Dependencies

Under the same hypotheses, suppose additionally that every periodic single-block chain space is contained in its matrix-product-vector line. Then

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

A finite block family has no gauge-phase-equivalent distinct blocks when, for any two distinct indices with the same bond dimension, the corresponding blocks are not gauge-phase equivalent.

For a fixed blocked length \(m\), the fixed-length form of the block-injective canonical-form condition is

Equivalently,

This is the fixed-length form of the block-injective canonical-form condition of [ 17 , lines 2113–2117 ] .

In the coefficient form of a basis of normal tensors, each block \(A_j\) is a normal tensor in the spectral sense (Definition 7.6.1), the MPV of \(A\) is given at every positive length by coefficients \(c_j^{(N)} \in \mathbb {C}\) with \(|V^{(N)}(A)\rangle = \sum _{j=1}^{g} c_j^{(N)} |V^{(N)}(A_j)\rangle \), and the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} \) are eventually linearly independent [ 13 , Section 2.3 ] .

For a basis of normal tensors \(A_1,\ldots ,A_g\), the vector space generated by their length-\(N\) matrix product vectors is \(\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \).

For \(N{\gt}0\) and \(L\le N\), define the periodic parent-Hamiltonian ground space

Here the condition \(\psi |_{[i,i+L-1]}\in G_L(A)\) means that, for every choice of physical indices outside the window, the restriction of \(\psi \) to that window lies in \(G_L(A)\). The one-index notation \(G_L(A)\) denotes the local ground space \(\mathcal G_L\) of [ 17 , Section IV.C ] . The two-index notation \(\mathcal G_{N,L}(A)\) used here denotes the periodic intersection of those local constraints over all translated length-\(L\) windows on the \(N\)-site ring. When \(N=0\) or \(L{\gt}N\), set \(\mathcal G_{N,L}(A):=\top \) by convention.

Let \(A\) have bond dimension \(D\). It is in the canonical form of [ 13 , Section 2.3 ] if there are positive integers \(D_1,\ldots ,D_r\), normal tensors \(A_k\) of bond dimension \(D_k\), nonzero weights \(\mu _k\in \mathbb {C}\), and, writing \(S=\sum _kD_k\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that \(S\le D\) and, for every physical index \(i\),

Thus the complement of the retained direct sum consists literally of zero bond-space coordinates. In particular, \(r=0\) is possible only when \(A=0\).

The convention in line 246 of [ 13 ] is a separate normalization of these data, not part of canonical-form membership:

Every weight \(\mu _k\) is nonzero. This is the reading of [ 13 , Section 2.3 ] adopted here: the remark in line 219 there places the degenerate case in the block dimensions (\(D_k=0\)), and lines 224–225 choose each \(\mu _k\) so that the transfer map of \(\mu _kA_k\) has spectral radius one, which a vanishing weight cannot satisfy. The reading loses nothing, since if \(\mu _{k_0}=0\) then \(\mu _{k_0}A_{k_0}^i=0\) is the zero matrix of size \(D_{k_0}\) and

so \(A^i\) is the same coisometric image of the direct sum over the remaining indices, with the \(D_{k_0}\) coordinates of the omitted summand moved to the zero complement. No ordering or separation condition is imposed.

The basis-of-normal-tensors refinement is Definition 8.1.1; it is not part of canonical form itself.

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 \(i\),

Let \(\iota _L\colon (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\simeq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) be the canonical computational-basis identification. The local ground space under this \(\ell ^2\) identification is \(G_L^{\mathrm{ES}}(A):=\iota _L(G_L(A)) \subseteq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).

The map \(\Gamma _L\) in (1) is a \(\mathbb {C}\)-linear map from \(M_{D}(\mathbb {C})\) to \((\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\). In tensor-network notation,

The chain denotes the word tensor \(A^\sigma \) with open physical indices \(\sigma _1,\ldots ,\sigma _L\), and the red capsule denotes the boundary matrix \(X\) inserted on the closing virtual bond before taking the trace. The brace records the chain length \(L\).

The local ground space is the image \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

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

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

An MPS tensor \(A=(A^i)_i\) is left-canonical when

Equivalently, its Kraus map is trace-preserving.

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.

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.

The subspace spanned by the MPS vector \(V^{(N)}(A)(\sigma )=\operatorname{tr}(A^{\sigma _0}\cdots A^{\sigma _{N-1}})\).

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

For \(L,N\in \mathbb {N}\), let

The compatible length-\(N\) open-chain parent Hamiltonian of interaction length \(L\) is

Thus no summand crosses the two endpoints of the interval. For \(L=l+1\), the starting sites are \(0,\ldots ,N-l-1\), in agreement with [ 225 , Equation (3.12) ] .

For two blocks \(A\) and \(B\), the length-\(S\) pair word tuple is \(w\mapsto (A^w,B^w)\). We also use the subalgebra of \(M_{D_A}(\mathbb {C})\times M_{D_B}(\mathbb {C})\) generated by the one-letter pairs \((A_i,B_i)\). The pair product-span condition says that the length-\(S\) tuples span \(M_{D_A}(\mathbb {C})\times M_{D_B}(\mathbb {C})\). Equivalently, in the trace-dual formulation of the same product-span condition, the only pair of test matrices \((\Delta _A,\Delta _B)\) whose trace pairing with every length-\(S\) tuple vanishes is \((0,0)\). The cumulative variant replaces one fixed length by all word lengths at most \(S\), and the all-length variant allows every finite word.

Fix a finite target set \(T\) of block indices. A length-\(S\) block-separating equation for \(k\) against \(T\) is a choice of coefficients \(c_\omega \) such that

The pairwise version requires this equation for each ordered pair of distinct blocks, with \(T=\{ j\} \).

For a periodic configuration \(\sigma \) on \(N\) sites and \(s\in \mathbb {Z}/N\mathbb {Z}\), define \((T_s\sigma )(k)=\sigma (k+s)\).

For a state \(\psi \) on a periodic chain, define \((T_s\psi )(\sigma )=\psi (T_s\sigma )\).

The chain Hamiltonian is

Denote by \(H_{N,L}^{\mathrm{ES}}(A)\) the conjugate of \(H_N(A,L)\) by the canonical identification between the coefficient space and \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).

Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if

where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. When \(\rho {\gt}0\), this implies strong irreducibility and uniform vector spreading (Definition 6.1.1.3) by Theorem 6.1.1.15.

If no nonzero trace functional vanishes on all finite pair words, then there is a finite \(S\) such that vanishing on every word of length at most \(S\) already forces the test pair to be zero. For a finite family of ordered block pairs, these bounds can be replaced by one common bound.

Let \(X_j,Y_j\in M_{D_j}(\mathbb {C})\), and suppose

If the simultaneous block words of length \(K\) span \(\prod _jM_{D_j}(\mathbb {C})\), then, for every block \(j\) and every word \(\alpha \) of length \(S\), \(X_jA^j_\alpha = A^j_\alpha Y_j\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\). If the periodic-boundary comparison with block-diagonal boundary conditions gives

then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\) and \(L\le N\), and assume that, for every block \(j\), the length-\(N-L\) word products of \(A_j\) span the full matrix algebra. Suppose that, for every block \(j\), every boundary-crossing interval beginning at \(i\), and every outside word \(\rho \) of length \(N-L\), there is a matrix \(E_{j,i,\rho }\) such that, for every word \(\beta \) before the cut of length \(i+L-N\), \(\mu _j^N X_j A^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies the normalization equation \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Suppose also that the boundary-crossing matrix identities of Lemma 13.3.1.21 hold for every outside word \(\rho \) of length \(N-L\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\) and \(L\le N\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every block \(j\), every cyclic interval beginning at \(i\) with \(N{\lt}i+L\), and every configuration \(\tau \),

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Fix block-diagonal boundary conditions \(X_j\). For every boundary-crossing interval beginning at \(i\), suppose there are matrices \(C^j_{i,\rho }\), indexed by outside words \(\rho \) of length \(N-L\), such that, for every word \(\beta \) before the cut of length \(i+L-N\), \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). After opening the periodic boundary at the chosen cut, this is the source comparison

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\) and \(L\le N\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every block \(j\), every boundary-crossing interval beginning at \(i\), and every configuration \(\tau \), there is a matrix \(Y_{j,i,\tau }\) such that, for every word \(\beta \) of length \(a=i+L-N\), \(\mu _j^N X_j A^j_\beta = A^j_\beta Y_{j,i,\tau }\). Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Fix complex numbers \(\mu _j\) and block-diagonal boundary conditions \(X_j\). For every boundary-crossing interval beginning at \(i\), suppose there are matrices \(C^j_{i,\rho }\), indexed by outside words \(\rho \) of length \(N-L\), such that, for every word \(\beta \) before the cut, outside word \(\rho \), and middle word \(w\),

Then, for every block \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). For block-diagonal boundary conditions \(X_j\),

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Write \(a=i+L-N\). For a local word \(\sigma \) of length \(L\), set

Then the corresponding coefficient of the \(j\)-th block component is

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). Suppose that there is a matrix \(E\) such that for every word \(\beta \) of length \(a\),

Then

Assume \(0{\lt}N\) and \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). Suppose that there is a matrix \(Y\) such that, for every word \(\beta \) of length \(a\), \(\mu _j^N X_j A^j_\beta = A^j_\beta Y\). Then

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\). For a local word \(\sigma \) of length \(L\), write

