Dependencies

Let \(A_0,\ldots ,A_{n-1}\) be an injective site-dependent closed chain on \(n\ge 3\) sites, with one physical dimension \(d\) and one positive bond dimension \(D\). If the closed-chain state is invariant under the cyclic shift \(A_v\mapsto A_{v+1}\), then there is an injective tensor \(B\), with bond dimension \(D\), such that, for every physical configuration \(\sigma \),

This is the uniform-bond-dimension target corresponding to the Applications-section corollary of [ 220 , lines 1803–1890 ] . The source also concludes that all bond dimensions of the original site-dependent representation are equal; in the present uniform-dimension statement that clause has already been built into the setting.

Let \(A_0,\ldots ,A_{n-1}\) be a site-dependent injective closed MPS on \(n\ge 3\) sites, with dimensions \(D_e\) on its individual bonds. If the closed-chain state is translationally invariant, in the finite closed-chain sense that it is invariant under the cyclic shift \(A_v\mapsto A_{v+1}\) of [ 220 , lines 1807–1827 ] , then there are a positive integer \(D\) and an injective tensor \(B\) of \(D\times D\) matrices such that \(D_e=D\) for every bond \(e\) and

for every physical configuration \(\sigma \).

This completes the at-least-three-site form of the translational-invariance corollary and its proof in [ 220 , lines 1803–1890 ] . The equality \(D_e=D\) is Corollary 24.9.4.13. Identifying each virtual space with \(\mathbb {C}^D\) gives a uniform closed-chain representation, to which the source’s \(L_i,R_i\) telescoping and repeated-tensor construction apply. The source section places its injective Fundamental Theorem in the regime of at least three sites; the omitted two-site case is false by Theorem 24.9.4.14, as recorded in [ 29 ] .

Let \(A\) be an injective PEPS on a discrete \(n\times m\) torus with \(n,m{\gt}1\). If the generated state is invariant under all lattice translations \(\tau _{a,b}\), that is, \(\operatorname{Coeff}_A(\sigma \circ \tau _{a,b})=\operatorname{Coeff}_A(\sigma )\) for every translation and every physical configuration \(\sigma \), then there are positive dimensions \(D_h,D_v\) such that the bond dimensions of \(A\) are orientation-uniform with these values, and there is an injective translation-invariant torus tensor \(B\) with the same bond-dimension function and the same represented state as \(A\).

This is the full two-dimensional Applications-section corollary of [ 220 , lines 1891–1894 ] .

Suppose a closed chain \(A_0,\ldots ,A_{n-1}\) is gauge equivalent to its cyclic shift. Then for every site \(v\) there are invertible matrices \(L_v\) and \(R_v\) such that \(A_v^i=L_vA_0^iR_v^{-1}\) for every physical index \(i\). This is the telescoping step in [ 220 , lines 1828–1862 ] , where the products denoted \(L_i\) and \(R_i\) express all tensors through the first one.

Let \(A_0,\ldots ,A_{n-1}\) be an injective site-dependent closed chain with positive bond dimension on \(n\ge 3\) sites. If its generated state is invariant under the cyclic translation of the sites, then for every site \(v\) there are invertible matrices \(L_v\) and \(R_v\) such that \(A_v^i=L_vA_0^iR_v^{-1}\) for every physical index \(i\). This is the uniform-bond-dimension form of the \(L_v,R_v\) step in the Applications-section proof of [ 220 , lines 1803–1890 ] .

Let \(A_0,\ldots ,A_{n-1}\) be an injective site-dependent closed chain with positive bond dimension on \(n\ge 3\) sites. If its generated state is invariant under the cyclic shift \(A_v\mapsto A_{v+1}\), then there are invertible matrices \(Z_0,\ldots ,Z_{n-1}\), one per bond, such that \(A_{v+1}^i=Z_vA_v^iZ_{v+1}^{-1}\) for every site \(v\) and physical index \(i\). This is the first comparison step in the Applications-section translation-invariant description corollary of [ 220 , lines 1807–1824 ] .

Let \(A_0,\ldots ,A_{n-1}\) and \(B_0,\ldots ,B_{n-1}\) be two site-dependent closed chains on \(n\ge 3\) sites, with common physical dimension and common positive bond dimension. Suppose every local tensor in both chains is injective and the two closed-chain states are equal. Then there are invertible matrices \(Z_0,\ldots ,Z_{n-1}\), one per bond of the closed chain, such that \(B_v^i=Z_vA_v^iZ_{v+1}^{-1}\) for every site \(v\) and every physical index \(i\), with indices read cyclically. This is the \(L=1\), uniform-dimension instance of [ 220 , Theorem thm:inj_MPS, lines 688–725 ] .

Let \(n\) be positive, let \(0 {\lt} L\) with \(2L + 1 \le n\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices with \(D{\gt}0\), over physical dimension \(d\), each \(L\)-block injective, whose matrix product vectors agree at system size \(n\): \(V^{(n)}(A)_\sigma = V^{(n)}(B)_\sigma \) for every configuration \(\sigma \). Then there are invertible matrices \(Z_v\), one per bond of the closed chain (\(v = 1, \ldots , n\), \(n + 1 \equiv 1\)), with \(B^i = Z_v^{-1}\, A^i\, Z_{v+1}\) for every site \(v\) and every \(i\). This is the first corollary following the general normal-PEPS theorem of [ 220 , Section 4 ] specialized to one site-independent tensor: the source states the corollary for site-dependent families, and its conclusion is the per-bond gauge family delivered here. The system size is the optimal \(n \ge 2L + 1\) of the source’s alternative proof (its Section “Normal MPS: alternative proof”), rather than the \(n \ge 3L\) of the blocking route. The source’s translation-invariant corollary (a single gauge \(Z\) and a constant \(\lambda \) with \(\lambda ^n = 1\)) refines this conclusion and is not part of this statement. A vanishing physical dimension is already excluded by block injectivity.

Let \(A\) and \(B\) be tensors on the closed chain of \(n\) sites (the cycle graph on \(n\) vertices, one tensor per site with two virtual legs), with the same positive physical dimension, and let \(0 {\lt} L\) with \(n \ge 3L\). Suppose \(A\) and \(B\) are normal MPS with blocking length \(L\) (Definition 24.9.1.2) and generate the same state. Then the defining tensors are related by a local gauge: there are invertible matrices, one on each bond, whose endpoint actions transform \(A_i\) into \(B_i\) at every site with no leftover scalar. This is the source’s family \(Z_i\) (\(i = 1, \ldots , n\), \(n + 1 \equiv 1\)) with \(B_i = Z_i^{-1} A_i Z_{i+1}\), the first corollary following the general normal-PEPS theorem in [ 220 , Section 4 ] , the statement for MPS without translation invariance at a fixed system size. The positive physical dimension is carried over from the general theorem, the nonzero bond spaces are part of the definition of a normal MPS, and \(0 {\lt} L\) is implicit in the source’s blocking of \(L\) consecutive sites; the equality of the bond dimensions is not assumed but derived, and the connectivity of the cycle is proved, not assumed.

In the setting of Corollary 24.9.1.3, for normal MPS \(A\) and \(B\) with blocking length \(L\) and equal bond dimensions, any two families of invertible bond matrices realizing the scalar-free gauge relation are proportional at every bond: \(Z'_i = c \cdot Z_i\) for a nonzero scalar \(c\) depending on the bond. This is the uniqueness clause of the first corollary following the general normal-PEPS theorem of [ 220 , Section 4 ] : the gauges \(Z_i\) are unique up to a multiplicative constant. Neither equality of the two states nor connectivity is needed, as in the general uniqueness clause.

Let \(n\) be positive, let \(0 {\lt} L\) with \(2L + 1 \le n\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices with \(D{\gt}0\), each \(L\)-block injective. If two families \(Z\) and \(Z'\) of invertible matrices on the bonds of the closed chain both satisfy \(B^i = Z_v^{-1} A^i Z_{v+1}\) at every site, then they are proportional by a single constant: there is a nonzero scalar \(c\) with \(Z'_v = c\, Z_v\) at every bond. This is the uniqueness clause of the first corollary following the general normal-PEPS theorem of [ 220 , Section 4 ] , for one site-independent tensor. Equality of the two states is not needed.

Let \(n\) be positive, let \(0 {\lt} L\) with \(2L + 1 \le n\) and \(0{\lt}D\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\), each \(L\)-block injective, whose matrix product vectors agree at system size \(n\). Then there are invertible matrices \(Z_v\), one per bond of the closed chain (\(v = 1, \ldots , n\), \(n + 1 \equiv 1\)), with \(B^i = Z_v^{-1}\, A^i\, Z_{v+1}\) for every site \(v\) and every \(i\). This is the per-bond gauge family of the closed-chain corollary, proved at the source’s strengthened bound \(n \ge 2L + 1\) ( [ 220 ] , the corollary after Lemma 5 in its closing section “Normal MPS: alternative proof”). The named matrix-level Corollary 24.9.2.1 delegates to it.

Let \(n\) be positive, let \(0 {\lt} L\) with \(2L + 1 \le n\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices with \(D{\gt}0\), over physical dimension \(d\), each \(L\)-block injective, whose matrix product vectors agree at system size \(n\): \(V^{(n)}(A)_\sigma = V^{(n)}(B)_\sigma \) for every configuration \(\sigma \). Then there are an invertible matrix \(Z\) and a constant \(\lambda \) with \(\lambda ^n = 1\) such that \(B^i = \lambda \, Z^{-1} A^i Z\) for every \(i\). This is the translation-invariant corollary of [ 220 , Section 4 ] , the statement for TI MPS following the first corollary after the general normal-PEPS theorem. The system size is the optimal \(n \ge 2L + 1\) of the source’s alternative proof.

Let \(0 {\lt} L\) and \(0{\lt}D\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices, each \(L\)-block injective. If invertible matrices \(Z\), \(Z'\) and constants \(\lambda \), \(\lambda '\) satisfy \(B^i = \lambda \, Z^{-1} A^i Z\) and \(B^i = \lambda '\, Z'^{-1} A^i Z'\) for every \(i\), then there is a nonzero scalar \(c\) with \(Z' = c\, Z\). This is the uniqueness clause of the translation-invariant corollary of [ 220 , Section 4 ] for TI MPS. No system size enters the statement: equality of the two states is not needed, nor are the root-of-unity conditions on \(\lambda \) and \(\lambda '\).

Let \(0 {\lt} L\) with \(2L + 1 \le n\) and \(0{\lt}D\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\), each \(L\)-block injective, whose matrix product vectors agree at system size \(n\): \(V^{(n)}(A)_\sigma = V^{(n)}(B)_\sigma \) for every configuration \(\sigma \). Then there are an invertible matrix \(Z\) and a constant \(\lambda \) with \(\lambda ^n = 1\) such that \(B^i = \lambda \, Z^{-1} A^i Z\) for every \(i\). This is the strengthened closed-chain corollary of [ 220 ] (the corollary after Lemma 5 in its closing section “Normal MPS: alternative proof”), proved through the overlapping windows at the optimal \(n \ge 2L + 1\) rather than the \(n \ge 3L\) of the blocking route; the single-gauge corollary 24.9.2.5 delegates to it. The uniqueness clause of the source corollary - the gauge \(Z\) is unique up to a multiplicative constant - needs no system size and is Corollary 24.9.2.6.

Let \(0{\lt}L\) with \(2L+1\le n\), and let \(A\) and \(B\) be site-dependent chains on \(n\) sites with a common positive bond dimension, each window injective at length \(L\), generating the same closed-chain state. Then the chains are gauge equivalent: there are invertible matrices \(Z_v\), one per bond of the closed chain, with \(B_v^i=Z_vA_v^iZ_{v+1}^{-1}\) for every site \(v\) and every \(i\), with indices read cyclically. This is the closed-chain corollary of the closing section of [ 220 ] obtained from its site-dependent Lemma 5 at every bond; the source displays the corollary for translation-invariant tensors (Corollary 24.9.3.6), with the site-dependent Lemma 5 as its engine.

Suppose \(A\) and \(B\) have the same bond dimension on every edge, and that these bond dimensions are positive. Let \(C\) be the gauge action of \(X\) on \(B\), reindexed along this bond-dimension equality, and assume that the \(R\)-blocked family of \(C\) is linearly independent. If the two two-block scalar proportionalities of the blocked weights of \(A\) and \(C\) over \(R\) and over \(S=R\cup \{ v\} \) hold, with ratios \(c_R\) and \(c_S\) and with \(c_R\ne 0\), then the per-vertex gauge relation

holds at every local configuration \(\eta \) at \(v\) and every physical index \(\sigma \).

Let \(A_0,\ldots ,A_{n-1}\) be a site-dependent injective closed MPS on \(n\ge 3\) sites, with one dimension \(D_e\) for each bond \(e\). If its generated state is invariant under the cyclic shift \(A_v\mapsto A_{v+1}\), then the injective Fundamental Theorem compares the family with its shift by invertible matrices on the corresponding bonds. Consequently the bond from \(v\) to \(v+1\) and the bond from \(v+1\) to \(v+2\) have the same dimension. Iterating around the cycle gives \(D_e=D_f\) for every pair of bonds \(e,f\).

This is the at-least-three-site form of the “all bond dimensions are the same” clause of the Applications-section corollary in [ 220 , lines 1803–1842 ] . As in the injective Fundamental Theorem, injectivity includes the convention that every virtual space is nonzero.

Let \(A\) and \(B\) be PEPS tensors on the discrete \(n\times m\) torus with \(n,m{\gt}1\), with equal bond dimensions and positive bond dimensions, generating the same state, and let \(X\) be a family of invertible matrices, one on each edge. Suppose that at every site \(v\) a region \(R_v\) not containing \(v\) is supplied together with the two two-block scalar proportionalities of the blocked weights of \(A\) and of the gauge-absorbed second tensor over \(R_v\) and over \(R_v\cup \{ v\} \), with ratios \(c_R(v)\) and \(c_S(v)\) satisfying \(c_R(v)\ne 0\) and whose quotient \(c_S(v)/c_R(v)\) is a single scalar \(\lambda \) independent of \(v\), and with the gauge-absorbed region block linearly independent. If, in addition, a single comparison region whose block and complement block are both blocked-tensor injective is supplied, then \(\lambda \) raised to the number of torus sites is one: \(\lambda ^{nm}=1\).

Let \(A\) be an injective PEPS on a discrete \(n\times m\) torus with \(n,m{\gt}1\). If its represented state is translation invariant, then its bond dimensions are orientation-uniform. Explicitly, with

at the reference right and up edges, every horizontal edge has dimension \(D_h\) and every vertical edge has dimension \(D_v\).

This is the first conclusion of the two-dimensional Applications-section corollary in [ 220 , lines 1891–1894 ] .

The PEPS tensor obtained by applying edge-gauge matrices to \(A\), with vertex tensors \(A^X_v\).

Let a closed chain on \(n \ge 1\) sites carry one tensor of \(D \times D\) matrices over physical dimension \(d\) per site, \(A_0, \ldots , A_{n-1}\). The arc product of a word \(x=x_1\cdots x_\ell \) read from site \(s\) is

with site indices read cyclically. The chain is window injective at length \(L\) when, for every site \(s\), the arc products of the length-\(L\) words read from \(s\) span the full matrix algebra. This is the site-dependent setting of the closing section of [ 220 ] (“Normal MPS: alternative proof”): an MPS on five sites in which the blocking of any two consecutive tensors is injective, for a general window length \(L\).

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.

For an edge \(e=(u,w)\) with \(u{\lt}w\) and gauge \(X_e\in \mathrm{GL}(D_e)\), the gauge matrix applied at vertex \(v\) is \(X_e\) if \(v=u\) and \((X_e^{-1})^\top \) if \(v=w\).

An edge-scalar family \(c=(c_e)_{e\in E}\) determines the endpoint scalar \(c_{v,e}\) by \(c_{v,e}=c_e\) at the lower endpoint of \(e\) and \(c_{v,e}=c_e^{-1}\) at the upper endpoint.

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

Two gauge families are equivalent modulo balanced edge scalars if there is a nonzero scalar \(c_e\) on each edge such that the corresponding endpoint actions satisfy \(X_{v,e}=c_{v,e}Y_{v,e}\), and \(\prod _{e\ni v}c_{v,e}=1\) for every vertex \(v\).

The preceding equivalence relation is the quotient relation on edge-gauge families modulo vertex-balanced edge scalars.

Two PEPS tensors \(A,B\) are gauge-equivalent if they have the same bond dimensions and \(B\) is obtained from \(A\) by applying invertible gauge matrices on each edge.

The gauged tensor at vertex \(v\) is \(A^X_v\). For each incident edge \(ie\), write \(M_{ie}\) for the matrix specified by Definition 24.1.2.1, applied at the vertex \(v\) to the gauge on the underlying edge of \(ie\). Then

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

An edge-scalar family is vertex-balanced when \(\prod _{e\ni v}c_{v,e}=1\) for every vertex \(v\).

A PEPS tensor \(A\) is vertex-injective if, at every vertex \(v\), the family of physical vectors \((A_v(\eta ,\cdot )\in \mathbb {C}^d)_{\eta }\) indexed by virtual configurations \(\eta \) on the edges incident to \(v\) is linearly independent in \(\mathbb {C}^d\). Equivalently, extending \(A_v\) by linearity to a map from the free vector space on virtual configurations into the physical space \(\mathbb {C}^d\) gives a linear map with trivial kernel. This matches the injectivity formulation of [ 220 , Section 3 ] , where the local tensor is interpreted as a map from virtual configurations to the physical space.

  \(\ker A_v=\{ 0\} \).

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

For a closed chain \(A_0,\ldots ,A_{n-1}\), its cyclic shift is the chain whose \(v\)-th tensor is \(A_{v+1}\), with indices read modulo \(n\). The generated state is cyclic-shift invariant when the closed-chain state of \(A_0,\ldots ,A_{n-1}\) agrees with that of \(A_1,\ldots ,A_{n-1},A_0\).

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

Fix a chain length \(n\). A non-translation-invariant periodic MPS chain consists of sitewise tensors \(A_k=\{ A_k^i\} _{i=0}^{d-1}\), where \(k=0,\ldots ,n-1\) and each \(A_k^i\in M_{D}(\mathbb {C})\). For a basis configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{n-1})\), its coefficient is

Two such chains generate the same chain state if these coefficients agree for every \(\sigma \). The chain is injective if each local tensor \(A_k\) is injective in the usual single-site sense. Finally, two chains are cyclically gauge equivalent if there are invertible bond matrices \(Z_0,\ldots ,Z_{n-1}\) such that, for every site \(k\) and physical index \(i\),

with indices understood modulo \(n\), so that \(Z_n=Z_0\). Locally, this gauge relation is represented by

\( = \)

Let \(f\) be a boundary edge of \(R\) and let \(Z\) be an invertible matrix on the bond space of \(f\). The orientation-adapted absorbing gauge is

It is the absorbing matrix on \(f\) for which the gauge action on the region coefficient reproduces the conjugation \(Z\, (\cdot )\, Z^{-1}\).

Given a tensor family \(B\) and an oriented gauge matrix \(X_e\) on every edge, the absorbed tensor family is \(\widetilde B=B^X\); at each vertex, its local tensor is obtained by inserting the oriented endpoint matrices on the incident virtual legs.

If \(R\subseteq S\), a physical configuration on \(R\) and a physical configuration on \(S\setminus R\) combine into a physical configuration on \(S\):

The blocked-middle local gauge formula at \(v\) is the assertion that there are matrices \(X_e\in \mathrm{GL}(D_A(e))\), one on each incident edge, such that for all local virtual configurations \(\eta \) and physical indices \(\sigma \),

Let \(f\) be an edge crossing the boundary of \(R\), and let \(M\in M_{D_f}(\mathbb {C})\). The bond-inserted region insert \(I_{A,R,f,M}\) is the insert on \(R\) defined by

Thus the complement boundary value \(\nu \) is fixed away from \(f\), while the matrix \(M\) couples the two \(f\)-labels. This is the one-bond insertion in the region/complement coefficient of [ 220 , Section 3 ] .

