-injective on a native torus with both periods at least three. For each native plaquette \(R_v\), let \(P_v\ge 0\) have kernel equal to the image of the regional PEPS map, and let \(H=\sum _v(P_v\otimes \mathbf1_{R_v^c})\). Then
where \(G\) acts by simultaneous conjugation. This specializes [ 235 , Theorems 5.7 and 5.9 ] to the regular representation and the indicated torus periods.
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 ] .
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 \(\operatorname {ran}\rho _R=\mathcal G_R\).
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 ] .
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.
- TNLean.PEPS.torusSingletonBoundaryEquiv
- TNLean.PEPS.range_regionReducedDensity_torus_singleton_of_regular
- TNLean.PEPS.rank_regionReducedDensity_torus_singleton_of_regular
- TNLean.PEPS.stateCoeff_groupBondTensor_torus_ne_zero_of_regular
- TNLean.PEPS.card_group_cube_le_physicalDimension_regular_torus
- TNLean.PEPS.entropy_normalizedRegionReducedDensity_torus_singleton_of_regular
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
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 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 \(S\).
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.
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.
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
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.
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.
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 \(\widetilde W_\delta \).
- TNLean.PEPS.PairEffect.EffectCircuit.nonprivateCount_rescaledOriginal
- TNLean.PEPS.PairEffect.EffectCircuit.participants_rescaledOriginal
- TNLean.PEPS.PairEffect.EffectCircuit.participationCount_rescaledOriginal
- TNLean.PEPS.PairEffect.EffectCircuit.isExpansionBounded_rescaledOriginal
- TNLean.PEPS.PairEffect.EffectCircuit.isMonomialBounded_rescaledOriginal
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 ] .
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 \({\max (D,1)}^\chi \).
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.
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.
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 ] .
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.
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.
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.
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
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 ] .
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 ] .
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 ] .
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
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.
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 ] .
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.
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 ] .
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 \(1\).
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.
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 27.147.3, 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 ] .
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 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.
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 \(X_e\).
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.
- TNLean.PEPS.graphIncidentMatrix_one
- TNLean.PEPS.graphIncidentMatrix_mul
- TNLean.PEPS.graphIncidentRepresentation_apply
- TNLean.PEPS.graphNumberedIncidentRepresentation_apply
- TNLean.PEPS.regularSiteMap_graphAveragingSite
- TNLean.PEPS.isGInjective_graphAveragingSite
- TNLean.PEPS.localTensorMap_numberedGraphDressedAveragingTensor_one
- TNLean.PEPS.isGInjective_numberedGraphDressedAveragingTensor_one
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
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.
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 \(|G|^k\).
Let \(G\) be a finite group and let \(\rho \) be a finite-dimensional complex representation. The following assertions are equivalent: \(\rho \) contains every irreducible representation; the operators \((\rho (g))_{g\in G}\) are linearly independent; and the induced algebra homomorphism \(\mathbb C[G]\to \operatorname {End}(V)\) is injective. This is an algebraic characterization of the occurrence condition in [ 235 , Definition 4.5 and Lemma 4.6 ] . More generally, for a representation of a monoid over a commutative semiring, independence of its operators is equivalent to injectivity of its monoid-algebra homomorphism.
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 \(U\).
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.
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
If \(U\) is semi-regular, then \(\operatorname{tr}[U_{g^{-1}}U_h\Delta ]=\delta _{g,h}\) for all \(g,h\in G\), with \(U_{g^{-1}}=U_g^\dagger \) for a unitary representation. For the left-regular representation \(\Delta =\frac{1}{|G|}\mathbb {1}\) [ 235 , Section “MPS: the \(G\)-injective case” ] .
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.
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 ] .
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 ] .
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 ] .
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 ] .
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 ] .
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 ] .
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.
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.
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 ] .
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 ] .
The four-leg coefficients are uniquely determined by the site map. For a physical matrix \(F\) and tensors \(A,B\), the coefficient identity \(FA=B\) is equivalent to \(F\mathcal P(A)=\mathcal P(B)\).
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.
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.
The two endpoints of \((g,e)\) are distinct and both belong to \(C_g\).
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.
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.
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.
- TNLean.PEPS.PairEffect.SourceInventory.Expands.refl
- TNLean.PEPS.PairEffect.SourceInventory.Expands.trans
- TNLean.PEPS.PairEffect.SourceInventory.Expands.cons
- TNLean.PEPS.PairEffect.SourceInventory.Expands.reverse
- TNLean.PEPS.PairEffect.SourceInventory.Expands.swap
- TNLean.PEPS.PairEffect.eval_swap_reversedSource
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 \(W:a\to b\) 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 27.140.3. 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 ] .