These are the word coordinates obtained by opening the periodic boundary at the chosen cut, in the comparison with the source expressions involving \(C^j_{i_1}\) and \(D^j_{i_{m+1}}\). If

then there are matrices \(C^j_{i,\tau }\) such that, for every \(\sigma \),

Assume that the physical dimension \(d\) is positive, \(0{\lt}N\), \(L\le N\), and \(\mu _j\ne 0\) for every block \(j\). Let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Let

For every outside word \(\rho \) of length \(N-L\), there are matrices \(C^j_{i,\rho }\) such that, for every word \(\beta \) of length \(i+L-N\) before the cut and every wrapped tail word \(\alpha \) of length \(N-i\),

Assume that the physical dimension \(d\) is positive, \(0{\lt}N\), \(L\le N\), and \(\mu _j\ne 0\) for every block \(j\). Let

Then there are matrices \(C^j_{i,\rho }\), indexed by the boundary-crossing start \(i\) and by an outside word \(\rho \) of length \(N-L\), such that whenever \(N{\lt}i+L\),

for every word \(\beta \) of length \(i+L-N\) before the cut and every wrapped tail word \(\alpha \) of length \(N-i\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), and fix block-diagonal boundary conditions \(X_j\). For every cyclic restriction \(R_{i,\tau }\) from \(N\) sites to \(L\) sites,

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Suppose that every vector \(\psi \in \mathcal G_{N,L}(B)\) can be written as

Then

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), with \(\mu _j\ne 0\), and suppose

Then every local constraint of \(\psi \) gives

Fix block-diagonal boundary conditions \(X_j\). If the cyclic window beginning at \(i\) stays inside the chosen linear interval, \(i+L\le N\), then for every block \(j\),

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for every \(j\). For every length \(L\) and every \(\psi \in \bigvee _{j=1}^gG_L(A_j)\), there are boundary matrices \(X_j\) such that

For every \(j\), one also has \(\Gamma _L^{A_j}(\mu _j^LX_j)\in G_L(A_j)\).

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), the corresponding boundary conditions satisfy the block-diagonal boundary-condition equations: for every boundary-crossing interval \(i\), every word \(\beta \) before the cut, and every outside word \(\rho \) of length \(N-L\), \(\mu _j^N X_jA^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Let

If every boundary-crossing interval has the corresponding tail-word spanning property, then there are block-diagonal boundary conditions \(X_j\) with the displayed representation of \(\psi \) and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\) for some block-diagonal boundary conditions \(X_j\). Suppose that whenever \(\psi \) has such a representation, there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) of length \(i+L-N\) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\). Assume also that, for some middle-word length \(m\), the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\), where \(w\) ranges over words of length \(m\) in \(\{ 0,\ldots ,d{-}1\} \), span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\) and \(L \ge (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))+1\). Put \(m = (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Suppose that whenever \(\psi =\Gamma _N^{\oplus _j\mu _jA_j}(\bigoplus _jX_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

Assume \(0{\lt}N\), \(L\le N\), and \(L+L_0\le N\). Suppose each block \(A_j\) is \(L_0\)-block-injective and satisfies \(\sum _a A^j_aA^{j\dagger }_a=\mathbb {1}\). Assume also that the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\) at the middle-word length \(m\) span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let

for some block-diagonal boundary conditions \(X_j\). Suppose that whenever \(\psi \) has such a representation, there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) of length \(i+L-N\) before the cut, and every word \(w\) of length \(m\),

Then there are block-diagonal boundary conditions \(X_j\) with

and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^NX_j) \in \mathcal G_{N,L}(A_j)\).

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Suppose every \(\psi \in \mathcal G_{N,L}(B)\) admits block-diagonal boundary conditions \(X_j\) such that \(\psi = \Gamma _N^B\! \left(\bigoplus _{j=1}^gX_j\right)\), and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then

Fix block-diagonal boundary matrices \(X_j\). Suppose the opened-boundary matrices \(C^j_{i,\rho }\) satisfy, for every boundary-crossing interval \(i\), every outside word \(\rho \), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \). Then, for every middle word \(w\),

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\). Fix block-diagonal boundary conditions \(X_j\). Suppose that, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). Suppose \(N{\gt}0\), \(L\le N\), and the periodic chain constraints for \(B\) split into the block constraints:

If every block chain space is contained in its periodic MPS line,

then the zero-energy space of the parent Hamiltonian is contained in the BNT vector span:

Let \(B=\bigoplus _j\mu _jA^j\), and assume \(\mu _j\ne 0\) for every block \(j\). Suppose

Assume also that the length-\(m\) simultaneous word tuples span the product algebra. Then, for every outside configuration \(\tau \), with \(\rho =\tau _{m+1}\cdots \tau _{M-1}\), there are matrices \(E_j\) such that \(A^j_b(A^j_\rho A^j_a(\mu _j^{M+1}X_j)) = A^j_bE_jA^j_a\) for every block \(j\) and all boundary letters \(a,b\).

In the setting of Lemma 13.3.1.13, assume that the length-\(m\) simultaneous word tuples span the product algebra and that

Then there are matrices \(E_j\) such that, for every block \(j\) and all boundary letters \(a,b\),

Let \(B=\bigoplus _j\mu _jA^j\), and assume \(\mu _j\ne 0\) for every block \(j\). Suppose

If the length-\(m\) simultaneous word tuples span the product algebra, then for every outside configuration \(\tau \) and every block \(j\),

In the setting of Lemma 13.3.1.13, assume that the length-\(m\) simultaneous word tuples span the product algebra and that

Then, for every block \(j\),

Write \(N=M+1\) and let the local length be \(m+2\le M+1\). Consider the cyclic interval beginning at the last site \(M\). For an outside configuration \(\tau \), put

Then the \(j\)-th component of the last crossing-window restriction has the left-boundary trace form

Write \(N=M+1\) and let the local length be \(m+2\le M+1\). Consider the cyclic interval beginning at the last site \(M\). For an outside configuration \(\tau \), put

Then the block sum of the last crossing-window restrictions has the left-boundary trace form

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\), and set \(S_M:=\bigvee _{j=1}^gG_M(A_j)\). If, for every \(M\ge L\), \(\mathbb {C}^d\otimes S_M\cap S_M\otimes \mathbb {C}^d =S_{M+1}\), then \(\mathcal G_{N,L}(B)\subseteq S_N\).

If \(A\) is \(L\)-block injective and \(X\neq 0\), then

Let \(A^1,\ldots ,A^g\) be block tensors whose length-\(m\) simultaneous word tuples span the product algebra. If, for every word \(w\) of length \(m\), two blockwise matrix families \(X_j\) and \(Y_j\) satisfy \(\sum _j\operatorname{tr}(X_jA^j_w) = \sum _j\operatorname{tr}(Y_jA^j_w)\), then \(X_j=Y_j\) for every block \(j\).

Under the hypotheses of Lemma 13.2.1.15, set \(S_n:=\bigvee _jG_{n+1}(A^j)\). Then

Under the hypotheses of Lemma 13.2.1.15, an \((n+2)\)-site vector satisfies \(\psi \in \bigvee _jG_{n+2}(A^j)\) if and only if

for every physical index \(a,b\).

Let \(A^1,\ldots ,A^g\) be block tensors with common word-span separation at length \(n\), and suppose that \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). If \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) satisfies

for every physical index \(a,b\), then \(\psi \in \bigvee _jG_{n+2}(A^j)\).

If every ordered pair of distinct blocks admits a common-length pairwise block-separating equation, then for each \(k\) there are coefficients \(c_\omega ^{(k)}\) with

Hence common block injectivity plus pairwise separation implies the finite-length blockwise physical–virtual correspondence of the block-injectivity proposition in [ 13 , Section 2.3 ] and finite-word trace separation for the block family.

Let \(A^1,\ldots ,A^g\) be block tensors with the following common word-span property: for a fixed \(m\), the simultaneous tuples \(w \longmapsto \left(A^1_w,\ldots ,A^g_w\right), \qquad |w|=m\), span the full product algebra \(\prod _j M_{D_j}(\mathbb {C})\). Suppose that matrices \(C^j_a,D^j_b\in M_{D_j}(\mathbb {C})\) satisfy, for all physical indices \(a,b\) and all words \(w\) of length \(m\),

Then \(A^j_bC^j_a = D^j_bA^j_a\) for every block \(j\) and all \(a,b\).

Let \(A^1,\ldots ,A^g\) be block tensors whose length-\(m\) simultaneous word tuples span the product algebra. Let \(C^j_a\in M_{D_j}(\mathbb {C})\) be matrices indexed by blocks \(j\) and physical letters \(a\). Suppose that, for every pair of boundary letters \(a,b\) and every word \(w\) of length \(m\), a vector \(\psi \) has the left-boundary trace form \(\psi (a,w,b) = \sum _j\operatorname{tr}\! \left(A^j_b C^j_a A^j_w\right)\). If \(\psi \in \bigvee _j G_{m+2}(A^j)\), then there are matrices \(E_j\) such that, for every block \(j\) and all \(a,b\), \(A^j_bC^j_a = A^j_bE_jA^j_a\).

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Suppose that \(C^j_\rho \in M_{D_j}(\mathbb {C})\) is indexed by a block \(j\) and a word \(\rho \) of some fixed length \(M\), and that \(D^j_\beta \in M_{D_j}(\mathbb {C})\) is indexed by a block \(j\) and a word \(\beta \) of some fixed length \(K\). If, for every word \(\rho \) of length \(M\), every word \(\beta \) of length \(K\), and every middle word \(w\) of length \(m\),

then \(A^j_\beta C^j_\rho = D^j_\beta A^j_\rho \) for every block \(j\) and all words \(\beta ,\rho \). In particular, taking \(D^j_\beta =X_jA^j_\beta \) gives \(A^j_\beta C^j_\rho = (X_jA^j_\beta )A^j_\rho \).