For \(f\in \partial R\), identify a boundary configuration \(\mu \) with \((\mu _f,\mu |_{\partial R\setminus \{ f\} })\). Denote the resulting bijection by \(E_f\).

Let \(v\notin R\). Given a crossing-edge label \(\mu \) for \(R\cup \{ v\} \) and a local virtual configuration \(\eta \) on the edges incident to \(v\), the crossing-edge label \(\beta (\mu ,\eta )\) of \(R\) is read off these two configurations: a crossing edge of \(R\) incident to \(v\) has become internal to \(R\cup \{ v\} \), so its label is taken from \(\eta \); a crossing edge of \(R\) not incident to \(v\) is also a crossing edge of \(R\cup \{ v\} \), so its label is taken from \(\mu \).

Suppose that every matrix \(X\in M_{D_A(f)}(\mathbb {C})\) has a matrix \(Y\in M_{D_B(f)}(\mathbb {C})\) with matching region-inserted coefficients, \(C_A(R,f,X;\sigma ,\tau )=C_B(R,f,Y;\sigma ,\tau )\). Choosing one such \(Y\) for each \(X\) defines a forward coefficient transfer \(T_f:M_{D_A(f)}(\mathbb {C})\to M_{D_B(f)}(\mathbb {C})\).

For a vertex \(v\), the complementary block \(V\setminus \{ v\} \) defines a two-block tensor on the same incident shared bonds. Its coefficient is the contraction of all tensors away from \(v\), with the incident bond indices left open at the complement endpoints: \(T_{v^c}(*,\eta ,\tau )=A_{v^c}(\eta ,\tau )\).

Let \(R\subseteq S\) be regions in a finite graph, and let \(C\) be a region insert on \(R\). Write \(\Phi _{R,S}\) for the coefficient obtained by contracting the added block \(S\setminus R\) between the boundary of \(R\) and the boundary of \(S\). The bare corner extension is

The normalized corner extension of \(C\) from \(R\) to \(S\) is the region insert

where \(P_{V\setminus S}\) is the product of bond dimensions over the edges not crossing the boundary of \(V\setminus S\). This is the open-boundary form of the extension used in the overlapping-window proof sketch of [ 220 , Section 4, proof of Theorem 3 ] .

On the cycle graph with \(n\ge 3\) sites, let \(r(v)=v+1\) be the one-site rotation. The cyclic shift of a site-dependent tensor family \(A_0,\ldots ,A_{n-1}\) is the transport along \(r^{-1}\), so its tensor at site \(v\) is \(A_{v+1}\). Thus the shifted family is \(A_1,\ldots ,A_{n-1},A_0\), exactly as in [ 220 , lines 1807–1827 ] . The generated state is cyclic-shift invariant when the original and shifted families generate the same closed-chain state.

For a finite simple graph \(G\), an edge is an ordered pair \((u,v)\) with \(u {\lt} v\) and \(u\) adjacent to \(v\). The ordering serves to avoid double counting.

The edge-blocked coefficient is the three-block contraction obtained from the left endpoint tensor, the blocked middle tensor, and the right endpoint tensor at the chosen edge.

For two PEPS tensors \(A\) and \(B\) and an edge \(e\), this property is

For a chosen edge \(e=(u,v)\), the edge-blocked three-site chain is injective when the left endpoint tensor map, the middle tensor family, and the right endpoint tensor map are all injective.

For an edge \(e=(u,v)\), the boundary data consist of the virtual index on \(e\), the remaining virtual indices incident to \(u\), and the remaining virtual indices incident to \(v\). These are the indices over which the endpoint tensors couple to the blocked middle region.

Ordinary edge-boundary data are equivalent to diagonal inserted-boundary data. This is the finite reindexing map used after the identity matrix has removed all off-diagonal inserted-boundary summands.

Ordinary edge-boundary data together with an independent left endpoint bond index are equivalent to inserted-edge boundary data. The ordinary boundary supplies the right bond index and the two residual endpoint boundaries.

A global virtual configuration is compatible with fixed edge-centered boundary data if it has the prescribed index on the distinguished edge and the prescribed residual indices at both endpoint stars.

A global virtual configuration determines edge-centered boundary data by reading the index on the distinguished edge and the residual endpoint indices.

Ordinary edge-boundary data together with an independent right endpoint bond index are equivalent to inserted-edge boundary data. The ordinary boundary supplies the left bond index and the two residual endpoint boundaries.

Ordinary edge-centered boundary data determine inserted-edge boundary data by using the same bond index at both endpoints. This is the diagonal boundary datum selected by the identity matrix in the edge-insertion coefficient.

The diagonal inserted-boundary configurations are those inserted-edge boundary data for which the two distinguished bond indices agree. This is the support of the identity matrix inside the inserted-boundary sum.

For an edge \(e=(u,v)\), inserted-edge boundary data consist of two independent indices on the bond \(e\), together with the residual virtual indices incident to \(u\) and to \(v\). These are the boundary indices used when a matrix is inserted on the distinguished bond.

For an edge \(e=(u,v)\) and a matrix \(X\) acting on the bond space of \(e\), the edge insertion coefficient is the contraction obtained by inserting \(X\) between the endpoint tensors and the open middle tensor:

The left endpoint local virtual configuration is reconstructed from the edge index and the residual data on the left endpoint star.

For fixed boundary data \(\beta \), the middle configurations are the subtype of global virtual configurations satisfying the compatibility condition with \(\beta \).

For fixed edge-boundary data, ordinary middle configurations are equivalent to compatible open-middle configurations. This is the finite reindexing used when the inserted matrix is the identity.

An ordinary middle configuration for the edge-blocked contraction restricts to a virtual configuration on all edges except the distinguished edge. This is the first reindexing map needed to compare the identity insertion with the ordinary edge-blocked middle tensor.

Let \(e=(u,v)\) and \(M_e=V\setminus \{ u,v\} \). An edge-middle kernel descent datum is indexed by boundary labels \(\rho \) consisting of the two residual endpoint configurations. It assigns to every finitely supported coefficient family \(c=(c_{\rho })_{\rho }\) a finite kernel descent datum \(K_c(S)\) such that

For an edge \(e=(u,v)\), the middle physical configurations are the physical indices on the vertices in \(V\setminus \{ u,v\} \). A global physical configuration restricts to such a middle configuration.

A comparison from finite-region injectivity to the edge-middle tensor records the following assertion: if the middle vertex region \(V\setminus \{ u,v\} \) is injective after blocking, then the middle tensor in the edge-blocked three-site chain is injective. This isolates the finite-region contraction claim in [ 220 , Section 3 ] .

For the edge-blocked three-site chain at \(e=(u,v)\), the middle tensor is the family obtained by fixing the two residual endpoint boundaries and evaluating the open middle contraction on the middle physical index. Its injectivity is the middle-block part of the assertion following the edge-blocking diagram.

For an edge \(e=(u,v)\), the middle region is the complement \(V\setminus \{ u,v\} \).

Fixing the edge-centered boundary data, the blocked middle tensor is the sum over global virtual configurations compatible with those data of the product of local tensor entries over the middle region \(V\setminus \{ u,v\} \). The graph-local contraction is read as a three-site chain:

\(\quad \rightsquigarrow \quad \)

The ordinary and open middle contractions may be written using only the middle physical configuration, rather than a physical configuration on all vertices. This is the middle tensor in the three-site chain obtained after the edge blocking in arXiv:1804.04964, Section 3.

Conversely, a compatible open middle configuration becomes an ordinary middle configuration after the fixed edge-boundary index is restored on the distinguished edge. Together with the restriction map, this gives the finite reindexing underlying the identity-insertion specialization.

Fixing the residual boundary configurations at the endpoint vertices \(u\) and \(v\) but leaving the bond \(e\) out of the middle region, the open middle tensor is the contraction over all virtual indices away from \(e\) that are compatible with those boundary configurations.

The right endpoint local virtual configuration is reconstructed from the edge index and the residual data on the right endpoint star.

Let \(\{ A_{1,\mu }\} _{\mu \in K}\) and \(\{ A_{3,\mu }\} _{\mu \in K}\) be two end families. The left physical operation \(O_1\) realizes a virtual matrix \(X\) when

The right physical operation \(O_3\), read in the transposed orientation \(O_3^{\mathsf T}\) of Lemma 5, realizes the same matrix when

This convention records the orientation of the displayed tensor relation; it does not choose a transpose on the abstract space of physical operations.

Let \(V\) be a vertex type with decidable equality. A finite kernel descent datum is a predicate \(K\) on finite subsets \(S\subseteq V\) together with implications \(j\in S\Longrightarrow K(S)\Longrightarrow K(S\setminus \{ j\} )\). In the edge-blocking proof, \(K(S)\) is the assertion that the boundary coefficients \(c_{\rho }\), with the virtual indices exposed at that stage, give zero after the tensors in \(S\) have been contracted.

For \(\Lambda =\{ 0,\ldots ,L-1\} ^2\), a forward edge is an ordered pair \(((x,y),(x+1,y))\) or \(((x,y),(x,y+1))\) with both endpoints in \(\Lambda \).

Fix \(A\), \(B\), a bond-dimension equality, and a vertex \(v\). Let \(\mathcal T^A_v\) be the local tensor map of \(A\), let \(L^A_v\) be the chosen left inverse of \(\mathcal T^A_v\), and let \(\Pi ^A_v=\mathcal T^A_vL^A_v\). The factorized local gauge datum consists of the following two identities for local virtual configurations \(\eta ,\eta '\):

with \(X_e\in \mathrm{GL}(D_A(e))\) on each incident edge. This is the local output required from the blocked-middle reduction in [ 220 , Section 3 ] .

Under the same geometric bounds, write \(\partial _{\mathrm{left}}S\) and \(\partial _{\mathrm{right}}S\) for the boundary edges of \(S\) which lie on the corresponding end-window boundary. Then the boundary-configuration spaces have canonical factorizations

These equivalences preserve every underlying lattice-edge label. Thus the reference edge is the only contracted virtual index shared by the two highlighted end windows; every other virtual index remains on the corresponding open boundary of \(S\). The constructions split each end-window configuration at the reference edge and reindex the remaining dependent products along the corresponding boundary-edge identifications.

Fix a horizontal reference edge and write the staircase corner as \(s=(a,b)\). For coordinate intervals write \(I(u,v)=\{ t\mid u\le t{\lt}v\} \). The two end windows of the overlapping-window chain are

Their end pair is \(S=W_{\mathrm{left}}\cup W_{\mathrm{right}}\). The staircase patch is the union

and the corner cut out by the two end windows is \(C=I(a,a+L)\times I(b+K,b+2K-1)\). Completing \(C\) by one row gives the \(L\times K\) rectangle \(\widehat C=I(a,a+L)\times I(b+K,b+2K)\). These cyclic rectangles are the regions used in the staircase chaining and corner-stripping step of [ 220 , proof sketch, lines 2320–2445 ] .

In the staircase coordinates \(s=(a,b)\), the reference edge is the horizontal edge joining \((a+L-1,b+K-1)\) to \((a+L,b+K-1)\). It is the edge between the left and right end windows.

For a vertex \(v\), an incident edge is an edge of \(G\) having \(v\) as one of its endpoints.

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 ] .

A contraction chain consists of the three contraction maps of Definition 24.5.4, together with the three implications used in Lemma 24.5.6.

Let \(K\), \(E_{012}\), \(E_2\), and \(E_{12}\) be complex vector spaces. A contraction-map family for the union-injectivity lemma in [ 220 , Section 4 ] consists of three linear maps \(\mathcal C_{\{ 0,1,2\} }:K\to E_{012}\), \(\mathcal C_{\{ 2\} }:K\to E_2\), and \(\mathcal C_{\{ 1,2\} }:K\to E_{12}\).

Let \(e=\{ u,w\} \) be an ordered edge whose two endpoints lie in a region \(R\), with \(u\) its ordered left endpoint, and let \(X\in M_{D_A(e)}(\mathbb {C})\). Write \(U=\{ u\} \) and let \(I_{A,U,e,X}\) be the boundary-bond insert on \(U\). The interior-bond insert on \(R\) is

Thus the genuine tensors at the vertices of \(R\setminus U\) are adjoined to a boundary insertion of \(X\) on \(e\), with the normalization changed from the non-boundary bond product of \(U\) to that of \(R\).

In the open \(7\times 7\) rectangle, let \(u=(0,2)\) and \(v=(1,2)\) be the endpoints of a left-boundary horizontal edge. Define a region relation by

A matrix on one edge incident to \(v\) acts on the coefficient space of local virtual configurations by applying the matrix to that edge coordinate and summing over the input edge index while keeping the residual virtual configuration fixed.

If the family of local physical vectors \((A_v(\eta ,\cdot ))_{\eta }\) indexed by the virtual configurations \(\eta \) at \(v\) is linearly independent, one may choose a linear map \(\Psi _v\) from \(\mathbb {C}^d\) to the coefficient space on local virtual configurations at \(v\) such that \(\Psi _v\circ \Phi _v=\mathbb {1}\).

The local projector is the physical realization of the identity virtual operator, \(P_v=\Phi _v\circ \Psi _v\).

For a fixed vertex \(v\), the family of local physical vectors \(\eta \mapsto A_v(\eta ,\cdot )\) extends by linearity to a map from the coefficient space on local virtual configurations at \(v\) into \(\mathbb {C}^d\).

Choosing one incident edge at \(v\) identifies the set of local virtual configurations with the product of the bond index on that edge and the residual indices on the remaining incident edges.

For a physical operator \(O\) on the local physical space at \(v\), define its virtual pullback by \(\Psi _v\circ O\circ \Phi _v\). This is the local injectivity step used in the physical-to-virtual direction of the insertion-algebra argument in [ 220 , Section 3 ] .

For indices \(i\) and \(j\), let \(E_{ij}\) denote the matrix whose only nonzero entry is the \((i,j)\) entry, where it is equal to \(1\).

For a family \(\mathcal I\) of regions in the \(w\times h\) torus, the \(L\times K\) cyclic-window hypothesis means that \(\mathcal R(s;L,K)\in \mathcal I\) for every starting vertex \(s\). Here \(\mathcal R(s;a,b)\) is the cyclic rectangle of side lengths \(a,b\) starting at \(s\). For a tensor, \(\mathcal I\) is its family of injective regions. This is the window hypothesis in the normal TI PEPS corollary of  [ 220 ] .

A PEPS tensor on the \(w\times h\) torus is normal at window size \(L\times K\) when its family of injective regions satisfies the \(L\times K\) cyclic-window hypothesis. As throughout, its virtual bond spaces are nonzero. Injectivity does not force this: with suitably placed zero-dimensional bonds every relevant local domain is empty, so the regions are injective and every state is zero. This is the hypothesis on each tensor in the normal TI PEPS corollary of  [ 220 ] .

A tensor on the closed chain of \(n\) sites is a normal MPS with blocking length \(L\) when blocking any \(L\) consecutive sites gives an injective tensor and every virtual bond space is nonzero. This is the hypothesis on each of the two MPS in the first corollary following the general normal-PEPS theorem in [ 220 , Section 4 ] . The nonzero bond spaces are the standing convention of that section; injectivity of the blocks does not imply them, and the convention is recorded in [ 67 ] .

Around one edge, the normal proof uses a blocking into three injective regions: a red region containing one endpoint, a blue region containing the other endpoint, and an injective complementary region. The three regions are pairwise disjoint and cover the whole vertex set. The edge-blocking hypotheses are precisely this one-edge datum at every edge; the red, blue, and complementary regions are read off from it. The local blocking picture used in the proof of [ 220 , Theorem 3 ] :

For each site, the normal theorem assumes two injective regions with injective complements that differ exactly at that site. This is the one-site comparison used after the edge gauges have been obtained. Diagrammatically, in the square-lattice proof this is the comparison between the source regions \(R\) and \(S=R\cup \{ v\} \):

\(\qquad \) \(\qquad R\subset S=R\cup \{ v\} \)

The general normal theorem assumes both the edge-centred blockings into three injective regions and the one-site comparison regions with injective complements. These two assumptions are collected as the normal PEPS blocking hypotheses; the same hypotheses give the corresponding edge and one-site consequences without restating the separate edge and one-site assumptions. The following two panels show their square-lattice source realization; they do not restrict the abstract definition to this lattice.

\(\qquad \text{edge blocking}\)

\(\text{and}\qquad \) \(\qquad \subset \qquad \) \(\qquad S=R\cup \{ v\} \)

Two PEPS tensors \(A\) and \(B\) on a finite simple graph are a pair of normal PEPS tensors if they satisfy the normal blocking hypotheses at the doubly injective assignment of the two tensors and every virtual bond space of each tensor is nonzero, that is, \(D_A(e){\gt}0\) and \(D_B(e){\gt}0\) for every edge \(e\). Normality is blockability into injective regions, as defined in [ 220 , Section 3 ] . The nonzero bond spaces are the standing convention of that section. Region injectivity does not imply them: a zero-dimensional boundary bond leaves a region with no boundary configurations, so its injectivity holds vacuously. The convention is recorded in [ 67 ] .

In the square-lattice normal theorem, every \(2\times 3\) and \(3\times 2\) rectangular region is assumed injective. The corresponding abstract hypotheses specify which finite regions have these two shapes and assert injectivity for all of them. In the coordinate square-lattice specialization, these two families are the contiguous coordinate \(2\times 3\) and \(3\times 2\) rectangles.

In the translationally invariant square-lattice proof, the source paper assumes that every \(2\times 3\) and \(3\times 2\) region is injective. The statement here specifies the corresponding abstract hypothesis: it records the rectangular injectivity assumptions, the size bound \(n,m\geq 7\) with \(|V|=nm\), and three finite regions \(R\), \(S\), and \(T\) whose injectivity is assumed directly. It does not by itself assert that these regions are the displayed square-lattice unions, nor that their translated variants give the edge-centred blocking around every edge. Conversely, for the coordinate square lattice with \(n,m\geq 7\), the rectangular hypotheses, union closure, and a rectangular cover of the displayed \(T\)-region yield this abstract \(R,S,T\) structure, with \(R\) and \(S\) obtained from their rectangular decompositions and \(T\) obtained from the supplied cover. The source regions are:

\(\qquad \text{and}\qquad \)

At a coordinate offset \((a,b)\) the comparison square is the \(3\times 3\) contiguous coordinate rectangle with lower-left corner \((a,b)\). The comparison region is this square with its lower-left corner \((a,b)\) removed, presented as the union of the \(2\times 3\) contiguous rectangle at \((a+1,b)\) and the \(3\times 2\) contiguous rectangle at \((a,b+1)\). This is an auxiliary one-site comparison pair carrying its omitted site at the lower-left corner; it is the reflected counterpart of the displayed regions \(R\) and \(S\), whose omitted site \(S\setminus R\) sits at the lower-right corner, placed at an arbitrary offset rather than at a fixed origin. The interior window at \((a,b)\) for a finite \(n\times m\) lattice is the placement margin \(4\le a\), \(a+9\le n\), \(4\le b\), \(b+9\le m\), under which every boundary edge of the comparison square and of the comparison region carries the interior margins of the open edge-blocking geometry.

A PEPS tensor on the open square lattice is normal with injective rectangles when its family of injective regions satisfies the coordinate rectangular injectivity hypotheses and every virtual bond space is nonzero. This is the hypothesis on each tensor in [ 220 , Theorem 3 ] , read on the open lattice. The nonzero bond spaces are the standing convention of that section; region injectivity does not imply them, and the convention is recorded in [ 67 ] .

