Dependencies

Grouping each consecutive \(p\)-tuple of physical indices defines a canonical computational-basis unitary satisfying

Its forward and inverse coordinate formulas are the corresponding reindexings by the blocked-configuration equivalence and its inverse.

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 \(A^{[L]}\).

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 \(N{\gt}0\) 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.

For a length-\(L\) cyclic window on \(\{ 0,\ldots ,N{-}1\} \) starting at \(i\), its support is \(S_i=\{ i,i+1,\ldots ,i+L-1\} \pmod N\), with repeated visits counted once.

For \(\Lambda \in M_{d}(\mathbb {C})\) and an operator \(h\) on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), the deformed interaction is

This is the operator written in [ 17 , Section IV.C ] as \(h'=(A^{-1}\otimes A^{-1})^\dagger h(A^{-1}\otimes A^{-1})\) for the two-site interaction of an isometric tensor network, and as \(h'=({\Lambda ^{-1}}^\dagger )^{\otimes k}h(\Lambda ^{-1})^{\otimes k}\) for a general physical perturbation.

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

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

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

For \(\sigma \in \{ 0,\ldots ,d{-}1\} ^{N}\) and a starting site \(i\), write \(\sigma |_{[i,i+L-1]}\in \{ 0,\ldots ,d{-}1\} ^{L}\) for the word defined by \((\sigma |_{[i,i+L-1]})(r)=\sigma (i+r\bmod N)\) for \(0\le r{\lt}L\).

The boundary map of [ 194 , Equation (5.5) ] defines a bounded linear operator

In the ground-space convention,

The weighted Gram operator on the virtual matrix Hilbert space is

For \(v=\{ v(\mu )\} _{\mu =0}^{d-1}\), write \(v^\sigma =v(\sigma (0))\cdots v(\sigma (N-1))\), with the empty word equal to \(I_D\). Define the complex-linear map \(F_N^v\colon M_{D}(\mathbb {C})\to (\{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C})\) by

This is the boundary map of [ 194 , Equation (5.5) ] , with the word order fixed by [ 194 , Equation (5.4) ] .

At length \(N\), let

This is the full open-chain ground-space range used in the orthogonality step of [ 194 , Lemma 6.2 ] .

For \(B\in M_{D}(\mathbb {C})\), define the special spectator families

These are the families obtained by splitting the length-\(\ell +m+r\) boundary vector at the two sides of the middle block.

Let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\). The normalized rank-one map associated with \(\rho \) is

If \(\operatorname{tr}(\rho )=1\), this becomes \(E_\infty ^\rho (B)=\operatorname{tr}(\rho B)I_D\), which is the limiting map in [ 194 , Equation (5.1) ] .

For \(B\neq 0\), define the boundary norm-square ratio

The lower boundary constant of [ 194 , Lemma 5.3 ] is

Let \(v(\mu )\) denote the product of the FNW letters along a word \(\mu \). For each pair \((\mu ^\ell ,\mu ^r)\), set

More generally, for words \(u\) and \(\tau \) of lengths \(K\) and \(L\),

Thus the left word moves to the left of the boundary matrix and the right word moves to its right.

Suppose that \(\rho \) is positive definite and \(\operatorname{tr}(\rho )=1\). For \(n\in \mathbb {N}\), define

This is the trace-one transfer-remainder quantity formed from the factor and weighted norm in [ 194 , Equations (5.9)–(5.10) ] .

For \(\Phi \in \mathscr H_\rho (\{ 0,\ldots ,d-1\} ^r)\) and \(\Psi \in \mathscr H_\rho (\{ 0,\ldots ,d-1\} ^\ell )\), define physical vectors \(L_{\ell ,m,r}\Phi \) and \(R_{\ell ,m,r}\Psi \) by

Let \(\rho \in M_{D}(\mathbb {C})\). Define

This is the factor multiplying the weighted remainder norm in [ 194 , Equations (5.9)–(5.10) ] .

Let \(A\) be an MPS tensor, with \(A^\mu =v(\mu )^\dagger \) in the notation of [ 194 , Equations (5.2)–(5.3) ] . Its FNW transfer map is

Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and let \(\operatorname{tr}(\rho )\neq 0\). Equip \(M_{D}(\mathbb {C})\) with \(\langle X,Y\rangle _\rho =\operatorname{tr}(\rho X^\dagger Y)\). For a linear map \(F\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), write

The weighted remainder spectral radius is

where the remainder is viewed as a bounded operator on this weighted Hilbert space.

For a finite configuration space \(S\), let

The canonical three-block bijection sends \(((\mu ^\ell ,\mu ^m),\mu ^r)\) to the concatenated configuration \(\mu ^\ell \mu ^m\mu ^r\) and has the evident inverse on such concatenations.

For a complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(R(\mathcal L)\) be the matrix with entries

The operator \(\mathscr R(\mathcal L)\) on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) is represented by the matrix \(R(\mathcal L)\) in the standard orthonormal basis. Thus the row–column pair \(((b,a),(e,c))\) is sent explicitly to the Choi row–column pair \(((c,e),(a,b))\).

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 boundary Gram operator is

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 \(L\).

Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization in column–row coordinates. The Hilbert-space boundary map is

Thus \(\Gamma _L^{\mathrm{ES}}\) has a Euclidean coordinate-space domain, whereas the original map \(\Gamma _L\) has matrix domain \(M_{D}(\mathbb {C})\).

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

For a tensor \(A\) and a half-chain length \(L\), the half-chain family is the matrix \(\Psi \) with rows indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^L\) and columns indexed by boundary pairs \((\alpha ,\beta )\), whose column at \((\alpha ,\beta )\) is the half-chain vector

The complementary family \(\Phi \) is the matrix with rows indexed by boundary pairs and columns by configurations, whose row at \((\alpha ,\beta )\) is the vector \(|\Psi _{\beta ,\alpha }\rangle \) with the two boundary indices swapped.

The half-chain Gram matrix is \(G=\Psi ^{\dagger }\Psi \), the matrix of overlaps \(G_{(\alpha ,\beta ),(\alpha ',\beta ')} =\langle \Psi _{\alpha ,\beta } | \Psi _{\alpha ',\beta '} \rangle \) of the half-chain vectors.

The coefficient kernel of the ring of \(2L\) sites is the matrix over the two half-chain configuration spaces whose entry at \((\sigma ,\tau )\) is the matrix product vector component \(V^{(2L)}(A)_{\sigma \tau }=\operatorname{tr}(A^{\sigma }A^{\tau })\) of the joined configuration.

For \(\lambda _\alpha \geq 0\) with \(\sum _\alpha \lambda _\alpha =1\), let \(\Lambda =\operatorname {diag}(\lambda _\alpha )\), \(G_\infty =\operatorname {diag}_{\alpha \beta }(\lambda _\alpha )\), and \(H_\infty =\Lambda \otimes \Lambda =\operatorname {diag}_{\alpha \beta }(\lambda _\alpha \lambda _\beta )\). The matrix \(H_\infty \) is positive semidefinite and has trace one.

For a Hermitian matrix \(M\) of size \(n\), let \(s_k(M)\) be its \(k\)th largest eigenvalue, counted with multiplicity, for \(0\leq k{\lt}n\), and set \(s_k(M)=0\) for \(k\geq n\). If \(M\geq 0\), all these values are nonnegative.

Write \(W_L=\Phi _L\Phi _L^\dagger =SG_L^{\mathsf T}S\) and define

Here \(\rho _L\) is the actual unnormalized physical reduced matrix. Both normalized matrices are defined as zero when \(Z_L=0\). The physical entanglement spectrum is \(s_k(\widehat\rho _L)\).

The unnormalized half-chain reduced density matrix of the ring state on \(2L\) sites is \(\rho _L=KK^{\dagger }\) with \(K\) the coefficient kernel. Entrywise this is the partial trace of the unnormalized ring state over the complementary half,

and \(\rho _L\) is positive semidefinite.

A finite-dimensional ground space \(S\) is one-dimensional if \(\dim S=1\).

A state \(\psi \) on \(L+1\) sites satisfies the left ground condition if, for every \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\).

A state \(\psi \) on \(L+1\) sites satisfies the right ground condition if, for every \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\).

A state \(\psi \) on \(K+L\) sites lies in the suffix ground condition if every fixed prefix \(u\) gives a suffix state in \(G_L(A)\).

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

For injective \(T\), define the operator \(Q_T:=T S_T T^\dagger \) on its codomain.

For an injective continuous linear map \(T\) with finite-dimensional domain, let \(S_T\) be the continuous endomorphism induced by the inverse linear equivalence of the Gram endomorphism \(T^\dagger T\).

A vector \(\psi \) is a frustration-free ground state of the parent Hamiltonian when each local interaction annihilates it, that is, when \(h_i\psi =0\) for every \(i\in \{ 0,\ldots ,N{-}1\} \).

Following [ 17 , Section IV.C, Definition (Parent Hamiltonian) ] , a parent interaction on \(L\) sites for the tensor \(A\) is a hermitian positive semidefinite operator \(h\ge 0\) on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) whose kernel equals \(G_L(A)\).

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.