Let \(A_1,\ldots ,A_g\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent, and let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\). Assume each block is injective at length \(L_0\), assume \(\sum _aA^j_aA^{j\dagger }_a=\mathbb {1}\) for every \(j\), and suppose

If every vector \(\psi \in \mathcal G_{N,L}(B)\) can be written as

then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\), and the sum \(\bigvee _jG_N(A_j)\) is internal.

With the hypotheses and notation of Lemma 13.3.8, suppose every vector \(\psi \in \mathcal G_{N,L}(B)\) has block-diagonal boundary conditions \(X_j\) such that \(\psi = \Gamma _N^B\! \left(\bigoplus _{j=1}^gX_j\right)\), and, for every \(j\), \(\Gamma _N^{A_j}(\mu _j^N X_j)\in \mathcal G_{N,L}(A_j)\). Then \(\mathcal G_{N,L}(B) = \bigvee _{j=1}^g\mathcal G_{N,L}(A_j)\), and the sum defining \(\bigvee _jG_N(A_j)\) is internal.

With the hypotheses and notation of Lemma 13.3.2.6, assume in addition that \(L+L_0\le N\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

the complementary-word matrix identities hold: for every block \(j\), boundary-crossing interval \(i\), word \(\beta \) before the cut, and outside word \(\rho \) of length \(N-L\), there is a matrix \(E_{j,i,\rho }\) such that \(\mu _j^N X_jA^j_\beta A^j_\rho = A^j_\beta E_{j,i,\rho }\). Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 13.3.2.6, assume in addition that \(L+L_0\le N\). Assume also that, for every boundary-crossing interval beginning at \(i\), the simultaneous products

span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum. The present statement is the span-dependent consequence, after opening the boundary, of the \(C^j,D^j\) compatibility input in [ 228 , Theorem 12, proof lines 1436–1451 ] in the length-\(L_0\) injectivity range \(L \ge (L_0+1)+3(g-1)(L_0+1)+1\). The source theorem has the shorter bound \(L \ge 3(g-1)(L_0+1)+1\).

With the hypotheses and notation of Lemma 13.3.2.10, one has

This records the ground-space equality conclusion of the conditional crossing-span version above. It is the corresponding part of [ 228 , Theorem 12 ] , but only under the visible tail-word span hypothesis and in the longer stated range. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 13.3.2.10. This intermediate statement remains conditional on the short tail-word spans. The source-range conclusion without them is supplied by the global change-of-cut theorem Theorem 13.3.1.41 under the unrestricted source normalization, with positive definite dual fixed points satisfying

The doubly normalized theorem with \(\Lambda _j=\mathbb {1}\) remains available as a specialization of that result.

With the hypotheses and notation of Lemma 13.3.8, every vector \(\psi \in \mathcal G_{N,L}(\bigoplus _{j=1}^g\mu _jA_j)\) has a unique family \((\psi _j)_{j=1}^g\) such that

With the hypotheses and notation of Lemma 13.3.8, every vector \(\psi \in \mathcal G_{N,L}(\bigoplus _{j=1}^g\mu _jA_j)\) admits block-diagonal boundary conditions \(X_j\) such that

for every \(j\).

Let \(g\ge 2\). Let \(A_j\) be irreducible blocks, pairwise inequivalent up to gauge and phase, and injective at the common length \(L_0{\gt}0\). Assume the canonical normalization of [ 228 ] :

Let every weight \(\mu _j\) be nonzero. If \(0{\lt}L\le N\) and \(L\ge 3(g-1)(L_0+1)+1\), then

In particular, every vector \(\psi \) in the left-hand side admits matrices \(X_j\) such that

This is the open-boundary decomposition used in [ 228 , Theorem 12, proof lines 1424–1456 ] .

With the hypotheses and notation of Lemma 13.3.8, one has

and the sum defining the right-hand side is internal.

With the hypotheses and notation of Lemma 13.3.8, set \(S_N:=\bigvee _{j=1}^gG_N(A_j)\). Then, for every \(N\ge L\), \(\mathcal G_{N,L}(B)\subseteq S_N\), and the sum defining \(S_N\) is internal: if

then \(\phi _j=0\) for every \(j\).

Let \(A_1,\ldots ,A_g\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). For nonzero scalars \(\mu _j\), set

If \(L\ge (L_0+1)+(g-1)3(L_0+1)+1\), then, for every \(N\ge L\), \(\mathcal G_{N,L}(B)\subseteq S_N\).

With the hypotheses and notation of Lemma 13.3.2.6, assume in addition that \(L+L_0\le N\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^N X_j)A^j_\beta )A^j_\rho \). This is the \(C^j,D^j\) compatibility input; the subsequent normalized \(E^j\) identities follow from Lemma 13.2.1.9. Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 13.3.2.8, one has

This records only the ground-space equality conclusion obtained after the \(C^j,D^j\) boundary-condition comparison input in [ 228 , Theorem 12, proof lines 1446–1456 ] ; the local normalized \(E^j\) calculation is Lemma 13.2.1.9. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 13.3.2.8. The statement is still in the length-\(L_0\) injectivity range displayed in that lemma, not the shorter range in [ 228 , Theorem 12 ] .

With the hypotheses and notation of Lemma 13.3.2.6, assume in addition that \(L+L_0\le N\) and \(L \ge (L_0+1)+(g-1)((L_0+1)+((L_0+1)+(L_0+1)))+1\). Put \(m = (L_0+1)+(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\). Assume that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

With the hypotheses and notation of Lemma 13.3.2.6, assume in addition that \(L+L_0\le N\). Assume also that, for some middle-word length \(m\), the simultaneous tuples \(w\mapsto (A^1_w,\ldots ,A^g_w)\), where \(w\) ranges over words of length \(m\) in \(\{ 0,\ldots ,d{-}1\} \), span the full product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Assume further that, for every \(\psi \in \mathcal G_{N,L}\! \left(\bigoplus _j\mu _jA_j\right)\) and every block-diagonal boundary representation

there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every word \(\beta \) before the cut, and every word \(w\) of length \(m\),

Then

and the \(N\)-site ground spaces \(G_N(A_j)=\mathcal G_{N,N}(A_j)\) have internal sum.

Under the hypotheses of Lemma 8.3.3, assume in addition the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Let

Then

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Assume \(L{\gt}1\), and assume that, for every block \(j\),

Set \(S=3L\) and \(q=(r-1)S\). Suppose that \(p{\gt}0\) and let \(s_0\) be a starting length. Assume that, for every block \(j\), \(\operatorname{span}\{ A^j_u:|u|=p\} =M_{D_j}(\mathbb {C})\), and that, for every \(0\le s{\lt}p+q\) and every block \(j\), \(\operatorname{span}\{ A^j_v:|v|=s_0+s\} =M_{D_j}(\mathbb {C})\). Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Then there is \(N\) such that, for every \(n\ge N\), if \(S_n:=\bigvee _jG_{n+1}(A^j)\), then

Let \((A_j)_{j=1}^g\) be irreducible left-canonical tensors with normalized self-overlaps and with no gauge-phase-equivalent distinct blocks. Suppose \(g\geq 2\), \(L_0{\gt}0\), and every \(A_j\) is block-injective at lengths \(L_0\), \(L_0+1\), and \(3(L_0+1)\). Then

This is the direct-sum induction length in [ 228 , lines 1346–1408 ] .

Let \(A^1,\ldots ,A^r\) be irreducible left-canonical representatives with normalized self-overlaps, and suppose that distinct same-dimensional representatives are not gauge-phase equivalent. Fix \(L_0{\gt}0\) such that every block is injective at lengths \(L_0\), \(L_0+1\), and \(3(L_0+1)\). Then each ordered pair \(k\ne j\) is trace-separated at the common length \(S=3(L_0+1)\): the map

is onto \(M_{D_k}(\mathbb {C})\oplus M_{D_j}(\mathbb {C})\). In particular, one may prescribe an arbitrary matrix on either block while making the same word combination vanish on the other block.

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Then there is a length \(N\) such that, for every \(n\ge N\), with \(S_n:=\bigvee _jG_{n+1}(A^j)\), one has

Let \(A^1,\ldots ,A^r\) be normalized representatives of a basis of normal tensors: each representative is irreducible, the family is left-canonical, the self-overlaps are normalized, and no two distinct same-dimensional representatives are gauge-phase equivalent. Suppose \(L_0{\gt}0\), and, for every block \(j\), the map

is injective. Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Set \(N=(L_0+1)+(r-1)3(L_0+1)\). For every \(n\ge N\), set \(S_n:=\bigvee _jG_{n+1}(A^j)\). The two consecutive sums are internal direct sums:

With this notation,

For a family \(A_0,\ldots ,A_{r-1}\) and a chain length \(N\), let \(M_j=\operatorname{span}\{ V^{(N)}(A_j)\} \). Then

Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). If every component vector has zero energy for the parent Hamiltonian, \(H_L^{(N)}(B)V^{(N)}(A_j)=0\) for \(1\le j\le g\), then

It is enough to assume the local frustration-free equations \(h_iV^{(N)}(A_j)=0\) for every \(i\) and \(j\). For \(N{\gt}0\) and \(L\le N\), in the block-diagonal case \(B=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\), the same inclusion follows because each local space \(G_L(A_j)\) is contained in \(G_L(B)\). This establishes the forward inclusion \(\operatorname{span}\{ V^{(N)}(A_j)\} _j \subseteq \ker H_L^{(N)}(B)\) of [ 13 , Definition 3.9 ] .