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.
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.
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.
- TNLean.PEPS.PairEffect.OriginalCircuit.rectangularTraceNorm_sourceOnly_density_sub_le
- TNLean.PEPS.PairEffect.OriginalCircuit.rectangularTraceNorm_sourceOnly_density_sub_le_half_of_count_le
- TNLean.PEPS.PairEffect.OriginalCircuit.rectangularTraceNorm_sourceOnly_wholeParty_density_sub_le_half
- TNLean.PEPS.PairEffect.sourceGateBudget_eq_of_pos
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.
- TNLean.PEPS.PairEffect.sum_norm_replacementCoefficients_le
- TNLean.PEPS.PairEffect.card_replacementLabels_le
- TNLean.PEPS.PairEffect.stackLength_le
- TNLean.PEPS.PairEffect.card_replacementLabels_stackLength_le
- TNLean.PEPS.PairEffect.OriginalCircuit.isExpansionBounded_replacement
- TNLean.PEPS.PairEffect.PreparedSourceGate.length_sources_branchWord_eq_choose
- TNLean.PEPS.PairEffect.SourceCircuit.slotCount_eq_choose
- TNLean.PEPS.PairEffect.OriginalCircuit.card_sourceLocations_replacement_le
- TNLean.PEPS.PairEffect.OriginalCircuit.participationCount_replacement
- TNLean.PEPS.PairEffect.OriginalCircuit.participationCount_replacement_le
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.
If \(P\) is finite and \(R\) has the pair coverage and distinctness in Definition 27.146.15, then \(|R|=\binom {|P|}{2}\le |P|^2\). This includes empty and singleton party sets. These are the pair slots used in [ 248 , proof of Theorem 5.2 ] , 04-compression.tex, lines 565–585.
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.
- TNLean.PEPS.PairEffect.Word.eval_eq_localPart_comp_prepare
- TNLean.PEPS.PairEffect.Word.localPart
- TNLean.PEPS.PairEffect.Word.sources_localPart
- TNLean.PEPS.PairEffect.Word.isAllowed_localPart
- TNLean.PEPS.PairEffect.Word.isNormalized_sources
- TNLean.PEPS.PairEffect.Word.eval_localPart_prepare
- TNLean.PEPS.PairEffect.SourceInventory.eval_frameList_prepare
- TNLean.PEPS.PairEffect.SourceInventory.eval_moveHead_prepare
- TNLean.PEPS.PairEffect.Word.castInput
- TNLean.PEPS.PairEffect.Word.sources_castInput
- TNLean.PEPS.PairEffect.Word.isAllowed_castInput
- TNLean.PEPS.PairEffect.Word.eval_castInput_of_heq
- TNLean.PEPS.PairEffect.Word.sources_frameList
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
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 \(1\).
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.
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 ] .
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 ] .
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 ] .
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 \(\varepsilon /4\).
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 27.154.29 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.
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.
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.
- TNLean.PEPS.PairEffect.SourceInventory.eval_prepare_nil_heq
- TNLean.PEPS.PairEffect.SourceInventory.vector_append_heq
- TNLean.PEPS.PairEffect.SourceInventory.eval_prepareFreeSlots_nil_heq
- TNLean.PEPS.PairEffect.SourceInventory.vector_append_eq
- TNLean.PEPS.PairEffect.SourceInventory.eval_framed_sourceVector
- TNLean.PEPS.PairEffect.SourceInventory.eval_framed_freeSourceVectors
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 \(\eta _e\).
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.
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.
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 ] .
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 ] .
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.
- TNLean.PEPS.AreaLaw.Geometry.abs_coordinate_sub_le_one_of_adj
- TNLean.PEPS.AreaLaw.Geometry.mem_closedSquareSample_of_adj
- TNLean.PEPS.AreaLaw.Geometry.abs_squareRadius_sub_le_one_of_adj
- TNLean.PEPS.AreaLaw.Geometry.mem_edgeBoundary_pair_iff
- TNLean.PEPS.AreaLaw.Geometry.mem_edgeBoundary_closedSquareSample_iff
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 \(K(c,t)\).
Let \(A_v\) be arbitrary local tensors with physical dimension \(q\) and edge-dependent virtual dimensions \(D_e\). Transport each virtual coordinate along the edge correspondence, and put the physical argument last when it was written first. The resulting graph tensor \(\widetilde A\) has dimensions \(\widetilde D_{\iota (e)}=D_e\). For every physical configuration \(s\),
The same equality holds when the source tensor already has its physical argument last. The physical-first assertion permits zero virtual dimensions.