A rectangular cover of a square-lattice region is a nonempty finite family of regions whose union is exactly the given region, and each member is a contiguous \(2\times 3\) or \(3\times 2\) rectangle. The two cases used in the proof would be a cover of the local displayed \(T\)-region and a cover of the finite-lattice complementary block \(A_3\). The panel below illustrates the two permitted shapes of a cover member.

A translated edge window records that a chosen edge is one of the translated horizontal or vertical edge pictures, together with a rectangular cover of the complementary block. Such a window gives the one-edge blocking datum for that edge, hence the three injective regions used in the edge-blocked chain. The coordinate right and upward edges obtain such windows directly from the corresponding translated picture. Equivalently, an actual coordinate right or upward edge obtains such a window when it has the necessary room around it for the translated \(5\times 7\) or \(7\times 5\) frame and the complementary cover is supplied. The same criterion applies to an arbitrary horizontal or vertical square-lattice edge after identifying it with its coordinate right or upward representative. The oriented margin-and-cover datum is the same criterion written as per-edge data in the rectangular coordinate model; it directly supplies the one-edge blocking datum, its three injective regions, and the identities \(u_e\in R_e\), \(v_e\in B_e\), \(R_e\cap B_e=R_e\cap C_e=B_e\cap C_e=\varnothing \), and \(R_e\cup B_e\cup C_e=V\). The boundary obstruction lemmas show that these translated-window and margin-cover criteria are interior criteria, not the final every-edge construction in the open square-lattice model.

A PEPS tensor on the discrete torus is normal with injective rectangles when its family of injective regions satisfies the torus rectangular injectivity hypotheses and every virtual bond space is nonzero. This is the hypothesis on each tensor in [ 220 , Theorem 3 ] . The nonzero bond spaces are the standing convention of that section; region injectivity does not imply them, and the convention is recorded in [ 67 ] .

Let \(R\) be an injective region and let \(C=\{ C(\mu )\} _{\mu \in \partial R}\) be any family in its physical space, indexed by the virtual boundary configurations. If \(L_{A,R}\) is a left inverse of the blocked tensor map, define

For an endomorphism \(T\) of the coefficient space on local virtual configurations at \(v\), define the local physical operator \(O_T=\Phi _v\circ T\circ \Psi _v\).

Let \(\Omega \) be a vector on the sites \(\Lambda \) of the square and \(R\subseteq \Lambda \). For configurations \(x\) on \(R\) and \(z\) on \(R^c\), let \(x\cup z\) be their joined configuration and set \(M_{x,z}=\Omega (x\cup z)\). The unnormalized regional density of \(\Omega \) is \(MM^\dagger \). For \(\Omega =\Phi \) it equals the partial trace in Theorem 24.13.4.

For tensor families \(A,\widetilde B\), the post-absorption comparison holds when \(h:D_A=D_{\widetilde B}\) and the following equality holds. For each edge \(e\), write \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C}) \to M_{D_{\widetilde B}(e)}(\mathbb {C})\) for the matrix-algebra transport induced by \(h_e:D_A(e)=D_{\widetilde B}(e)\). Then, for every matrix \(X\in M_{D_A(e)}(\mathbb {C})\) and every physical configuration \(\sigma \), \(C_A(e,X;\sigma ) =C_{\widetilde B}(e,\phi _{h_e}(X);\sigma )\).

An equality \(R=S\) identifies their physical and boundary spaces. Write \(O|_S\) and \(C|_S\) for a physical operation and a boundary-indexed physical tensor under these identifications.

Let \(O\) be a physical operator on the physical space of a region \(R\). Its action on the genuine open-boundary blocks defines the insert

A region insertion transfer along a boundary edge \(f\in \partial R\) consists of maps \(T_f:M_{D_A(f)}(\mathbb {C})\to M_{D_B(f)}(\mathbb {C})\) and \(S_f:M_{D_B(f)}(\mathbb {C})\to M_{D_A(f)}(\mathbb {C})\). For every inserted matrix and every pair of physical configurations, these maps satisfy

The forward map is required to satisfy \(T_f(XY)=T_f(X)T_f(Y)\) and \(T_f(\mathbb {1})=\mathbb {1}\).

Suppose coefficient transfers exist in both directions:

Assume also that the two tensors have the same state and the same bond dimensions, all \(B\)-bond dimensions are positive, and the region and complement-region blocked maps of \(B\) are injective. If the forward choice is multiplicative, \(T_f(XY)=T_f(X)T_f(Y)\), then \(T_f\) and \(S_f\) form a region insertion transfer.

Suppose the forward and reverse boundary matrix transfers are realized on the two tensors, the two tensors have the same state and the same bond dimensions, and the region and complement-region blocked maps of \(B\) are injective. Then the maps \(T_f\) and \(S_f\) form a region insertion transfer:

together with \(T_f(XY)=T_f(X)T_f(Y)\) and \(T_f(\mathbb {1})=\mathbb {1}\).

The out-of-region endpoint of \(f\in \partial R\) is the endpoint outside \(R\). It is the in-region endpoint of the same edge viewed as an edge of \(\partial (V\setminus R)\). The blocked-region map is \(c\longmapsto \sum _{\mu \in \partial R}c_\mu \, T_{A,R}^{\mu }\), and injectivity of the blocked tensor gives a left inverse. In particular, \(L_{A,R}\! \left(T_{A,R}^{\mu }\right)=e_\mu \).

For a region \(R\) and a physical configuration \(\tau \) on its complement, define the partial state \(\psi ^\tau _{A,R}\) on the physical space of \(R\) by

The incident-form condition associated with the region resonance asserts that, for every matrix \(X\) inserted on \(f\), the virtual pullback \(W_X\) of the in-region endpoint operator is in incident-matrix form, namely \(\exists Y\colon W_X=\mathcal{I}_f(Y)\).

Let \(v\) be the endpoint of \(f\) lying in \(R\). A matrix transfer \(T_f\) is realized on the region if, for every matrix \(X\) inserted on \(A\)’s boundary bond and every local virtual coefficient \(c\) for \(B\) at \(v\),

This is the region version of the physical-to-virtual insertion relation in [ 220 , Section 3, lines 254–582 ] .

For a finite vertex region \(R\subseteq V\), its complement is \(V\setminus R\). Equivalently, a vertex lies in the complement exactly when it does not lie in \(R\).

The edges crossing the boundary of \(R\) are the same edges crossing the boundary of \(V\setminus R\). Thus a boundary configuration \(\eta \) for \(R\) determines the boundary configuration \(\eta ^c\) for the complement by reading the same edge labels through this identification.

For a finite region \(R\subseteq V\), the complement block \(V\setminus R\) is a two-block tensor over the same crossing bonds: \(B_{A,R^c}(*,\eta ,\tau )=T_{A,V\setminus R}^{\eta ^c}(\tau )\). This is the complementary block in the region/complement comparison of [ 220 , Section 3 ] .

A region-injectivity hypothesis assigns to each finite vertex region the assertion that the tensor obtained by blocking that region is injective. This predicate is used for regions such as \(R\), \(S\), \(T\), and their complements in the normal PEPS proof. The union-closure hypothesis records the conclusion supplied by the source union-of-injective-regions lemma: the union of two injective regions is injective.

The region-injectivity predicate of a fixed PEPS tensor \(A\) sends a finite region \(R\) to the assertion that the blocked-region tensor of \(R\) is injective.

For two region-injectivity assignments on the same vertex set, a region is doubly injective when it is injective for each of the two. The general normal theorem blocks two states by one shared geometry, and its final comparison contracts both tensors over the same regions, so its blocking hypotheses are instantiated at the doubly injective assignment of the two tensors.

Let \(f\) be an edge crossing the boundary of \(R\), and let \(M\in M_{D_f}(\mathbb {C})\). The region insertion coefficient is

It is the coefficient obtained by inserting \(M\) on the boundary bond \(f\) between the region \(R\) and its complement.

For a finite region \(R\subseteq V\), the blocked tensor of \(R\) may be read as a two-block tensor with a one-point external boundary and shared bonds the edges crossing the boundary of \(R\): \(B_{A,R}(*,\eta ,\sigma )=T_{A,R}^{\eta }(\sigma )\). This is the region block used in the one-block-against-complement step of [ 220 , Section 3 ] .

Let \(V\) be a finite vertex set. For two regions \(A,B\subseteq V\), define the four regions \(A\setminus B\), \(A\cap B\), \(B\setminus A\), and \(V\setminus (A\cup B)\). These are the four regions used in the proof of Lemma 24.5.7. They also form an indexed family over \(\{ 0,1,2,3\} \) in the order \(A\setminus B\), \(A\cap B\), \(B\setminus A\), \(V\setminus (A\cup B)\).

The row-cut factorization assertion is the following statement. Let \(A\) and \(B\) be MPS tensors with physical alphabet \(\{ 0,\ldots ,15\} \) and bond space \(\mathbb {C}^4\). If \(A\) and \(B\) are \(1\)-block injective and generate the same MPV family, then there exist \(Z\in \mathrm{GL}_4(\mathbb {C})\) and \(\lambda \in \mathbb {C}\) such that \(Z\) is a per-edge product matrix and \(B^i=\lambda Z^{-1}A^iZ\) for all physical indices \(i\in \{ 0,\ldots ,15\} \). This is the additional product conclusion needed by a row-wise one-dimensional reduction before it could imply the per-edge conclusion of [ 220 , Theorem 3 ] .

A matrix \(M\in M_{4}(\mathbb {C})\) is a per-edge product on two collected vertical bonds if, under the identification \((r,s)\longmapsto 2r+s\) between \(\{ 0,1\} \times \{ 0,1\} \) and \(\{ 0,1,2,3\} \), there are matrices \(Y_0,Y_1\in M_{2}(\mathbb {C})\) such that, for all \(r,s,r',s'\in \{ 0,1\} \),

Two PEPS tensors represent the same state if all their state coefficients agree for every physical configuration.

For a singleton block \(\{ v\} \), the blocked physical index is the original physical index and the blocked tensor is \(T_{A,\{ v\} }(\eta ,\sigma )=A_v(\eta ,\sigma )\). Its injectivity is \(\operatorname{inj}(T_{A,\{ v\} })\). The comparison with a region-injectivity predicate \(\kappa \) records the implication \(\operatorname{inj}(T_{A,\{ v\} })\Longrightarrow \kappa (\{ v\} )\).

Fix a target vector \(\Omega \). The physical-first approximation constraint asks for positive \(D_e\le CL^c\), a nonzero contraction \(\Phi \), and a phase \(\theta \) with \(\| \| \Phi \| ^{-1}\Phi -e^{i\theta }\Omega \| \le L^{-1}\). The virtual-first phase-error predicate concerns vectors only: for vectors \(x,\Omega \) and \(\varepsilon \in \mathbb R\) it asks for a phase \(\theta \) that minimizes the normalized error of \(x\) against \(e^{i\varphi }\Omega \) over all real \(\varphi \) and attains error at most \(\varepsilon \). Bond-dimension bounds and nonvanishing of the contraction are separate hypotheses in that presentation.

The forward edges are in bijection with the lexicographically oriented edges of the open square graph by preserving both endpoints. For every vertex this gives a bijection of incident edges, and for every family of dimensions \((D_e)_e\) it gives a bijection of assignments \(\prod _e\{ 0,\ldots ,D_e-1\} \), with each coordinate unchanged.

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 \(\max (\{ D_e:e\in E\} \cup \{ 0\} )\).

The finite \(n\times m\) square lattice is represented by pairs of coordinates. The corresponding square-lattice graph has horizontal and vertical nearest-neighbor edges; the two elementary adjacency lemmas record the right-neighbor and upper-neighbor cases used for translated edge blockings, and the corresponding coordinate edges name those two nearest-neighbor edges as elements of the edge set. These named right and upward edges have the expected horizontal and vertical orientations. Every edge of this graph is horizontal or vertical, but not both. Since edges are represented by ordered endpoint pairs, a horizontal edge is oriented from left to right and is exactly the coordinate right edge from its left endpoint; similarly, a vertical edge is oriented from bottom to top and is exactly the coordinate upward edge from its lower endpoint. A coordinate rectangle is a product of a horizontal coordinate set and a vertical coordinate set. The \(2\times 3\) and \(3\times 2\) coordinate-product predicates assert only that these two coordinate sets have cardinalities \(2,3\) and \(3,2\), respectively. Contiguous coordinate rectangles are products of contiguous coordinate intervals, and the contiguous \(2\times 3\) and \(3\times 2\) predicates name the rectangular blocks used in the normal square-lattice theorem. The displayed region \(R\) is the union of a \(2\times 3\) contiguous rectangle with the \(3\times 2\) contiguous rectangle occupying its upper two rows and extending one column to the right. The displayed region \(S\) is the union of two \(2\times 3\) contiguous rectangles shifted by one horizontal coordinate, hence the full \(3\times 3\) square spanned by them. The displayed local-window model for \(T\) is the complement, inside a local \(5\times 6\) coordinate window, of the vertical \(2\times 3\) edge block and the horizontal \(3\times 2\) edge block shown in the source picture. The finite-lattice complementary block is the whole finite square lattice with those two edge blocks removed; this is the block denoted \(A_3\) in the proof of [ 220 , Theorem 3 ] . The source shapes are:

\(\qquad \text{and}\qquad \)

Under the hypotheses of Theorem 24.7.3.36, the algebra isomorphism \(M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) is \(F_{AB}\), and its inverse is \(F_{BA}\).

Under the hypotheses of Theorem 24.7.3.36, in the paper’s notation the input and uniquely recovered matrices are called \(X\) and \(Y\). Writing \(Y_{A\to B}(X)\) only to distinguish the two tensor families, define

to be the unique matrix satisfying (43) and (44) for every left and right external boundary configuration. This definition makes no choice of an external boundary configuration.

Under the same hypotheses, the region-insertion transfer on \((W,e)\) has forward map \(F_{AB}\) and reverse map \(F_{BA}\). Its coefficient identities are (52) and the corresponding identity with \(A,B\) exchanged; the forward map preserves products and the identity.

For \(X\in M_{D_A(e)}(\mathbb {C})\), define \(O_1^A(X)=O_{L+K-1}^A(X)\) after identifying the last staircase window with the left end window.

For \(X\in M_{D_A(e)}(\mathbb {C})\), define \(O_3^A(X)=O_0^A(X)\) after identifying the first staircase window with the right end window. The relation (44) has the paper’s transposed \(O_3^{\mathsf T}\) orientation; no transpose is chosen on the abstract physical endomorphism space.

Let \(e\) be the horizontal reference edge and let \(X\in M_{D_B(e)}(\mathbb {C})\). Define an insert \(C_j(X)\) on each staircase window \(W_j\), for \(0\le j{\lt}L+K\), by

The first window contains only the ordered right endpoint of \(e\), the middle windows contain both endpoints, and the remaining windows contain only the ordered left endpoint. The transpose in the first case makes all three boundary orientations represent the same matrix \(X\).

For a physical configuration \(\sigma \) on the vertices, the PEPS state coefficient is obtained by summing over all virtual configurations and, for each such configuration, multiplying the local tensor entries over all vertices.

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:

Let \(\phi :G\simeq G'\) be a graph isomorphism. The transported tensor \(A^\phi \) on \(G'\) has bond dimension \(D^\phi _{e'}=D_{\phi ^{-1}(e')}\) at every edge \(e'\) of \(G'\). Its local tensor at \(w\in G'\) is the local tensor of \(A\) at \(\phi ^{-1}(w)\), with virtual indices moved through the incident-edge bijection induced by \(\phi \).

For three virtual leg spaces \(\alpha ,\beta ,\gamma \) and an endomorphism \(Z\) of \(\beta \otimes \gamma \), define the induced endomorphism of \(\alpha \otimes \beta \otimes \gamma \) by acting as the identity on the \(\alpha \) leg and as \(Z\) on the remaining two legs. This is the \(\mathbb {1}_\alpha \otimes Z\) form appearing after the first tensor is inverted in [ 220 , Section 3 ] .

For an endomorphism \(W\) of \(\alpha \otimes \gamma \), define the induced endomorphism of \(\alpha \otimes \beta \otimes \gamma \) by acting as \(W\) on the outer two legs and as the identity on the middle \(\beta \) leg, with matrix entries \(\delta _{bb'}W_{ac,a'c'}\) for \((a,b,c),(a',b',c')\in \alpha \times \beta \times \gamma \). This is the third residual form in [ 220 , Section 3 ] .

For an endomorphism \(U\) of \(\alpha \otimes \beta \), define the induced endomorphism of \(\alpha \otimes \beta \otimes \gamma \) by acting as \(U\) on the first two legs and as the identity on the \(\gamma \) leg. This is the \(U\otimes \mathbb {1}_\gamma \) form in [ 220 , Section 3 ] .

The coordinate exchange \(S(x,y)=(y,x)\) is a graph isomorphism from the \(w\times h\) torus to the \(h\times w\) torus. Its inverse is the coordinate exchange in the opposite direction.

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 \(\phi (v)\).

A PEPS tensor \(A\) on the discrete torus represents a translation-invariant state if

for every lattice translation \(\tau _{a,b}\) and every physical configuration \(\sigma \). This condition concerns the contracted state; the local tensor family itself may depend on the lattice site.

A PEPS tensor \(A\) on the discrete torus is translation invariant if \(A^{\tau _{a,b}}=A\) for every translation \(\tau _{a,b}\). This is the setting of [ 220 , Theorem 3 ] , where one tensor is repeated at every site of a periodic lattice.

A bond-dimension function on the torus is orientation-uniform with horizontal dimension \(D_h\) and vertical dimension \(D_v\) if \(D_{e_h}=D_h\) for every horizontal edge \(e_h\) and \(D_{e_v}=D_v\) for every vertical edge \(e_v\). The reference horizontal edge is the right edge at the origin, and the reference vertical edge is the up edge at the origin.

On the discrete torus, the rectangular injectivity hypotheses assert that every contiguous \(2\times 3\) and every contiguous \(3\times 2\) coordinate rectangle is injective for the given region-injectivity assignment. This is the torus counterpart of the square-lattice rectangular injectivity hypotheses.

For a vertex \(v\), the edges crossing the boundary of \(V\setminus \{ v\} \) are precisely the edges incident to \(v\). Hence the boundary configurations of \(V\setminus \{ v\} \) identify with the local virtual configurations at \(v\). Similarly, the vertices in \(V\setminus \{ v\} \) identify with the vertices distinct from \(v\), giving the corresponding identification of physical configurations.

Fix a vertex \(v\). The single-vertex tensor \(A_v\) is regarded as a two-block tensor whose shared bonds are the edges incident to \(v\), whose external boundary is a one-point space, and whose physical index is the physical index at \(v\): \(T_v(*,\eta ,\sigma )=A_v(\eta ,\sigma )\).

A virtual configuration assigns to every edge \(e\) one basis index in the corresponding bond space of dimension \(D_e\).

Given two virtual configurations and a predicate on edges, their piecewise merge reads the first configuration on selected edges and the second configuration elsewhere. For a vertex region, the regionwise merge selects precisely the edges incident to that region.

A predicate on the edges splits a virtual configuration canonically into its restrictions to the selected and unselected edges. An equality between two bond-dimension families also induces a canonical reindexing between their virtual configurations. For a vertex \(v\), selecting the incident edges splits a virtual configuration into its local labels at \(v\) and the labels on all nonincident edges.

A global virtual configuration is equivalently a choice of edge-centered boundary data together with a middle configuration in the corresponding fibre.

Write \({\lbrack a,b)}\) for the vertices whose chosen coordinate values lie between \(a\) and \(b-1\). Define

These are the one-site comparison regions in the final argument of [ 220 , proof of Theorem 3 ] , with arbitrary positive window lengths.

