Dependencies

Let \(A\) and \(B\) be MPS tensors of bond dimension \(D\), and fix a chain length \(n \ge 1\). Assume that \(A\) is injective and that the combined tensors of the two constant chains \((A, \ldots , A)\) and \((B, \ldots , B)\) generate the same MPV family. Then there exist an invertible matrix \(Z \in \mathrm{GL}_D(\mathbb {C})\) and a scalar \(\lambda \in \mathbb {C}^{\times }\) with \(\lambda ^n = 1\) such that

for every physical index \(i\).

Given \(r\) blocks with bond dimensions \((D_k)_{k=1}^r\), per-block tensors \(\{ A_k^i\} \), and scaling factors \((\mu _k)_{k=1}^r\), the associated block-diagonal tensor is the tensor of total bond dimension \(D = \sum _{k=1}^r D_k\) defined by

For an MPS tensor \(A\), a blocking length \(L\), and a chain length \(n\), the blocked chain is the constant \(n\)-site chain whose local tensor is the blocked tensor \(A^{[L]}\).

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

Let \(N\geq 1\) and let \(A_0,\ldots ,A_{N-1}\) be a periodic chain of \(D\times D\) matrices. For \(k\in \mathbb Z/N\mathbb Z\), write \((A^{(k)})_j=A_{j+k}\) for the cyclic rotation by \(k\) sites. Let \(P\) be the permutation matrix on the block coordinate satisfying \(P(e_a\otimes e_k)=e_a\otimes e_{k-1}\). For each physical index \(i\), set

Thus the \((k,k+1)\) block of \(B^i\) is \(N^{-1/N}A_k^i\), with block indices taken modulo \(N\), and every other block vanishes. The tensor \(B\) depends on the fixed length \(N\).

Let \(A\) be an injective MPS tensor. The decomposition map \(\Psi _A\colon M_{D}(\mathbb {C})\to \mathbb {C}^d\) is a fixed choice of right inverse of \(\Phi _A\).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (8) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

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

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

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

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 \(A_k, B_k\) be MPS tensors of bond dimension \(D_k\) with \(A_k\) injective and \(\mathcal{V}(A_k) = \mathcal{V}(B_k)\) for each block \(k \in \{ 0, \ldots , r{-}1\} \). The per-block linear extension \(T_k : M_{D_k}(\mathbb {C}) \to M_{D_k}(\mathbb {C})\) is the unique \(\mathbb {C}\)-linear map satisfying \(T_k(A_k^i) = B_k^i\) for every physical index \(i\).

For a fixed period \(m\), a site-periodic tensor family is a family of local tensors indexed by \(\{ 0,\ldots ,m{-}1\} \), with local physical dimension \(d\) and bond dimension \(D\).

For site-periodic tensors \(A,B\) at fixed period \(m\):

• equality of the induced site-periodic states;

• cyclic gauge equivalence.

Let \(A=(A_i)_i\) be a tensor of bond dimension \(D\) and let \(L\ge 0\). For \(\alpha ,\beta \in \{ 0,\ldots ,D{-}1\} \) the left cut vector and the right cut vector of length \(L\) are the vectors

For a word split into a left part of length \(k\) and a right part of length \(m\), the coefficient of the ring state factors across the cut as

and extending the left word by one site multiplies the left cut vectors by the matrix of that site, \(\Phi _{\alpha ,\beta }(i_1\cdots i_k\, i) =\sum _\gamma \Phi _{\alpha ,\gamma }(i_1\cdots i_k)\, (A_i)_{\gamma \beta }\). These are the vectors introduced in the proof of the proposition on condition C1 in [ 228 ] .

Let a window of matrix families on the doubled bond space be given, \(A^{[j]}_i\in M_{D^2}(\mathbb {C})\) for \(i\in \{ 0,\ldots ,d{-}1\} \) and \(0\le j\le m-1\). A solution of the quadratic reconstruction system over this window consists of matrices \(Y_0,\ldots ,Y_{m-1}\) and \(Z_0,\ldots ,Z_{m-1}\) in \(M_{D^2}(\mathbb {C})\) together with matrices \(B_i\in M_{D}(\mathbb {C})\) satisfying

In the system (S) of [ 228 ] the window consists of the \(m=D^4\) matrices of the unique open-boundary canonical form at the sites \(L_0+1,\ldots ,L_0+D^4\); the unknown \(Y_j\) above is the source unknown \(Y_{L_0+1+j}\), while \(Z_j\) is the source unknown \(Z_{L_0+2+j}\) attached to the right of that site.