For each block index \(j\) with \(\mu _j\neq 0\), and whenever \(0{\lt}N\) and \(L\le N\), the periodic-boundary ground space of the block \(A_j\) embeds into the corresponding ground space of the direct-sum tensor:

Let \(A=(A^i)\) and \(C=(C^i)\) have the same physical alphabet, and suppose there is a coisometry \(U:\mathbb {C}^D\to \mathbb {C}^n\) such that \(UU^\dagger =I_n\) and \(A^i=U^\dagger C^iU\) for every \(i\). Then, for every \(L\geq 1\),

In particular, if \((A_j)_{j=1}^g\) are the distinct BNT representatives selected from literal canonical-form data for \(A\), then, for \(p,L\geq 1\),

Identity (195) is a derived trace identity for an exact coisometric reconstruction. It is not the literal direct-sum canonical-form equation of CPSV16.

Let \(A_\beta \) be the products over wrapped words of length \(K\), let \(A_\rho \) be the products over complementary words of length \(M\), and let \(X\in M_D(\mathbb {C})\) be a matrix. Suppose that the complementary-word products satisfy \(\sum _\rho A_\rho A_\rho ^\dagger = \mathbb {1}\) and that there are matrices \(C_\rho \), indexed by complementary words, such that, for every wrapped word \(\beta \) and complementary word \(\rho \), \(A_\beta C_\rho = (X A_\beta )A_\rho \). Then, for every complementary word \(\rho \), with

one has, for every wrapped word \(\beta \), \((X A_\beta )A_\rho = A_\beta E_\rho \).

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix matrices \(X_j\). Suppose that \(C^j_\rho \) is indexed by a block \(j\) and a complementary word \(\rho \) of length \(M\). Assume that, for every wrapped word \(\beta \) of length \(K\), every complementary word \(\rho \), and every middle word \(w\) of length \(m\),

If each block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a = \mathbb {1}\), then, for every block \(j\) and every complementary word \(\rho \), there is a matrix \(E_{j,\rho }\) such that, for every wrapped word \(\beta \), \((X_jA^j_\beta )A^j_\rho = A^j_\beta E_{j,\rho }\).

If \(A\) and \(B\) are length-\(L\) block-injective, the map \(Z\mapsto (t\mapsto \operatorname{tr}(A_tZ))\) has image dimension \(D_A^2\), and similarly for \(B\). Therefore an inclusion \(\mathcal G_L^A\subseteq \mathcal G_L^B\), together with \(D_B\le D_A\), forces \(D_A=D_B\) and \(\mathcal G_L^A=\mathcal G_L^B\). In particular, the three-block trace relation from Lemma 13.2.4 gives this equality whenever the ordering \(D_B\le D_A\) is available. If the inclusion comes from the strictly larger block, \(D_B{\lt}D_A\), then the same dimension step already contradicts the existence of the three-block trace relation. More generally, let \(H\) be a source hypothesis which excludes the simultaneous conclusion

Then the three-block intersection vanishes:

With injectivity at the three-block length, the strict-size branch gives

One such source hypothesis is non-proportionality of the periodic MPS vectors at a sufficiently long chain length: \(\mathbb {C}\, V^{(N)}(A)\ne \mathbb {C}\, V^{(N)}(B)\). Indeed,

by equality of the periodic-boundary constraint spaces, followed by injective uniqueness. For same-dimensional normalized BNT representatives that are not gauge-phase equivalent, the non-equivalence condition supplies this non-proportionality at arbitrarily large chain lengths; the direct-sum injectivity hypotheses remain explicit. Combining the equal-dimension and unequal-dimension branches over all ordered block pairs gives one common positive length \(S_0\) with

For an ordered pair of blocks, the trace-separation conclusion has the form

Equivalently, the homogeneous pair span is full:

Concatenating words propagates this equality to positive multiples of \(S\), and hence to a finite period window for every ordered pair. Combining that window with the conditional period-window homogenization theorem gives a single length \(S_0\) such that

The same conclusion applies to the corresponding normal canonical-form hypotheses with a basis of normal tensors once one-site injectivity is supplied explicitly, since those hypotheses carry the irreducibility and normalization assumptions used in the direct-sum comparison.

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix a complementary word \(\rho \) of length \(M\) and matrices \(C^j_\rho ,X_j\in M_{D_j}(\mathbb {C})\). If, for every wrapped word \(\beta \) of length \(K\) and every middle word \(w\) of length \(m\),

then \(A^j_\beta C^j_\rho = (X_jA^j_\beta )A^j_\rho \) for every block \(j\) and every wrapped word \(\beta \).

If \(A\) and \(B\) are gauge equivalent and \(A\) is \(L\)-block injective, then \(B\) is \(L\)-block injective.

Gauge equivalent replacements on both sides preserve gauge-phase equivalence, including after identifying equal bond dimensions.

Let \(A,A',B,B'\) have the same bond dimension. If \(A\) is gauge equivalent to \(A'\), \(A'\) is gauge-phase equivalent to \(B'\), and \(B\) is gauge equivalent to \(B'\), then \(A\) is gauge-phase equivalent to \(B\).

If \(\psi \in G_{L+1}(A)\), i.e. if \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\) for some \(X\), then, for each \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\) with witness \(A^jX\).

If \(\psi \in G_{L+1}(A)\) with \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\), then, for each \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\) with witness \(XA^i\).

Suppose the length-\(L\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose the simultaneous identity tuple \((I,\ldots ,I)\) lies in the span of the length-\(S\) simultaneous block-word tuples. Then the length-\(L+S\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). In particular, two full homogeneous spans at lengths \(L\) and \(S\) give a full homogeneous span at length \(L+S\).

Assume \(\mu _j\neq 0\) for every block index \(j\). Whenever \(0{\lt}N\) and \(L\le N\), the linear sum of the blockwise ground spaces with periodic boundary conditions is contained in the direct-sum tensor’s ground space with periodic boundary conditions:

where \(j\) ranges over all block indices.

Let \(A\) be a block tensor, and let \(C_a,E\in M_D(\mathbb {C})\) satisfy \(A_b C_a = A_bEA_a\) for all physical indices \(a,b\). For a word \(\sigma =(\sigma _1,\ldots ,\sigma _{n+2})\), define

Then \(\alpha \in G_{n+2}(A)\).

Let \(A^1,\ldots ,A^g\) be block tensors, and let \(C^j_a,E^j\in M_{D_j}(\mathbb {C})\) satisfy \(A^j_b C^j_a = A^j_bE^jA^j_a\) for every block \(j\) and all physical indices \(a,b\). For a word \(\sigma =(\sigma _1,\ldots ,\sigma _{n+2})\), define

Then \(\sum _j \alpha _j \in \bigvee _j G_{n+2}(A^j)\).

Let \(A^1,\ldots ,A^g\) be block tensors with common word-span separation at length \(n\), and suppose that \(\sum _a A^j_a A^{j\, \dagger }_a = \mathbb {1}\) for every block \(j\). Suppose a vector \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) has the left-boundary trace decomposition

and suppose that the two trace decompositions agree for every middle word:

Then \(\psi \in \bigvee _j G_{n+2}(A^j)\).

Let \(B=\bigoplus _{k=1}^g\mu _kA_k\). If \(\mu _j\ne 0\), then, for every length \(L\), one has \(G_L(A_j)\subseteq G_L(B)\).

The Hilbert-space parent Hamiltonian is the sum of the Hilbert-space local terms, \(H_{N,\mathrm{ES}}=\sum _i h_{i,\mathrm{ES}}\), and each local term admits the exact cyclic-averaging formula

For \(L\le N\), a Hilbert-space local term \(h_{i,\mathrm{ES}}\) annihilates a vector \(v\) if and only if every restriction of \(v\) to the cyclic length-\(L\) window starting at \(i\), with the outside configuration fixed, lies in \(G_L^{\mathrm{ES}}(A)\).

For \(v\in \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), one has \(v\in G_L^{\mathrm{ES}}(A)\) if and only if \(\iota _L^{-1}v\in G_L(A)\).

Suppose that a finite family of nonzero-dimensional MPS tensors is block injective at a common positive word length \(L_0\), and that its members are pairwise inequivalent under virtual conjugation and nonzero scalar multiplication. Then each tensor admits a normalized primitive representative, with positive definite invariant matrix, which remains block injective at length \(L_0\). The representatives remain pairwise inequivalent. For any nonzero weights, the original and normalized block sums have exactly the same periodic parent Hamiltonians at every interaction range and chain length.

Let \((A_a)_{a\in I}\), \((C_a)_{a\in I}\), and \((D_b)_{b\in I}\) be matrices in \(M_D(\mathbb {C})\), indexed by a finite alphabet. If

for all \(a,b\in I\), and if \(E=\sum _a C_a A_a^\dagger \), then

for all \(a,b\in I\).

In particular, the same calculation applies when the alphabet is the finite set of words of a fixed length and \(A_\beta \) denotes \(A_{\beta _1}\cdots A_{\beta _K}\).

The same argument also applies when the normalized family is indexed by a finite set \(J\), while \((A_b)_{b\in I}\) and \((D_b)_{b\in I}\) are indexed by a set \(I\). If \((B_a)_{a\in J}\) and \((C_a)_{a\in J}\) are indexed by \(J\), and if, for all \(a\in J\) and \(b\in I\),

then, with \(E=\sum _a C_aB_a^\dagger \), one has

This includes the case where \(I\) and \(J\) are word sets of different lengths.

Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that every non-wrapping length-\(L\) interval of \(\psi \), for every fixed choice of the complementary sites, lies in \(S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then \(\psi \in S_N\).

If one of the two blocks is injective, then the corresponding coordinate projection of the pair algebra generated by the simultaneous one-letter pairs is the full matrix algebra. Every finite pair word lies in this pair algebra, and the all-word linear span is the submodule generated by the pair algebra. If the pair algebra is the graph of an algebra automorphism of the matrix algebra, Skolem–Noether turns this graph case into gauge-phase equivalence, so under non-gauge-equivalence the graph branch is impossible. For two blocks of unequal bond dimensions, the graph branch is also impossible because it would identify matrix algebras with different finite dimensions over \(\mathbb {C}\).

Let \((A^\ell )_{\ell =1}^r\) be a finite family of tensors. Suppose that the length-\(S\) pair evaluations of blocks \(k\) and \(j\) span their product matrix algebra. For every \(X\in M_{D_k}(\mathbb {C})\), there is a tuple \(M=(M_\ell )_\ell \) in the span of the simultaneous length-\(S\) word evaluations such that \(M_k=X\) and \(M_j=0\).

Homogeneous identity-padding witnesses multiply by concatenation of words. Therefore, if one has a positive period and a full residue window of homogeneous pair spans, then the simultaneous identity pair lies in all sufficiently large homogeneous pair spans. Combined with all-length pair trace separation, this period-window input gives one homogeneous pair trace-separation length.

At a fixed length \(S\), the pair trace-separation criterion is dual to full pair product-span. Therefore a pair trace-separation witness supplies a linear combination of word evaluations that is identity on the first block and zero on the second. A common witness for all ordered distinct pairs gives pairwise block-separating words at the same length.

If \(\beta \) has length \(K\) and \(\alpha \) has positive length \(S\), then \(T_K(\beta \alpha )=\alpha \beta \) as configurations on the periodic chain of length \(K+S\).

If \(\psi \in \mathcal G_{N,L}(A)\), then \(T_s\psi \in \mathcal G_{N,L}(A)\) for every cyclic translation \(s\).

Let \(p{\gt}0\). Suppose the length-\(p\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose that, for every \(0\le r{\lt}p\), the length-\((s+r)\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). Assume also the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) for every block \(j\). Then there is a length \(N\) such that, for every \(n\ge N\), if \(S_n:=\bigvee _jG_{n+1}(A^j)\), then

Let \(p{\gt}0\). Suppose the length-\(p\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\), and suppose that for every \(0\le r{\lt}p\), the length-\(s+r\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). Then there is a length \(N\) such that, for every \(n\ge N\), the length-\(n\) simultaneous block-word tuples span \(\prod _j M_{D_j}(\mathbb {C})\). More concretely, one may take \(N=s\): writing \(n=s+r+qp\), with \(0\le r{\lt}p\), the span at length \(s+r\) propagates to the span at length \(n\) by multiplying \(q\) times by the period-\(p\) span.

Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that \(G_L(A)\subseteq S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then the periodic local constraints imply \(\mathcal G_{N,L}(A)\subseteq S_N\).

Let \(A^1,\ldots ,A^g\) be block tensors. If \(\psi \in (\mathbb {C}^d)^{\otimes (n+2)}\) satisfies

for every physical index \(a,b\), then there are matrices \(C^j_a\) and \(D^j_b\) such that

and, for every middle word \(w\) of length \(n\),

Let \(A^1,\ldots ,A^g\) be block tensors with the common word-span property at length \(m\). Fix matrices \(X_j\), and put \(D^j_\beta =X_jA^j_\beta \). Suppose that \(C^j_\rho \) is indexed by a block \(j\) and a complementary word \(\rho \) of length \(M\), and assume that, for every wrapped word \(\beta \) of length \(K\), every complementary word \(\rho \), and every middle word \(w\) of length \(m\),

If each block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a = \mathbb {1}\), then, for every block \(j\), there is a matrix \(E^j = \sum _\rho C^j_\rho \left(A^j_\rho \right)^\dagger \) such that, for every wrapped word \(\beta \) and complementary word \(\rho \),

This is the calculation in [ 228 , Theorem 12, proof lines 1446–1451 ] , with the adjoint on \(A^j_\rho \) required by the displayed normalization.

Suppose \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} = \prod _jM_{D_j}(\mathbb {C})\). Then the sum of the local spaces \(G_n(A^j)\) is direct: if

then \(\phi _j=0\) for every \(j\). In particular, for the normalized BNT hypotheses above, this directness holds for every \(n\ge (L_0+1)+(r-1)3(L_0+1)\).

Let \(J\) be a finite set of blocks, and let \((A_j)_{j\in J}\) be a family of tensors such that every \(A_j\) is \(L\)-block injective. Suppose that for each block \(k\) there are coefficients \(c_\omega ^{(k)}\), indexed by words of length \(S\), such that

Then the simultaneous length-\((L+S)\) block-word evaluations span \(\prod _{j\in J} M_{D_j}(\mathbb {C})\).

If \(\sum _a A_aA_a^\dagger =\mathbb {1}\), then, for every length \(M\), \(\sum _\rho A_\rho A_\rho ^\dagger = \mathbb {1}\), where \(\rho \) ranges over all words of length \(M\).

Under the hypotheses of Lemma 13.2.3, the length-\(L\) spaces \(\mathcal G_L^A\) and \(\mathcal G_L^B\) satisfy \(\mathcal G_L^A\subseteq \mathcal G_L^B\). If the corresponding nonzero middle matrix exists on both sides, then \(\mathcal G_L^A=\mathcal G_L^B\).

Let \(A\) and \(B\) be tensors with virtual bond dimensions \(D_A\) and \(D_B\). Assume that \(A\) is \(L\)-block injective. Let \(\Delta _A\in M_{D_A}(\mathbb {C})\) with \(\Delta _A\neq 0\), and let \(\Delta _B\in M_{D_B}(\mathbb {C})\). Suppose that, for all words \(u,v,t\) of length \(L\),

Then for every \(Z\in M_{D_A}(\mathbb {C})\) there is a matrix \(W\in M_{D_B}(\mathbb {C})\) such that, for all words \(t\) of length \(L\), \(\operatorname{tr}(ZA_t)+\operatorname{tr}(WB_t) =0\). This is the trace-dual algebraic step in the two-block case of the direct-sum lemma of [ 228 ] .

If \(\sigma \) is positive definite and \(B\) is defined by (1), then \(A\) and \(B\) are gauge equivalent.

Let \(A\) have positive bond dimension and irreducible transfer map. If \(\Lambda {\gt}0\), then the square-root-gauged tensor \(\Lambda ^{1/2}A\Lambda ^{-1/2}\) is irreducible.

Let \(A^1,\ldots ,A^r\) be blocks satisfying the normalization \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\) and assume that every block is injective at length \(L\): \(\operatorname{span}\{ A^j_u:|u|=L\} =M_{D_j}(\mathbb {C})\). If there are block-separating equations of length \(S\), then, for every \(n\ge L+(r-1)S\), \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} = \prod _j M_{D_j}(\mathbb {C})\). In particular, for normalized BNT representatives satisfying the displayed unital normalization, if \(L_0{\gt}0\) and \(\Gamma _{L_0}^{A^j}\) is injective for every block \(j\), the same conclusion holds with \(S=3(L_0+1)\) and \(n\ge (L_0+1)+(r-1)3(L_0+1)\).

Suppose that the length-\(q\) simultaneous products span the product algebra, \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=q\} = \prod _jM_{D_j}(\mathbb {C})\), and that every block satisfies \(\sum _a A^j_aA^{j\, \dagger }_a=\mathbb {1}\). Then the simultaneous products span the product algebra at every length \(n\ge q\). This is the simultaneous-tuple form of the unital propagation in [ 228 , lines 893–898 ] .

On the periodic chain, for a full configuration \(\rho \) supplying the complementary values, put

The restriction to the cyclic interval beginning at \(i\) is

with all site labels read modulo \(N\). If \(L\le N\), this is the same configuration as \(\rho ^{[i,i+L-1]\leftarrow \omega }\). If two boundary conditions agree whenever \(\delta _i(k)\ge L\), their restrictions are equal. In particular, filling the interval inside the outside configuration leaves the restriction unchanged:

For every \(r{\lt}L\), the inserted word is recovered as \(\rho ^{\mathrm{cyc}}_{i,L,\omega }(i+r)=\omega (r)\) with site labels taken modulo \(N\). Reconstructing a configuration from its local restriction returns the original configuration: taking \(\rho =\sigma \) and \(\omega (r)=\sigma (i+r)\) for \(r{\lt}L\),

Modifying a single inserted site commutes with the cyclic extension: for \(r{\lt}L\) and a letter \(x\), writing \(\omega [r\mapsto x]\) for the word \(\omega \) with its value at \(r\) replaced by \(x\),

Deleting the final site of an \((L+1)\)-interval gives the \(L\)-interval beginning at \(i\); for \(L+1\le N\), deleting the first site gives the \(L\)-interval beginning at \(i+1\); and the cyclic formula agrees with the contiguous restriction when the interval does not wrap around the ring.

Let \(P_L^{\mathrm{ES}}(A)\) denote the orthogonal projector onto \((G_L^{\mathrm{ES}}(A))^\perp \) on the canonical \(\ell ^2\) \(L\)-site Hilbert space. For a cyclic window starting at \(i\) and an outside configuration \(\tau \), the map \(R_{i,\tau }\) restricts an \(N\)-site vector to that length-\(L\) cyclic window with the outside configuration fixed by \(\tau \). The local term \(h_{i,\mathrm{ES}}\) is the corresponding \(N\)-site local projector, and \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is the basic positive summand in its cyclic-averaging formula.