Let \(W\) be the left end window of a horizontal staircase with parameters \((L,K)\) and lower-left coordinate \((a,b)\), and let \(e\) be its reference edge. Suppose that

Then \(e\in \partial W\). Suppose also that the blocked tensors of \(B\) on \(W\) and \(V\setminus W\) are injective, that all bond dimensions of \(B\) are positive, and that the bond dimensions of \(A\) and \(B\) at \(e\) are identified by \(\iota _e:M_{D_A(e)}(\mathbb {C})\simeq M_{D_B(e)}(\mathbb {C})\). If \(Z\in \mathrm{GL}(D_B(e))\) satisfies, for every matrix \(M\) and physical configurations \(\sigma ,\tau \),

then these objects form the coefficient-identity witness at \(e\) with witness region \(W\) and with both gauge entries equal to \(Z\).

Let \(W\) be the left end window around the reference edge \(e\), placed with lower-left coordinate \((a,b)\). Suppose that every cyclic \(L\times K\) window of \(B\) is blocked-tensor injective, unions of injective regions are injective, all bond dimensions of \(B\) are positive, and

Suppose also that the bond dimensions of \(A\) and \(B\) at \(e\) are identified by \(\iota _e:M_{D_A(e)}(\mathbb {C})\simeq M_{D_B(e)}(\mathbb {C})\), and let \(Z\in \mathrm{GL}(D_B(e))\). If the conjugation identity

holds for every \(M,\sigma ,\tau \), then the coefficient-identity witness of Definition 24.7.3.57 is obtained for the region \(W\). The hypotheses supply the injectivity of both \(W\) and its complement at the minimal torus size.

Two tensors \(A\) and \(B\) (of the same bond dimension) generate the same MPV family if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every system size \(N\).

Let \(0{\lt}L\) with \(2L+1\le n\), and let \(A\) and \(B\) be site-dependent chains on \(n\) sites with a common positive bond dimension, each window injective at length \(L\), generating the same closed-chain state. Then for every site \(p\) there is an invertible matrix \(Z\) with \(B_{(p)}^w=ZA_{(p)}^wZ^{-1}\) for every word \(w\) of length \(n\). The statement for one repeated tensor is Lemma 24.9.3.4.

Let \(0{\lt}L\) with \(2L+1\le n\), let \(A\) and \(B\) be site-dependent chains on \(n\) sites, each window injective at length \(L\), generating the same closed-chain state, and fix a site \(r\). Then there is a linear map \(\Phi \) of the matrix algebra, unital and multiplicative, such that \(\operatorname{tr}(XA_{(r+L)}^w)=\operatorname{tr}(\Phi (X)B_{(r+L)}^w)\) for every matrix \(X\) and every word \(w\) of length \(n\), and \(\Phi \) is the unique linear map with this pairing property. This is the closing clause of Lemma 5 of [ 220 ] —the maps \(O_1\mapsto X\) and \(O_3^T\mapsto X\) are uniquely defined and are algebra homomorphisms—in the site-dependent setting of the source, with the algebra structure carried by the insertions: \(\Phi \) sends an insertion on the bond \((r+L-1,r+L)\) of the first chain to the bond operator extracted on the second.

Let \(0{\lt}L\) with \(2L+1\le n\), let the closed chain on \(n\) sites carry one tensor of \(D \times D\) matrices per site, window injective at length \(L\), and fix a site \(r\). Let \(C_0, \ldots , C_L\) be deformed window tensors around the bond \((r+L-1,r+L)\): each \(C_j\) assigns a matrix to every word of length \(L\), read as replacing the blocked tensor of the \(L\)-site window starting at site \(r+j\). Suppose the deformed states agree for consecutive window positions: for every \(j{\lt}L\), every prefix \(p\) of \(j\) sites read from \(r\), every assignment \(u\) on the \(L+1\) sites covered by the two windows, and every suffix \(s\) on the remaining sites,

Then the deformation is a virtual operation on the bond: there is a matrix \(X\) with \(C_0(a)=A_{(r)}^aX\) and \(C_L(b)=XA_{(r+L)}^b\) for all length-\(L\) words \(a\), \(b\), and \(X\) is unique. This is Lemma 5 of [ 220 ] in the site-dependent setting in which the source states it, with the deformed window tensors arising as physical operators contracted with the blocked windows they act on; Lemma 24.9.3.3 is its constant specialization.

Let two site-dependent chains \(A\), \(B\) on \(n\) sites generate the same closed-chain state, let \(k+q=n\), and suppose the \(A\)-arc products of length \(k\) read from a site \(s\) and of length \(q\) read from \(s+k\) each span the full matrix algebra. Then there is a linear map \(\Lambda \) of the matrix algebra with \(\Lambda (A_{(s)}^x)=B_{(s)}^x\) for every word \(x\) of length \(k\).

Let \(0 {\lt} L\) with \(2L + 1 \le n\) and \(0{\lt}D\), and let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\), each \(L\)-block injective, whose matrix product vectors agree at system size \(n\). Then there is an invertible matrix \(Z\) such that, for every word \(w\) of length \(n\),

This captures the insertion correspondence of Lemma 5 of [ 220 ] and its algebra-homomorphism clause for one site-independent tensor: the map sending an insertion on one network to the corresponding insertion on the other is an automorphism of the matrix algebra, hence inner.

Let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\), let \(n = k + q\), and suppose the length-\(k\) and length-\(q\) word products of \(A\) each span the full matrix algebra. If the matrix product vectors of \(A\) and \(B\) agree at system size \(n\), then there is a linear map \(\Lambda \) of the matrix algebra with \(\Lambda (A^{x_1} \cdots A^{x_k}) = B^{x_1} \cdots B^{x_k}\) for every word \(x\) of length \(k\). The same holds when, instead of the traces, the word products of the two tensors are themselves equal at length \(n\). This is the operator-transport mechanism underlying Lemma 5 of [ 220 ] : an operator expanded over the window products of one network is re-read in the other.

Let \(n\) be positive, let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\) with \(A\) being \(L\)-block injective, and let \(Z_v\) and \(Z'_v\) (\(v = 1, \ldots , n\), \(n + 1 \equiv 1\)) be two families of invertible matrices with \(B^i = Z_v^{-1} A^i Z_{v+1}\) and \(B^i = Z'^{-1}_v A^i Z'_{v+1}\) at every site \(v\) and every \(i\). Then for every bond \(v\) there is a nonzero scalar \(c\) with \(Z'_v = c\, Z_v\). No condition on the system size is needed.

Let \(n\) be positive, let \(A\) and \(B\) be tensors of \(D \times D\) matrices over physical dimension \(d\) with \(A\) being \(L\)-block injective, and let \(Z_v\) (\(v = 1, \ldots , n\), \(n + 1 \equiv 1\)) be invertible matrices with \(B^i = Z_v^{-1} A^i Z_{v+1}\) at every site \(v\) and every \(i\). Then for every site \(v\) there is a nonzero scalar \(c\) with \(Z_{v+1} = c\, Z_v\).

Let \(A\) be a tensor of \(D \times D\) matrices over physical dimension \(d\), \(L\)-block injective with \(0 {\lt} L\). Then \(A\) is \(m\)-block injective for every \(m \ge L\). This is the chain form of the statement that any region of at least the blocking size is injective, derived in the closing section of [ 220 ] (“Normal MPS: alternative proof”) from its union lemma.

Let a site-dependent chain on \(n \ge 1\) sites be window injective at length \(L\) with \(0{\lt}L\). Then for every \(m \ge L\) and every site \(s\), the arc products of the length-\(m\) words read from \(s\) span the full matrix algebra. This is the site-dependent form of the remark that any region of at least the blocking size is injective ( [ 220 ] , closing section, cf. Lemma 24.9.3.1).

If every tensor in a closed chain is injective, then the chain is window injective at length \(1\). Indeed, the length-one arc products at site \(s\) are precisely the matrices of the tensor at site \(s\).

Let \(0 {\lt} L\) with \(2L + 1 \le n\), let \(B\) be a tensor of \(D \times D\) matrices over physical dimension \(d\), \(L\)-block injective, and let \(C_0, \ldots , C_L\) be deformed window tensors: each \(C_j\) assigns a matrix to every word of length \(L\), read as replacing the blocked tensor of the \(L\)-site window starting \(j\) sites left of a fixed bond of the closed chain on \(n\) sites. Suppose the deformed states agree for consecutive window positions: for every \(j {\lt} L\), every prefix \(p\) of \(j\) sites, every assignment \(u\) on the \(L + 1\) sites covered by the two windows, and every suffix \(s\) on the remaining sites,

Then the deformation is a virtual operation on the bond: there is a matrix \(X\) such that, for all length-\(L\) words \(a\) and \(b\),

and \(X\) is unique. This is Lemma 5 of [ 220 ] (its closing section, “Normal MPS: alternative proof”), specialized to one site-independent tensor; the source states it for site-dependent tensors on five sites with \(L = 2\), with the deformed window tensors arising as physical operators contracted with the blocked windows they act on; the site-dependent form, in the setting in which the source states the lemma, is Lemma 24.9.4.4, and this statement is its constant specialization. The algebra-homomorphism clause of the source lemma is established with the insertion correspondence in Lemma 24.9.3.4.

Let \(\omega \) be a physical configuration on the full vertex set. If \(\omega _R\) and \(\omega _{V\setminus R}\) are its restrictions to \(R\) and to the complement, then \(\operatorname {Assemble}_R(\omega _R,\omega _{V\setminus R})=\omega \).

If the virtual configuration space on the complement of the distinguished edge is empty, then for every residual boundary label \(\rho \) the vector \(T_{A,M_e}^{\rho }\) in the edge-middle tensor family is zero.

For a finite kernel descent datum and any finite \(R\subseteq V\), \(K(R)\Longrightarrow K(\varnothing )\). More generally, if \(K(\varnothing )\Longrightarrow Q\), then \(K(R)\Longrightarrow Q\).

Under the bounds \(1{\lt}W\), \(1{\lt}H\), \(0{\lt}L\), \(0{\lt}K\), \(1\le a\), \(a+2L\le W\), and \(b+2K-1\le H\), a boundary edge of the end pair is a boundary edge of one of the two end windows, and it is not the reference edge. Conversely, any boundary edge of one end window other than \(e\) is a boundary edge of \(S\).

If \(2L\le W\), then the two end windows of the horizontal staircase are disjoint: \(W_{\mathrm{left}}\cap W_{\mathrm{right}}=\varnothing \).

If \(2L\le W\), the complement of either end window splits as the opposite end window and the complement of the end pair:

If \(L,K{\gt}0\), \(2L\le W\), and \(2K\le H\), then the corner is disjoint from the end pair and the patch decomposes as \(P=S\cup C\), with \(P\setminus S=C\).

Under the one-orientation \(L\times K\) torus-window injectivity hypotheses, the three regions \(W_{\mathrm{left}}\), \(W_{\mathrm{right}}\), and \(\widehat C\) are injective.

Let \((R_i)_{i\in s}\) be a nonempty finite family of possibly overlapping regions. If the blocked tensor of each \(R_i\) is injective and every virtual bond dimension is positive, then the blocked tensor of their union \(\bigcup _{i\in s} R_i\) is injective. This is the finite iteration of the overlapping union lemma, with no disjointness assumption.

Let \((R_i)_{i\in s}\) be a nonempty finite family of pairwise-disjoint regions. If the blocked tensor of each \(R_i\) is injective and every virtual bond dimension is positive, then the blocked tensor of their union \(\bigcup _{i\in s} R_i\) is injective.

Let \(V\) be a finite vertex set. Write \(\operatorname{inj}(R)\) for the assertion that the PEPS tensor blocked to the region \(R\subseteq V\) is injective. For two possibly overlapping regions \(A,B\subseteq V\), \(\operatorname{inj}(A)\wedge \operatorname{inj}(B)\Longrightarrow \operatorname{inj}(A\cup B)\). The union-injectivity proof in [ 220 , Section 4 ] blocks the network into four parts \(A\setminus B\), \(A\cap B\), \(B\setminus A\), and \((A\cup B)^c\), and applies the two available left inverses in sequence:

Taking a boundary tensor \(X\) on \((A\cup B)^c\) with \(\mathcal C_{\{ 0,1,2\} }(X)=0\), injectivity of \(A\) gives \(\mathcal C_{\{ 2\} }(X)=0\); reinserting \(A\cap B\) and applying injectivity of \(B\) then gives \(X=0\):

 \(=0\) \(\xrightarrow {A^{-1}}\)   \(=0\) \(\xrightarrow {A\cap B}\)   \(=0\) \(\xrightarrow {B^{-1}}\)   \(=0\).

Let \(P_0=A\setminus B\), \(P_1=A\cap B\), \(P_2=B\setminus A\), and \(P_3=(A\cup B)^c\). Let \(K\) be the boundary vector space attached to \(P_3\), and let \(\mathcal C_{\{ 0,1,2\} }:K\to E_{012}\), \(\mathcal C_{\{ 2\} }:K\to E_2\), and \(\mathcal C_{\{ 1,2\} }:K\to E_{12}\) be the three contraction maps in the proof of the union-injectivity lemma in [ 220 , Section 4 ] . If

and \(\mathcal C_{\{ 1,2\} }(X)=0\Longrightarrow X=0\), then \(\ker \mathcal C_{\{ 0,1,2\} }=\{ 0\} \).

Let \(S,T\subseteq V\) be two disjoint regions. If the blocked tensors of \(S\) and of \(T\) are each injective and every virtual bond dimension is positive, then the blocked tensor of their union \(S\cup T\) is injective.

Fix an edge \(e\) and a partition of the vertex set into the three edge-centred blocks \(R\), \(B\), \(C\) (red, blue, complementary) of the normal PEPS proof, with the red block carrying one endpoint of \(e\) and the blue block the other. The injectivity of the blue and complementary blocks is part of the edge-centred blocking data; if every virtual bond dimension is positive, then their union \(B\cup C=V\setminus R\) is injective.

Let \(A,B\subseteq V\) be two possibly overlapping regions. If the blocked tensors of \(A\) and of \(B\) are each injective and every virtual bond dimension is positive, then the blocked tensor of their union \(A\cup B\) is injective. This extends Lemma 24.5.7 to arbitrary overlapping regions when all virtual bond dimensions are positive.

If \(A\) is vertex-injective and \(R\) is a coefficient function on the local virtual configurations at \(v\) satisfying \(\sum _{\eta }R(\eta )A_v(\eta ,\tau )=0\) for every physical index \(\tau \), then \(R=0\).

Fix a region \(R\) and a site \(v\notin R\), and let \(\mu \) be a boundary label of \(S=R\cup \{ v\} \). Two local configurations at \(v\) that are both consistent with \(\mu \) and carry the same \(R\)-boundary label coincide.

Assume rectangular injectivity and union closure. On a finite \(n\times m\) lattice, if \(2\le a\), \(2\le b\), \(a+5\le n\), and \(b+5\le m\) then the complement of the comparison square at \((a,b)\) is injective, being the union of its four surrounding bands, and the complement of the comparison region at \((a,b)\) is injective, being that union together with one further corner rectangle.

Assume rectangular injectivity and union closure. On a finite \(n\times m\) lattice, if \(a+3\le n\) and \(b+3\le m\) then the comparison region at \((a,b)\) is injective, being the union of a \(2\times 3\) and a \(3\times 2\) rectangle, and the comparison square at \((a,b)\) is injective, being a \(3\times 3\) rectangle.

On a finite \(n\times m\) lattice, if \(2\le a\), \(1\le b\), \(a+3\le n\), and \(b+3\le m\) then the complement of the comparison region at \((a,b)\) is the union of the same four bands of Lemma 24.6.34 together with the \(3\times 2\) corner rectangle \(\{ (x,y):a-2\le x{\lt}a+1,\ b-1\le y{\lt}b+1\} \), which contains the omitted corner \((a,b)\).

For any offset \((a,b)\) a vertex with coordinates \((x,y)\) lies in the comparison region at \((a,b)\) if and only if

Let \(v\) be a lattice vertex and place the comparison region at \(v\)’s own coordinates. Then \(v\) lies outside that region, and inserting \(v\) back in recovers the full \(3\times 3\) comparison square placed at \(v\)’s coordinates.

On a finite \(n\times m\) lattice, if \(a+3\le n\) and \(b+3\le m\) then the complement of the comparison square at \((a,b)\) is the union of the four coordinate bands

The finite-lattice complementary block around the edge is disjoint from the two displayed edge blocks, and its union with them is the full finite square lattice. The local-window \(T\) region is contained in this finite-lattice complementary block.

In the normalized horizontal-edge \(5\times 7\) frame with lower-left corner \((0,0)\), any rectangular cover of the local \(T\)-region extends to a rectangular cover of the edge-complementary block \(A_3\) by adjoining the two top-collar contiguous \(3\times 2\) rectangles. In the normalized vertical-edge \(7\times 5\) frame with lower-left corner \((0,0)\), any rectangular cover of the rotated local \(T\)-region extends to a rectangular cover of the vertical edge-complementary block by adjoining the two right-collar contiguous \(2\times 3\) rectangles.

If the finite-lattice complementary block \(A_3\) has a nonempty finite cover by contiguous \(2\times 3\) and \(3\times 2\) rectangles, then rectangular injectivity and finite union closure imply that \(A_3\) is injective. The same criterion applies to every vertical edge-complementary block: a nonempty finite cover of that block by contiguous \(2\times 3\) and \(3\times 2\) rectangles implies injectivity of the vertical edge-complementary block. In the normalized horizontal-edge \(5\times 7\) frame, it is enough to give such a cover of the local \(T\)-region, since adjoining the two top-collar \(3\times 2\) rectangles gives a cover of \(A_3\).

In the normalized horizontal-edge \(5\times 7\) frame, the finite-lattice complementary block \(A_3\) is the local-window \(T\)-region together with two top-collar contiguous \(3\times 2\) rectangles. Consequently, if the local \(T\)-region is injective, then rectangular injectivity and union closure imply that \(A_3\) is injective. There is also a transposed \(7\times 5\) horizontal-frame collar identity with two contiguous \(2\times 3\) right-collar rectangles. For the rotated vertical edge-blocking regions, the corresponding complement is the rotated local \(T\)-region together with the same two right-collar rectangles; hence rotated \(T\)-injectivity, and hence a rectangular cover of rotated \(T\), implies injectivity of the vertical complementary block.

\( = \)

In the normalized \(5\times 7\) horizontal-edge frame, the red block is the vertical \(2\times 3\) block, the blue block is the horizontal \(3\times 2\) block, and the complementary block is the finite-lattice edge complement \(A_3\). The distinguished edge is a horizontal edge of the square-lattice graph. The left endpoint of the distinguished edge lies in the red block, the right endpoint lies in the blue block, the three regions are pairwise disjoint, and their union is the whole finite square lattice. If the local \(T\)-region is injective, rectangular injectivity and union closure give injectivity of all three blocks; in particular a rectangular cover of \(T\) suffices. The normalized \(7\times 5\) vertical-edge frame has the analogous red and blue rectangular blocks, a distinguished vertical edge, and the same set-theoretic blocking data; at this stage its complementary-block injectivity is either kept as an explicit hypothesis or obtained from rotated \(T\)-injectivity, and therefore from a rectangular cover of rotated \(T\), by the vertical collar lemma. In both cases the named blocking datum supplies the corresponding three injective regions for the edge-blocked chain.

For any coordinate origin whose horizontal blocking window lies in the finite square lattice, there is a distinguished horizontal edge with the same relative position as in the normalized picture. The translated red region is the \(2\times 3\) block, the translated blue region is the \(3\times 2\) block, and the complementary region is the finite-lattice complement of their union. If this complementary region is injective, these three regions form the one-edge blocking datum; if it has a rectangular cover, rectangular injectivity and union closure supply that complementary injectivity; and two rows and columns of margin below and to the left together with eight rows and columns of total room around the frame supply the same datum directly. In each form the datum supplies the three injective regions used in the edge-blocked chain, and the distinguished edge is the only edge crossing between the red and blue regions. The distinguished edge is the corresponding coordinate right edge. At coordinate origin \(0,0\), the translated edge and the three translated regions specialize to the normalized horizontal picture.