For each site index \(i\in \{ 0,\ldots ,N{-}1\} \), the translated local term is determined by

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.

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 local dimension \(d\) and chain length \(N\), the \(N\)-site Hilbert space is \(\mathcal H_{N}^{(d)}:=\{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C}\), identified with functions on the computational basis.

For \(\Lambda \in M_{d}(\mathbb {C})\) and \(L\ge 0\), the tensor power \(\Lambda ^{\otimes L}\) is the matrix on \(L\)-site configurations with entries

It acts on the \(L\)-site space \(\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C}\) by matrix multiplication and on its Hilbert-space realization \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) in the same way.

For every \(L\in \mathbb {N}\), let \(\mathcal U_L(A)\) be the subspace of the \((L+1)\)-site Hilbert space whose corresponding functions satisfy the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\). For every \(K,L\in \mathbb {N}\), let \(\mathcal V_{K,L}(A)\) be the subspace of the \((K+L)\)-site Hilbert space whose corresponding functions satisfy the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\). These definitions include \(L=0\). Let \(P^{\mathrm{left}}_L(A)\) and \(P^{\mathrm{tail}}_{K,L}(A)\) be the orthogonal projectors onto \(\mathcal U_L(A)\) and \(\mathcal V_{K,L}(A)\), respectively.

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

The chain Hamiltonian is

Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). The source parent-Hamiltonian spanning condition is that, for every \(N{\gt}L\), \(\ker H_L^{(N)}(B)=\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). This is the ground-space clause in [ 13 , Definition 3.9 ] . It is separate from the commutation equation \([\tau _j(P_L),P_L]=0\) and from the zero-energy equation for a single periodic MPS vector.

The parent interaction at range \(L\) is the canonical orthogonal projector specified in (14).

For a matrix \(\Lambda \in M_{d}(\mathbb {C})\) and a tensor \(A=(A^i)_{i=0}^{d-1}\), the physically rotated tensor \(\Lambda A\) has matrices

Precomposition with \(r_N\) defines a complex-linear isomorphism

This is a linear map of coefficient spaces and does not require an inner product.

For a length-\(N\) configuration \(\sigma \), define

where \(0\leq j{\lt}N\). This is an involutive permutation of \(\{ 0,\ldots ,d-1\} ^{N}\).

Precomposition with the reversal \(r_N\) defines a unitary operator

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.

Define the tail boundary map in the common \((K+L+Q)\)-site coordinates by

If \(L\le N\) and \(\tau \in \{ 0,\ldots ,d{-}1\} ^{L}\), write \(\sigma ^{[i,i+L-1]\leftarrow \tau }\) for the configuration defined by

The index \(r\) in the first case is unique because \(L\le N\).

Fixing the first physical index \(i\) gives the state \(\sigma \mapsto \psi (i,\sigma _1,\ldots ,\sigma _L)\).

Fixing the last physical index \(j\) gives the state \(\sigma \mapsto \psi (\sigma _1,\ldots ,\sigma _L,j)\).

Given a matrix \(M \in \mathbb {C}^{m\times d}\) and an MPS tensor \(A\) with physical dimension \(d\), the physical-index rotation is the tensor with physical dimension \(m\) given by \((A_M)^i := \sum _j M_{ij}\, A^j\) for \(i=0,\ldots ,m-1\). When the matrix is a unitary \(u\), the rotation is written \(A_u\).

Transport the tail and left boundary maps through the canonical Euclidean identifications on their virtual matrix families and physical coefficient spaces. Denote the resulting continuous maps by \(\Gamma ^{\mathrm{tail}}_{K,L}\) and \(\Gamma ^{\mathrm{left}}_{K,L}\).

Fix a prefix \(u\in \{ 0,\ldots ,d-1\} ^K\). The suffix restriction of an \((K+L)\)-site state \(\psi \) is the \(L\)-site state given by \((R^{\mathrm{tail}}_u\psi )(\sigma )=\psi (u,\sigma )\).

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

Let \(A_i,B_i\) be square virtual blocks, and let \(P_i,Q_i,R_i,L_i\) be arbitrary perturbation blocks of the matching sizes. Define

The regular interpolant \(U(\varepsilon )\) has diagonal blocks \((A_i+\varepsilon P_i)(A_j+\varepsilon P_j)+\varepsilon ^2R_iL_j\) and \((B_i+\varepsilon Q_i)(B_j+\varepsilon Q_j)+\varepsilon ^2L_iR_j\); its off-diagonal blocks are those of \(U\) plus respectively \(\varepsilon (P_iR_j+R_iQ_j)\) and \(\varepsilon (L_iP_j+Q_iL_j)\). In particular \(U(0)=U\).

The word span at length \(n\) is the subspace

This is  [ 233 , Section II ] .

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), where \(\mu _j\ne 0\). Suppose that, for every \(N{\gt}L\), \(G_L^{(N)}(B) =\bigvee _{j=0}^{r-1}\operatorname{span}\{ V^{(N)}(A_j)\} \). Then the source parent-Hamiltonian spanning equation holds: \(\ker H_N(B,L) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \). It is enough to prove the two equations

Let \(A\) be an MPS tensor and let \(L_0,K\ge 0\). Let \(X\) be a matrix, and let \(Y_{c_b}\) be a matrix for each iterated block index \(c_b\) of the complement. Suppose that, for every block letter \(b\) of the alphabet \(\{ 0,\ldots ,d{-}1\} ^{L_0}\) and every iterated block index \(c_b\) of the complement, with \(w(b)\) the length-\(L_0\) word of \(b\) and \(\widetilde{w(c_b)}\) the length-\(L_0K\) complement word obtained by concatenating the blocks of \(c_b\), \(X A^{w(b)} A^{\widetilde{w(c_b)}} = A^{w(b)} Y_{c_b}\). Then there is a family \(Y'_c\), indexed by the length-\(L_0K\) words \(c\), with \(X A^s A^c = A^s Y'_c\) for every length-\(L_0\) word \(s\) and every length-\(L_0K\) word \(c\).

For every \(x\in \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), the Euclidean boundary maps intertwine with the blocked-configuration isometry:

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 ] .

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Suppose two length-\((L_0+1)\) restrictions of a state \(\psi \) at the same cyclic support are represented by \(\operatorname{Res}^\rho _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\rho )\) and \(\operatorname{Res}^\tau _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\tau )\). If these two restrictions are equal, then \(Y_\rho A^j=Y_\tau A^j\) for every physical letter \(j\). In particular, the last-site boundary-crossing condition and the second boundary-crossing condition satisfy, respectively,

Consequently, for every word \(\sigma \) of length \(L_0\), both implications remain true after right multiplication by \(A^\sigma \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\). Suppose that matrices \(Y_r(\rho )\), for \(0\le r\le L_0-1\), represent the length-\((L_0+1)\) cyclic restrictions beginning at the sites \(M+1-L_0+r\): \(\operatorname{Res}^\rho _{M+1-L_0+r,L_0+1}\psi =\Gamma _{L_0+1}(Y_r(\rho ))\). Then

For \(L_0=1\), both products are empty.

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Let \(\rho ^+\) and \(\rho ^-\) be two boundary conditions satisfying \(\rho ^+_{k+L_0}=\mu _k\) and \(\rho ^-_{k+1}=\mu _k\). If the adjacent-window comparison gives, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\), \(Y_M(\rho ^+)A^jA^\sigma =Y_{M+1-L_0}(\rho ^-)A^jA^\sigma \), then the same product equation holds for \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\):

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, and let \(Y_v\) be a matrix for each complementary word \(v\) of length \(K\). Suppose that, for every word \(u\) of length \(L_0\) and every word \(v\) of length \(K\), \(X A^u A^v = A^u Y_v\). Then, for every physical letter \(j\), \(X A^j = A^j X\). This establishes the boundary-matrix commutation step of [ 17 , Section IV.C, lines 2049–2090 ] .

Under the hypotheses of Theorem 11.10.4, one has \(X A^i = A^i X\) for every physical index \(i\).

Let \(L_0{\gt}0\) and \(L_0\le M\), and set

With \(R_j\) as in Lemma 11.11.11,

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, let \(W_\eta \) and \(V_\eta \) be matrix families indexed by a physical letter \(\eta \), and let \(\mu \) be a word. Suppose that, for every physical letter \(j\), \(X A^j = A^jX\). Suppose also that, for every \(\eta \) and every \(j\),

Then, for every \(\eta \) and every \(j\), \(W_\eta A^j = V_\eta A^j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(L_0\le M\). Let \(\psi =\Gamma _{M+1}(X)\), and suppose that matrices \(Y_i(\rho )\) represent all length-\((L_0+1)\) cyclic restrictions of \(\psi \). Then, for every boundary condition \(\rho \) and every physical letter \(j\),

For \(\rho =\tau ^-_\eta (\mu )\), this is the transport identity supplied by the adjacent-window argument, with the single factor \(A^j\) following \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))\). The padded identity, whose corresponding factor is \(A^jA^\sigma \), is supplied separately.

If \(0{\lt}N\), \(L\le N\), and \(\psi \in \mathcal G_{N,L}(A)\), then, for every cyclic window and every outside configuration, there is a boundary matrix \(Y\) such that the corresponding restricted \(L\)-site state is \(\Gamma _L(Y)\).