A tensor \(A\) of bond dimension \(D\) is a translation-invariant canonical representation when its bond indices split into finitely many blocks of positive size, and there are weights \(1\ge \lambda _j{\gt}0\) and tensors \(A^j\) on the blocks with

such that every block satisfies the three conditions of Theorem 7.5.6: \(\sum _iA_i^j(A_i^j)^\dagger =\mathbb {1}\), there is a diagonal positive definite \(\Lambda ^j\) with \(\sum _i(A_i^j)^\dagger \Lambda ^jA_i^j=\Lambda ^j\), and the identity is the only fixed point of \(X\mapsto \sum _iA_i^jX(A_i^j)^\dagger \) up to scalars. This is the block form of [ 228 , Theorem 4 ] , which lists the blocks consecutively.

Let \(A\) be an injective tensor. For \(X\in M_{D}(\mathbb {C})\), define the \(d\times d\) matrix \(O_A(X)\in M_{d}(\mathbb {C})\) by

so that row \(i\) of \(O_A(X)\) contains the decomposition coefficients of \(A^iX\) in the spanning set \(\{ A^j\} \). Diagrammatically, the inserted virtual matrix is re-expressed as a physical operation on the same local tensor:

\( = \)

The map \(X\mapsto O_A(X)\) is \(\mathbb {C}\)-linear.

The product algebra automorphism is the \(\mathbb {C}\)-algebra equivalence

defined by \(T(M)_k = T_k(M_k)\), where each \(T_k\) is the per-block linear extension. It is an algebra automorphism since each \(T_k\) is a multiplicative, unital, bijective linear map. Diagrammatically, this is the componentwise map on the product algebra obtained by applying the corresponding extension to each summand.

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 \(N\).

For tensors \(A\) and \(B\) of possibly different bond dimensions \(D_1\) and \(D_2\), we write \(\mathcal{V}(A) = \mathcal{V}(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every word length \(N\), including the empty word. This algebraic relation is used for exact direct-sum identities.

Two tensors \(A\) and \(B\) of the same bond dimension satisfy finite-length MPV agreement from \(N_0\) onward if they generate the same MPV family for all system sizes \(N \ge N_0\), meaning that \(V^{(N)}(A)_\sigma = V^{(N)}(B)_\sigma \) for every \(N \ge N_0\) and \(\sigma \in [d]^N\).

Fix a chain length \(N\) and dimensions \(D_0,\ldots ,D_N\leq D\) with \(D_0=D_N=1\). A varying-bond open chain consists of matrices \(A_k^i\in \mathbb C^{D_k\times D_{k+1}}\) for \(k=0,\ldots ,N-1\) and \(i=0,\ldots ,d-1\). Intermediate dimensions may vanish. The number \(D\) is a common upper bound and need not be the least such bound.

This is the open-boundary representation of [ 228 , Section II.B ] . The source takes \(D=\max _k D_k\); allowing an arbitrary common upper bound does not change the rectangular chain.

For a physical configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\), define

Since \(D_0=D_N=1\), this is the scalar matrix product in [ 228 , Section II.B ] . If an intermediate dimension vanishes, the virtual-path set is empty. For \(N=0\), there is one empty path and its empty product is \(1\).

For \(N\geq 1\), a varying-bond open chain represents a translation-invariant state at length \(N\) if, for every \(s\in \mathbb Z/N\mathbb Z\) and every physical configuration \(\sigma \),

This condition compares coefficients at one fixed length. It does not rotate the bond dimensions or identify open cuts. At \(N=0\), the condition is vacuous.

Let \(N\geq 1\). For \(s\in \mathbb Z/N\mathbb Z\), the cyclic translation of a physical configuration is

Only the physical configuration is translated. The dimensions \(D_0,\ldots ,D_N\), the site matrices, and the distinguished open cut remain fixed. At \(N=0\), there is no translation parameter.

For each site \(k\), embed \(A_k^i\) into the upper-left \(D_k\times D_{k+1}\) corner of a \(D\times D\) matrix \(\widetilde A_k^i\) and set every other entry to zero. The matrices \(\widetilde A_k^i\) form a square site-dependent closed chain.

This square padding embeds the rectangular open chain into square matrices, whereas [ 228 , Theorem 3 ] places the rectangular matrices into the off-diagonal blocks of an \(ND\times ND\) cyclic matrix.

If \(A_k\) and \(B_k\) generate the same MPV family for each block \(k\), then the block-diagonal tensors \(\bigoplus _k \mu _k A_k\) and \(\bigoplus _k \mu _k B_k\) also generate the same MPV family.

If \(A\) is \(L\)-block injective, then the constant blocked chain built from \(A^{[L]}\) is injective at every site.