For an interior horizontal or vertical edge with the corresponding interior margins, the translated edge-blocking datum is built with no rectangular cover: its complementary-block injectivity is the interior \(A_3\) injectivity. A family of these cover-free interior data over all edges establishes the normal edge-blocking hypotheses, with the injective chain, endpoint membership, pairwise disjointness, and coverage at every edge.

This establishes the edge-blocking hypotheses for edges with interior margins [ 78 ] .

Let the two edge blocks be removed at an interior position of a sufficiently large lattice, with one row and column of room below and to the left and further room above and to the right. Then the complementary block \(A_3\) is the union of four full-length bands around the removed blocks — below, above, left, and right — together with two filler rectangles inside the bounding box of the removed blocks. Each band is a large contiguous rectangle and each filler is a source rectangle, all avoiding the removed blocks, so rectangular injectivity and union closure prove \(A_3\) injective with no rectangular-cover hypothesis. The transposed statement holds for the rotated vertical complementary block.

Scope restriction (open-boundary interior edges): These edge-blocking constructions apply only to edges with interior margins in the open rectangular model, whereas [ 220 , Theorem 3 ] obtains every edge via periodic translations on the torus; documented in [ 78 ] .

If a square-lattice region has a nonempty finite cover by contiguous \(2\times 3\) and \(3\times 2\) rectangles, then rectangular injectivity and finite union closure imply that the region is injective.

For the displayed regions \(R\) and \(S\) with the same lower-left corner, \(R\) is contained in \(S\). The difference \(S\setminus R\) is exactly the lower-right site of the \(3\times 3\) square: in coordinates, the site one obtains by shifting two steps horizontally and zero steps vertically from the lower-left corner.

If region injectivity is closed under unions, then the coordinate square-lattice rectangular injectivity hypotheses imply that the displayed regions \(R\) and \(S\) are injective.

If the displayed \(3\times 3\) window containing \(R\) and \(S\) lies in the finite square lattice, then \(R\) is the union of a contiguous \(2\times 3\) rectangle and a contiguous \(3\times 2\) rectangle, while \(S\) is the union of two contiguous \(2\times 3\) rectangles.

Fix a square-lattice PEPS tensor with every bond dimension positive whose blocked tensor is injective on every contiguous \(2\times 3\) and \(3\times 2\) rectangle. Then the displayed regions \(R\) and \(S\) are injective. If moreover the displayed region \(T\) has a cover by such rectangles, then \(T\) is injective too. The union closure used here is supplied by the overlapping union lemma, so no single-vertex injectivity is assumed.

For the displayed local \(T\)-region inside its \(5\times 6\) coordinate window, if the smaller edge-block bounds \(x_0 + 5 \leq n\) and \(y_0 + 5 \leq m\) hold in the finite \(n\times m\) square lattice, then the coordinate square-lattice rectangular injectivity hypotheses imply that the vertical \(2\times 3\) block and horizontal \(3\times 2\) block removed from the window are injective. If, in addition, region injectivity is closed under unions, then the union of these two removed edge blocks is injective.

If the displayed local \(T\)-region has a nonempty finite cover by contiguous \(2\times 3\) and \(3\times 2\) rectangles, then rectangular injectivity and finite union closure imply that \(T\) is injective. The rotated vertical local \(T\)-region has the same conditional injectivity criterion.

The present origin-based local-window model for the displayed \(T\)-region admits no nonempty finite cover by contained contiguous \(2\times 3\) and \(3\times 2\) rectangles whenever the ambient lattice contains the \(5\times 6\) window. The origin-based rotated local-window model used for vertical edges has the analogous obstruction whenever the ambient lattice contains the \(6\times 5\) window. The normalized \(5\times 6\) and \(6\times 5\) cases are recorded as corollaries. The present origin-based horizontal edge-complement model has the analogous obstruction whenever the ambient lattice contains the \(5\times 7\) frame; the normalized \(5\times 7\) case is again recorded as a corollary. The present origin-based vertical edge-complement model has the corresponding obstruction whenever the ambient lattice contains the \(7\times 5\) frame; the normalized \(7\times 5\) case is recorded as a corollary.

The two edge blocks removed from the displayed local \(T\)-region lie inside the local \(5\times 6\) coordinate window. They are disjoint from each other and from this local \(T\) region, and their union with it is exactly that local window.

If the two edge blocks cut out of the displayed \(T\)-region lie in the finite square lattice, then they are respectively a contiguous \(2\times 3\) rectangle and a contiguous \(3\times 2\) rectangle.

For any coordinate origin whose vertical blocking window lies in the finite square lattice, there is a distinguished vertical edge with the same relative position as in the normalized vertical picture. The translated red region is the \(3\times 2\) block, the translated blue region is the \(2\times 3\) block, and the complementary region is the finite-lattice complement of their union. If this complementary region is injective, these three regions form the one-edge blocking datum; if it has a rectangular cover, rectangular injectivity and union closure supply that complementary injectivity; and two rows and columns of margin below and to the left together with eight rows and columns of total room around the frame supply the same datum directly. In each form the datum supplies the three injective regions used in the edge-blocked chain, and the distinguished edge is the only edge crossing between the red and blue regions. The distinguished edge is the corresponding coordinate upward edge. At coordinate origin \(0,0\), the translated edge and the three translated regions specialize to the normalized vertical picture.

A contiguous rectangle whose width is at least two and whose height is at least three is the union of the source \(2\times 3\) rectangles sliding over it, so rectangular injectivity and union closure prove it injective. The transposed statement holds for a contiguous rectangle of width at least three and height at least two.

If the complement configuration space is empty and the residual boundary-label set is nonempty, then the edge-middle tensor family is not injective.

If \(j=0\) or \(L\le j\), then the reference edge \(e\) is a boundary edge of \(W_j\): exactly one of its two endpoints lies in the window.

If \(1\le j{\lt}L\), then both endpoints of \(e\) lie in \(W_j\).

Let \(L,K{\gt}0\), let \(s=(a,b)\) be a staircase corner on the torus, and assume \(a+2L\le n\) and \(2K\le m\). Let \(e\) be the horizontal reference edge from \((a+L-1,b+K-1)\) to \((a+L,b+K-1)\), and let \(W_j\) be the \(j\)th staircase window based at \(s\). Then the left endpoint \(u=(a+L-1,b+K-1)\) lies in \(W_j\) if and only if \(1\le j\).

If \(1\le j{\lt}L\), then \(e\) is not a boundary edge of \(W_j\).

In the same notation, the right endpoint \(w=(a+L,b+K-1)\) lies in \(W_j\) if and only if \(j{\lt}L\).

Under the union-closure hypothesis for region injectivity, any nonempty finite union of injective finite regions is injective.

Under the union-closure hypothesis for region injectivity, if the indexed subfamilies \(\{ 0,1\} \) and \(\{ 1,2\} \) in the four-region decomposition are injective, then the inside subfamily \(\{ 0,1,2\} \) is injective. Equivalently, injectivity of the two source-paper regions \((A\setminus B)\cup (A\cap B)\) and \((A\cap B)\cup (B\setminus A)\) gives injectivity of \((A\setminus B)\cup (A\cap B)\cup (B\setminus A)=A\cup B\).

In the four-region decomposition, the left-only and right-only parts are disjoint from the overlap and from each other. Each inside part, and hence the union of the three inside regions, is disjoint from the outside region. In indexed form, the pieces forming \(A\) are disjoint from the pieces forming \(A^c\), and the pieces forming \(B\) are disjoint from the pieces forming \(B^c\), both as explicit binary unions and as finite unions over the corresponding selected indexed subfamilies. The indexed four-region family is pairwise disjoint.

Let \(V\) be a finite vertex set and let \(A,B\subseteq V\). The left-only and overlap regions reconstruct \(A\), the overlap and right-only regions reconstruct \(B\), the three inside regions reconstruct \(A\cup B\), and all four regions reconstruct \(V\). The same identities also hold in the indexed four-region family: pieces \(0\) and \(1\) reconstruct \(A\), pieces \(1\) and \(2\) reconstruct \(B\), and pieces \(0,1,2\) reconstruct \(A\cup B\). These same reconstructions are available as finite unions over the corresponding subfamilies of the indexed four-region family. Piece \(3\) is the complement of \(A\cup B\), equivalently the complement of the three inside indexed pieces, and also \(A^c\cap B^c\). The singleton subfamily \(\{ 3\} \) gives the outside block. Pieces \(2,3\) reconstruct \(A^c\), equivalently the complement of pieces \(0,1\), and pieces \(0,3\) reconstruct \(B^c\), equivalently the complement of pieces \(1,2\), again both as explicit unions and as finite unions over the indexed subfamilies. Conversely, pieces \(0,1\) are the complement of pieces \(2,3\), pieces \(1,2\) are the complement of pieces \(0,3\), and adding all four pieces reconstructs \(V\). Equivalently, each vertex lies in one of the four regions, and the indexed four-region family has union \(V\). In fact, every vertex belongs to exactly one indexed piece.

Membership in a coordinate rectangle is coordinatewise membership, the contiguous coordinate interval has the expected coordinate inequalities, membership in a contiguous coordinate rectangle is the conjunction of the four bounding inequalities, and membership in \(R\) is the disjunction of membership in the \(2\times 3\) lower-left piece or in the shifted \(3\times 2\) upper piece. Membership in \(S\) is the disjunction of membership in the left \(2\times 3\) piece or in the horizontally shifted \(2\times 3\) piece. Membership in the local-window \(T\) region is membership in the local \(5\times 6\) window together with nonmembership in the union of the two displayed edge blocks. Bounded contiguous coordinate rectangles of sizes \(2\times 3\) and \(3\times 2\) satisfy, respectively, the predicates for contiguous \(2\times 3\) and contiguous \(3\times 2\) square-lattice rectangles. The cardinality of a coordinate rectangle is the product of the two coordinate cardinalities, and the full coordinate rectangle is the whole finite vertex set.

Under the coordinate square-lattice rectangular injectivity hypotheses, every bounded contiguous \(2\times 3\) rectangle and every bounded contiguous \(3\times 2\) rectangle is injective.

The coordinate square-lattice rectangular injectivity hypotheses determine abstract rectangular injectivity hypotheses, with the two abstract rectangular families taken to be the contiguous coordinate \(2\times 3\) and \(3\times 2\) rectangles.

The end members of the family are

At a frame origin \((x_0,y_0)\) with at least two columns and one row of margin below and to the left (one column and two rows in the rotated case), and with the removed blocks either touching the seam (\(x_0+5=n\), \(y_0+5=m\)) or keeping a further margin of at least two, the horizontal reference edge \(e_h\) is the right edge leaving \((x_0+1,y_0+2)\), and the vertical reference edge \(e_v\) is the up edge leaving \((x_0+2,y_0+1)\). Around each reference edge the rectangular injectivity hypotheses and union closure supply the one-edge blocking datum: the red block, the blue block, and the complement are injective, and the reference edge is the only edge crossing between the red and blue blocks,

and similarly with \(e_v\) at the rotated frame.

Let \(B_1,K_1\subseteq V\setminus T\) and \(B_2,H_2\subseteq T\). Fix boundary data \(c_1\) on \(V\setminus K_1\) and \(c_2\) on \(H_2\), and physical labels \(\sigma _1\) on \(B_1\) and \(\sigma _2\) on \(B_2\). Set

Then the boundary-compatible double sum collapses to the multiplicity of the interior bonds of \(T\) times the single merged sum:

Let \(B\) be a PEPS tensor on a torus whose periods \(W,H\) satisfy \(1{\lt}W\) and \(1{\lt}H\). Assume the one-orientation \(L\times K\) window-injectivity hypotheses for \(B\), union closure for injective regions, positive bond dimensions, and

Suppose also that

Let \(W_{\mathrm L}\) be the left end window and let \(e\) be the reference edge. If \(M,M'\in M_{D_B(e)}(\mathbb {C})\) satisfy

for every physical configuration \(\sigma \) on \(W_{\mathrm L}\) and \(\tau \) on \(V\setminus W_{\mathrm L}\), then \(M=M'\).

Let \(0 {\lt} L\) with \(2L + 1 \le n\) and \(0 {\lt} D\), and let \(A\) and \(C\) be tensors of \(D \times D\) matrices over physical dimension \(d\), each \(L\)-block injective, with equal word products at length \(n\): \(C^{w_1} \cdots C^{w_n} = A^{w_1} \cdots A^{w_n}\) for every word \(w\) of length \(n\). Then there is a constant \(\lambda \) with \(\lambda ^n = 1\) and \(C^i = \lambda A^i\) for every \(i\).

Let \(0 {\lt} L {\lt} n\) with \(n \ge 3\), let the bond dimension be positive, and suppose \(A\) is \(L\)-block injective. Then for every arc of \(L\) consecutive sites of the closed chain, the blocked tensor of the arc of the cycle tensor of \(A\) is injective.

For \(n \ge 3\) and every configuration \(\sigma = (i_1, \ldots , i_n)\), the state coefficient of the cycle tensor of \(A\) at \(\sigma \) is the matrix product vector component \(V^{(n)}(A)_\sigma = \operatorname{tr}\! (A^{i_1} \cdots A^{i_n})\).

Applying a gauge to a PEPS tensor preserves the state coefficients. Along each internal edge, the two endpoint insertions cancel:

Inverting an edge-scalar family gives \((c^{-1})_{v,e}=c_{v,e}^{-1}\) on every oriented endpoint.

If two edge-scalar families are multiplied, then their endpoint scalars satisfy \((cd)_{v,e}=c_{v,e}d_{v,e}\) on every oriented half-edge.

Every oriented endpoint scalar satisfies \(c_{v,e}\ne 0\).

Under the positive-bond-dimension hypotheses of Corollary 24.9.4.10, suppose \(Z_v\) and \(Z'_v\) are two families of invertible matrices satisfying \(B_v^i=Z_vA_v^iZ_{v+1}^{-1}\) and \(B_v^i=Z'_vA_v^i(Z'_{v+1})^{-1}\) for every \(v,i\). Then there is a single nonzero scalar \(c\) such that \(Z'_v=cZ_v\) for every bond \(v\). This is the uniqueness clause in [ 220 , line 724 ] .

Suppose two normal PEPS generate the same state, can be blocked into three-site injective chains around every edge, and for every site admit two injective regions with injective complements that differ exactly at that site. Then their defining tensors are related by a local gauge, and the gauges are unique up to the scalar freedom allowed by the graph. This is the general normal-PEPS theorem in [ 220 , Section 4 ] , following its Theorem 3. A connected-graph form, with the blocking hypothesis stated as the source’s proof uses it (one blocking shared by the two states, each distinguished edge the entire bond of its chain), is Theorem 24.8.7; its per-edge uniqueness clause is Theorem 24.8.8. The source’s remark following the theorem, that for a translationally invariant system the gauges are also translationally invariant when the proportionality constants are not absorbed into them, is realized on the torus by Theorem 24.7.1.10: its gauge family is carried onto every edge by the translations from one reference matrix per orientation class, and its scalar \(\lambda \) is kept separate rather than absorbed.

Let \(A\) and \(B\) be a pair of normal PEPS tensors on a connected finite simple graph with the same positive physical dimension, generating the same state. Thus the normal blocking hypotheses hold at the doubly injective assignment of the two tensors: around every edge the network blocks into three doubly injective regions with the edge endpoints in the first two, and every site admits two doubly injective regions, with doubly injective complements, differing exactly at that site. Suppose moreover that at every edge the distinguished edge is the only edge of the graph joining its two adjacent blocks. Then the defining tensors are related by a local gauge: there are invertible matrices, one on each edge, whose endpoint actions transform \(A_v\) into \(B_v\) at every vertex, with no leftover scalar. The shared blocking gives a common tripartition for the two tensors, with all three regions injective for both. For an edge \(e\) between the first two regions, the coefficient transfer \(T_e\) supplied by the three-site isomorphism argument satisfies

for an invertible \(Z_e\) and every virtual insertion \(X\) on \(e\). The single-crossing hypothesis identifies \(e\) with the entire bond between \(R_1\) and \(R_2\), so \(Z_e\) acts on the original edge. The equality of the bond dimensions is not assumed; it follows from the blocking hypotheses by Theorem 24.8.6. The nonzero bond spaces of the pair are used in the injectivity argument, while connectivity permits the per-vertex constants to be absorbed into the edge gauges; see Theorem 24.4.1.24. This is the blocking argument of [ 220 , Section 4 ] .

Let \(A\) and \(B\) be a pair of normal PEPS tensors on a finite simple graph with equal bond dimensions. Suppose two families of invertible matrices, one on each edge, each realize the scalar-free gauge relation of Theorem 24.8.7: at every vertex the endpoint actions of the family transform \(A_v\) into \(B_v\). Then the two families are proportional at every edge: \(X'_e = c\cdot X_e\) for a nonzero scalar \(c\) depending on the edge. This is the uniqueness clause of the general normal-PEPS theorem in [ 220 , Section 4 ] , following its Theorem 3, read against the scalar-free relation; neither equality of the two states nor connectivity is needed.

Suppose PEPS tensors \(A\) and \(B\) on a connected graph are injective, generate the same state, and \(A\) satisfies the normal blocking hypotheses. Then their defining tensors are related by a local gauge. This is the vertex-injective specialization of the general normal-PEPS theorem: when each single tensor is already injective, the gauge follows from the injective Fundamental Theorem, without requiring the region blocking hypotheses.

Let \(A\) and \(B\) be translationally invariant normal square-lattice PEPS tensors such that every \(2\times 3\) and \(3\times 2\) region is injective. If they generate the same state on an \(n\times m\) region with \(n,m\geq 7\), then they are related by horizontal and vertical gauges \(X,Y\), up to a scalar \(\lambda \) satisfying \(\lambda ^{nm}=1\). This is [ 220 , Theorem 3 ] . Theorem 24.6.51 proves a related interior-window theorem on an open rectangle, without translation invariance and with per-vertex scalars in place of the single \(\lambda \); the gauge-equivalence conclusion on a general connected graph, from region injectivity alone, is Theorem 24.8.7. The periodic statement of the source is established in Theorem 24.7.1.10. Extending the open-rectangle theorem to its boundary is a separate question.

Let \(A\) and \(B\) be PEPS tensors on the open \(n\times m\) square lattice with the same positive physical dimension and equal bond dimensions, generating the same state, and both normal with injective rectangles (Definition 24.6.13): every contiguous \(2\times 3\) and \(3\times 2\) coordinate rectangle of either tensor is injective, and every virtual bond space is nonzero. Then there is a family of invertible matrices, one on each edge, such that:

1. at every edge with interior margins (for a horizontal edge with left endpoint \((x,y)\): \(3\le x\), \(x+7\le n\), \(4\le y\), \(y+6\le m\); for a vertical edge the mirrored margins), inserting any matrix on the edge of \(A\) gives the same coefficient as inserting it on the corresponding edge of the gauge-absorbed \(B\), for every physical configuration; and

2. every vertex \(v=(x,y)\) of the interior comparison window \(4\le x\le n-9\), \(4\le y\le m-9\) carries a nonzero scalar \(\lambda _v\) such that \(A_v\) equals \(\lambda _v\) times the gauge action of the family on \(B\) at \(v\), on every virtual configuration of the incident edges and every physical index.

This is the open-lattice form of [ 220 , Theorem 3 ] , restricted in two ways. First, the conclusion reaches only the interior: the absorbed equality holds at the edges with interior margins and the per-vertex relation at the window vertices, while vertices near the lattice boundary are not reached, the blocking frames of the open lattice requiring interior margins. Second, the scalars of distinct vertices are not asserted equal, and no condition \(\lambda ^{nm}=1\) is asserted: the single \(\lambda \) of the source comes from translation invariance, which is not assumed here, and without it a single scalar is false — rescaling the components of \(A\) at two interior vertices by \(\mu \) and \(\mu ^{-1}\) preserves the state and the rectangular injectivity but separates the two scalars. No lower bound on \(n\) or \(m\) is assumed; both conclusions are empty when the lattice is too small to contain interior edges or window vertices.