- TNLean.PEPS.Approximation.stateCoeff_pinnedTensorToGraphTensor
- TNLean.PEPS.Approximation.stateCoeff_vectorTensorToGraphTensor
- TNLean.PEPS.Approximation.pinnedTensorToGraphTensor
- TNLean.PEPS.Approximation.vectorTensorToGraphTensor
- TNLean.PEPS.Approximation.pinnedTensorToGraphTensor_component
- TNLean.PEPS.Approximation.vectorTensorToGraphTensor_component
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 ] .
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 ] .
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 ] .
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.
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
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 \(U_e\).
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 \(\mathcal K_{A,T,C_0}=\mathcal O_A\).
For \(Q\subseteq T\) with all uncut edges internal,
If a complementary configuration \(\xi _0\) exists, then \(\mathcal K_{A,Q,C}=\mathcal G_{A,Q,C}\).
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
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 ] .
Let \(A\) be invariant under the virtual group action and let \(gh=hg\). For any column \(c\) and row \(r\), put
Contracting \(A\) with these bond operators gives \(|\mathcal M(A\mid g,h)\rangle \), independently of \(c\) and \(r\). This is the displacement of commuting loops described after the string-deformation display of [ 235 ] , local source lines 1638–1644; neither isometry nor semi-regularity is needed. The identity holds on every nonempty rectangular torus, including dimensions equal to one.
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 ] .
Let \(G\) be a group and let \(a_{x,y},b_{x,y}\in G\) be rightward and upward transports on \(\mathbb Z_w\times \mathbb Z_h\), where \(w,h{\gt}0\). Suppose the connection is flat: \(b_{x+1,y}a_{x,y}=a_{x,y+1}b_{x,y}\). Write \(A_m=a_{m-1,0}\cdots a_{0,0}\) and \(B_{x,n}=b_{x,n-1}\cdots b_{x,0}\), with empty products equal to the identity. The holonomies \(A_w\) and \(B_{0,h}\) commute. For the least nonnegative representatives of \(x,y\), set \(k_{x,y}=B_{x,y}A_x\). Then
If \(w,h\ge 3\) and the transports come from operators \(u_e\) on the ordered edges of the torus graph, the same gauge transforms \(u_e\) into the closure assignment with \(g=B_{0,h}^{-1}\) and horizontal closure element \(A_w\). No trivial-holonomy assumption is made. This is the group-valued seam deformation of [ 235 , Section 5 ] .
- TNLean.PEPS.zmodTransport_step
- TNLean.PEPS.IsTorusFlat.zmodTransport_rectangle
- TNLean.PEPS.IsTorusFlat.commute_holonomies
- TNLean.PEPS.torusTreeGauge
- TNLean.PEPS.IsTorusFlat.torusTreeGauge_right
- TNLean.PEPS.IsTorusFlat.torusTreeGauge_up
- TNLean.PEPS.torusNativeRightTransport
- TNLean.PEPS.torusNativeUpTransport
- TNLean.PEPS.torusNativeRightTransport_closure
- TNLean.PEPS.torusNativeUpTransport_closure
- TNLean.PEPS.torusEdgeAssignment_ext
- TNLean.PEPS.exists_regularVertexGauge_eq_torusClosure
Let \(q:\mathbb Z_w\times \mathbb Z_h\to X\) be labels in any set \(X\), where \(w,h{\gt}0\). Suppose that \(q_{x+1,y}=q_{x,y}\) whenever \(x+1\ne 0\), and \(q_{x,y}=q_{x,y+1}\) whenever \(y+1\ne 0\). Then \(q_{x,y}=q_{0,0}\) at every vertex. This is the synchronization away from the closure seams used in [ 235 , Theorem 5.9 ] .
Let \(U\) be a finite-dimensional representation of a finite group \(G\), and let \(\Pi \) be the four-leg averaging projector. Write \(e(v)\) and \(n(v)\) for the neighbours east and north of \(v\), and let \(s_v=(t_v,r_v,b_v,l_v)\) denote the physical basis label of the projector site. For arbitrary bond operators \(O^H_v,O^V_v\), set
The actual torus contraction of the projector sites has coefficient
No semi-regularity or unitarity assumption is required for this identity. It is the expansion of the projector contraction in [ 235 , Theorem 5.9 ] .