Let \(A_0,\ldots ,A_{m-1}\) be injective MPS tensors with compatible bond dimensions. Then the blocked tensor

is again injective.

For a canonical-form record \(C\), the assembled tensor \(T_C\) is definitionally equal to the weighted block-diagonal tensor \(\bigoplus _k \mu _k^C\, A_k^C\), where \(\mu _k^C\) are the block weights and \(A_k^C\) are the block tensors stored in \(C\).

If a matrix \(Z \in M_{D}(\mathbb {C})\) commutes with every element of \(M_{D}(\mathbb {C})\), then there exists \(\lambda \in \mathbb {C}\) such that \(Z = \lambda \mathbb {1}_D\).

On fixed-length periodic chains, cyclic gauge equivalence is reflexive, symmetric, and transitive.

Let \(\mathbf{A} = (A_0, \ldots , A_{n-1})\) be an \(n\)-site chain and let \(\zeta \in \mathbb {C}\). If \(\zeta \mathbf{A}\) denotes the chain obtained by replacing each site tensor \(A_k\) with \(\zeta A_k\), and if \(\widetilde{\mathbf{A}}\) denotes the combined tensor of \(\mathbf{A}\), then \(\widetilde{\zeta \mathbf{A}}=\zeta \, \widetilde{\mathbf{A}}\).

For a configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\), let \(w_\sigma =[\sigma _0,\ldots ,\sigma _{N-1}]\). Then \(A^{w_\sigma } =A^{\sigma _0}A^{\sigma _1}\cdots A^{\sigma _{N-1}}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for any word \(w\), \((\zeta A)^w = \zeta ^{|w|} A^w\), where \(|w|\) is the length of \(w\).

If \(A\) is injective, then for every \(X\in M_{D}(\mathbb {C})\) there exist coefficients \(c\in \mathbb {C}^d\) such that \(X=\sum _{i=0}^{d-1}c_iA^i\).

If \(A\) is injective, then there exists a \(\mathbb {C}\)-linear map \(\Psi \colon M_{D}(\mathbb {C})\to \mathbb {C}^d\) satisfying \(\Phi _A\circ \Psi =\operatorname{id}_{M_{D}(\mathbb {C})}\).

Assume that every block dimension is nonzero. Under the hypotheses of Lemma 23.9.1.7, the product-algebra automorphism determined by the per-block linear extensions decomposes into a block permutation together with inner conjugation on each block: there exist a permutation \(\sigma \) of the blocks, compatible block-dimension equalities, and invertible matrices \(X_k\) such that, after reindexing the target block \(\sigma (k)\) back to block \(k\), the action on the \(k\)th matrix block is \(M \mapsto X_k M X_k^{-1}\).

Under the hypotheses of Lemma 23.9.1.7, there exist invertible matrices \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\) such that \(B_k^i = X_k\, A_k^i\, X_k^{-1}\) for every block \(k\) and every physical index \(i\).

Let \(\{ A_k\} _{k=0}^{r-1}\) and \(\{ B_k\} _{k=0}^{r-1}\) be two families of MPS tensors with block dimensions \(D_k\), each \(A_k\) injective, and \(\mathcal{V}(A_k) = \mathcal{V}(B_k)\) for all \(k\). Then:

1. Per-block gauge equivalence: \(B_k^i = X_k\, A_k^i\, X_k^{-1}\) for some \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\), for every \(k\) and \(i\).

2. Global gauge equivalence: the block-diagonal tensors \(\bigoplus _k \mu _k A_k\) and \(\bigoplus _k \mu _k B_k\) are gauge equivalent.

Explicitly, for \(X=\bigoplus _k X_k\),

Let \(A_k, B_k\) be block families, and let \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\) and scalars \(\zeta _k \in \mathbb {C}\) satisfy \(B_k^i = \zeta _k \cdot X_k A_k^i X_k^{-1}\) for every block \(k\) and physical index \(i\). If the weights satisfy \(\mu _A(k) = \mu _B(k) \zeta _k\), then the weighted block-diagonal tensors \(\bigoplus _k \mu _A(k) A_k\) and \(\bigoplus _k \mu _B(k) B_k\) are gauge equivalent.

A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.

Let \(A\) and \(B\) be left-canonical irreducible tensors of common positive bond dimension satisfying \(B^i=\zeta XA^iX^{-1}\) for an invertible \(X\) and a nonzero scalar \(\zeta \). There is a unitary matrix \(U\), with the same scalar \(\zeta \), such that \(|\zeta |=1\) and \(B^i=\zeta UA^iU^\dagger \) for every \(i\).

If two block-dimension families agree pointwise, the corresponding weighted block-diagonal tensors generate the same \(\mathrm{SameMPV}_2\) family after transporting the blocks across the dimension equalities.