Let \(A,B\) be translation-invariant PEPS tensors generating the same state on an \(n\times m\) torus, both normal at window size \(L\times K\), where \(L,K{\gt}0\), \(n\geq 2L+1\) and \(m\geq 2K+1\). Then their bond dimensions agree and there are invertible horizontal and vertical gauges \(X,Y\) and a scalar \(\lambda \) such that

The matrices \(X,Y\) are unique up to nonzero scalar factors. This is the normal translation-invariant PEPS corollary of [ 220 ] . The displayed relation abbreviates the horizontal and vertical virtual-leg contractions. In the chosen torus edge orientations, the corresponding edge gauge family is translation covariant; seam reversals replace a matrix by its transposed inverse.

Let \(A\) and \(B\) be translationally invariant normal PEPS on the discrete torus such that every \(2\times 3\) and \(3\times 2\) region is injective. If they generate the same state on an \(n\times m\) region with \(n,m\geq 7\), then they are related by horizontal and vertical gauges \(X,Y\), up to a scalar \(\lambda \) satisfying \(\lambda ^{nm}=1\), and \(X\) and \(Y\) are unique up to a multiplicative constant. This is [ 220 , Theorem 3 ] on the torus geometry, where every edge is interior and translation acts freely. Theorem 24.7.1.10 proves this at the source’s sizes \(n,m\geq 7\) with the gauge family described by its translation covariance, and the uniqueness of \(X\) and \(Y\) up to a multiplicative constant is proved at the same sizes in Theorem 24.7.1.12 per edge, Theorem 24.7.1.13 on the two orientation classes, and Theorem 24.7.1.14 against the per-vertex relation. In the torus edge convention the two class matrices \(X\) and \(Y\) are read through this translation-covariant family: seam edges carry the transposed inverse of the corresponding class matrix, so this is the orientation-faithful form of the source’s horizontal and vertical gauges.

Let \(A\) and \(B\) be translationally invariant PEPS tensors on the discrete \(n\times m\) torus with \(n,m\geq 7\), with positive physical dimension, generating the same state, and both normal with injective rectangles (Definition 24.7.1.6). Then the bond dimensions of \(A\) and \(B\) agree on every edge, and there are a family of invertible matrices, one on each edge, carried onto each edge by the torus translations from a single matrix on a reference horizontal edge and a single matrix on a reference vertical edge, and a single scalar \(\lambda \) with \(\lambda ^{nm}=1\), such that at every site the tensor \(A\) equals \(\lambda \) times the gauge action of the family on \(B\). The family also realizes the absorbed inserted-coefficient equality at every edge: inserting any matrix on an edge of \(A\) gives the same coefficient as inserting it on the corresponding edge of the gauge-absorbed \(B\). This is [ 220 , Theorem 3 ] on the torus at the source’s sizes \(n,m\geq 7\), with the horizontal and vertical gauges \(X\), \(Y\) described by translation covariance: the stored orientation of a wraparound edge is reversed, so on the seam the family carries the transposed inverse of the class matrix, and a family that is literally one matrix per orientation class cannot realize the absorbed equality there. The translation covariance of the family, with the scalar \(\lambda \) kept separate rather than absorbed, realizes the remark of [ 220 ] following the general normal theorem: for a translationally invariant system the gauges are also translationally invariant, provided the proportionality constants are not absorbed into the gauges. The uniqueness of \(X\) and \(Y\) up to a multiplicative constant is not part of this statement; the uniqueness clause is Theorem 24.7.1.12 (per edge, against the absorbed equality), Theorem 24.7.1.13 (one constant per orientation class), and Theorem 24.7.1.14 (against the per-vertex relation itself).

Let \(A\) and \(B\) be injective PEPS tensors on a finite simple graph, and suppose that the graph is connected. If they give the same PEPS state, then their corresponding virtual edge spaces have the same dimensions and, after identifying these spaces, there are invertible matrices \(X_e\), one for each edge, such that \(B_v\) is obtained from \(A_v\) by the oriented endpoint action of the matrices incident to \(v\).

Suppose that the corresponding virtual edge spaces of \(A\) and \(B\) are already identified and the graph is connected. If \(A\) and \(B\) are injective PEPS tensors and give the same PEPS state, then there are invertible matrices \(X_e\), one for each edge, such that at every vertex the tensor \(B_v\) is obtained from \(A_v\) by the oriented endpoint action of the incident matrices.

If two gauge families relate the same injective PEPS tensor to the same target tensor, then their edge gauges differ by nonzero scalars \(c_e\) whose oriented product at every vertex is \(1\). Thus the correct graph quotient is by balanced edge scalars.

Suppose \(A\) and \(B\) are vertex-injective PEPS tensors with the same PEPS state, that the corresponding virtual bond dimensions are equal, that every virtual bond of \(A\) has positive dimension, and that the graph is connected. The connectivity assumption is necessary: on a disconnected graph the per-vertex scalars produced by the reduction below cannot be absorbed into edge gauges, since on the empty graph on two vertices the products of the vertex scalars can agree while no edge gauge relates the two tensors. Choose an edge \(e=(u,v)\) and block all other vertices into one middle tensor. The three resulting tensors form an injective chain, so the one-dimensional injective comparison assigns an invertible matrix to the edge \(e\). Repeating this for every edge and absorbing these matrices into the second tensor family gives the post-absorption insertion equality of [ 220 , Section 3 ] : for every edge and every matrix, inserting that matrix on the edge in the first PEPS gives the same state as inserting it on the same edge in the modified second PEPS. Applying the local extraction at each vertex then gives one edge gauge family whose endpoint actions transform the first tensor family into the second:

If \(A \sim B\) are gauge-equivalent PEPS tensors, then they represent the same state: for every physical configuration \(\sigma \),

If two gauge families \(X\) and \(Y\) differ by a balanced edge-scalar reweighting, then applying them to the same PEPS tensor \(A\) produces the same gauged tensor:

If two gauge families \(X\) and \(Y\) differ by a balanced edge-scalar reweighting, then the PEPS tensors obtained by applying those gauge families represent the same state: for every physical configuration \(\sigma \),

Gauge equivalence modulo balanced edge scalars is reflexive, symmetric, and transitive.

If two gauge families \(X\) and \(Y\) differ by a balanced edge-scalar reweighting, then for every vertex \(v\), virtual configuration \(\eta \), and physical index \(\sigma \),

Every edge-gauge family is equivalent to itself modulo balanced edge scalars, witnessed by \(c_e=1\) for every edge \(e\).

If one gauge family differs from another by balanced edge scalars, then the second differs from the first by the inverse balanced edge-scalar family.

If \(c\) and \(d\) witness the equivalences of \(X\) with \(Y\) and of \(Y\) with \(Z\) modulo balanced edge scalars, then their pointwise product scalar family \((cd)_e = c_e d_e\) witnesses that \(X\) is equivalent to \(Z\) modulo balanced edge scalars.

The pointwise inverse of a vertex-balanced edge-scalar family is again vertex-balanced.

The pointwise product of two vertex-balanced edge-scalar families is again vertex-balanced.

Under the factorized local gauge hypothesis, the local gauge formula follows. What remains open is to upgrade equality of the edge-blocked coefficients to the blocked-middle formula using the corresponding three-site injective-chain argument. The physical-realization and physical-to-virtual steps in that argument are recorded separately as Theorems 24.3.6 and 24.3.14. After Theorem 24.3.30 supplies the factorized local gauge hypothesis, injectivity on the star neighborhood provides a left inverse that isolates the local edge gauges. Thus there are incident-edge matrices \(X_e\) such that

If the full matrix algebras \(M_{D}(\mathbb {C})\) and \(M_{E}(\mathbb {C})\) are isomorphic as \(\mathbb {C}\)-algebras, then \(D=E\).

Let \(\varphi :M_{D}(\mathbb {C})\to M_{E}(\mathbb {C})\) be a \(\mathbb {C}\)-algebra isomorphism. Then \(D=E\); let \(h\) denote this equality and \(\iota _h:M_{D}(\mathbb {C})\to M_{E}(\mathbb {C})\) the canonical index-relabelling isomorphism. There is \(X\in \mathrm{GL}_E(\mathbb {C})\) with \(\varphi (M)=X\iota _h(M)X^{-1}\) for all \(M\in M_{D}(\mathbb {C})\).

Let \(A_0,\ldots ,A_{n-1}\) and \(B_0,\ldots ,B_{n-1}\) be two site-dependent closed chains with a cyclic gauge relation \(B_v^i=Z_vA_v^iZ_{v+1}^{-1}\) at every site. Then they generate the same closed-chain state. More generally, the product along any arc is conjugated by the two gauges on the boundary bonds of that arc; for a full turn around the chain the two boundary bonds coincide, and cyclicity of trace removes the remaining conjugation.

Let \(A,B\) be translation-invariant tensors on a torus with \(w,h{\gt}2\), matched bond dimensions, and positive bond dimensions for \(A\). Let \(e_{\mathrm h}\) and \(e_{\mathrm v}\) be distinguished horizontal and vertical edges. Suppose that at each of these edges there is a coefficient-identity witness: its region and complement are injective for \(B\), all bond dimensions of \(B\) are positive, and its two invertible matrices \(Z,Z_{\mathrm{ref}}\) satisfy \(C_A(R,e,M;\sigma ,\tau )=C_B(R,e,Z M Z^{-1};\sigma ,\tau )\) for both choices \(Z\) and \(Z_{\mathrm{ref}}\), all matrices, and all physical configurations, after identifying the matched matrix sizes. Require that the first stored endpoint of each distinguished edge belongs to its witnessing region. Then there is a translation-covariant family of invertible edge matrices \(X_e\) such that, for every edge \(e\), global physical configuration \(\sigma \), and matrix \(M\),

Here \(X\cdot B\) denotes the tensor obtained by the edge gauges, and the matrix on the right is identified by the matched bond dimensions. The witnessing regions need not be rectangles.

Let \(w,h{\gt}1\), and let \(A,B\) have matched bond dimensions, positive for \(B\). Let \(Q\) and \(Q^c\) be injective for \(B\), and let \(e\in \partial Q\). Suppose two invertible edge gauge families \(X,X'\) both satisfy \(C_A(e,M;\sigma )=C_{X\cdot B}(e,M;\sigma )\) at this edge, with the second family substituted for the first as well, for every \(M,\sigma \). Then \(X'_e=cX_e\) for some \(c\in \mathbb C^\times \).

The absorbed tensor family has \(D_{\widetilde B}(e)=D_B(e)\) for every edge \(e\).

If \(M_{ie}\) is the oriented endpoint matrix attached to an incident edge \(ie\) at \(v\), then \(\widetilde B_v(\eta ,\sigma ) =\sum _{\eta '}(\prod _{ie}M_{ie}(\eta (ie),\eta '(ie))) B_v(\eta ',\sigma )\).

For \(a{\lt}w\) and \(b{\lt}h\), the complement of \(\mathcal R(s;a,b)\) is

Let \(\mathcal I\) be closed under finite nonempty unions and satisfy the \(L\times K\) cyclic-window hypothesis, where \(L,K{\gt}0\). If \(L\leq a\leq w\) and \(K\leq b\leq h\), then \(\mathcal R(s;a,b)\in \mathcal I\) for every \(s\).

Let \(A\) and \(B\) be translation-invariant PEPS tensors on the \(w\times h\) torus with the same contracted state, both normal at window size \(L\times K\) (Definition 24.7.5.2), where \(L,K{\gt}0\), \(2L+1\leq w\), and \(2K+1\leq h\). Then \(D_A(e)=D_B(e)\) at every edge \(e\).

Let \(A,B\) be translation-invariant PEPS tensors on a \(w\times h\) torus with the same state, both normal at window size \(L\times K\), where \(L,K{\gt}0\), \(2L+1\leq w\) and \(2K+1\leq h\). Then their bond dimensions agree and there are a translation-covariant invertible edge gauge family \(X\) and \(\lambda \in \mathbb C\) such that

The first identity holds at every vertex, after identifying equal bond dimensions. This is the existence and normalization part of the normal translation-invariant PEPS corollary of [ 220 ] .

The two restrictions recover the original configurations: \(\operatorname {As}_{R,S}(\sigma _R,\sigma _Q)|_R=\sigma _R\) and \(\operatorname {As}_{R,S}(\sigma _R,\sigma _Q)|_{S\setminus R}=\sigma _Q\).

Let \(A,B\) be tensors on a finite graph with matched bond dimensions, and put \(C=X\cdot B\) for an invertible edge gauge family \(X\). Suppose \(A\) and \(C\) are injective on both \(Q\) and \(Q^c\), the boundary of \(Q\) is nonempty, and \(C_A(e,M;\sigma )=C_C(e,M;\sigma )\) for every edge, inserted matrix and global physical configuration. Then \(A_Q=c_Q C_Q\) for some \(c_Q\ne 0\).

Let \(w,h{\gt}1\), let \(A,B\) be translation invariant on the torus with matched bond dimensions, and let \(X\) be a translation-covariant invertible edge gauge family. Put \(C=X\cdot B\). If \(A_Q=c C_Q\), then \(A_{tQ}=c C_{tQ}\) for every translation \(t\).

For every \(c\in \mathbb C\) and every boundary configuration \(\mu \), \(I_{A,R,f,M+N}=I_{A,R,f,M}+I_{A,R,f,N}\), \(I_{A,R,f,cM}=cI_{A,R,f,M}\), and \(I_{A,R,f,1}(\mu )=T_{A,R}(\mu )\).

If \(A\) is injective on \(R\) and \(V\setminus R\), and its bond dimensions are positive, then \(I_{A,R,f,M}=I_{A,R,f,N}\) implies \(M=N\).

For every exterior configuration \(\eta \) and every bond index \(j\),

Let \(A\) be translation invariant on the discrete torus, let \(f\) be a crossing edge of \(R\), and let \(T_{a,b}\) be a translation. The bond dimension of \(A\) at the translated crossing edge equals its bond dimension at the reference crossing edge: \(D_A(T_{a,b}(f))=D_A(f)\).

Let \(A\) and \(B\) be a pair of normal PEPS tensors on a finite simple graph with the same positive physical dimension, generating the same state, with each distinguished edge the only edge of the graph joining its two adjacent blocks. Then the bond dimensions of \(A\) and \(B\) agree on every edge. The one-edge form assumes only the blocking datum at a single edge, and takes the positive bond dimensions as explicit hypotheses. In the source the equality is part of the isomorphism lemma of [ 220 , Section 3 ] : the insertion correspondence \(X\mapsto Y\) on the chosen bond is an algebra isomorphism between the two full bond matrix algebras, so the bond dimensions on the two sides agree; the normal case inherits the argument through the blocked three-site chains. The single-crossing condition is the source’s blocking around an edge made explicit: the chosen edge is the entire bond between the first two parties of the blocked chain.

Let \(A\) and \(B\) be translationally invariant PEPS tensors on the \(n\times m\) torus with \(n,m\geq 7\) and positive physical dimension \(d{\gt}0\), generating the same state. Suppose both tensors are normal with injective rectangles (Definition 24.7.1.6): every contiguous \(2\times 3\) and \(3\times 2\) torus rectangle is injective, and every virtual bond space is nonzero. Then the bond dimensions of \(A\) and \(B\) agree on every edge:

The equality is derived from the one-edge blocking data built from the rectangular injectivity — the source’s Lemma inj_isomorph ( [ 220 , Section 3, lines 560–583 and 1407–1500 ] ) — together with the translation invariance and same-state assumptions; the bond dimension is not assumed as a hypothesis.

Let \(A\) and \(B\) be PEPS tensors on the \(n\times m\) square lattice with positive physical dimension \(d{\gt}0\) and positive bond dimensions at every edge, generating the same state. Suppose both tensors satisfy the coordinate rectangular injectivity hypotheses (every contiguous \(2\times 3\) and \(3\times 2\) rectangle is injective) and the associated region-injectivity predicates are closed under finite unions. Then the bond dimensions of \(A\) and \(B\) agree at every edge with interior margins: for every edge \(e\) with a margin of at least one row and column of room below and to the left, and at least three rows and columns above and to the right (avoiding the lattice boundary),

The equality is derived from the one-edge blocking data built from the rectangular injectivity — the source’s Lemma inj_isomorph ( [ 220 , Section 3, lines 560–583 and 1407–1500 ] ) — together with the same-state and positive-bond assumptions; the bond dimension at the edge is not assumed.

Let \(R_1\) and \(R_2\) be two regions for which the same edge \(e\) is a boundary edge. Let \(M_1,M_2\in M_{D_B(e)}(\mathbb {C})\), and suppose that the two oriented insertions agree, \(\mathcal O_{R_1,e}(M_1)=\mathcal O_{R_2,e}(M_2)\). Then for every physical configuration \(\omega \) on the full vertex set,

For a complement physical configuration \(\tau \), let \(\psi ^\tau _{A,R}\) be the partial state across the cut. Then

For every inserted matrix \(X\) and every pair of physical configurations, the chosen coefficient transfer satisfies \(C_A(R,f,X;\sigma ,\tau ) =C_B(R,f,T_f(X);\sigma ,\tau )\). If the tensors have the same state and the same bond dimensions, all \(B\)-bond dimensions are positive, and the region and complement-region blocked maps of \(B\) are injective, then \(T_f(\mathbb {1})=\mathbb {1}\).

The image of the blocked tensor map on \(R\) is \(\operatorname {span}\{ T_{A,R}(\mu ):\mu \text{ a boundary configuration}\} \). If \(A\) and \(B\) generate the same closed state, both tensors are injective on \(V\setminus R\), and their bond dimensions are positive, these two images coincide.

Let \(w,h{\gt}1\), and suppose \(A,B\) generate the same torus state, have matched bond dimensions, and have positive bond dimensions for \(A\). Suppose \(A\) is injective on \(Q\) and \(Q^c\) for some \(Q\). If an invertible edge gauge family \(X\) satisfies \(A_v=\lambda (X\cdot B)_v\) at every vertex, then \(\lambda ^{wh}=1\).

Let \(A\) and a comparison tensor \(C\) share their bond dimensions and have positive bond dimensions, and suppose the \(R\)-blocked family of \(C\) is linearly independent. If the blocked weights satisfy the two region proportionalities \(A_R=c_R C_R\) and \(A_S=c_S C_S\) over \(R\) and over \(S=R\cup \{ v\} \), with \(c_R\ne 0\), then the inserted-site tensors are proportional with ratio \(c_S/c_R\) at every local configuration \(\eta \) at \(v\) and every physical index \(\sigma \):

Let a finite set map to the virtual configurations so that every fiber has cardinality \(n\). For every function \(g\) into an additive commutative monoid, \(\sum _x g(m(x))=n\mathbin {\boldsymbol {\cdot }}\sum _{\eta }g(\eta )\), where the first sum ranges over the finite set and the second over all virtual configurations.

For nested regions \(R\subseteq S\subseteq P\), the coefficient kernels satisfy

This is the two-block merge collapse in the proof of [ 220 , Section 4, proof of Theorem 3 ] .

The corner extension is linear in the inserted boundary coefficients. For nested regions \(R\subseteq S\subseteq P\), the bare extension satisfies

without a positivity assumption. If all bond dimensions are positive, the normalized corner extension is transitive:

If all bond dimensions are positive, then corner extension from \(R\subseteq S\) preserves the deformed region state: \(\Psi _{A,R,C}=\Psi _{A,S,\operatorname{Ext}_{R\subseteq S}(C)}\).

Under the hypotheses of Theorem 24.7.3.36, the assignment \(F_{AB}\) preserves addition, complex scalar multiplication, and the identity. It also satisfies \(F_{BA}(F_{AB}(X))=X\) for every \(X\in M_{D_A(e)}(\mathbb {C})\).