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Suppose that the block tensors have the common word-span property at length \(N-i\) and that

Then each block component of the fixed boundary-crossing restriction lies in its local block space:

Assume \(0{\lt}N\), \(L\le N\), and let the cyclic interval beginning at \(i\) cross the boundary cut, so \(N{\lt}i+L\). Put \(a=i+L-N\) and \(\rho =\tau _a\cdots \tau _{i-1}\). Suppose also that the block tensors have the common word-span property at length \(N-i\). If

then there are matrices \(C^j_{i,\tau }\) such that, for every block \(j\) and every word \(\beta \) of length \(a\),

Let

Assume that the physical alphabet is nonempty, that every \(\mu _j\) is nonzero, that \(0{\lt}N\), and that \(L\le N\). For every boundary-crossing interval beginning at \(i\), assume that the simultaneous products

span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Then there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every outside word \(\rho \) of length \(N-L\), every block \(j\), and every word \(\beta \) before the cut, \(A^j_\beta C^j_{i,\rho } = ((\mu _j^NX_j)A^j_\beta )A^j_\rho \).

Let \(g\ge 2\), assume the physical alphabet is nonempty, and let \(A_j\) be irreducible blocks with positive bond dimensions, pairwise inequivalent up to gauge and phase, and injective at a common length \(L_0{\gt}0\). Assume normalized self-overlap and the two normalization identities

Let every \(\mu _j\) be nonzero, and suppose

If \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\), then there are matrices \(X_j\) such that, for every \(j\),

This theorem is restricted to the case in which the positive dual fixed matrices of the source normalization are all the identity, with normalized self-overlap imposed separately.

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\) and that every boundary-crossing interval beginning at \(i\) has the simultaneous tail-word spanning property: the products

span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

With the hypotheses and notation of Lemma 13.3.12, assume in addition that \(L+L_0\le N\). Thus

Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

No simultaneous spanning assumption is imposed at the short crossing tails.

With the hypotheses and notation of Lemma 13.3.11, assume in addition that \(L+L_0\le N\). Assume also that, for every boundary-crossing interval beginning at \(i\) whose tail length lies below the block-separation bound, \(N-i {\lt} (L_0+1) +(g-1)((L_0+1)+((L_0+1)+(L_0+1)))\), the simultaneous products

span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). Let \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\). Then there are block-diagonal boundary conditions \(X_j\) such that the displayed representation holds and, for every \(j\), the displayed component lies in the corresponding single-block ground space:

This derives the boundary-condition comparison of [ 228 , Theorem 12, proof lines 1446–1456 ] at every crossing tail whose length reaches the displayed block-separation bound.

Let \(g\ge 2\). Let \(A_j\) be irreducible blocks, pairwise inequivalent up to gauge and phase, and injective at a common length \(L_0{\gt}0\). Assume \(\sum _aA^j_aA^{j\, \dagger }_a=\mathbb {1}\) and, for positive-definite \(\Lambda _j\), \(\sum _aA^{j\, \dagger }_a\Lambda _jA^j_a=\Lambda _j\). Let \(\mu _j\ne 0\) for every \(j\), and assume

Then

as asserted in [ 228 , Theorem 12, proof lines 1424–1456 ] .

Let \(g\ge 2\), assume the physical alphabet is nonempty, and let \(A_j\) be irreducible blocks with positive bond dimensions, pairwise inequivalent up to gauge and phase, and injective at a common length \(L_0{\gt}0\). Assume normalized self-overlap and the two normalization identities

Let every \(\mu _j\) be nonzero, and suppose

Then

Under the same hypotheses,

The open-boundary spaces \(G_N(A_j)\) also form an internal sum.

Under the hypotheses of Theorem 13.3.1.42,

and the open-boundary spaces \(G_N(A_j)\) form an internal sum.

Let \(g\ge 2\), assume the physical alphabet is nonempty, and let \(A_j\) be irreducible blocks with positive bond dimensions, pairwise inequivalent up to gauge and phase, and injective at a common length \(L_0{\gt}0\). Assume the source normalization

Let every \(\mu _j\) be nonzero. If \(0{\lt}L\le N\) and \(L\ge 3(g-1)(L_0+1)+1\), then every \(\psi \in \mathcal G_{N,L}(\bigoplus _j\mu _jA_j)\) has a unique decomposition

Under the hypotheses of Theorem 13.3.1.42, suppose in addition that, for every \(j\),

Then

This theorem is restricted to the doubly normalized case in which every positive dual fixed matrix is the identity and normalized self-overlap is assumed.

Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\). Then, for every length \(L\),

The second identity is the image of the first under the canonical identification \(\iota _L\), using \(G_L^{\mathrm{ES}}(A)=\iota _L(G_L(A))\).

Suppose the length-\(S\) word tuples of a finite family \(A^0,\ldots ,A^{r-1}\) span the full product matrix algebra. If every \(\zeta _j\) is nonzero, the word tuples of \(\zeta _0A^0,\ldots ,\zeta _{r-1}A^{r-1}\) also span that algebra at the same length \(S\).

Let \(S{\gt}0\), and suppose the length-\(S\) word tuples of a finite family \(A^0,\ldots ,A^{r-1}\) with positive bond dimensions span the full product matrix algebra. There are normalized primitive representatives \(B^0,\ldots ,B^{r-1}\), with positive definite fixed points, whose word tuples still span the full product algebra at length \(S\). For every \(0\leq j{\lt}r\), every length \(L\), and every chain length \(N\),

For any nonzero weights \(\mu _j\), the periodic parent Hamiltonians of \(\bigoplus _j\mu _j A^j\) and \(\bigoplus _j\mu _j B^j\) are equal at every interaction range and chain length.

Let \(g\ge 2\), assume the physical alphabet is nonempty, and let \(A_j\) be irreducible blocks with positive bond dimensions, pairwise inequivalent up to gauge and phase, and injective at the common length \(L_0{\gt}0\). Assume normalized self-overlap and

Let every weight \(\mu _j\) be nonzero. If \(0{\lt}L\le N\) and \(L\ge 3(g-1)(L_0+1)+1\), then

In particular, every vector \(\psi \) in the left-hand side admits matrices \(X_j\) such that

for every \(j\).

Let \(A\) be a canonical tensor with positive physical dimension and positive bond dimension \(D\). There are distinct normal representatives \((A_j)_j\) and a positive integer \(R\leq 3D^5\) such that, for every \(R\leq L\leq N\), every collection of positive operators \(H_i\) with the same kernels as the canonical range-\(L\) parent terms satisfies

The equality includes \(N=L=R\), and in particular holds at \(N=L=3D^5\) for the canonical parent Hamiltonian. Repeated copies of the normal representatives are allowed in \(A\).

If \(A\) is injective, \(D \ge 1\), \(N \ge 2\), and \(1 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

If \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension, left-canonical normal tensors. There is a positive integer \(L\) such that every \(A_j\) is \(L\)-block injective.

Assume \(s+L\leq N\). For every boundary matrix \(X\) and complementary configuration \(\tau \),

Let \((A_j)_{j=1}^{g}\) be a basis of normal tensors with bond dimensions \(D_j\), or more generally pairwise inequivalent normalized normal tensors. If \(K{\gt}0\) and \(\sum _jD_j\leq K\), then there is a positive integer \(L\) with \(L+1\leq 3K^5\), in particular \(L\leq 3K^5\), such that

Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Suppose that, for some nonzero \(\zeta \in \mathbb {C}\), \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) for \(N\geq 0\). Then the bond dimensions are equal and there is an invertible matrix \(X\) and a scalar \(\eta \) of unit modulus such that \(B^i=\eta X A^iX^{-1}\) for every physical letter \(i\). This is the gauge recovery used in [ 13 , Appendix A ] .

Let \(N\geq 1\), let \(s\in \mathbb Z/N\mathbb Z\), and suppose matrices \(Y_i\) satisfy \(Y_iA^a=A^aY_{s+i}\) for every \(i\) and physical label \(a\). Then, for every word \(w\) of length \(N\) and every \(i\),

If a cyclic \((L+1)\)-window is represented by a boundary matrix \(Y\), then deleting one endpoint gives

Hence two adjacent cyclic windows whose restrictions agree on the common length-\(L\) overlap have boundary matrices satisfying \(Y_1A^a=A^bY_2\). Consequently, for a finite chain of adjacent overlaps indexed by \(0\le r{\lt}n\), satisfying \(Y_rA^{a_r}=A^{b_r}Y_{r+1}\), one has \(Y_0A^{a_0}\cdots A^{a_{n-1}} = A^{b_0}\cdots A^{b_{n-1}}Y_n\). In particular, when the boundary words are given by \(a_r=\rho (i_0+r)\) and \(b_r=\rho (i_0+r+L+1)\) modulo \(N\), this identity reads \(Y_0A^a=A^bY_n\).

Let \(A\) and \(B\) be tensors of the same bond dimension, and let \(c,e\in \mathbb {C}\) be nonzero. If \(cA\) and \(eB\) are gauge-phase equivalent, then \(A\) and \(B\) are gauge-phase equivalent.