The \(\mathrm{SameMPV}_2\) family of a weighted block-diagonal tensor is invariant under permuting the block index set and transporting the weights correspondingly.

Let \(e : \{ 0,\ldots ,r'-1\} \simeq \{ 0,\ldots ,r-1\} \) be a bijection of block indices. For weights \(\mu _k\), block tensors \(A_k\), word length \(N\), and configuration \(\sigma \), the coefficient of the block-diagonal MPV is unchanged by reindexing the blocks along \(e\):

If all block tensors \(A_k\) are injective and the per-block same-MPV condition holds (\(A_k\) and \(B_k\) generate the same MPV family for each \(k\)), then each pair \((A_k, B_k)\) is gauge equivalent.

If each block \(A_k\) is injective and each pair \((A_k, B_k)\) generates the same MPV family, then the block-diagonal tensors \(\bigoplus _k \mu _k A_k\) and \(\bigoplus _k \mu _k B_k\) are gauge equivalent.

Let \(\{ A_k\} \) be a family of injective block tensors. Then \(\mathcal{V}(A_k) = \mathcal{V}(B_k)\) for all \(k\) if and only if \(A_k\) and \(B_k\) are gauge equivalent for all \(k\).

Each \(T_k\) is bijective.

For each block \(k\), the per-block linear extension \(T_k\) is multiplicative: \(T_k(MN) = T_k(M)T_k(N)\) for all \(M, N \in M_{D_k}(\mathbb {C})\).

Let \(T,S\) be linear maps defined on the same vector spaces, and let \(Y_0,\ldots ,Y_n\) be vectors such that \(T(Y_k)=S(Y_{k+1})\) for every \(0\le k\le n-1\), the vectors \(Y_0,\ldots ,Y_{n-1}\) are linearly independent, and

Let \(x\neq 0\) be a solution of the polynomial equation

and define \(\mu _j=\sum _{k=0}^{j}\lambda _k x^{j+1-k}\). Then the combination \(Y=\sum _{k=0}^{n-1}\mu _k Y_k\) is nonzero and satisfies \(T(Y)=\tfrac 1x S(Y)\). This is the first auxiliary result before the proof of [ 228 , Theorem 7 ] , written with indices starting at zero.

Let \(B\) and \(C\) be tensors of bond dimension \(D\) whose coefficients agree on every word of length \(k+m\), and suppose that the left cut vectors of \(B\) of length \(k\) and the right cut vectors of \(B\) of length \(m\) are linearly independent. Then there is an invertible matrix \(W\) on the doubled bond space with

where the index pairs are written column first. If \(W\) and \(W'\) realize (24) at the cuts after \(k\) and after \(k+1\) sites, and the left cut vectors of \(B\) of length \(k\) are linearly independent, then

This is the square case of the freedom theorem for open-boundary representations [ 228 , Theorem 2 ] invoked in the proof of [ 228 , Theorem 7 ] , obtained here by a direct comparison of the two bipartite decompositions (23).

If \(A\) is \(L\)-block injective, then the \(D^2\) left cut vectors of length \(L\) are linearly independent, and so are the \(D^2\) right cut vectors of length \(L\). If \(\sum _iA_iA_i^\dagger =\mathbb {1}\) as well, then \(A\) is \(L'\)-block injective for every \(L'\ge L\), so both statements hold at every length at least \(L\).

Let \(B\) and \(C\) satisfy \(\sum _iB_iB_i^\dagger =\mathbb {1}\) and \(\sum _iC_iC_i^\dagger =\mathbb {1}\), let \(B\) be \(L_0\)-block injective, and let \(R\neq 0\) and \(x\neq 0\) satisfy (26). Then \(R\) is invertible, \(|x|=1\), and there is a unitary \(U\) with

Consequently the length-\(N\) coefficients of \(B\) and \(C\) satisfy \(V^{(N)}(B)=x^NV^{(N)}(C)\) for every \(N\).

Let \(B,C\in M_{n}(\mathbb {C})\) and let \(W\) act on the doubled space. Then

holds if and only if every multiplicity slice \(X_{ab}\) of \(W\), with entries \((X_{ab})_{pq}=W_{(p,a),(q,b)}\), satisfies \(X_{ab}C=BX_{ab}\). Consequently a nonzero simultaneous solution \(W\) of the amplified equations for a family of pairs \((B_i,C_i)\) yields a nonzero matrix \(X\) with \(XC_i=B_iX\) for every \(i\). This realizes the second auxiliary result before the proof of [ 228 , Theorem 7 ] , which phrases the solution space of (17) as \(\mathcal{S}\otimes M_n\) for \(\mathcal{S}\) the solution space of \(XC=BX\).