Under the hypotheses of Theorem 24.7.3.36, write \(F_{AB}(X)=Y_{A\to B}(X)\). For every boundary configuration \(\nu \), \(O_1^A(X)T_{B,W_{\mathrm L}}(\nu )=I_{B,W_{\mathrm L},e,F_{AB}(X)}(\nu )\). Moreover, for every pair of physical configurations \(\sigma ,\tau \),

Let \(L,K{\gt}0\), and choose the displayed non-wrapping horizontal staircase coordinates \((a,b)\) with

Suppose that \(A\) and \(B\) are translation invariant, generate the same closed state, every cyclic \(L\times K\) window of either tensor is injective, and every bond dimension of either tensor is positive. For \(X\in M_{D_A(e)}(\mathbb {C})\), there is a unique \(Y\in M_{D_B(e)}(\mathbb {C})\) such that, for every external boundary configuration \(\eta _{\mathrm L}\) and \(\eta _{\mathrm R}\), and every reference-edge index \(k\),

Thus the relation (44) has the paper’s \(O_3^{\mathsf T}\) orientation. No equality between \(D_A(e)\) and \(D_B(e)\) is assumed or concluded, and this statement contains no multiplication or gauge assertion.

The bond-uniformity implication is false for two sites. More precisely, there are injective tensors

whose closed-chain coefficients are invariant under the cyclic shift, but whose positive virtual dimensions are unequal. Consequently the condition \(n\ge 3\) in Corollary 24.9.4.13 cannot be removed. This corrects the unqualified printed corollary in [ 220 , lines 1803–1805 ] ; see [ 29 ] .

A closed chain of \(n\) sites is the cycle graph on \(n\) vertices, and a block of consecutive sites is an arc: the vertices reached clockwise from a starting site. Suppose \(0 {\lt} L\), \(3L \le n\), every arc of \(L\) consecutive sites is injective for a region-injectivity assignment, and injectivity is closed under unions. Then the chain satisfies the normal blocking hypotheses: around every edge it blocks into the \(L\) sites on one side, the \(L\) sites on the other, and the remaining \(n - 2L \ge L\) sites, all injective and covering the chain; and at every site the \(L + 1\) sites starting there and the \(L\) sites starting at its successor are injective regions with injective complements differing exactly at that site. Moreover the only nearest-neighbour pair joining the two blocks around an edge is the edge itself, since the far ends of the two blocks are separated by the remaining \(n - 2L \ge L\) sites.

For every physical configuration \(\sigma \) on \(R\) and \(\tau \) on \(V\setminus R\), the deformed state of the bond-inserted region insert is the region insertion coefficient: \(\Psi _{A,R,I_{A,R,f,M}}(\sigma ,\tau ) =C_A(R,f,M;\sigma ,\tau )\).

Let \(R\) be a region, let \(f\) be a boundary edge of \(R\), let \(M\in M_{D_B(f)}(\mathbb {C})\), and let \(\omega \) be a physical configuration on the full vertex set. Then the assembled deformed state of the bond-inserted region insert is the non-boundary bond product times the corresponding edge-inserted coefficient:

Here \(p_B(R)=\prod _{g\notin \partial R}D_B(g)\), and \(\mathcal O_{R,f}\) is the boundary-edge orientation.

Let \(w,h{\gt}1\), let \(A,B\) have matched bond dimensions on the torus, and let \(X\) be an invertible edge gauge family. If \(A_v=\lambda (X\cdot B)_v\) at every vertex and \(\lambda ^{wh}=1\), then \(C_A(e,M;\sigma )=C_{X\cdot B}(e,M;\sigma )\) for every edge, inserted matrix and global physical configuration, after identifying the equal bond dimensions.

Given edge gauges \(X_e\), set \(\widetilde B=B^X\). Then \(D_{\widetilde B}=D_B\) and, for every vertex \(v\),

where \(M_{ie}=X_e\) or \(M_{ie}=(X_e^{-1})^t\) according to the orientation of the endpoint \(ie\). After this absorption, arbitrary insertions of the same matrix on the same edge are compared in the first PEPS and the modified second PEPS.

Let \(\phi \) be a graph isomorphism between two tori whose side lengths exceed one, and let \(A'=A^\phi \) and \(B'=B^\phi \). Let \(e\) be an edge of the original torus and put \(f=\phi (e)\). Write \(e_-\) and \(f_-\) for their first endpoints in the chosen edge ordering, and assume \(\phi (e_-)=f_-\). Suppose \(f\in \partial R'\), \(f_-\in R'\), all bond dimensions of \(B'\) are positive, and \(B'\) is injective on \(R'\) and \((R')^c\). Suppose also \(D_{A'}(f)=D_{B'}(f)\) and there is an invertible matrix \(Z\) such that

for every inserted matrix \(M\) and every physical configuration on \(R'\) and its complement, after identifying the equal bond spaces. Then \(R=\phi ^{-1}(R')\) has \(e\in \partial R\) and \(e_-\in R\); all bond dimensions of \(B\) are positive, \(B\) is injective on \(R\) and \(R^c\), and \(D_A(e)=D_B(e)\). There is an invertible matrix \(Y\) such that

for every inserted matrix \(N\) and every physical configuration on \(R\) and its complement, with the equal bond spaces identified.

For every edge \(e\), the edge-blocked three-block coefficient is exactly the original PEPS state coefficient.

If the original PEPS tensor \(A\) is vertex-injective, then for each edge \(e=(u,v)\), the two endpoint tensor maps in the edge-blocked three-site chain are injective:

Suppose the two edge-blocked three-site chains associated to \(A\) and \(B\) at the edge \(e\) are injective, suppose every virtual bond of \(A\) and \(B\) has positive dimension, and suppose the two PEPS tensors define the same state. Then

This is the edge-blocked form of the algebra isomorphism \(X\mapsto Y\) in [ 220 , Section 3 ] :

\(=\)

For a tensor map or blocked tensor family \(T\), write \(\operatorname{inj}(T)\) for the assertion that \(T\) is injective. If \(A\) is vertex-injective and every bond dimension is positive, then for every edge \(e=(u,v)\), \(\operatorname{inj}(T_{A,u})\), \(\operatorname{inj}(T_{A,V\setminus \{ u,v\} })\), and \(\operatorname{inj}(T_{A,v})\). Here \(T_{A,u}\) and \(T_{A,v}\) are the two endpoint tensor maps, and \(T_{A,V\setminus \{ u,v\} }\) is the blocked middle tensor family. This is the precise assertion after the edge-blocking diagram in [ 220 , Section 3 ] . The positive-bond hypothesis is the source’s standing assumption that injective PEPS have nonzero virtual bond spaces. Without it, an empty complement configuration space can make the middle tensor vanish identically; Lemmas 24.2.21 and 24.2.22 record this obstruction.

If every edge-middle tensor is injective, then a vertex-injective PEPS tensor \(A\) gives an injective edge-blocked three-site chain at every edge \(e=(u,v)\):

Suppose every edge-middle region \(V\setminus \{ u,v\} \) is injective after blocking, and suppose finite-region injectivity has been identified with injectivity of the corresponding edge-middle tensor family. Then a vertex-injective PEPS tensor gives an injective edge-blocked three-site chain at every edge.

If \(M_e=V\setminus \{ u,v\} \) is nonempty for every edge \(e=(u,v)\), then Theorem 24.2.37 applies at every edge. Thus vertex injectivity of \(A\) gives injectivity of every edge-blocked three-site chain.

Assume singleton comparison, union closure for \(\kappa \), the comparison \(\kappa (M_e)\Longrightarrow \operatorname{inj}(T_{A,M_e})\), and vertex injectivity of \(A\). If \(2{\lt}|V|\), then \(V\setminus \{ u,v\} \ne \varnothing \) for every edge \(e=(u,v)\), and these hypotheses give \(\operatorname{inj}(T_{A,V\setminus \{ u,v\} })\) and edge-blocked three-site injectivity at every edge.

For a vertex-injective PEPS tensor \(A\), injectivity of the edge-blocked middle tensor gives the full injectivity statement for the edge-blocked three-site chain at \(e=(u,v)\):

where \(M_e = V\setminus \{ u,v\} \).

Suppose the middle region \(V\setminus \{ u,v\} \) is injective after blocking, and suppose this region-injectivity assertion has been identified with injectivity of the corresponding edge-middle tensor family. Then a vertex-injective PEPS tensor gives an injective edge-blocked three-site chain at the edge \(e=(u,v)\).

Let \(e=(u,v)\) and put \(M_e=V\setminus \{ u,v\} \). Under the hypotheses of Theorem 24.2.36, vertex injectivity of \(A\) gives injectivity of the edge-blocked three-site chain at \(e\).

Let \(e=(u,v)\). If \(P_uO_1P_u=\Phi _uT_{u,e}(M^{\mathsf T})\Psi _u\) and \(P_vO_2P_v=\Phi _vT_{v,e}(M)\Psi _v\), then \(\Psi _uO_1\Phi _u=T_{u,e}(M^{\mathsf T})\) and \(\Psi _vO_2\Phi _v=T_{v,e}(M)\). Here \(T_{w,e}(N)\) denotes the virtual operator on the legs incident to \(w\) which acts by \(N\) on the leg belonging to \(e\) and by the identity on the other incident legs.

Suppose the edge insertion correspondence is a \(\mathbb {C}\)-algebra isomorphism between the two full matrix algebras on the chosen bond. Then the two bond dimensions on that edge are equal. If \(h_e:D_A(e)=D_B(e)\) is this equality, and if \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) is the induced algebra isomorphism, then, for an invertible edge matrix \(Z_e\), the insertion correspondence is \(Y=Z_e\phi _{h_e}(X)Z_e^{-1}\). This is the finite-dimensional matrix-algebra step in [ 220 , Section 3 ] .

If the PEPS tensor is vertex-injective, then for every physical configuration \(\sigma \), each inserted edge matrix is realized by physical operators at the endpoint tensors: the left endpoint realizes \(M^{\mathsf T}\), the right endpoint realizes \(M\), and both realizations reproduce the same inserted-edge coefficient at \(\sigma \) in the edge-centered three-site decomposition.

Let \(R\) be a region with boundary edge \(f\), assume the bond-dimension equality \(h:D_A=D_B\) and that every bond dimension of \(A\) is positive, and suppose the absorbed region equality holds: for every matrix \(M\) on the bond space of \(f\) and all physical configurations \(\sigma \) on \(R\) and \(\tau \) on \(V\setminus R\),

Then the edge-level absorbed equality holds at \(f\): for every global physical configuration \(\sigma \) and every matrix \(N\) on the bond space of \(f\),

The edge-level identity no longer mentions the region.

If the inserted matrix on the distinguished edge is the identity, then the edge insertion coefficient is the ordinary edge-blocked coefficient.

On the diagonal boundary datum selected by the identity matrix, the inserted-edge summand is the ordinary edge-blocked summand. Thus the remaining identity-insertion step is purely a finite summation statement: the off-diagonal inserted boundary data vanish and the diagonal data reindex the ordinary edge-boundary sum.

If the inserted matrix on the distinguished edge is the identity, then the edge insertion coefficient is the original PEPS state coefficient.

If the two inserted bond indices are distinct, then the corresponding summand in the identity insertion is zero. This is the pointwise off-diagonal vanishing used before the inserted-boundary sum is restricted to its diagonal part.

Let the edge-blocked three-site chain of \(B\) at \(e\) be injective, and assume every virtual bond of \(B\) has positive dimension. If two matrices \(M,M'\in M_{D_B(e)}(\mathbb {C})\) give the same inserted coefficient at every physical configuration, \(C_B(e,\sigma ,M)=C_B(e,\sigma ,M')\) for every \(\sigma \), then \(M=M'\).

Let \(e=(u,v)\) and let \(\sigma \) be a physical configuration. If a physical operator on \(u\) realizes the one-edge virtual action of \(M^{\mathsf T}\) on the distinguished incident edge, then the inserted-edge coefficient for \(M\) at \(\sigma \) is obtained by applying that physical operator to the left endpoint tensor and leaving the open middle contraction and the right endpoint explicit.

Let \(e=(u,v)\) and let \(\sigma \) be a physical configuration. If a physical operator on \(v\) realizes the one-edge virtual matrix action on the distinguished incident edge, then the inserted-edge coefficient at \(\sigma \) is obtained by leaving the left endpoint and open middle contraction explicit and applying that physical operator to the right endpoint tensor.

If a global virtual configuration is compatible with boundary data, then its restriction to the left endpoint star is the local configuration reconstructed from those data.

Let \(e=(u,v)\) and suppose the family of local physical vectors at the left endpoint \(u\) is linearly independent. A physical operator at that endpoint then pulls back to the one-edge virtual action of \(M^{\mathsf T}\) if and only if its compressed physical action is the canonical realization of that one-edge virtual action.

Suppose every edge-middle region \(V\setminus \{ u,v\} \) is injective after blocking, and suppose finite-region injectivity has been identified with injectivity of the corresponding edge-middle tensor family. Then every edge-middle tensor family is injective.

Let \(e=(u,v)\) and \(M_e=V\setminus \{ u,v\} \). If an edge-middle kernel descent datum is given, then \(T_{A,M_e}\) is injective. Consequently, if \(A\) is vertex-injective, the edge-blocked three-site chain at \(e\) is injective.

Let \(e=(u,v)\) and put \(M_e=V\setminus \{ u,v\} \). Assume \(M_e\ne \varnothing \). Assume also singleton comparison, union closure for \(\kappa \), and the comparison \(\kappa (M_e)\Longrightarrow \operatorname{inj}(T_{A,M_e})\). If \(A\) is vertex-injective, then the edge-middle tensor family \(T_{A,M_e}\) is injective.

If \(2{\lt}|V|\), then for every edge \(e=(u,v)\) the middle region \(V\setminus \{ u,v\} \) is nonempty.

Restoring the fixed distinguished-edge index identifies the ordinary blocked middle tensor with the open middle tensor. This is the finite reindexing used before the identity matrix collapses the two inserted bond indices to the diagonal.

The middle contractions written with a global physical configuration are the corresponding middle-indexed contractions evaluated on the restricted middle physical configuration.

The ordinary middle tensor and the open middle tensor agree after restoring the fixed distinguished-edge index and reindexing the finite sum, with the physical index already restricted to the middle block.

Let \(e=(u,v)\). Suppose that \(P_uO_1P_u=\Phi _uT_{u,e}(M^{\mathsf T})\Psi _u\) and \(P_vO_2P_v=\Phi _vT_{v,e}(M)\Psi _v\). Suppose also that \(O_1\) maps the image of \(\Phi _u\) into itself and \(O_2\) maps the image of \(\Phi _v\) into itself. Then

If a global virtual configuration is compatible with boundary data, then its restriction to the right endpoint star is the local configuration reconstructed from those data.

Let \(e=(u,v)\) and suppose the family of local physical vectors at the right endpoint \(v\) is linearly independent. A physical operator at that endpoint then pulls back to the one-edge virtual action of \(M\) if and only if its compressed physical action is the canonical realization of that one-edge virtual action.

For an edge \(e=(u,v)\), the three regions \(\{ u\} \), \(V\setminus \{ u,v\} \), and \(\{ v\} \) cover the whole vertex set.

Suppose the tensor is translation invariant and the staircase satisfies the stated size bounds. Then for every \(X\in M_{D_B(e)}(\mathbb {C})\) and every global physical configuration \(\omega \),

Let \(A\) and \(B\) be vertex-injective PEPS tensors with the same PEPS state. Suppose the corresponding virtual bond dimensions are equal and every virtual bond of \(A\) has positive dimension. Then there is a family \(X_e\in \mathrm{GL}(D_A(e))\) such that, for every edge \(e\), there is a \(\mathbb {C}\)-algebra isomorphism \(\Phi _e:M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) satisfying \(C_A(e,\sigma ,M)=C_B(e,\sigma ,\Phi _e(M))\) for every physical configuration \(\sigma \) and matrix \(M\). Writing \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) for the algebra isomorphism induced by the dimension equality on \(e\), one has \(\Phi _e(M)=\phi _{h_e}(X_eMX_e^{-1})\).

Let \(A\) and \(B\) be normal PEPS on the open \(W\times H\) square lattice that satisfy the rectangular-injectivity hypotheses with union closure, have positive physical dimension \(d{\gt}0\), have equal and everywhere-positive virtual bond dimensions, and define the same PEPS state. Fix a translated horizontal interior frame with two rows and two columns of margin, that is \(2\le x_0\), \(2\le y_0\), \(x_0+8\le W\), and \(y_0+8\le H\), and let \(e\) be the distinguished horizontal edge of that frame. Then the bond dimensions agree on \(e\), \(D_A(e)=D_B(e)\), and there are an invertible edge matrix \(Z\in \mathrm{GL}(D_B(e))\) and a forward map \(\mathrm{fwd}\) given by the conjugation \(\mathrm{fwd}(M)=Z\phi _{h_e}(M)Z^{-1}\), where \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) is the reindexing induced by the bond-dimension equality on \(e\). No single-vertex injectivity is used; the only injectivity hypotheses are the blocked-region injectivities of the interior blocking data and the red and host injectivities of \(B\).

The coordinate-swap counterpart of Theorem 24.3.22. Under the same hypotheses on \(A\) and \(B\), for a translated vertical interior frame with two rows and two columns of margin, that is \(2\le x_0\), \(2\le y_0\), \(x_0+8\le W\), and \(y_0+8\le H\), with distinguished vertical edge \(e\), the bond dimensions agree on \(e\), \(D_A(e)=D_B(e)\), and there are an invertible edge matrix \(Z\in \mathrm{GL}(D_B(e))\) and a forward map \(\mathrm{fwd}\) given by the conjugation \(\mathrm{fwd}(M)=Z\phi _{h_e}(M)Z^{-1}\), with \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C})\to M_{D_B(e)}(\mathbb {C})\) the reindexing induced by the bond-dimension equality on \(e\).

Let \(K\) be the basis of the virtual space on the bond between two injective blocks, and let \(\{ A_{1,\mu }\} _{\mu \in K}\) and \(\{ A_{2,\mu }\} _{\mu \in K}\) be linearly independent families. Suppose that the two end physical operations \(O_1\) and \(O_3\) produce families \(C_1\) and \(C_2\) satisfying the open contraction identity

Then there is a unique matrix \(X\in M_{|K|}(\mathbb {C})\) such that

Thus the two end operations are obtained by inserting the same virtual operation \(X\) on their common bond. No injectivity is required of the modified families \(C_1\) and \(C_2\). This is the “compare the two ends” step in Lemma 5 of [ 220 ] .

Suppose the tensor is translation invariant, all bond dimensions are positive, and the two genuine highlighted end-window blocked tensors are injective. If arbitrary inserts \(C_{\mathrm L}\) and \(C_{\mathrm R}\) on those windows have equal extensions to the open end pair, then there is a unique matrix \(X\) on the reference edge such that

No injectivity is assumed for either modified insert, a single vertex, the end pair, or its complement. In particular, when the two inserts are obtained by applying the physical end operations \(O_1\) and \(O_3\), these relations say that \(O_1\) and \(O_3^{\mathsf T}\) realize the same unique matrix \(X\).

Let \(R\subseteq S\), and suppose that all bond dimensions are positive and that \(S\setminus R\) is blocked-tensor injective. If two inserts on \(R\) have the same corner extension to \(S\), \(\operatorname{Ext}_{R\subseteq S}(C_1)=\operatorname{Ext}_{R\subseteq S}(C_2)\), then \(C_1=C_2\).

Let \(R\subseteq S\) be finite regions. Suppose that all bond dimensions are positive and that the added block \(Q=S\setminus R\) is blocked-tensor injective. If a region insert \(C\) on \(R\) has zero corner extension, \(\operatorname{Ext}_{R\subseteq S}(C)=0\), then \(C=0\).