Let \(A\) be an MPS tensor, let \(L_0{\gt}0\), and let \(L_0\le M\). Suppose the two boundary-crossing restrictions of length \(L_0+1\) are represented by boundary matrices \(Y^+_{\eta ,\mu }\) and \(Y^-_{\eta ,\mu }\):

If, for every physical letter \(j\), \(Y^+_{\eta ,\mu }A^j = Y^-_{\eta ,\mu }A^j\), then \(B^+_{\eta ,\mu }(\psi ) = B^-_{\eta ,\mu }(\psi )\).

Let \(L_0{\gt}0\) and \(L_0\le M\). Let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). If, for every boundary letter \(\eta \) and physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\), then, for every \(\eta \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Suppose that, for every outside letter \(\eta \),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j=A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations

and the product equation

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every word \(\sigma \) of length \(L_0\),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) of length \(M+1-(L_0+1)\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

Let \(A\) be a tensor with virtual dimension at least one whose single-site matrices span the full matrix algebra. Let \(L_0{\gt}0\) and \(L_0{\lt}M\), and let \(\psi =\Gamma _{M+1}(X)\). If every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\), then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

Let \(A\) be a tensor with virtual dimension at least one, and let \(L_0{\gt}0\) and \(L_0{\lt}M\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every physical letter \(j\) and every word \(\alpha \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),

Each side has \(M+2-L_0\) one-site factors, with site labels read modulo \(M+1\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),

For \(L_0=1\), both products are empty. The same equation remains true after multiplying both sides on the right by any matrix \(R\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Then, for every physical letter \(j\) and boundary condition \(\tau \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and suppose the two length-\((L_0+1)\) boundary restrictions are represented by \(\Gamma _{L_0+1}(Y_M(\tau ^+_\eta (\mu )))\) and \(\Gamma _{L_0+1}(Y_{M+1-L_0}(\tau ^-_\eta (\mu )))\). If these two restrictions are equal, then for every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). For every boundary letter \(\eta \), every physical letter \(j\), and every complementary word \(\mu \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let

For every boundary letter \(\eta \) and complementary word \(\mu \), the two restrictions crossing the periodic cut satisfy

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) with \(\psi =\Gamma _{M+1}(X)\). Let \(R_j\) denote restriction to the first physical letter \(j\). For every outside letter \(\eta \), complementary word \(\mu \), and physical letter \(j\), one has

This provides the single-letter product form of the periodic-boundary comparison in [ 17 , Section IV.C, lines 2078–2079 ] .

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),

Then, for every \(\eta \), \(j\), and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every word \(\sigma \) of length \(L_0\),

Then, for every \(\eta \) and \(j\), \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j = A^\mu A^jX\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Assume that for every boundary letter \(\eta \),

Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

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.

Suppose a single family of boundary matrices \(Y_\mu \), indexed by words \(\mu \) on the complementary sites, satisfies the two one-sided identities

for every pair of physical letters \(a,b\). Then \(X\) commutes with every word product \(A^aA^\mu A^b\) obtained by adjoining one letter on each side of the word \(A^\mu \).

If an \(N\)-site state satisfies the \(L\)-site ground condition on every non-wrapping contiguous window, with \(L \ge 2\), then it lies in \(G_N(A)\).

Let \(0{\lt}N\), \(L\le N\), let \(i\in \{ 0,\ldots ,N{-}1\} \), and let \(\tau \in \{ 0,\ldots ,d{-}1\} ^{N}\) be an outside configuration. For every \(\phi \colon \{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C}\), the restriction of \(\Lambda ^{\otimes N}\phi \) to the cyclic window \([i,i+L-1]\) with outside configuration \(\tau \) is

where the sum runs over the configurations \(\omega \in \{ 0,\ldots ,d{-}1\} ^{N}\) that agree with \(\tau \) on the window, and \(c_\tau (\omega )=\prod _{k\notin [i,i+L-1]}\Lambda _{\tau _k\omega _k}\) is the product of the entries of \(\Lambda \) over the sites outside the window.

On a nonempty periodic chain, longer cyclic-window constraints imply all shorter cyclic-window constraints: if \(L'\le L\le N\), then \(\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L'}(A)\).

The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is positive. Consequently, every summand \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is positive, and so is the finite sum over \(\tau \).

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

If \(L\le N\), then restricting to the same cyclic interval after prescribing its values recovers the inserted word: \(\left(\sigma ^{[i,i+L-1]\leftarrow \tau }\right)|_{[i,i+L-1]}=\tau \).

Let \(R_j\) be restriction to first letter \(j\):

For \(\phi ,\psi \in (\mathbb {C}^d)^{\otimes (L+1)}\),

For every \(L\), one has \(\dim G_L(A)\le D^2\).

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 \(XA^i\).

If \(\psi \in G_{K+L}(A)\), then every fixed-prefix suffix restriction of \(\psi \) lies in \(G_L(A)\). For a vector in \(G_{L+1}(A)\), fixing either endpoint gives the expected left or right multiplication of its boundary matrix.

For every boundary matrix \(X\in M_{D}(\mathbb {C})\) and every length-\(L\) word \(\sigma \), one has \(\Gamma _L(X)(\sigma )=\operatorname{tr}(A^\sigma X)\).

If \(d^L{\gt}D^2\), then \(G_L(A)\) is a proper subspace of the local Hilbert space, namely \(G_L(A)\neq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

For a diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\) with \(\operatorname{tr}(\Lambda )\neq 0\), the limiting overlaps form a diagonal matrix in the boundary pairs:

For \(\operatorname{tr}(\Lambda )=1\) this is the display \(\lambda _\alpha \delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}\) of [ 228 ] .

The half-chain overlaps are matrix elements of the \(L\)-fold transfer map on matrix units:

The coefficient kernel factors as the product \(\Psi \Phi \) of the half-chain family with its complementary family. Equivalently,

Assume \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every fixed prefix of length \(K\). Then there exist matrices \((Z_j)_j\) and \((Y_u)_u\) with the two trace representations below and, for all \(j\) and \(u\), the compatibility relation

For each site \(i\), one has \(h_iV^{(N)}(A)=0\).

If \(L\le N\), then the translated term is obtained by applying the local parent interaction on the chosen cyclic window:

For every site \(i\), one has \(h_i(A,L)^2=h_i(A,L)\).

Let \(A\) be an \(e\times d\) matrix and \(B\) a \(d\times e\) matrix, with \(d\leq e\). Then

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

Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the second boundary-crossing reduced cyclic interval has boundary matrices \(Y_\tau \), then \(X A^j C^-_\tau =A^jY_\tau \) for every physical letter \(j\), where \(C^-_\tau \) is the complementary word seen from the second cyclic position.

For every chain length \(N\), one has \(V^{(N)}(A)=\Gamma _N(\mathbb {1})\).

If \(0{\lt}N\) and \(L\le N\), then \(V^{(N)}(A)\in \mathcal G_{N,L}(A)\).

\(V^{(N)}(A)\in G_N(A)\) for every \(N\), with boundary matrix \(X=\mathbb {1}\).

For \(M\in \mathbb {C}^{m\times d}\) and a length-\(N\) physical configuration \(s=(s_0,\ldots ,s_{N-1})\in \{ 0,\ldots ,m-1\} ^N\), physical rotation satisfies

For any window of \(L\) consecutive sites starting at position \(i\) in an \(N\)-site periodic chain with \(L\le N\), the restriction of the periodic MPS vector \(V^{(N)}(A)\) to that window lies in \(G_L(A)\).

The ambient space of \(L\)-site tensors satisfies \(\dim (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})=d^L\).

For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\) one has \((\Lambda \Lambda ')^{\otimes L}=\Lambda ^{\otimes L}\Lambda '^{\otimes L}\), \(\mathbb {1}^{\otimes L}=\mathbb {1}\), and \((\Lambda ^{\otimes L})^\dagger =(\Lambda ^\dagger )^{\otimes L}\). In particular, \(\Lambda ^{\otimes L}\) is unitary when \(\Lambda \) is. If \(\Lambda \) is invertible, then \((\Lambda ^{-1})^{\otimes L}\) is a two-sided inverse of \(\Lambda ^{\otimes L}\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If two boundary matrices have the same product with every one-site tensor on one fixed side, then the two boundary matrices are equal.

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 \(\psi \in S_N\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). If an outside configuration \(\rho \) has the same word \(\mu \) as \(\tau ^+_\eta (\mu )\) on the sites outside the last-site cyclic window, then the two matrices representing that restriction agree: \(Y_\rho = Y_{\tau ^+_\eta (\mu )}\). The same assertion holds for the second boundary-crossing window: if \(\rho \) has the same outside word as \(\tau ^-_\eta (\mu )\), then \(Y_\rho = Y_{\tau ^-_\eta (\mu )}\). This is the boundary-matrix independence step used to choose one representative outside configuration for each outside word in the closure-property matrix comparison.

If \(A\) is \(L_0\)-block-injective, \(q \ge 1\), and \(k \le qL_0\), then the two one-sided annihilation statements are

The second implication is the left-handed form.