In any solution of the quadratic reconstruction system, every matrix \(Y_{j+1}\) and every matrix \(Z_j\) appearing in (12) is invertible.

Let two solutions of the quadratic reconstruction system over the same window be given, with row unknowns \(Y_j\), \(Y'_j\), column unknowns \(Z_j\), \(Z'_j\), and tensors \(B_i\), \(C_i\). For every window site \(j\) carrying an inversion equation, the matrix

is invertible, and whenever the neighbouring site also carries an inversion equation,

Thus a window of \(m\) sites supplies exactly \(m-1\) internal intertwiners and \(m-2\) relations between consecutive internal intertwiners. This yields the internal part of the chain in [ 228 , Theorem 8 ] ; the full linear dependence required by Lemma 23.2.3 incorporates the boundary cuts.

The weighted direct sum \(\bigoplus _k\mu _kA_k\) of positive-dimensional blocks produced by Theorem 7.5.6, with positive weights \(\mu _k\le 1\), is a translation-invariant canonical representation with the blocks in consecutive order.

Let \(A\) be a translation-invariant canonical representation with the block form (28). Every product \(A_{i_1}\cdots A_{i_L}\) is block diagonal, with \(\lambda _j^LA^j_{i_1}\cdots A^j_{i_L}\) on block \(j\), and

If \(A\) has two distinct blocks, then \(A\) is not \(L\)-block injective for any \(L\). Consequently, if \(D\ge 1\) and \(A\) is \(L\)-block injective for some \(L\), then \(A\) has a single block, and \(\sum _iA_iA_i^\dagger =\lambda ^2\, \mathbb {1}\) with \(0{\lt}\lambda \le 1\). This is the first assertion of the proposition on condition C1 in [ 228 ] .

Let \(B\) and \(C\) be tensors of bond dimension \(D\) with \(\sum _iB_iB_i^\dagger =\lambda _B^2\, \mathbb {1}\) and \(\sum _iC_iC_i^\dagger =\lambda _C^2\, \mathbb {1}\) for some \(\lambda _B,\lambda _C{\gt}0\), let \(B\) be \(L_0\)-block injective, let \(N{\gt}2L_0+D^4\), and let \(B\) and \(C\) represent the same nonzero state on the ring of \(N\) sites. Then \(\lambda _B=\lambda _C\), and there are a unitary \(U\) and a scalar \(\xi \in \mathbb {C}\) with

For every \(X\in M_{D}(\mathbb {C})\) and every physical index \(i\),

Let \(M = (M_0, \ldots , M_{r-1}) \in \prod _{k} M_{D_k}(\mathbb {C})\). If \(\sum _{k} \operatorname{tr}(M_k N_k) = 0\) for all \(N = (N_0, \ldots , N_{r-1})\), then \(M = 0\).

Let \(\{ A_k\} \) be a family of injective block tensors. The product Gram map \(M \mapsto (k, i \mapsto \operatorname{tr}(M_k \cdot A_k^i))\) is injective.

If, for every block \(k\) and physical index \(i\), \(B_k^i=X_kA_k^iX_k^{-1}\), then the weighted direct sums satisfy

where \(X\) is the flattened block-diagonal gauge from Definition 9.3.1. Hence the two assembled tensors are gauge equivalent.

Let \(A_1,A_2\) and \(B_1,B_2\) be injective MPS tensors with the same positive bond dimension \(D\). Suppose that, for all \(X\in M_{D}(\mathbb {C})\) on the internal bond, all \(Y\in M_{D}(\mathbb {C})\) on the external bond, and all physical indices \(i,j\),

Diagrammatically, inserting \(X\) on the bond between the two sites, as in (31), gives

\( = \)

while the insertion \(Y\) on the outer trace bond, as in (32), is

\( = \)

so the two hypotheses distinguish the internal and external virtual loops of the two-site periodic chain.

Then there exists a nonzero scalar \(\lambda \in \mathbb {C}^{\times }\) such that, for all physical indices \(i,j\),

Let \(A=(A_0,A_1,A_2)\) and \(B=(B_0,B_1,B_2)\) be two 3-site injective periodic chains with common bond dimension \(D\ge 1\) and \(\mathcal{V}(A)=\mathcal{V}(B)\) (same MPV family). Then at the middle bond there exists an invertible matrix \(Z\in \mathrm{GL}_D(\mathbb {C})\) such that, for all \(X\in M_{D}(\mathbb {C})\) and all physical indices \(i_0,i_1,i_2\), the virtual-insertion trace coefficients agree after conjugation:

In [ 220 , Lemma 1 ] , the statement is given for arbitrary chain length \(n\ge 3\) and any bond position, with uniqueness of \(Z\) up to scalar; the three-site formulation here isolates the local algebraic content.