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and let \(m \ge L_0\). If a boundary matrix \(X\) commutes with every length-\(m\) word product, then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(d\ge 1\) and \(g\ge 2\). For \(0\le j{\lt}g\), let \(D_j\ge 1\) and let \(A_j=(A^j_a)_{a=0}^{d-1}\) be an irreducible left-canonical block with normalized self-overlap. Assume that no two distinct blocks are equivalent up to gauge and phase. For every \(0\le j{\lt}g\), let \(\mu _j\ne 0\). Suppose there is a common \(L_0{\gt}0\) such that the words of length \(L_0\) in each block \(A_j\) span \(M_{D_j}(\mathbb {C})\). In this doubly-normalized specialization, assume for every \(0\le j{\lt}g\) that

In the general canonical form, (22) is replaced by the dual fixed-point condition with a positive definite matrix \(\Lambda _j\). If

then the parent-Hamiltonian kernel is exactly the span of the BNT component vectors:

Retain the same nonzero dimensions and weights, irreducibility, gauge-phase inequivalence, common block-injectivity length, and source-length bounds, but do not assume left-canonicality. For every \(0\le j{\lt}g\), assume there is a positive definite matrix \(\Lambda _j\) such that

Then

If an MPS tensor is irreducible, then its transfer map has no nontrivial invariant face.

Let \(D{\gt}0\). If \(A\) is algebraically normal and left-canonical, then

Let \(A\) and \(B\) be gauge equivalent. For every \(L\) one has \(G_L(A)=G_L(B)\).

Let \(\zeta \neq 0\). For every \(L\) one has \(G_L(\zeta A)=G_L(A)\).

Let \(A^0,\ldots ,A^{r-1}\), \(r\geq 2\), be MPS tensors with positive physical and bond dimensions, block injective at a common positive word length \(L_0\), and pairwise inequivalent under virtual conjugation and nonzero scalar multiplication. Let \(B=\bigoplus _j\mu _j A^j\), with every \(\mu _j\ne 0\). For every prescribed range

there is a constant \(\gamma _R{\gt}0\) such that the canonical periodic parent Hamiltonian \(H_{R,N}\) of \(B\) satisfies

for every chain length \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}\). This is the block-injective parent-Hamiltonian gap of [ 17 , Section IV.C ] at the interaction bound of [ 228 , Theorem 12 ] .

Let \(A^0,\ldots ,A^{r-1}\) be normal MPS tensors with positive physical and bond dimensions, pairwise inequivalent under virtual conjugation and nonzero scalar multiplication. Let \(B=\bigoplus _j\mu _j A^j\), where every \(\mu _j\) is nonzero. There is a positive integer \(R_0\) such that, for every \(R\geq R_0\), there are constants \(\gamma _R{\gt}0\) and \(N_R\in \mathbb N\) for which the canonical periodic parent Hamiltonian \(H_{R,N}\) satisfies

for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}\). In particular, at least one positive interaction range has a uniform eventual periodic gap.

Let \((A^i)_{i\in I}\) be normal tensors with positive bond dimensions. Suppose that no two distinct blocks are related by a virtual gauge and a nonzero complex scalar. Then there are tensors \((B^i)_{i\in I}\) of the same respective bond dimensions and positive definite matrices \(\rho _i\) such that each \(B^i\) is primitive and trace preserving, \(\rho _i\) is its fixed point, and \(G_L(A^i)=G_L(B^i)\) for every \(i\) and every length \(L\). The tensors \(B^i\) remain pairwise inequivalent under a virtual gauge and a nonzero scalar.

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then \(A\) is normal (eventually full Kraus rank).

Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist a scalar \(\zeta \neq 0\), a tensor \(B\), and a matrix \(\rho {\gt}0\) such that \(\zeta A\) and \(B\) are gauge equivalent, the pair \((B,\rho )\) is primitive, and

for every chain length \(N\) and every configuration \(\sigma \). The local ground spaces, the parent interactions, and the periodic chain ground spaces of \(A\) and \(B\) also coincide:

for all \(L\) and \(N\), where \(h_L\) is the canonical parent interaction and \(\mathcal G_{N,L}\) the periodic chain ground space.

This is the normalization taken without loss of generality in [ 225 , equations (3.1)–(3.2b), lines 1394–1435 ] and at the start of the proof of [ 228 , Theorem 4, lines 765–770 ] . The gauge here is built from the Perron eigenvector of the adjoint transfer map, giving the trace-preserving orientation. That orientation is what conditions 1 and 2 of [ 228 , Theorem 4, lines 752–758 ] , \(\sum _iA^i(A^i)^\dagger =\mathbb {1}\) and \(\sum _i(A^i)^\dagger \Lambda A^i=\Lambda \) with \(\Lambda {\gt}0\), become after the further gauge by \(\Lambda ^{1/2}\): condition 2 turns into \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\) and condition 1 into the statement that \(\Lambda \) is a positive definite fixed point of the transfer map of \(B\).

For every \(A\), \(L\), and \(N\),

If \(H^{\mathrm{open}}_{N,L}(A)v=0\), then, for every \(i\in I_{L,N}\),

For offsets \(s,t\in \mathbb Z/N\mathbb Z\) and a state \(\psi \) on the periodic chain,

Under the canonical \(\ell ^2\) identification of the \(N\)-site state space, the parent ground space is exactly the kernel of the corresponding parent Hamiltonian.

Assume \(N{\gt}0\) and \(L\le N\). Then \(\ker H_N(A,L)=G_L^{(N)}(A)\). Equivalently, after the canonical Hilbert-space identification, \(\mathcal K_N^{\mathrm{ES}}(A,L)=e_N^{-1}G_L^{(N)}(A)\), where \(\mathcal K_N^{\mathrm{ES}}(A,L)\) is the zero-energy space of \(H_{N,L}^{\mathrm{ES}}(A)\).

Assume \(N{\gt}0\) and \(L\le N\). If a vector \(\psi \) has zero energy for the finite periodic parent Hamiltonian, \(H_N(A,L)\psi =0\), then every cyclic length-\(L\) restriction of \(\psi \) lies in the local MPS ground space. Equivalently, \(\ker H_N(A,L)\subseteq G_L^{(N)}(A)\).

Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) is tensor-irreducible and the transfer map is primitive, then \(O_{AA}(N)\to 1\).

Under the hypotheses of Theorem 13.2.2.14, the spaces \(G_n(A^j)\) form an internal sum for every \(n\ge 3(r-1)(L_0+1)\).

Under the hypotheses of Theorem 13.2.2.11, the spaces \(G_n(A^j)\) form an internal sum for every \(n\ge 3(r-1)(L_0+1)\).

Under the same hypotheses, if \(m\ge 3(r-1)(L_0+1)+1\), then

Under the hypotheses of Theorem 13.2.2.11, let \(m\ge 3(r-1)(L_0+1)+1\). Then

Theorem 13.2.2.12 also shows that the sums at lengths \(m\) and \(m+1\) are internal. This is the direct-sum lemma in the proof of [ 228 , Theorem 12, proof lines 1346–1456 ] .

Let \(r\ge 2\), and let the bond dimensions \(D_1,\ldots ,D_r\) be positive. Suppose that every block \(A^j\) is irreducible and that no two distinct blocks of the same bond dimension are gauge-phase equivalent. Let \(L_0{\gt}0\), and suppose that, for every \(j\),

For every \(j\), let \(\Lambda ^j\) be positive definite and satisfy

Then

Diagonality of \(\Lambda ^j\) is not needed.

Let \(r\ge 2\), and let the bond dimensions be positive. Suppose that the blocks are irreducible, have normalized self-overlap, are pairwise inequivalent up to gauge and phase, and are injective at a common length \(L_0{\gt}0\). Assume both normalization identities

Then for every \(n\ge 3(r-1)(L_0+1)\),

Let \(r\ge 2\), and let the bond dimensions \(D_1,\ldots ,D_r\) be positive. Suppose that every block \(A^j\) is irreducible and that no two distinct blocks of the same bond dimension are gauge-phase equivalent. Let \(L_0{\gt}0\), and suppose every block is injective at length \(L_0\):

For every \(j\), let \(\Lambda ^j\) be positive definite and assume the canonical identities from [ 228 ] :

For every \(n\ge 3(r-1)(L_0+1)\),

No identity \(\sum _a(A^j_a)^\dagger A^j_a=\mathbb {1}\) is assumed for the source blocks. This is the simultaneous spanning estimate used in [ 228 , Theorem 12, proof lines 1424–1456 ] .

Let \(N\geq 2\). Suppose that the simultaneous words of length \(N-1\) in the tensors \(A^j\) span the product of their matrix algebras. Let \(\psi \) be a vector such that, for every cyclic cut \(i\in \mathbb Z/N\mathbb Z\), there are boundary matrices \(X_i^j\) with

Then \(\psi \) belongs to the span of the periodic MPS vectors of the blocks. This is the block-diagonal cyclic closure argument in [ 17 , Section IV.C ] .

Let \(A^0,\ldots ,A^{r-1}\) have a common positive physical dimension and positive bond dimensions. Suppose their length-\(S\) word tuples span the full product matrix algebra, where \(S{\gt}0\), and let \(B=\bigoplus _j\mu _j A^j\) with all \(\mu _j\ne 0\). For every interaction range and chain length satisfying

the periodic local constraints have ground space

This is the kernel of the canonical parent Hamiltonian. It is also the kernel of the sum of any positive terms whose cyclic local kernels agree with those of the canonical parent terms. The result includes the minimal ring \(N=S+1\) and gives the block-injective closure statement of [ 17 , Section IV.C ] at the interaction range specified by the preceding intersection argument.