Let \(U\) be a semi-regular representation of a finite group \(G\) on a virtual space with finite basis \(V\). Let \(\Lambda \) be a rectangular torus with positive width and height, and let \(A_0\) be the averaging- projector site. There is one explicit linear map \(E:\mathbb {C}^{\Lambda \to V^4}\to \operatorname {Mat}_{V\times V}(\mathbb {C})\), independent of \(g,h\), such that
Here the tensor product is written in the product basis, with the horizontal factor first. The coefficient \(|G|^{-|\Lambda |}\) is strictly positive. This is an explicit contraction realizing the independence argument of [ 235 , Theorem 5.9 ] .
- TNLean.PEPS.matrixLinearFunctional_eq_sum
- TNLean.PEPS.torusOutputBondPairing
- TNLean.PEPS.torusOutputBondPairing_bondProduct
- TNLean.PEPS.torusDeltaPairing
- TNLean.PEPS.torusSeamPairing
- TNLean.PEPS.torusDeltaPairing_apply_rep
- TNLean.PEPS.torusSeamPairing_apply_rep
- TNLean.PEPS.torusMatrixEntry
- TNLean.PEPS.torusHorizontalExtractionPairing
- TNLean.PEPS.torusVerticalExtractionPairing
- TNLean.PEPS.torusProjectorExtraction
- TNLean.PEPS.torusProjectorExtraction_bondProduct
- TNLean.PEPS.torusDeltaPairing_apply_mul_inv
- TNLean.PEPS.torusHorizontalExtractionPairing_constant
- TNLean.PEPS.torusVerticalExtractionPairing_constant
- TNLean.PEPS.torusProjectorExtraction_constant_bondProduct
- TNLean.PEPS.torusHorizontalExtractionPairing_nonseam
- TNLean.PEPS.torusVerticalExtractionPairing_nonseam
- TNLean.PEPS.torusProjectorExtraction_label_bondProduct
- TNLean.PEPS.torusProjectorExtraction_sum_label_bondProduct
- TNLean.PEPS.torusProjectorExtraction_torusGClosure
- TNLean.PEPS.torusProjectorExtraction_normalization_pos
A bounded coordinate rectangle with positive side lengths is connected in the ordinary grid, hence also in the periodic lattice. If both side lengths are strictly smaller than the torus dimensions, its complement is connected even without the wrap bonds. This last assertion also holds for coordinate intervals clipped by the bounds of the grid. A nonempty bounded rectangle starting at a positive horizontal coordinate has a crossing bond on its left side.
Work on a native torus with both dimensions at least three and an original regular \(G\)-isometric site map. Assume the actual closed-cell realization \(X_R\) is simply connected and the induced occupied graph is connected. Let \(b\) be the number of crossing bonds. There exists one physical reduced density \(\rho _R\) for every nonzero coherent superposition of commuting torus closures, with coefficient matrix \(M\). It satisfies
The Rényi formula holds for every nonnegative finite order, including zero and the von Neumann value at one. The exterior walks, spanning trees, roots, and boundary enumeration are derived internally. This is a specialization of [ 235 , Theorem 6.9 ] to regular native tori with periods at least three. Occupied graph connectivity expresses the connected block in the source, lines 1931–1933 and 1349–1354; simple connectedness of the closed-cell union alone does not force four-neighbor connectivity of the occupied sites.
Under the hypotheses of 26.1.4.80, let \(A\) and \(B\) be the physical coefficient matrices of any two nonzero coherent superpositions of commuting torus closures. Every operator \(O\) on the occupied physical region has equal normalized expectation in these states. There exists a unitary \(U\) on the complementary physical factor satisfying
This is a specialization of [ 235 , Theorem 6.7 and Corollary 6.8 ] to connected simply connected regions of regular native tori with periods at least three. The width-one stripe construction is separate.
Consider an original regular \(G\)-isometric tensor on a native torus whose two periods are at least three. Let \(R\) be the nonwrapping rectangle of sites with both coordinates nonzero. Its complement is the union of two width-one stripes wrapping around the torus:
For any two nonzero coherent superpositions of commuting torus closures, let \(A\) and \(B\) be their physical coefficient matrices across \(R\mid S\). There exists a unitary \(U\) on the physical factor of \(S\) satisfying
Thus \(U\) transforms the first normalized physical state into the second while acting only on the stripes (217). This is the closure-vector specialization of [ 235 , Theorem 6.7 ] to regular native tori with periods at least three. Identification of the closure span with a parent-Hamiltonian ground space is separate.
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 \(C_g\neq \varnothing \).