Each block ideal \(I_k\) is an atom in the lattice of two-sided ideals of \(\prod _{j=0}^{r-1} R_j\).

Let \(N\geq 1\) and let \(B_0,\ldots ,B_{N-1}\) be a square site-dependent chain of bond dimension \(D\). For every physical configuration \(\sigma \),

For every basis configuration \(\sigma \in \{ 0,\ldots ,d-1\} ^N\), the length-\(N\) coefficient of the tensor in Definition 23.1.14 is

Reindexing the finite sum and cyclically permuting factors under the trace identifies this with the physical-configuration rotation in [ 228 , Theorem 3 ] . This identity concerns only the fixed length \(N\).

Let \(C\) be a canonical-form record with block weights \(\mu _k^C\) and block tensors \(A_k^C\). Let \(\{ B_k\} \) be a family of MPS tensors on the same block dimensions, and assume that each \(A_k^C\) is injective and generates the same MPV family as \(B_k\). Then \(T_C\) and \(\bigoplus _k \mu _k^C\, B_k\) are gauge equivalent.

Let \(A\) be an injective MPS tensor and \(B\) a tensor of the same bond dimension. If \(A\) and \(B\) generate the same matrix-product vectors, then there is \(X \in \mathrm{GL}_D(\mathbb {C})\) with \(B^i = X A^i X^{-1}\) for all \(i\). In tensor-network notation the gauge conjugates the local tensor on its virtual legs, with the physical index \(i\) on the upper leg:

Let \(A\) be an injective tensor of bond dimension \(D \ge 1\), and let \(B\) be a tensor of the same bond dimension. If \(A\) and \(B\) agree on all MPV coefficients for system sizes \(N \ge N_0\) (for any threshold \(N_0\)), then \(A\) and \(B\) are gauge equivalent.

Let \(A\) and \(B\) be MPS tensors of positive bond dimension \(D\), let \(L\) be a blocking length, and let \(n{\gt}0\) be a chain length. Assume that \(A\) is \(L\)-block injective and that the combined tensors of the two constant blocked chains built from \(A^{[L]}\) and \(B^{[L]}\) generate the same MPV family. Then the two blocked chains are gauge equivalent.

Let \(\mathbf{A} = (A_0, \ldots , A_{n-1})\) and \(\mathbf{B} = (B_0, \ldots , B_{n-1})\) be two chains with \(n{\gt}0\) sites and common positive bond dimension \(D\), and assume that \(\mathbf{A}\) is injective. Suppose that, after combining the site index and physical index into a single MPS tensor, the resulting two tensors generate the same MPV family. Then \(\mathbf{A}\) and \(\mathbf{B}\) are cyclically gauge equivalent.

Let \(\mathbf{A} = (A_0, \ldots , A_{n-1})\) and \(\mathbf{B} = (B_0, \ldots , B_{n-1})\) be two chains with \(n{\gt}0\) sites and common positive bond dimension \(D\), and assume that \(\mathbf{A}\) is injective. Suppose that the combined tensors \(\widetilde{\mathbf{A}}\) and \(\widetilde{\mathbf{B}}\) are gauge-phase equivalent. Then there exist invertible matrices \(Z_0, \ldots , Z_{n-1} \in \mathrm{GL}_D(\mathbb {C})\) and a nonzero scalar \(\zeta \in \mathbb {C}^{\times }\) such that

for all sites \(k\) and physical indices \(i\), where \(k+1\) is interpreted cyclically modulo \(n\).

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

The MPV of a block-diagonal tensor decomposes as

Equivalently, for each configuration \(\sigma \) one has

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 ] .

Let \(B\) be a tensor of bond dimension \(D\ge 1\) with \(\sum _iB_iB_i^\dagger =\mathbb {1}\) that is \(L_0\)-block injective, let \(N\ge 2L_0+D^4\), and let \(C\) be a tensor of bond dimension \(D\) with the same length-\(N\) coefficients as \(B\). Then there are a nonzero matrix \(R\in M_{D}(\mathbb {C})\) and a scalar \(x\neq 0\) such that

This is the intertwiner obtained in the proof of [ 228 , Theorem 7 ] before the unitarity argument, with the intertwiner count corrected to the \(D^4+1\) cuts of the window [ 132 ] .

Consider a translation-invariant state on an \(N\)-site ring with unique open-boundary canonical form such that the rank of each reduced density operator of successive spins is bounded by \(D^2\), and form the quadratic system of Definition 23.2.1 over the window of \(D^4\) open-boundary canonical matrices starting after the injectivity onset. Write \(A=(A_i)_i\) for its canonical translation-invariant tensor.