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 \(s\).

For every \(N\) with \(L\le N\), the periodic MPS vector satisfies \(H_N(A,L)V^{(N)}(A)=0\).

The periodic MPS vector is a frustration-free ground state: for every site \(i\), one has \(h_iV^{(N)}(A)=0\).

For any \(v\in G_L(A)\), one has \(h_L(A)v=0\).

The Hilbert-space local interaction \(P_L^{\mathrm{ES}}(A)\) annihilates exactly the Hilbert-space local ground space: \(P_L^{\mathrm{ES}}(A)v=0\) if and only if \(v\in G_L^{\mathrm{ES}}(A)\).

The local parent interaction is a projector: \(h_L(A)^2=h_L(A)\).

Let \(B\) and \(C\) have the same physical alphabet and satisfy \(G_L(B)=G_L(C)\). Their periodic chain spaces agree for every \(N\):

Their parent-Hamiltonian kernels agree for every \(N{\gt}L\):

Consequently, any fixed BNT-vector family satisfying the parent-Hamiltonian spanning equation for \(C\) also satisfies it for \(B\).

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 \(\sum _iA_iA_i^\dagger =\mathbb {1}\), let the dual map \(X\mapsto \sum _iA_i^\dagger XA_i\) have a positive definite fixed point, and let every fixed point of \(X\mapsto \sum _iA_iXA_i^\dagger \) be a multiple of the identity. Then no orthogonal projection other than \(0\) and \(\mathbb {1}\) is invariant under every \(A_i\).

For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\),

Rotation by the identity is trivial: \(\mathbb {1}A=A\).

For \(\Lambda \in M_{d}(\mathbb {C})\), \(|V^{(N)}(\Lambda A)\rangle =\Lambda ^{\otimes N}|V^{(N)}(A)\rangle \) and \(\Gamma _L^{\Lambda A}(X)=\Lambda ^{\otimes L}\, \Gamma _L^{A}(X)\) for every boundary matrix \(X\in M_{D}(\mathbb {C})\).

If \(L\le N\), then prescribing on a cyclic interval the values already carried by \(\sigma \) leaves \(\sigma \) unchanged: \(\sigma ^{[i,i+L-1]\leftarrow \sigma |_{[i,i+L-1]}}=\sigma \).

If \(L\le N\), then two prescriptions on the same cyclic interval reduce to the second prescription:

If \(A\) is left canonical, then, for every bond matrix \(X\),

Fix a physical letter \(\eta \) used outside the complementary sites. Suppose the two one-sided boundary matrix families \(Y^+\) and \(Y^-\) have been extracted from the cyclic windows. If, for every word \(\mu \) on the complementary sites, these boundary matrices agree on the boundary conditions built from \(\eta \), \(Y^+_{\tau ^+_\eta (\mu )}=Y^-_{\tau ^-_\eta (\mu )}\), then there is a single family \(Y_\mu \) satisfying

for all letters \(a,b\).

Let \(T_x\) be a family of finite-dimensional boundary maps continuous at \(x_0\). If \(T_{x_0}\) is injective, its range projector is continuous at \(x_0\). Consequently, fix a window length \(L\) and a tensor family continuous at \(x_0\). If the products of exactly \(L\) tensor matrices at \(x_0\) span the full virtual matrix algebra, the canonical parent interaction on \(L\) sites is continuous at that point.

The interpolant is continuous, and for every \(\varepsilon \ne 0\), \(U(\varepsilon )=\mathcal L_{\varepsilon ^{-1}} (C(\varepsilon )\mathbin {\circ }C(\varepsilon ))\).

For \(z\ne 0\), the linear map \(\mathcal L_z\bigl(\begin{smallmatrix} X & Y \\ Z & W \end{smallmatrix}\bigr) =\bigl(\begin{smallmatrix} X & zY \\ zZ & W \end{smallmatrix}\bigr)\) is invertible and preserves the one-site physical span of a tensor.

If a boundary matrix \(X\) commutes with every word product of length \(m\), then it commutes with every word product whose length is a multiple \(q\, m\).

One has \(S_L(A)=M_{D}(\mathbb {C})\) if and only if \(A\) is \(L\)-block injective.

On a chain of length \(N=M+1\), with \(L_0\le M\), fix a physical letter \(\eta \) used outside the complementary sites and a word \(\mu \) of length \(M+1-(L_0+1)\). Write \(\tau ^+_\eta (\mu )\) for the boundary condition at the support crossing the last site and \(\tau ^-_\eta (\mu )\) for the boundary condition at the second boundary-crossing support. Their exposed complements are both exactly \(\mu \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Fix two matrix families \(Y^+\) and \(Y^-\) and the two reindexed boundary conditions \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\). If, for every physical letter \(j\), \(Y^+_{\tau ^+_\eta (\mu )} A^j = Y^-_{\tau ^-_\eta (\mu )} A^j\), then \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the reduced cyclic interval crossing the last site has boundary matrices \(Y_\tau \), then, for every physical letter \(j\), one has

Here \(C^+_\tau \) is the complementary word seen by that cyclic interval.

For a non-wrapping interval \([s,s+M-1] \subseteq \{ 0,\ldots ,N-1\} \) and a full configuration \(\rho \) supplying the complementary values, the restriction map is

These identities include the boundary cases \(s=0\) and \(M=N\), the deletion of the first or last site, and the composition of fixing \(K\) prefix sites followed by restricting to \([s+K,s+K+L-1]\).

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.

Let \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and

then \(K\) is invertible and

The same conclusion follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).

Let \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and (152) holds, then

The same estimate follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).

Let \(B=\bigoplus _{j=1}^{g}\mu _jA_j\), where every \(\mu _j\) is nonzero and every bond dimension is positive. Suppose that the simultaneous one-letter evaluations span the product algebra:

Then, for every \(N{\gt}2\),

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 \(G_L^{\mathrm{ES}}(A)=\iota _L(G_L(A))\).

Let \(A\) be \(L_0\)-block-injective. Let \(X\) be a boundary matrix, and let \(Y_\nu \) be a matrix for each complementary word \(\nu \) of length \(K\). Suppose that, for every word \(\alpha \) of length \(L_0\), every word \(\beta \) of length \(L_0\), and every complementary word \(\nu \),

Then, for every word \(\alpha \) of length \(L_0\) and every complementary word \(\nu \), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) word product, then \(X\) commutes with every matrix in \(M_{D}(\mathbb {C})\).

If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) product \(A^\omega \), then \(X\) already commutes with every length-\(L_0\) product \(A^u\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). Let \(\{ Z_u\} _{|u|=L_0}\) be a family indexed by words of length \(L_0\). If, for some \(K \ge 0\) and every suffix word \(w\) of length \(K\), there exists \(Y_w \in M_{D}(\mathbb {C})\) such that \(Z_u A^w = A^u Y_w\) for every word \(u\) of length \(L_0\), then there exists \(X \in M_{D}(\mathbb {C})\) with \(Z_u = A^u X\) for every word \(u\) of length \(L_0\).

The Euclidean local space of the blocked tensor is carried exactly to the original Euclidean local space:

Hence their orthogonal projectors satisfy

These statements include \(N=0\) or \(p=0\); in those cases the same explicit configuration isometry and boundary-map calculation apply. In particular, the projector identity holds for three blocked sites and \(3p\) original sites. These identities concern the full local ground spaces and their orthogonal projectors under blocking.

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Suppose that, for each physical letter \(j\) and word \(\sigma \) of length \(L_0\), boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfy, for every complementary position \(k\),

If, for every \(j\) and \(\sigma \),

then, for every \(j\) and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every letter \(\eta \) used in the two boundary conditions, every word \(\mu \) on the complementary sites, and every physical letter \(j\),

This is the one-site product equality used to compare the two boundary matrices in the periodic-boundary comparison.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every boundary letter \(\eta \), complementary word \(\mu \), physical letter \(j\), and word \(\sigma \) of length \(L_0\),

Let \(A\) be injective with \(D\ge 1\), and let \(\psi =\Gamma _N(X)\) for some boundary matrix \(X\in M_{D}(\mathbb {C})\). If every cyclic restriction of length \(L\) of \(\psi \) lies in \(G_L(A)\), with \(N\ge 2\) and \(1{\lt}L\le N\), then \(X A^j=A^jX\) for every \(j=0,\ldots ,d-1\).

The canonical parent interaction \(P_L^{\mathrm{ES}}(A)\), the orthogonal projector onto \(G_L(A)^\perp \), is a parent interaction on \(L\) sites for \(A\).