Assume that \(\kappa \) satisfies singleton comparison and union closure: \(\operatorname{inj}(T_{A,\{ v\} })\Longrightarrow \kappa (\{ v\} )\) for every vertex \(v\), and \(\kappa (R)\wedge \kappa (S)\Longrightarrow \kappa (R\cup S)\). If \(A\) is vertex-injective, then every nonempty finite region \(R\) satisfies \(\kappa (R)\).

Assume that the blocked tensors of \(B\) on \(R\) and \(R^c\) are injective and that all virtual bonds of \(B\) have positive dimension. Suppose that the equality \(D_A(f)=D_B(f)\) fixes the canonical identification \(\iota :M_{D_A(f)}(\mathbb {C})\simeq M_{D_B(f)}(\mathbb {C})\). For \(i=1,2\), suppose that the invertible matrix \(Z_i\in \mathrm{GL}(D_B(f))\) realizes the region-insertion identity

Then the two conjugations agree on every boundary matrix: \(Z_1\, \iota (M)\, Z_1^{-1}=Z_2\, \iota (M)\, Z_2^{-1}\).

Let \(L,K{\gt}0\), \(1\leq a\), \(a+2L\leq \mathrm{width}\), and \(b+2K-1\leq \mathrm{height}\). Suppose that \(OT_{A,W_{\mathrm L}}(j,\eta )=\sum _i M_{ij}T_{A,W_{\mathrm L}}(i,\eta )\) for every \(j\) and every exterior boundary configuration \(\eta \). Then \(C^O_{A,W_{\mathrm L}}=I_{A,W_{\mathrm L},e,M}\). Injectivity of \(A\) is not required for this identity.

Let \(\phi :G\simeq G'\) and \(\psi :G'\simeq G''\) be graph isomorphisms between graphs with linearly ordered vertex sets. For every edge \(e\) of \(G\), one has \(\psi (\phi (e))=(\psi \circ \phi )(e)\). For every tensor \(A\) on \(G\), one also has \((A^\phi )^\psi =A^{\psi \circ \phi }\).

If the blocked-middle local gauge formula holds at \(v\) with matrices \(X_e\), then the factorized local gauge datum holds at \(v\) with the same matrices.

Assume that \(A\)’s and \(B\)’s components at the in-region endpoint of \(f\) are linearly independent, and that \(D_B(e){\gt}0\) for every edge \(e\). Let \(W\) be the virtual pullback, at the in-region endpoint of \(f\), of the physical operator obtained by inserting \(X^T\) on \(f\). If \(W=\mathcal I_f(P)\) is in incident-matrix form on the boundary leg \(f\), then the transfer matrix read off from \(W\) reinserts to \(W\).

Under the displayed horizontal staircase hypotheses, the highlighted edge has a coefficient-identity witness whose region is the left end window and contains the first stored endpoint of the edge.

Assume the one-orientation \(L\times K\) window-injectivity hypotheses, union closure for injective regions, positive bond dimensions, and the bounds \(2\le L\), \(2\le K\), \(2L+1\le W\), and \(2K+1\le H\). Suppose two horizontally consecutive \(L\times K\) windows carry inserts \(C_1\) and \(C_2\) whose deformed states agree. Let \(U\) be their \((L+1)\times K\) union. Then their extensions to \(U\) agree:

Let \(B\) be translation invariant on a torus whose periods \(W,H\) satisfy \(1{\lt}W\) and \(1{\lt}H\). Assume the one-orientation \(L\times K\) window-injectivity hypotheses, union closure for injective regions, positive bond dimensions, and the bounds \(2\le L\), \(2\le K\), \(2L+1\le W\), and \(2K+1\le H\). Let \(j{\lt}L\), let \(U_j=W_j\cup W_{j+1}\) be the horizontal consecutive-window union, and let \(g\) be a boundary edge of both \(W_j\) and \(W_{j+1}\). If the matrices obtained from \(M_1\) and \(M_2\) by the orientation convention at \(g\) agree, then, with \(P_j=P_B(W_j)\),

Under the bounds \(1{\lt}W\), \(1{\lt}H\), \(0{\lt}L\), \(0{\lt}K\), \(1\le a\), \(a+2L\le W\), and \(b+2K-1\le H\), an edge is a boundary edge of the staircase end pair \(S\) if and only if it is a boundary edge of one of the two end windows and is not the reference edge:

Moreover each edge in \(\partial S\) lies on exactly one of the two window boundaries.

Let \(C_{\mathrm L}\) and \(C_{\mathrm R}\) be arbitrary inserts on the two end windows. In the boundary coordinates above, their extensions to the end pair are

Hence, for a translation-invariant tensor with positive bond dimensions, equality of the two extensions implies the single-bond identity

Write the boundary coordinates of the two end windows and their union as

where \(k\) is the coordinate on the reference edge. In the extension of a left-window insert to the end pair, the normalized coupling to the right window vanishes unless \(\eta _{\mathrm L}=\eta '_{\mathrm L}\). On that fibre it is

where \(A_{\mathrm R}\) is the genuine blocked tensor of the right window and \(\Lambda _{\mathrm L}\) is the product of the exterior interior-bond factor and the bond dimensions on the external legs of the left window. The analogous statement holds after interchanging left and right. If the tensor is translation invariant and every bond dimension is positive, then \(\Lambda _{\mathrm L}=\Lambda _{\mathrm R}\).

Assume positive bond dimensions and the bounds \(0{\lt}L\), \(0{\lt}K\), \(2L\le W\), and \(2K\le H\). If the deformed states of the two end-window inserts agree, then their extensions to the staircase patch \(P\) have equal deformed states:

Assume \(1{\lt}W\), \(1{\lt}H\), \(0{\lt}L\), \(0{\lt}K\), \(1\le a\), \(a+2L\le W\), and \(b+2K-1\le H\). Then an edge crosses from the left end window to the right end window if and only if it is the reference edge \(e\):

Thus \(e\) is a boundary edge of each end window, but both endpoints of \(e\) lie in the end pair \(S\), so \(e\notin \partial S\).

Let \(A\) and \(B\) be injective PEPS tensors on the same finite simple graph, and suppose that they represent the same state. Then

for every edge \(e\). This conclusion does not require the graph to be connected. It is the bond-dimension consequence of the virtual-operation algebra isomorphism in [ 220 , lines 279–282 and 571–582 ] .

Suppose all bond dimensions are positive. For every global physical configuration \(\omega \),

Let \(v\notin R\). A crossing edge of \(R\) whose two endpoints both differ from \(v\) is also a crossing edge of \(R\cup \{ v\} \).

If \(A\) is vertex-injective and every virtual bond space has positive dimension, then the complement two-block tensor at every vertex is injective.

If the blocked tensor family of \(V\setminus R\) is injective, then the complement-region two-block tensor is injective: \(\operatorname{inj}(T_{A,V\setminus R})\Longrightarrow \operatorname{inj}(B_{A,R^c})\).

If \(A\) is vertex-injective, then the single-vertex two-block tensor at every vertex is injective.

The relation \(\operatorname{inj}_{\partial }\) admits no three-partite injective MPS blocking around the edge \(u\)–\(v\).

Every contiguous \(2\times 3\) and \(3\times 2\) rectangle satisfies \(\operatorname{inj}_{\partial }\). Indeed, a rectangle of either shape that contains \(u\) necessarily contains its right neighbor \(v\).

The relation \(\operatorname{inj}_{\partial }\) is closed under unions.

Suppose that the two genuine end families are linearly independent and that the open contraction identity for \(O_1\) and \(O_3\) holds. Then there is a unique matrix \(X\) simultaneously realized by \(O_1\) and by \(O_3^{\mathsf T}\).

If \(O_1\) and \(P_1\) realize \(X\) and \(Y\), respectively, then \(O_1\circ P_1\) realizes \(XY\). If \(O_3^{\mathsf T}\) and \(P_3^{\mathsf T}\) realize \(X\) and \(Y\), then their product realizes \(XY\); in terms of the underlying right operations this product is \(P_3\circ O_3\). Uniqueness identifies the virtual matrix of either composite with \(XY\). This is precisely the composition step in the final sentence of Lemma 5 of [ 220 ] ; it does not by itself identify operations belonging to two different PEPS tensors.

Suppose that the edge-centred three-site reduction supplies invertible matrices \(X_e\) on the incident edges of \(v\) such that, for all local virtual configurations \(\eta \) and physical indices \(\sigma \),

Then the same family \(X_e\) is a local gauge at \(v\):

If the family of local physical vectors at \(v\) is linearly independent, then a matrix acting on one edge incident to \(v\) is realized by a physical operator on the physical space at \(v\).

If the coefficient function is the unit basis function at one local virtual configuration, and the matrix acts on the distinguished incident edge, the result is the sum over the new distinguished edge index with the residual local boundary held fixed.

Applying the local tensor map after a one-edge matrix action on a basis virtual configuration gives the corresponding linear combination of local tensor components.

The compressed physical action \(P_vOP_v\) is the canonical physical realization of a virtual operation \(T\) if and only if the virtual pullback of \(O\) is \(T\).

Suppose the family of local physical vectors at \(v\) is linearly independent. Two physical endpoint operations then have the same virtual pullback exactly when their projected actions agree on every vector in the image of the local tensor map. For every coefficient function \(c\),

Suppose that the compressed physical action \(P_vOP_v\) agrees with the canonical physical realization of a virtual operation \(T\). Then the virtual pullback of \(O\) is \(T\).

Suppose the family of local physical vectors at \(v\) is linearly independent. If a physical operator \(O\) realizes a virtual operation \(T\) on every vector in the image of \(\Phi _v\), then pulling back \(O\) recovers \(T\).

Pulling back the canonical physical realization of a virtual operation recovers the original virtual operation.

If \(O\) sends every vector in the image of \(\Phi _v\) back into that image, then its virtual pullback realizes exactly the action of \(O\) on \(\Phi _v(c)\).

For every coefficient function \(c\), applying the local tensor map after the virtual pullback gives the projection of \(O(\Phi _v(c))\) onto the image of \(\Phi _v\).

A family of one-edge blocking data is precisely the normal edge-blocking hypotheses. If \(H\) is formed from the family \(d_e\), then the one-edge datum that \(H\) associates with \(e\) is \(d_e\). Write the endpoints of \(e\) as \(u_e{\lt}v_e\) in the canonical vertex ordering. If the three regions are \(R_e\), \(B_e\), and \(C_e\), then

and the underlying PEPS tensor is injective on \(R_e\), \(B_e\), and \(C_e\).

Under the square-lattice normality hypotheses of [ 220 , Theorem 3 ] , the regions \(R\), \(S\), and \(T\), together with their translated variants, give an edge-centred blocking into red, blue, and complementary injective regions. These are the regions on which the three-site injective-chain insertion argument is applied. Here the translations are those of the periodic square lattice; the cited proof does not give separate boundary blockings for an open rectangle:

\(\quad \Rightarrow \quad \)

For the two regions associated to a site \(v\), a vertex \(w\) belongs to the region containing \(v\) if and only if either \(w=v\), or \(w\) belongs to the comparison region omitting \(v\).

Given rectangular injectivity hypotheses, finite union closure, oriented margin-and-cover hypotheses at every edge, and one-site separation, the finite square lattice satisfies the general normal blocking hypotheses. The edge-blocking hypotheses are exactly the margin-cover construction, the one-edge datum at an edge is the datum supplied there, and the one-site separation is exactly the supplied one-site separation. Hence these hypotheses give the three injective regions, the endpoint-disjoint-cover facts at each edge, and the one-site membership formula as consequences of the same normal blocking hypotheses.

Given rectangular injectivity hypotheses, finite union closure, and a translated edge window at every edge of the finite square lattice, the corresponding one-edge blockings determine the normal edge-blocking hypotheses. This is the conditional construction; constructing such windows for all edges of the \(7\times 7\) lattice remains separate; the current open rectangular translated-window criterion does not by itself supply such a family. The one-edge datum recovered from the resulting hypotheses is the datum supplied by the corresponding translated window, and the resulting hypotheses give the three injective regions at every edge, together with \(u_e\in R_e\), \(v_e\in B_e\), \(R_e\cap B_e=R_e\cap C_e=B_e\cap C_e=\varnothing \), and \(R_e\cup B_e\cup C_e=V\). Equivalently, it is enough to supply the oriented margin-and-cover data at every edge; the resulting hypotheses recover the corresponding datum, give the same three injectivity conclusions, and give the same endpoint, disjointness, and covering conclusions. The obstruction theorem records that this interior margin-cover criterion does not by itself supply data at every edge of the open \(7\times 7\) lattice.

Let the shared bond type and the two external boundary spaces be nonempty. After the edge gauges have been absorbed into the second tensor family \(\widetilde B\), block one vertex \(v\) against its complement. Assume that \(A_v,A_{v^c},\widetilde B_v,\widetilde B_{v^c}\) are injective as two-block tensors. The post-absorption insertion equality gives, for every shared bond \(b\), every matrix \(X\in M_{D_b}(\mathbb {C})\), and all external and physical indices,

Then \(\exists \lambda _v\in \mathbb {C}^\times \) such that \(A_v=\lambda _v\widetilde B_v\).

Under the equal-state, complement-injectivity, and positive-dimension hypotheses of Theorem 24.7.3.45, if \(OT_{A,R}(\mu )=PT_{A,R}(\mu )\) for every \(\mu \), then \(OT_{B,R}(\nu )=PT_{B,R}(\nu )\) for every \(\nu \).

For every virtual boundary configuration \(\mu \),

Thus the insert is obtained from the genuine region block by one physical operator before the region is contracted with its complement.

If \(A\) and \(B\) generate the same closed state, then

The two virtual boundary spaces need not be identified.

Let \(e=(u,v)\). Assume that the edge-blocked three-site chain at \(e\) is injective and that every virtual bond has positive dimension. Let \(O_1,O_2\) be physical operators at the endpoint tensors, and write \(\Phi _u,\Phi _v\) for the two local tensor maps. Suppose that, for every physical configuration \(\sigma \),

Then

The same implication is represented diagrammatically by the \(O_1,O_2\mapsto W\) step:

\( = \) \( \Longrightarrow \)

This is the converse part of the insertion-algebra lemma in [ 220 , Section 3 ] , where the recovered matrix is denoted \(W\).

Suppose the family of local physical vectors at \(v\) is linearly independent. If two virtual operations at \(v\) have the same canonical physical realization, then they agree.

Let \(P_v=\Phi _v\circ \Psi _v\) be the local projector, and let \(T_O=\Psi _v\circ O\circ \Phi _v\) be the virtual pullback of a physical operator \(O\). Then \(\Phi _v\circ T_O\circ \Psi _v=P_v\circ O\circ P_v\).

For every coefficient function \(c\) on the local virtual configurations at \(v\), one has \(O_T(\Phi _v(c))=\Phi _v(Tc)\).

Let a finite family of legs be given, each a nonzero finite index set, with invertible matrices \(M_i\) and \(N_i\) on each leg. If \(\prod _i(M_i)_{p_iq_i}=\prod _i(N_i)_{p_iq_i}\) for every configuration pair \((p,q)\), then there are nonzero scalars \(c_i\) with \(M_i=c_iN_i\) for each leg and \(\prod _i c_i=1\).

Let \(B\) have positive bond dimensions and injective cyclic \(L\times K\) windows, with union closure for injective regions. Assume \(L,K{\gt}0\), \(2L+1\leq w\) and \(2K+1\leq h\). Write \(W_j\) for the staircase windows and \(U_j=W_j\cup W_{j+1}\). If \(j{\lt}L\) and inserts \(C_j,C_{j+1}\) have the same deformed state, then

The equality of deformed states is an equality at every physical configuration on the torus.

Under the same hypotheses, suppose inserts \(C_j\) on the staircase windows have one common deformed state for \(0\leq j{\lt}L+K\). Then for each \(n{\lt}L+K\),

In particular, the extensions of the two end-window inserts agree, by taking \(n=L+K-1\). These comparisons implement the overlapping-window argument of  [ 220 ] for every positive pair \(L,K\).

Under the same hypotheses, let \(P\) be the staircase patch. For \(j+1{\lt}L+K\), equality of the deformed states of two consecutive inserts implies

Under the same tensor and size hypotheses, if \(L\leq j\) and \(j+1{\lt}L+K\), equality of the deformed states of \(C_j,C_{j+1}\) again implies (64).

Under the same hypotheses, \(\mathcal R(s;L,K+1)^c\in \mathcal I\) for every \(s\). In particular, this conclusion applies to blocked-tensor injectivity.

Let \(\mathcal I\) be a family of torus regions closed under unions, containing every cyclic \(L\times K\) window. Assume \(L,K{\gt}0\), \(2L+1\leq w\) and \(2K+1\leq h\). Then \(\mathcal R(s;L+1,K)^c\in \mathcal I\) for every starting vertex \(s\). In particular, this conclusion applies to blocked-tensor injectivity.

Let \(A\) and \(B\) be vertex-injective PEPS with the same state, assume every bond dimension is positive, and assume the bond-dimension equality \(h:D_A=D_B\). After the edge gauges have been absorbed into the second tensor family, there are gauges \(Z_e\) and \(\widetilde B=B^Z\) such that \(D_{\widetilde B}=D_A\) and, for every edge \(e\), write \(\phi _{h_e}:M_{D_A(e)}(\mathbb {C})\to M_{D_{\widetilde B}(e)}(\mathbb {C})\) for the matrix-algebra transport induced by the corresponding equality \(h_e:D_A(e)=D_{\widetilde B}(e)\). Then, for every matrix \(X\in M_{D_A(e)}(\mathbb {C})\) and every physical configuration \(\sigma \), \(C_A(e,X;\sigma ) =C_{\widetilde B}(e,\phi _{h_e}(X);\sigma )\). On the five-site graph used in the proof, this equality reads

\( = \)

as in [ 220 , Section 3, lines 1037–1065 ] . The positive-bond hypothesis is the source’s standing assumption that injective PEPS have nonzero virtual bond spaces; it is needed because the edge gauges come from blocking the PEPS around each edge into a three-site injective chain.

Let \(v\notin R\). At a fixed global virtual configuration \(\zeta \) and a physical configuration \(\sigma \) on \(R\cup \{ v\} \), the product of the local tensors over the vertices of \(R\cup \{ v\} \) factors as the local tensor at the inserted site \(v\) times the product over the vertices of \(R\):

For any edge \(e=(u,v)\) and any function \(f\) on the vertex set with values in a commutative monoid \(M\),

If \(R=S\), then \(C^O_{A,R}|_S=C^{O|_S}_{A,S}\).

There is a union-closed region relation on the open \(7\times 7\) rectangle that satisfies all contiguous \(2\times 3\) and \(3\times 2\) rectangle conditions, but admits no three-partite injective MPS blocking around the edge \(u\)–\(v\).

This counterexample for abstract region relations shows only that the rectangular and union hypotheses do not entail a boundary blocking on an open rectangle. It neither gives a PEPS tensor nor concerns the periodic translationally invariant statement of  [ 220 , Theorem 3 ] . The later general theorem in the source takes the every-edge three-partite injective MPS blocking as a hypothesis.

Assume that \(A\) and \(B\) define the same PEPS state, that \(B\) is vertex-injective, that \(D_B(e){\gt}0\) for every edge \(e\), and that \(D_A(e)=D_B(e)\) for every edge \(e\). Assume also that \(A\)’s and \(B\)’s components at the in-region endpoint of \(f\) are linearly independent. Suppose that for every matrix \(X\) on \(A\)’s boundary bond there is a matrix \(Y\) on \(B\)’s boundary bond such that \(C_A(R,f,X;\sigma ,\tau )=C_B(R,f,Y;\sigma ,\tau )\) for all \(\sigma ,\tau \). Then the virtual pullback of \(X\) is in incident-matrix form on \(f\).