1. If the state admits a translation-invariant matrix product representation satisfying injectivity (condition C1), then the system has a solution.

2. Let \(B=(B_i)_i\) be the tensor of a solution and suppose in addition that its length-\(N\) trace-MPS representation is the given state. Then there are a unitary \(U\) and a scalar \(\xi \in \mathbb {C}\) such that \(|\xi |=1\), \(\xi ^N=1\), and

   \begin{align} A_i & =\xi UB_iU^{\dagger } \qquad \text{for every }i. \label{eq:pgvwc07_quadratic_reconstruction_phase_conjugacy} \end{align}

   Consequently the explicitly rephased tensor \(\widehat B_i:=\xi B_i\) still represents the same length-\(N\) state and is exactly unitarily conjugate to \(A\).

The unrestricted representation claim and exact conjugacy in [ 228 , Theorem 8 ] carry a residual finite-ring phase defect [ 60 ] , which is resolved by the explicit phase \(\xi \) in (20).

This is the cyclic-block construction in [ 228 , Theorem 3 ] , conditional on its open-chain input. Let \(N\geq 1\). Suppose a translation-invariant state on the \(N\)-site ring has a varying-bond open-chain representation whose virtual dimensions are at most \(D\). Then it has a site-independent trace-MPS representation of bond dimension \(ND\) with the same length-\(N\) coefficients. The constructed tensor depends on \(N\), and the equality is asserted only at this ring length.

Let \(A\) be a translation-invariant MPS representation of bond dimension \(D\). There is a positive scalar \(s\) and a finite family of blocks \(A_k\), possibly empty, such that the globally rescaled tensor \(s^{-1}A\) has the same MPV coefficients on every positive-length ring as \(\bigoplus _k \mu _k A_k^i\). The weights are positive and satisfy \(1\ge |\mu _k|\); if the family is nonempty, then some weight has norm \(1\). For each block,

Here \(\Lambda _k\) is diagonal, positive, and full-rank, and each block has positive bond dimension. Moreover, the only fixed points of \(\mathcal{E}_{A_k}(X)=\sum _i A_k^iX(A_k^i)^\dagger \) are the scalar multiples of \(\mathbb {1}\): for every \(X\), the identity \(\mathcal{E}_{A_k}(X)=X\) implies that there exists \(c\in \mathbb {C}\) such that \(X=c\mathbb {1}\). The total bond dimension of the decomposition is at most \(D\).

This is the positive-length form of [ 228 , Theorem 4 ] , with the global spectral-radius normalization step made explicit.

Let \(B=(B_i)_i\) and \(C=(C_i)_i\) be translation-invariant canonical representations of bond dimension \(D\) in the sense of Definition 23.3.6. Assume that \(B\) is \(L_0\)-block injective, that \(N{\gt}2L_0+D^4\), and that \(B\) and \(C\) represent the same nonzero state on the ring of \(N\) sites. Then there are a unitary \(U\) and a scalar \(\xi \in \mathbb {C}\) satisfying (30). The rephased representation \(\widehat C_i:=\xi C_i\) represents the same length-\(N\) state and satisfies \(B_i=U\widehat C_iU^\dagger \) exactly; equivalently, \(C\) and \(B\) are gauge-phase equivalent with the unitary gauge \(U\) and the unit-modulus phase \(\xi \).

Uniqueness of the open-boundary canonical representation is not needed; Theorem 23.3.4 obtains the intertwiners directly from injectivity, yielding the conclusion recorded in [ 102 ] .

Let \(\lvert \psi \rangle =\sum \operatorname {tr}(B_{i_1}\cdots B_{i_N}) \lvert i_1\cdots i_N\rangle \) be a translation-invariant canonical \(D\)-MPS such that condition C1 holds at some length \(L_0\), the open-boundary canonical representation of \(\lvert \psi \rangle \) is unique, and \(N{\gt}2L_0+D^4\). Then, if \(\lvert \psi \rangle \) admits another translation-invariant canonical \(D\)-MPS representation with matrices \(C_i\), there is a unitary \(U\) with \(B_i=UC_iU^\dagger \) for every \(i\). This is [ 228 , Theorem 7 ] as printed. Its conclusion is false: for \(d=D=1\), \(B_0=1\) and \(C_0=\eta \) with \(\eta \neq 1\) an \(N\)th root of unity are two canonical representations of the same nonzero state on \(N{\gt}3\) sites, and unitary conjugation acts trivially in bond dimension one [ 60 ] . The corrected statement is Theorem 23.3.11.