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 \(\Lambda \in M_{d}(\mathbb {C})\) be invertible. Then, for all \(L\) and \(N\),

and for \(0{\lt}N\) and \(L\le N\) the kernels of the parent Hamiltonians satisfy \(\ker H_N(\Lambda A)=\Lambda ^{\otimes N}\ker H_N(A)\). This is the matrix-product-state instance, on the periodic chain, of the one-to-one correspondence, through \(R^{\otimes N}\), between the ground states of the parent Hamiltonians of \(|\psi \rangle \) and \(|\psi '\rangle =R^{\otimes N}|\psi \rangle \) for an invertible map \(R\) stated in [ 17 , Section IV.C ] for tensor network states on a general graph. Without invertibility, the inclusion \(\Lambda ^{\otimes N}\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L}(\Lambda A)\) still holds.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \) be a state on the chain of length \(M+1\) with open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every nonempty complementary word \(\mu \) there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\), depending on the physical letter \(j\) and the word \(\sigma \) of length \(L_0\), such that, for every complementary position \(k\),

and, for every \(j\) and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\sigma ,\alpha \),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations

and the product equation

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\) and every word \(\beta \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation. Assume that, for all \(i\) and \(\tau \), \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )\in G_{L_0+1}(A)\). In local coordinate notation, fix a nonempty word \(\mu \) on the sites outside the two cyclic windows. For every outside letter \(\eta \), choose the two outside configurations so that, for each index \(k\) of this outside word,

and assign the letter \(\eta \) to the remaining sites, following the coordinate parametrization of the periodic-boundary comparison in [ 17 , Section IV.C, lines 2078–2079 ] . The corresponding boundary-crossing restriction equality is

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0\le M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Then, for every boundary letter \(\eta \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites and let \(D \ge 1\). Assume \(N \ge 2\) and \(L_0 {\lt} L \le N\). Suppose that, for every physical letter \(\eta \) used in the two boundary conditions and every boundary matrix obtained from an \(N\)-site chain ground state, the identities

hold for every physical letter \(j\) and every complementary word \(\mu \), and are supplemented, for every \(\mu \), by \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\). Then \(\mathcal G_{N,L}(A)\subseteq \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be in literal CPSV canonical form with positive ambient bond dimension \(D\). There are a basis of normal tensors \((A_j)_{j=1}^{g}\) and an integer \(p\) satisfying \(1\leq p\leq 3D^5\) such that \((A_j^{[p]})_{j=1}^{g}\) is a basis of normal tensors for \(A^{[p]}\) and, for every \(N{\gt}2\),

Thus the actual blocked tensor has a nearest-neighbor parent Hamiltonian on the blocked lattice. This is distinct from Theorem 11.6.6, which supplies a nonzero bounded-range interaction and the parent-Hamiltonian spanning identity on every sufficiently long original chain. The two conclusions and the argument closing the short-ring window are recorded in [ 34 ] .

Let \((A_j)_{j=1}^g\) be a basis of normal tensors, with bond dimensions \(D_j\). There is a positive integer \(L\) such that the one-letter evaluations of the blocked tensors span the full product algebra:

Moreover, there is \(L_0\) such that, for every \(L\geq L_0\), the unblocked word tuples have the simultaneous span

This establishes simultaneous block injectivity following [ 13 , lines 317–345 ] .

Let literal canonical-form data of ambient bond dimension \(D\) have phase classes \(j\) and chosen representatives of dimensions \(D_j\). Then

If \((A_j)_{j=1}^{g}\) is a basis of normal tensors for \(A\) and \(p\geq 1\), then \((A_j^{[p]})_{j=1}^{g}\) is a basis of normal tensors for \(A^{[p]}\).

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

be literal CPSV canonical-form data. There are phase classes \(j\), copy multiplicities \(r_j\), and an equivalence

Each class has a representative \(C_j\). If the copy \((j,q)\) corresponds to the block index \(k\), then its bond dimension equals that of \(C_j\) and there are \(\zeta _k\in \mathbb {C}\), with \(|\zeta _k|=1\), and \(X_k\in \mathrm{GL}_{D_k}(\mathbb {C})\) such that

The copy retains the original coefficient \(\mu _k\), and the representatives \((C_j)_j\) form a basis of normal tensors for \(A\).

Enumerate \(\mathcal G\). The induced equivalence \(\Phi \) between the dependent grouped block coordinates and the original flattened listed coordinates gives

For \(x=(j,q)\), let \(\widehat\mu _x\) be the original weight of that copy and let \(\widehat B_x^i=\zeta _xC_j^i\), with the required dimension identification. Define the grouped tensor by

If \(G=\bigoplus _kX_k\) is the block-diagonal gauge, then, letter by letter,

and

Let \(A\) be in literal CPSV canonical form with positive ambient bond dimension \(D\), and suppose that the physical dimension satisfies \(1{\lt}d\). There are a basis of normal tensors \((A_j)_{j=1}^{g}\) and a positive integer \(L\) such that

and, for every original periodic chain of length \(N{\gt}L\),

Thus the orthogonal complement of the local MPS space is nonzero, and the corresponding bounded-range interaction is a parent Hamiltonian. This is the construction of [ 13 , Definition 3.9 and line 527 ] , with its condition \(d^L{\gt}D^2\) made explicit.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D \ge 1\). If an \(N\)-site periodic-chain state satisfies all cyclic ground-space constraints at some range \(L_0+1\le L\le N\), then it lies in the open-chain ground space \(G_N(A)\).

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 \(\Lambda \in M_{d}(\mathbb {C})\) be invertible and let \(h\) be a parent interaction on \(L\) sites for \(A\). Then the deformed interaction \(h'\) of (217) is a parent interaction on \(L\) sites for \(\Lambda A\): \(h'\ge 0\) and \(\ker h'=G_L(\Lambda A)\).

Let \(I\) be a finite index set and let \(B=(B_{ij})_{i,j\in I}\) be a complex matrix. Then the squared operator norm of \(B\) on \(\ell ^2(I)\) satisfies

Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then \(S_{D^4}(A)=M_{D}(\mathbb {C})\); equivalently, \(A\) is \(D^4\)-block injective.

Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and define \(P_\rho (X):=(\operatorname{tr}X/\operatorname{tr}\rho )\rho \). Then

If \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, then \(R(P_\rho )\) is positive definite and \(\mathscr R(P_\rho )\) is invertible, hence injective.

Suppose that \(\operatorname{tr}(\rho )=1\), \(A\) is left canonical, \(\mathcal{E}_A(\rho )=\rho \), and both \(L_{\ell ,m,r}\Phi \) and \(R_{\ell ,m,r}\Psi \) are orthogonal to \(\mathcal G_{\ell +m+r}\). Then, for every \(B\in M_{D}(\mathbb {C})\),

Consequently,

Under the hypotheses of Theorem 11.3.4.2, if \(a_-(m){\gt}0\), then

This is the quadratic contribution in the proof of [ 194 , Lemma 6.2 ] .

Under the hypotheses of Theorem 11.3.4.2, if \(a_-(m){\gt}0\), then

For all \(B,C\in M_{D}(\mathbb {C})\),

For all \(B,C\in M_{D}(\mathbb {C})\),

These are the two forms of the scalar-product identity in [ 194 , Equation (5.8) ] . Identity (61) preserves the source orientation \(B^\dagger E_{ik}C\).

For every \(B\in M_{D}(\mathbb {C})\) and every length-\(N\) configuration \(\sigma \),

Set \(A_\mu =v(\mu )^\dagger \). Then

For \(N{\gt}0\), the word on the right is \(A_{\sigma (N-1)}\cdots A_{\sigma (0)}\).

With the exact convention \(A_\mu =v(\mu )^\dagger \), one has

For every \(B,C\in M_{D}(\mathbb {C})\),

In particular,

No sign assumption on \(a(N)\) is needed for the last inequality.

For all \(B,C\in M_{D}(\mathbb {C})\),

This is the estimate in [ 194 , Equation (5.9) ] . It contains no factor depending only on the dimension.

For every \(B\in M_{D}(\mathbb {C})\),

This recursion follows from the boundary formula in [ 194 , Equation (5.5) ] and is used in the proof of [ 194 , Lemma 5.3 ] . Here \(A^\mu =v(\mu )^\dagger \).

Suppose that \(\operatorname{tr}(\rho )=1\), \(A\) is left canonical, \(\mathcal{E}_A(\rho )=\rho \), and \(a_-(m){\gt}0\). Then

This is the linear estimate in [ 194 , Equation (6.5) ] . Positivity of the actual lower boundary constant is assumed directly.

Suppose that \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \). Then

Suppose that \(\rho \) is stationary for the Schrödinger transfer map, \(\mathcal{E}_A(\rho )=\rho \). Then

For every \(B\in M_{D}(\mathbb {C})\),

As an endomorphism of \(M_{D}(\mathbb {C})\), the limiting map satisfies

Define the fixed-point projection

Then \(E_\infty ^\rho =P_\rho ^\sharp \), where \(\sharp \) denotes the adjoint for the bilinear trace pairing. For the trace-one Hermitian density matrix used by FNW, the same map also agrees with the Hilbert-space adjoint for the unweighted Frobenius inner product. This Frobenius adjoint is generally different from the adjoint for the \(\rho \)-weighted inner product of [ 194 , Equation (5.6) ] .

Suppose that \(A\) is block injective at length \(L_0\). Then

If \(\mathcal{E}_A(\rho )=\rho \), then for every \(m\geq L_0\),

This is the positivity input used in the proof of [ 194 , Lemma 6.2 ] .

The lower boundary constant satisfies

Consequently, \(a_-(N){\gt}0\) whenever \(a(N){\lt}1\). This is the lower estimate in [ 194 , Lemma 5.3 ] . No equality \(a_-(N)=1-a(N)\) is asserted.