- TNLean.PEPS.PairEffect.SourceCircuit.choiceAt_projectAllChoices
- TNLean.PEPS.PairEffect.SourceCircuit.choiceAt_choicesOfTouchedLabels
- TNLean.PEPS.PairEffect.SourceCircuit.choicesOfTouchedLabels_choiceAt
- TNLean.PEPS.PairEffect.SourceCircuit.isTouched_all_iff
- TNLean.PEPS.PairEffect.SourceCircuit.allChoicesNonemptyLabelsEquiv_apply
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
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.
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.
- TNLean.PEPS.PairEffect.SourceCircuit.integrable_sampledSourceMatrix
- TNLean.PEPS.PairEffect.SourceCircuit.integral_sampledSourceMatrix
- TNLean.PEPS.PairEffect.SourceCircuit.exists_unbiased_sampledSourceMatrix
- TNLean.PEPS.PairEffect.SourceCircuit.integral_sourceGaussianCorrection
- TNLean.PEPS.PairEffect.SourceCircuit.sourceGaussianCorrection_eq_transport
In the decomposition (359), each row has dimension \(d_i\), and the character of a row occurs in the full representation with multiplicity exactly \(m_i\). Thus the orthonormal decomposition gives the dimensions and multiplicities appearing in [ 235 , Lemma 4.4 ] .
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.
- TNLean.PEPS.VertexVirtualConfig
- TNLean.PEPS.EdgePairVirtualConfig
- TNLean.PEPS.vertexVirtualConfigEquivEdgePair
- TNLean.PEPS.fullRegionVertexConfigEquiv
- TNLean.PEPS.fullRegionVertexConfigEquivEdgePair
- TNLean.PEPS.IsVertexVirtualConsistent
- TNLean.PEPS.isVertexVirtualConsistent_iff_edgePair_diagonal
- TNLean.PEPS.vertexVirtualConfigOfBonds
- TNLean.PEPS.vertexVirtualConfigEquivEdgePair_ofBonds
- TNLean.PEPS.isVertexVirtualConsistent_iff_exists_bonds
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.
- TNLean.PEPS.vertexImageProjection_mulVec_eq_self_iff
- TNLean.PEPS.globalVertexTensorMap_leftInverse_eq_self_of_mem
- TNLean.PEPS.globalVertexTensorMap_mem_vertexImageGroundSpace
- TNLean.PEPS.vertexImageGroundSpace_eq_range_globalVertexTensorMap
- TNLean.PEPS.vertexImageGroundSpace_map_fullRegionPhysicalEquiv
- TNLean.PEPS.globalPhysicalMap_one
- TNLean.PEPS.globalPhysicalMap_comp
- TNLean.PEPS.globalPhysicalMap_oneVertex_apply
- TNLean.PEPS.globalPhysicalMap_fixed_of_oneVertex_fixed
- TNLean.PEPS.vertexPhysicalSlice_globalPhysicalMap_oneVertex
- TNLean.PEPS.globalPhysicalMap_vertexImageProjection
- TNLean.PEPS.oneVertexPhysicalMap_vertexImageProjection_eq_self_of_mem
- TNLean.PEPS.oneVertexPhysicalMap_comp_globalVertexTensorMap
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.
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.
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 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.
- TNLean.PEPS.virtualBondProduct_update
- TNLean.PEPS.virtualBondProduct_mem_virtualBondGroundSpace
- TNLean.PEPS.eq_zero_of_mem_virtualBondGroundSpace_off_diagonal
- TNLean.PEPS.virtualBondGroundSpace_diagonal_update_eq
- TNLean.PEPS.virtualBondGroundSpace_diagonal_eq
- TNLean.PEPS.virtualBondGroundSpace_eq_span
- TNLean.PEPS.virtualBondProduct_ne_zero_iff
- TNLean.PEPS.finrank_virtualBondGroundSpace_eq_one
For finite coefficient families \(c_\alpha ,d_\beta \), rectangular matrices \(K_\alpha ,L_\beta \), and a compatible rectangular matrix \(R\),
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.
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 \(O\).
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 ] .
Let \(S,T\subseteq I\). If \(A\) on \((\mathbb {C}^d)^{\otimes I}\) is supported on \(S\) and \(B\) on \((\mathbb {C}^e)^{\otimes I}\) is supported on \(T\), then the matrix \(A\otimes B\), expressed in the coordinates \(p\), is supported on \(S\cup T\) in \((\mathbb {C}^{de})^{\otimes I}\). This is the support identification for the doubled local interactions in [ 248 ] , 02-information.tex, lines 416–424 and 513–524.
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 \(C_A(q,J,\Delta )=C(q,1,J,\Delta )\).
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.
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.
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 ] .