Let \(A\) be an injective MPS tensor. For all \(X,Y\in M_{D}(\mathbb {C})\), \(O_A(XY)=O_A(X)O_A(Y)\).

Assume \(D_k \ge 1\) for all \(k\). Every \(\mathbb {C}\)-algebra automorphism \(T\) of \(\prod _{k=0}^{r-1} M_{D_k}(\mathbb {C})\) decomposes as per-block inner automorphisms followed by a block permutation \(\pi \): there exist \(\pi \in S_r\) with \(D_{\pi (k)} = D_k\) and invertible matrices \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\) such that, for every tuple \(M = (M_0,\ldots ,M_{r-1})\) and every block index \(k\), \(T(M)_{\pi (k)} = X_k M_k X_k^{-1}\), where \(T(M)_j\) denotes the \(j\)-th block component of \(T(M)\).

Let \(I\) be finite, let \(r_i\leq s_i\) be nonnegative integers for \(i\in I\), and let \(f\) take values in an additive commutative monoid. Then the sum of \(f\) over \(\prod _{i\in I}\{ 0,\ldots ,r_i-1\} \) equals its sum over \(\prod _{i\in I}\{ 0,\ldots ,s_i-1\} \) after extending \(f\) by zero outside the smaller Cartesian product.

The product algebra automorphism \(T\) decomposes as a block permutation composed with per-block inner automorphisms: there exist a permutation \(\sigma \in S_r\) (with \(D_{\sigma (k)} = D_k\) for all \(k\)) and invertible matrices \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\) such that

for all \(M \in M_{D_k}(\mathbb {C})\) (viewed as supported in block \(k\)) and all \(k\). Equivalently, the product-algebra automorphism first permutes the simple matrix blocks and then acts inside each block by an inner conjugation.

If the per-block linear extension \(T_k\) is positive, then there is a unitary matrix \(U_k\) such that \(T_k(M)=U_kMU_k^\dagger \) for every \(M\in M_{D_k}(\mathbb {C})\).

Let \(\{ R_i\} _{i\in I}\) and \(\{ S_j\} _{j\in J}\) be two finite families of simple rings. Every ring isomorphism \(T : \prod _{i\in I}R_i \to \prod _{j\in J}S_j\) determines an equivalence \(\sigma :I\simeq J\) such that \(T\) maps the \(i\)-th block ideal onto the \(\sigma (i)\)-th block ideal. In particular, an automorphism of one product permutes its block ideals.

Let \(A\) be an injective tensor of bond dimension \(D \ge 1\), and let \(B\) be a tensor of the same bond dimension. If \(A\) and \(B\) generate the same MPV family for every system size \(N \ge N_0\), then they generate the same MPV family for every system size.

A nonzero multiplicative \(\mathbb {C}\)-linear endomorphism of \(M_{D}(\mathbb {C})\) is bijective.

Let \(A\) and \(B\) be MPS tensors of bond dimension \(D\) with \(A\) injective. Consider the two constant translation-invariant chains \((A, \ldots , A)\) and \((B, \ldots , B)\) of length \(n \ge 1\). If the combined tensors of the two chains generate the same MPV family, then:

1. \(B\) is gauge equivalent to \(A\): there exists \(X \in \mathrm{GL}_D(\mathbb {C})\) with \(B^i = X\, A^i\, X^{-1}\) for all \(i\).

2. \(B\) is injective.

Let \(A\) and \(B\) be MPS tensors of bond dimension \(D\), and fix a chain length \(n \ge 1\). Assume that \(A\) is injective and that the combined tensors of the two constant chains \((A, \ldots , A)\) and \((B, \ldots , B)\) generate the same MPV family. Then there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that \(B^i = X A^i X^{-1}\) for every physical index \(i\).

For a nonempty closed chain, the trace of the word \(A^{\sigma _0}\cdots A^{\sigma _{L-1}}\) is the sum over cyclic virtual-index configurations:

At \(N=0\), the open-chain coefficient is \(1\).

If \(D_k=0\) for some \(k\in \{ 0,\ldots ,N\} \), then \(c_A(\sigma )=0\) for every physical configuration \(\sigma \).

Let \(N\geq 1\). For every physical configuration \(\sigma \),

No positivity assumption is imposed on the intermediate dimensions. The endpoint conditions already imply \(D\geq 1\) whenever such a chain exists. The equality concerns only this fixed length.

At \(N=0\), the coefficient of the padded square chain is \(D\).

Let \(N\geq 1\). If \(D_k=0\) for some \(k\in \{ 0,\ldots ,N\} \), then the padded square-chain coefficient vanishes for every physical configuration.