If \(\mathcal{E}_A(\rho )=\rho \), then

for every \(N\in \mathbb {N}\). Hence \(N\mapsto a_-(N)\) is nondecreasing. The same conclusion holds for every primitive MPS tensor with positive-definite fixed point \(\rho \).

For every \(B\in M_{D}(\mathbb {C})\),

More generally, a real number \(c\) satisfies

The operator \(G_N\) is self-adjoint and positive. For every nonzero \(B\in M_{D}(\mathbb {C})\),

Let \(m=\dim _{\mathbb {C}}M_{D}(\mathbb {C})\). There is an index \(i\in \{ 0,\ldots ,m-1\} \) such that

and \(a_-(N)\leq \sigma _j(F_N)^2\) for every \(j\in \{ 0,\ldots ,m-1\} \). The indices range only over the finite virtual domain dimension. No zero-padded sequence indexed by all natural numbers is used.

The real number \(a_-(N)\) is a spectral value of \(G_N\), and every \(z\in \operatorname {spec}(G_N)\) satisfies

Thus \(a_-(N)\) is the least spectral value in the sense of real parts. The spectrum is a subset of \(\mathbb {C}\), where no compatible order is fixed, so the assertion is stated as spectral membership together with (76).

Let \(A\) be primitive with nonzero bond dimension and positive-definite trace-one fixed point \(\rho \). Let \(\lambda \geq 0\) satisfy \(r_\rho (A){\lt}\lambda \). There is a constant \(C_\lambda {\gt}0\) such that, for every integer \(n\geq 1\),

This proves the prescribed-rate transfer-remainder estimate in [ 194 , Lemma 5.2 ] . The condition \(\lambda {\lt}1\) is needed only when one subsequently deduces decay to zero.

The weighted operator norm is nonnegative, and hence so is \(a(n)\) for every \(n\in \mathbb {N}\).

The weighted matrix identity

gives the source aggregates

Then

Consequently, the difference of these two pairings is the sum of the length-\(m\) boundary defects over all spectator pairs.

Under the hypotheses of Theorems 11.3.3.10 and 11.3.4.4,

The two source contributions are deliberately left unfactored.

If \(\operatorname{tr}(\rho )=1\), then

If, in addition, \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \), then

The overlap norms satisfy

The optimal lower-boundary inequality also gives

If \(\mathcal{E}_A(\rho )=\rho \) and \(a_-(m){\gt}0\), monotonicity from \(m\) to \(\ell +m\) and from \(m\) to \(m+r\) gives

Under the hypotheses of Theorem 11.3.4.6,

This is the coefficient in [ 194 , Lemma 6.2 ] .

Under the hypotheses of Theorem 11.3.4.6, suppose in addition that \(a(m)\leq c\lambda ^m{\lt}1\). Then

Let \(A\) have nonzero bond dimension, let \(\rho \) be positive definite with \(\operatorname{tr}(\rho )=1\), and suppose that \(A\) is left canonical and \(\mathcal E_A(\rho )=\rho \). Suppose that \(L_0\geq 1\) is an injective length. For every \(r\geq 0\), every \(\ell \geq 1\) and every \(m\geq L_0\),

Let \(A\) be a primitive MPS tensor of nonzero bond dimension with fixed-point witness \(\rho \), and set \(R_A^\rho =E_{\mathrm{FNW},A}-E_\infty ^\rho \). If \(\nu \) is an eigenvalue of \(R_A^\rho \), then

This is the norm-independent spectral step in [ 194 , Lemma 5.2 ] .

Suppose that \(A\) is left canonical and that \(\mathcal{E}_A(\rho )=\rho \). For every integer \(n\geq 1\),

The restriction \(n\geq 1\) is essential. At \(n=0\), the two sides would be \(I\) and \(I-E_\infty ^\rho \), respectively, and \(E_\infty ^\rho (I_D)=I_D\).

Suppose that \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \). For every \(N\geq 0\) and \(X\in M_{D}(\mathbb {C})\),

Consequently,

If \(\rho \in M_{D}(\mathbb {C})\) is positive definite and \(D{\gt}0\), then \(c_\rho {\gt}0\).

If \(A\) is left canonical, then the FNW observable map in (17) satisfies

If \(A\) is left canonical, then

Under \(A^\mu =v(\mu )^\dagger \), this is the normalization \(\sum _\mu v(\mu )v(\mu )^\dagger =I_D\) from [ 194 , Equation (5.3b) ] .

Write \(\mathcal{E}_A^\sharp \) for the adjoint of \(\mathcal{E}_A\) with respect to the bilinear trace pairing, characterized by

Then \(E_{\mathrm{FNW},A}=\mathcal{E}_A^\sharp \). Here \(\sharp \) denotes the adjoint for \((X,Y)\mapsto \operatorname{tr}(XY)\), so the theorem is phrased as a bilinear trace-adjoint identity. For the unweighted Frobenius inner product, the same map also agrees with the Hilbert-space adjoint of \(\mathcal{E}_A\). This Frobenius adjoint is generally different from the adjoint for the \(\rho \)-weighted inner product in [ 194 , Equation (5.6) ] .

Let \(A\) be primitive with nonzero bond dimension and positive-definite fixed-point witness \(\rho \). Then

Let \(A\) be any tensor. If

then \(\Gamma _{L_0}(XA^j)=\Gamma _{L_0}(A^jY)\) for every physical letter \(j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(\psi \in \mathcal G_{L_0+1,L_0+1}(A)\) satisfy \(\psi =\Gamma _{L_0+1}(X)\). Then \(XA^j=A^jX\) for every physical letter \(j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If

then \(XA^j=A^jY\) for every physical letter \(j\).

If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), then

Let \(V\) be a complex Banach space, let \(T\colon V\to V\) be a bounded linear operator, and let \(\lambda \geq 0\). If \(\rho _{\mathrm{spec}}(T){\lt}\lambda \), then there is a constant \(C_\lambda {\gt}0\) such that

for every \(n\in \mathbb {N}\) and \(x\in V\).

Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that

for every \(n\in \mathbb {N}\) and \(x\in V\).

Let \(A\) be a complex Banach algebra, let \(a\in A\), and let \(\lambda \geq 0\). If \(\rho _{\mathrm{spec}}(a){\lt}\lambda \), then there is a constant \(C_\lambda {\gt}0\) such that

for every \(n\in \mathbb {N}\). This operator-theoretic estimate provides the prescribed-rate geometric decay used for the peripheral-spectrum remainder of the transfer map in [ 194 , Lemma 5.2 ] .

Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that

for every \(n\in \mathbb {N}\).

Let \(c{\gt}0\), let \(0{\lt}r{\lt}1\), and let \(l,K\in \mathbb N\). If \(cr^l{\lt}1\), then

For every complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the sum of squared entry norms of \(R(\mathcal L)\) equals that of the rectangular Choi matrix:

The two matrices differ only by a reshuffling of indices; the equality is entrywise Frobenius, not operator norm.

Let \(I\) be a finite index set, equip \(\mathbb C^{I\times I}\) with its \(L^2\) operator norm, and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Denote the operator norm of \(\mathcal L\) for these matrix norms by \(\lVert \mathcal L\rVert _{\mathrm{op},L^2}\). Then

Let \(I\) be a finite index set and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Then

Let \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be complex-linear and let \(\mathbf e_{(j,i)}\) be the standard unit vector at \((j,i)\). For all \(a,b,c,e\in \{ 0,\ldots ,D{-}1\} \),

This is the coordinate identity in (35); it is not a Hilbert–Schmidt pairing of \(E_{ab}\) with \(\mathcal L(E_{ce})\).

For complex-linear maps \(\mathcal L,\mathcal M\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),

Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then the physical orthogonal projector onto the local ground space is

Thus the displayed operator is the physical range projector on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If an element \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every prefix of length \(K\), then \(\psi \in G_{K+L_0+1}(A)\).

If \(A\) is injective and \(L \ge 1\), then \(\dim G_L(A) = D^2\).

Let \(\mathbf e_{(j,i)}:=\operatorname {single}_{(j,i)}(1)\) denote the standard unit vector in \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), and let \(\mathcal E_A\) be the transfer map. Then

Under Frobenius vectorization, \(\mathbf e_{(b,a)}=U(E_{ab})\) and \(\mathbf e_{(e,c)}=U(E_{ce})\), explaining the reversed coordinate order. Thus the Gram operator is the indicated Choi reshuffling; in particular it is not obtained by replacing the right-hand side with \(\langle E_{ab},\mathcal E_A^L(E_{ce})\rangle _{\mathrm{HS}}\).

The ground-space Gram operator is exactly the operator-level reshuffling of the Choi matrix of the iterated transfer map:

The equality is an equality of operators on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\). Its entries are those in (37), rather than Hilbert–Schmidt pairings with \(\mathcal E_A^L\).

Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the fixed-point projection

Suppose that \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and

Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),

No positivity, injectivity, or uniqueness hypothesis is imposed.

Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the corresponding fixed-point projection. If \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and \(\rho _{\mathrm{spec}}(\mathcal E_A-P_\rho ){\lt}1\), then

in operator norm as \(n\to \infty \).