Let \(B=\bigoplus _j\mu _j A^j\) have nonzero block coefficients and positive physical dimension. Suppose that each block is trace preserving and that the simultaneous words of length \(S{\gt}0\) span the product matrix algebra. For every \(S+1\leq L\leq N\), the periodic parent Hamiltonian satisfies

Let \(A^j\) and \(B^j\) have the same respective bond dimensions, and let all coefficients \(\mu _j,\nu _j\) be nonzero. If \(G_L(A^j)=G_L(B^j)\) for every \(j\), then the assembled tensors \(A=\bigoplus _j\mu _jA^j\) and \(B=\bigoplus _j\nu _jB^j\) satisfy \(G_L(A)=G_L(B)\). Their canonical periodic parent Hamiltonians of range \(L\) coincide at every volume: \(H^{\mathrm{ES}}_{N,L}(A)=H^{\mathrm{ES}}_{N,L}(B)\).

Let \(A\) be a canonical MPS tensor with positive physical dimension, including arbitrary multiplicities of its distinct normal blocks \(B^0,\ldots ,B^{g-1}\). Suppose that their simultaneous words of length \(S{\gt}0\) span the product matrix algebra. For every \(S+1\leq L\leq N\), every periodic parent Hamiltonian with positive interactions and the prescribed length-\(L\) local kernels satisfies

The conclusion includes the minimal ring \(N=L=S+1\) and the canonical orthogonal-projection interaction. This is the block-injective closure conclusion of [ 17 , Section IV.C ] .

Let \(A\) be in canonical form, and let \(B^0,\ldots ,B^{g-1}\) be its distinct normal representatives. Repeated copies and their nonzero coefficients do not change the positive-length local spaces: for every \(L{\gt}0\),

This is the local-space consequence of the canonical decomposition in [ 17 ] .

Let \(A^0,\ldots ,A^{r-1}\) be tensors, let \(p{\gt}0\), and suppose that every \(A^j\) is block injective at length \(p\). Suppose that for every \(k\) there are coefficients \(c_\omega ^{(k)}\) with

Then there is \(L_0\) such that, for every \(L\ge L_0\), the tuples \((A^{0,w},\ldots ,A^{r-1,w})\), with \(|w|=L\), span \(\prod _jM_{D_j}(\mathbb {C})\).

Let \(d{\gt}0\) and \(B=\bigoplus _j\mu _jA^j\), with nonzero coefficients \(\mu _j\). Suppose that the blocks have positive bond dimensions, are trace-preserving and primitive with positive definite fixed points, and are pairwise inequivalent under conjugation and a nonzero scalar. There is \(L_0{\gt}0\) such that for every \(L_0\le L\le N\),

Let \(d{\gt}0\), \(r\ge 2\), and \(B=\bigoplus _{j=0}^{r-1}\mu _jA^j\), with all coefficients nonzero. Suppose that the blocks have positive bond dimensions, are trace-preserving and primitive with positive definite invariant matrices, and are pairwise inequivalent under conjugation and multiplication by a nonzero scalar. Suppose that each block is block injective at the supplied common length \(L_0{\gt}0\). Then for

the open-chain parent Hamiltonian satisfies \(\ker H^{\mathrm{open}}_{N,R}(B)=G_N(B)\). This is the interaction threshold of [ 228 , Theorem 12 ] in trace-preserving coordinates.

Let \(A^0,\ldots ,A^{r-1}\) be trace-preserving primitive tensors with positive bond dimensions and positive definite fixed points. Suppose that no two distinct blocks are related by conjugation and a nonzero complex scalar. Then there is \(L_0\) such that, for every \(L\ge L_0\), the tuples \((A^{0,w},\ldots ,A^{r-1,w})\), with \(|w|=L\), span \(\prod _jM_{D_j}(\mathbb {C})\).

Let \(A=\bigoplus _j\mu _jA^j\) have nonzero coefficients and normal blocks of positive bond dimensions. Suppose that no two distinct blocks are related by conjugation and a nonzero scalar. There are normalized primitive blocks \(B^j\), with positive definite invariant matrices, which remain pairwise inequivalent and satisfy \(H^{\mathrm{ES}}_{N,L}(A)=H^{\mathrm{ES}}_{N,L}(B)\) for every \(L,N\), where \(B=\bigoplus _j\mu _jB^j\).

Let \(A^0,\ldots ,A^{r-1}\) be tensors with positive bond dimensions such that each \(A^j\) is irreducible, satisfies \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\), and has self-overlap tending to \(1\) with the chain length. Suppose that no two distinct blocks are related by a virtual gauge and a nonzero complex scalar. Let \(L_0{\gt}0\) and suppose that every \(A^j\) is block injective at the lengths \(L_0\), \(L_0+1\) and \(S=3(L_0+1)\). Then for all \(k\ne j\) there are coefficients \(c_\omega \) with

Let \(A^0,\ldots ,A^{r-1}\) satisfy \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\) for each \(j\). Suppose that the tuples \((A^{0,w},\ldots ,A^{r-1,w})\), with \(|w|=n\), span \(\prod _jM_{D_j}(\mathbb {C})\). Writing \(S_m=\sum _jG_m(A^j)\), one has

Let \(d{\gt}0\), and let \(B=\bigoplus _j\mu _j A^j\), where every \(\mu _j\) is nonzero and \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\) for every \(j\). Suppose that the simultaneous word tuples span the full product matrix algebra at every length \(n\ge L_0\). For \(L_0+1\le L\le N\), the open-chain parent Hamiltonian with interaction range \(L\) satisfies

Suppose that \(\sum _b (A^b)^\dagger A^b=\mathbb {1}\) and that \(A^b C_a=D_b A^a\) for every pair of physical letters \(a,b\). Then \(C_a=EA^a\), where \(E=\sum _b(A^b)^\dagger D_b\). This is the trace-preserving form of the boundary calculation in [ 228 , Theorem 12 ] .

Suppose that \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\) for every block \(j\). If the simultaneous length-\(L\) word tuples span the product matrix algebra, the same holds at every length \(n\ge L\).

Let \(B=\bigoplus _{j=0}^{r-1}\mu _j A_j\), where \(r\geq 2\), all coefficients are nonzero, and the normalized primitive tensors \(A_j\) are pairwise inequivalent under similarity and phase. Assume positive physical and bond dimensions, positive definite recorded invariant densities, and a common block-injectivity length \(L_0\geq 1\). For every \(R\geq 3(r-1)(L_0+1)+1\), there exists \(\gamma _R{\gt}0\) such that, for every \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}(B)\),

Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Assume that no two blocks are related by a virtual similarity and a phase. Let \(\mu _j\ne 0\) for every \(j\), and set \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). For every prescribed lower bound \(R_*\), there exist \(R\geq \max \{ 1,R_*\} \), \(\gamma {\gt}0\), and \(N_0\) such that the periodic range-\(R\) parent Hamiltonian satisfies

for every \(N\geq N_0\) and every \(v\perp \ker H_{R,N}\). In particular, the block sum admits a parent interaction with a uniform positive periodic gap.

Let \(B=\bigoplus _j\mu _j A_j\), where the coefficients are nonzero and \((A_j)_j\) is a finite family of normalized primitive tensors of positive bond dimension, pairwise inequivalent under similarity and phase. Assume that each recorded invariant density is positive definite and that the physical dimension is positive. There exists \(R_0\geq 1\) such that, for every \(R\geq R_0\), there exist \(\gamma _R{\gt}0\) and \(N_R\) for which

for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}(B)\).

Let \(\sigma \) be positive definite and satisfy \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \). Define

Then \(B\) is trace-preserving: \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

If an MPS tensor \(A\) is irreducible, then its transfer map \(\mathcal{E}_A\) is irreducible.

Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space. A positive operator \(P\) annihilates a vector \(v\) as soon as \(\operatorname {Re}\langle Pv,v\rangle =0\), and consequently, for real weights \(w_i{\gt}0\),

equivalently \((\sum _{i\in I}w_iP_i)v=0\) exactly when \(P_iv=0\) for every \(i\in I\). For the empty family both sums are zero and all three kernels are the full space. This is the analytic kernel step of the blocked-sum comparison in Lemma block-obc of Rozmán–Molnár–Schuch ¹ , in which the blocked parent Hamiltonian is a weighted sum \(\sum _i\gamma _ih_i\) of positive local terms with \(\gamma _i{\gt}0\).

Let \(A\) be a tensor whose length-\(K\) word products span \(M_{D}(\mathbb {C})\). Let \(\{ F_b\} _{b\in B}\) and \(\{ Z_b\} _{b\in B}\) be two families of matrices. If, for every length-\(K\) word \(w\), there is a matrix \(Y_w\) such that \(Z_b A^w = F_bY_w\) for every \(b\in B\), then there is a single matrix \(Y\) such that \(Z_b = F_bY\) for every \(b\in B\).

Let \((A_j)_{j=1}^g\) and \((B_j)_{j=1}^g\) be tensor families with the same physical dimension and the same bond dimension \(D_j\) in each labelled block. Suppose that for every \(j\) there is an arbitrary invertible matrix \(X_j\in \mathrm{GL}_{D_j}(\mathbb {C})\) such that, for every physical index \(i\),

Then, at every fixed length \(L\),

Both directions hold for general invertible gauges without unitarity constraints, allowing span certificates in left-canonical form to be transported to unital gauges.

For every \(n\),

Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This combines the block-injectivity discussion in [ 13 , Section II ] with [ 13 , Appendix C.3, Lemma L ] .

Suppose

and \(S_L(A)=M_{D}(\mathbb {C})\). Then \(S_m(A)=M_{D}(\mathbb {C})\) for every \(m\ge L\). Equivalently, \(L\)-block injectivity propagates to every larger homogeneous length.