Let \(\psi \) be a vector on \(K+1\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K}(Y_i)(\tau )\) for all letters \(i\) and words \(\tau \) of length \(K\), and \(A^jY_i=Z_jA^i\) for all letters \(i,j\). If matrices \(R_j\in M_{D}(\mathbb {C})\) satisfy \(\sum _jR_jA^j=\mathbb {1}\), then \(\psi \in G_{K+1}(A)\).

For injective \(A\) and \(L \ge 2\), \(G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}} = G_{L+1}(A)\), equivalently, \(\psi \in G_{L+1}(A)\) if and only if \(\psi \in G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}}\).

If \(A\) is injective and \(L \ge 2\), then the nontrivial inclusion in the intersection property holds:

Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization, with the target indexed in column–row order, and let \(\iota _L\) be the computational-basis identification on the physical coefficient space. Then, for every \(X\in M_{D}(\mathbb {C})\), \(\Gamma _L^{\mathrm{ES}}(U(X))=\iota _L(\Gamma _L(X))\).

If \(A\) is \(L\)-block injective, then \(\Gamma _L^{\mathrm{ES}}\) is injective.

Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then \(\Gamma _L^{\mathrm{ES}}\) is injective.

If \(A\) is \(L\)-block injective, then the injective range projector of \(\Gamma _L^{\mathrm{ES}}\) is the physical orthogonal projector onto the local ground space:

In particular, this is the physical range projector on the local Hilbert space, not the virtual transfer fixed-point projector.

Let \(e_{(b,a)}\) denote the Frobenius coordinate vector representing the virtual matrix unit \(E_{ab}\). Then \(\Gamma _L^{\mathrm{ES}}(e_{(b,a)})=\iota _L(\Gamma _L(E_{ab}))\).

If \(A\) is injective and \(L \ge 1\), then \(\Gamma _L\) is injective.

If \(A\) is \(L_0\)-block injective, then \(\Gamma _{L_0}\) is injective.

If the length-\(L\) products \(\{ A^\sigma : |\sigma | = L\} \) span \(M_{D}(\mathbb {C})\), then \(\Gamma _L\) is injective.

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites. Suppose there are matrices \(Y_\mu \) such that

for every pair of physical letters \(a,b\). Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

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 \(A\) be injective after blocking \(L_0{\gt}0\) sites. If \(X\) commutes with every word product of some positive length \(m\), then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and fix a physical letter \(\eta \) used outside the complementary sites. Suppose that, for every boundary condition \(\tau \) and every physical letter \(j\),

where \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic windows. Suppose also that the two boundary matrices satisfy \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\), for every word \(\mu \) on the complementary sites. Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

For every \(\Lambda \in M_{d}(\mathbb {C})\) and \(L\ge 0\),

and the same identity holds for the Hilbert-space realizations of the two local ground spaces.

Let \(\psi \) be a vector on \(K+2\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K+1}(Y_i)(\tau )\) and \(\psi (\tau ,j)=\Gamma _{K+1}(Z_j)(\tau )\) for all letters \(i,j\) and all words \(\tau \) of length \(K+1\). Then

for all letters \(i,j\).

For an injective tensor \(A\) with \(D \ge 1\), if \(N \ge 2\) and \(1 {\lt} L \le N\), then the periodic chain ground space is one-dimensional.

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

For \(B_i=A_i^\dagger \), the half-chain Gram matrices satisfy \(G_L(B)_{(\beta ',\alpha '),(\beta ,\alpha )} =G_L(A)_{(\alpha ,\beta ),(\alpha ',\beta ')}\) at every length.

The characteristic polynomials of the half-chain reduced density matrix and of the boundary comparison matrix \((SG^{\mathsf T}S)G\) of size \(D^2\) agree after matching the trivial kernel factors:

In particular the nonzero spectrum of \(\rho _L\), with multiplicities, is that of the boundary comparison matrix.

Let \(A\) have trace-preserving transfer map with diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\), \(\operatorname{tr}(\Lambda )\neq 0\), satisfying the spectral condition (249). Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) the boundary comparison matrix satisfies, entrywise,

The limiting matrix is the diagonal matrix \(\Lambda \otimes \Lambda \) up to the trace normalization. Together with Theorem 11.12.8, the nonzero spectrum of the half-chain reduced density matrix on the ring of \(2L\) sites is that of a \(D^2\times D^2\) matrix converging entrywise, geometrically in \(L\), to \(\Lambda \otimes \Lambda /\operatorname{tr}(\Lambda )^2\). Theorem 11.12.1.7 proves convergence of the normalized physical eigenvalues.

For a TI MPS satisfying Condition C2, the source asserts convergence of the half-chain reduced spectrum to the doubled canonical fixed-point spectrum  [ 228 , Theorem 6 ] . For a general weighted canonical representation, passing to its dominant block and controlling the subleading physical contribution remains separate from Theorem 11.12.2.3; see [ 63 ] .

Let \(A\) have trace-preserving transfer map \(\mathcal{E}_A\) with fixed point \(\Lambda \), \(\operatorname{tr}(\Lambda )\neq 0\), and suppose

where \(P_{\Lambda }\) is the fixed-point projection. Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) and all boundary pairs,

The hypothesis (249) is the spectral gap condition of [ 228 , Theorem 6 ] in the trace-preserving gauge: apart from the simple eigenvalue one, the spectrum of \(\mathcal{E}_A\) lies strictly inside the unit disc. The rate \(r\) is chosen strictly between the spectral radius of \(\mathcal{E}_A-P_{\Lambda }\) and one.

Suppose \(\mathcal{E}_A\) is trace preserving, fixes \(\Lambda \), and satisfies (249). Then \(Z_L\to 1\), so \(Z_L{\gt}0\) for all sufficiently large \(L\). The actual normalized physical eigenvalues obey

Repeated eigenvalues are counted with multiplicity, and both lists use the zero-extension convention of Definition 11.12.1.1. No invertibility of \(\Lambda \) or finite-length normalization is assumed.

Let \(M\geq 0\) have size \(n\) and \(N\geq 0\) have size \(m\). If \(X^m\chi _M(X)=X^n\chi _N(X)\), then \(s_k(M)=s_k(N)\) for every \(k\geq 0\).

For every finite half-chain length \(L\), \(H_L\geq 0\) and

The same characteristic-polynomial identity holds before normalization. Both normalized matrices are positive semidefinite, and \(s_k(\widehat\rho _L)=s_k(\widehat H_L)\) for every \(k\). In particular, the physical spectrum vanishes for \(k\geq D^2\). These statements include \(d^L{\lt}D^2\), singular Gram matrices, and zero-norm lengths. If \(Z_L{\gt}0\), then \(\operatorname {tr}\widehat\rho _L=1\).

Let \(S\) denote the swap of the two boundary indices. The unnormalized half-chain reduced density matrix factors exactly as

where \(SG^{\mathsf T}S\) is the Gram matrix of the complementary family.

Let \(A\) satisfy the three one-block canonical conditions of [ 228 , Theorem 4 ] : \(\sum _i A_iA_i^\dagger =I\), a faithful diagonal dual fixed point \(\Lambda =\operatorname {diag} (\lambda _\alpha ){\gt}0\), and \(\{ X:\mathcal{E}_A(X)=X\} =\mathbb C I\). Normalize \(\operatorname {tr}\Lambda =1\) and the transfer spectral radius to one. Assume Condition C2: every unit-modulus transfer eigenvalue is one. Then \(A\) is normal. For every \(k\in \mathbb N\), the ordered physical half-chain eigenvalues, with multiplicities and zero extension, satisfy \(s_k(\widehat\rho _L)\to s_k(\Lambda \otimes \Lambda )\). This gives the one-block canonical half-chain conclusion of [ 228 , Theorem 6 ] without an additional normality or complementary-gap assumption. The hypothesis that this is a single canonical block is explicit; reduction of a general weighted canonical representation is not asserted here.

Let \(A\) be a normal tensor in the unital canonical convention, \(\sum _i A_iA_i^\dagger =I\), and let \(\Lambda =\operatorname {diag}(\lambda _\alpha )\geq 0\) have trace one and satisfy \(\sum _i A_i^\dagger \Lambda A_i=\Lambda \). Then \(Z_L\to 1\) and \(Z_L{\gt}0\) for all sufficiently large \(L\). For every \(k\in \mathbb N\), the actual physical reduced density obeys \(s_k(\widehat\rho _L)\to s_k(\Lambda \otimes \Lambda )\). This statement assumes normality explicitly. The source canonical conditions and Condition C2 are treated in Theorem 11.12.2.3. The index and decay conventions are detailed in [ 63 ] . The intermediate Gram limit is \(\operatorname {diag}_{\alpha \beta }(\lambda _\beta )\).

An increasing real function acts termwise on the ordered eigenvalues of a Hermitian matrix. In particular, adding \(tI\) adds \(t\) to each eigenvalue, and multiplication by \(t\geq 0\) multiplies each eigenvalue by \(t\). If \(M\) and \(N\) are Hermitian matrices of the same size, then \(|s_k(M)-s_k(N)|\leq \lVert M-N\rVert _{\mathrm{op}}\) for every \(k\). Consequently, convergence of Hermitian matrices of fixed size implies convergence of every zero-extended ordered eigenvalue.

For finite-dimensional \(S\), \(\dim S=1\) if and only if there exists a nonzero \(\psi _0\in S\) such that every \(\psi \in S\) is of the form \(c\, \psi _0\).

If \(T\) is an injective map from a finite-dimensional Hilbert space into a complete Hilbert space, then

Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. If \(T\colon E\to F\) is an injective continuous linear map, then the inverse Gram operator is a two-sided inverse of the Gram endomorphism:

Consequently, \(S_T\) is both the canonical inverse and the ring inverse of \(T^\dagger T\). The two pseudoinverse factors satisfy

Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. Let \(T\colon E\to F\) be an injective continuous linear map. If \(a\in \mathbb R\) satisfies \(a{\lt}1\) and

then

Let \(K_i\) be a net in a normed ring with convergent geometric series. If \(K_i\to K_0\) and \(K_0\) is invertible, then

Let \(I\) be a finite index set. For every \(i,j\in I\), the matrix unit \(E_{ij}\) satisfies

Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal U_L(A)\) exactly when the corresponding function satisfies the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\).

Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\).

For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),

where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.

If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(N\ge L_0+1\), then \(V^{(N)}(A)\ne 0\) in the \(N\)-site space.

If \(\lVert t\rVert \le a{\lt}1\) in a normed ring with convergent geometric series and \(\lVert 1\rVert =1\), then

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

Let \(A\) be \(L_0\)-block-injective with \(D \ge 1\) and \(L_0 {\gt} 0\). If an \(N\)-site state satisfies the ground-space constraint on every non-wrapping contiguous interval of length \(L_0+1\), for every fixed choice of the complementary sites, with \(N \ge L_0+1\), then it lies in \(G_N(A)\).

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 \(\mathcal G_{N,L}(A) \subseteq \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

If \(D\ge 1\), \(A\) is \(L_0\)-block-injective, and \(L_0{\gt}0\), then

In Nachtergaele’s notation, with \(n=K+L_0\) and \(l=L_0\), this is the common range identity for \(G_{\Lambda _n}\) and \(G_{\Lambda _{n+1}\setminus \Lambda _{n-l}}\) inside \(\Lambda _{n+1}\).

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

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 \(H_{N,L}^{\mathrm{ES}}(A)\).

If \(D \ge 1\), \(A\) is injective after blocking \(L_0{\gt}0\) sites, and \(L_0 {\lt} L \le N\), then the parent Hamiltonian of interaction range \(L\) has a unique periodic ground state. These inequalities imply \(N\ge 2\) and \(N\ge L_0+1\), extending the finite-size range \(N\ge 2L_0\) of [ 228 , Theorem 10 ] .

Let \(A\) be an irreducible MPS tensor in unital gauge \(\sum _iA^i(A^i)^\dagger =\mathbb {1}\), with \(D{\gt}0\). Then there exist a unitary \(U\) and a diagonal positive definite \(\Lambda \) such that, for \(B^i:=U^\dagger A^iU\),

Moreover, \(A\) and \(B\) generate the same MPV family. This is the dual-map fixed-point and unitary-diagonalization step of [ 228 , Theorem 4 ] .

Let \(\Lambda :\mathbb C^d\to \mathbb C^m\) be a linear map. For every word \(\sigma \) of length \(N\) in the \(m\)-letter alphabet,

where \(\tau \) ranges over words in the \(d\)-letter alphabet.

For every coefficient function \(\psi \) and every configuration \(\sigma \),

The fixed-point projection \(P\) depends only on the ray through \(\rho \): for every nonzero scalar \(c\) with \(\operatorname{tr}(c\rho )\neq 0\), the projection associated to \(c\rho \) equals the projection associated to \(\rho \). Write

for the normalized representative of the ray through a matrix of nonzero trace. Then \(\operatorname{tr}(\sigma )=1\); if \(\rho {\gt}0\) then also \(\sigma {\gt}0\); and if \(A\) is primitive with positive-definite fixed point \(\rho \), then \(A\) is primitive with fixed point \(\sigma \).

A one-site suffix increment needs no reassociation, so the open-chain projector defect at overlap length \(l\) and prefix length \(K\) equals the whole-increment defect with \(r=K\), \(m=l\), and \(\ell =1\). Consequently, for a primitive tensor with positive-definite fixed point there are \(c{\gt}0\), \(0{\lt}\lambda {\lt}1\), and an injective length \(L_0\geq 1\) such that, whenever \(l\geq L_0\) and \(c\lambda ^l{\lt}1\), for every prefix length \(K\),

This establishes the projector defect bound (132) for primitive matrix product states.

Let \(A\) be primitive with positive-definite fixed point \(\rho \) and \(\operatorname{tr}(\rho )=1\). Then there is an injective length \(L_0\geq 1\) such that the bound (128) holds whenever \(m\geq L_0\) and \(\ell \geq 1\), for every \(r\geq 0\).

Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There are constants \(c{\gt}0\) and \(0{\lt}\lambda {\lt}1\) and an injective length \(L_0\geq 1\) such that, whenever \(m\geq L_0\) and \(c\lambda ^m{\lt}1\), the bound (133) holds for every prefix length \(r\) and every \(\ell \geq 1\).

Let \(A\) be a primitive MPS tensor of nonzero bond dimension with fixed-point witness \(\rho \). Then, for every integer \(n\geq 1\),

Let \(A\) be a primitive MPS tensor whose fixed-point witness \(\rho \) is positive definite, and put \(\mathcal K_\infty =\mathscr R(P_\rho )\). For every \(0{\lt}a{\lt}1\), all sufficiently large \(n\) satisfy

Let \(A\) be a primitive MPS tensor with fixed-point witness \(\rho \). Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),

Let \(A\) be a primitive MPS tensor of nonzero bond dimension whose fixed-point witness \(\rho \) is positive definite. Put

There are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that, for every \(n\geq 1\),

Whenever \(cr^n{\lt}1\), the operator \(\mathcal K_n\) is invertible and

Under the same primitive-MPS and positive-definite fixed-point hypotheses,

in operator norm.

If \(A\) is a primitive MPS tensor with fixed-point witness \(\rho \), then

in operator norm as \(n\to \infty \).

Under the same hypotheses, for every \(0{\lt}a{\lt}1\) and all sufficiently large \(n\), the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective and its inverse Gram operator is \(\mathcal K_n^{-1}\), satisfying

Under the hypotheses of Theorem 11.4.19, there are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that (164) holds for every \(n\geq 1\). If \(cr^n{\lt}1\), then the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective, its inverse Gram operator equals \(\mathcal K_n^{-1}\), and

The range of \(\Gamma _L^{\mathrm{ES}}\) is exactly \(G_L^{\mathrm{ES}}(A)\).

A one-site suffix increment needs no reassociation:

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Suppose \(\psi \in \mathcal G_{N,L}(A)\), \(N\ge 2\), \(L_0{\lt}L\le N\), and \(\psi =\Gamma _N(X)\) for a boundary matrix \(X\). Then there are two families of boundary matrices \(Y^+_\tau \) and \(Y^-_\tau \), indexed by fixed complement values \(\tau \), such that, for every physical letter \(j\),

Here \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic intervals, both of reduced length \(L_0+1\).

If \(\mathcal{E}_A(\rho )=\rho \), then every \(B\in M_{D}(\mathbb {C})\) satisfies

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If matrices \((Z_j)_j\) satisfy \(Z_jA^\sigma =A^jY_\sigma \) for every word \(\sigma \) of length \(K\), then there exists a matrix \(X\) such that \(Z_j=A^jX\) for all \(j\).

Equip each virtual matrix fiber with the unweighted Hilbert–Schmidt inner product and flatten the finite family of fibers into a Euclidean space. The resulting continuous tail and left boundary maps have ranges \(\mathcal V_{K,L}(A)\) and, at suffix length one, \(\mathcal U_K(A)\). If \(J^{\mathrm{tail}}_K(X)_u=XA^u\) and \(J^{\mathrm{left}}_L(X)_\tau =A^\tau X\), then

Let \(E:\mathbb C^d\to \mathbb C^m\) be a linear map and let \((EA)^i=\sum _jE_{ij}A^j\). Write \(T_L=E^{\otimes L}\). Then \(\Gamma _L^{EA}(X)=T_L\Gamma _L^A(X)\) and \(G_L(EA)=T_LG_L(A)\) for every boundary matrix \(X\) and every \(L\ge 0\). If \(E^\dagger E=I\), then \(T_L^\dagger T_L=I\). More generally, adjoints distribute over finite tensor products: \((\bigotimes _j E_j)^\dagger =\bigotimes _j E_j^\dagger \). These identities apply to the physical inclusions in [ 236 , Section II.F.2 ] .

Let \(I\) be a finite index set, let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix, and let \(j\in I\). Then

Let \(I\) be a finite index set and let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix. Then

For all \(\rho ,X\in M_{D}(\mathbb {C})\),

This is the explicit trace-pairing form of the convention \(A^\mu =v(\mu )^\dagger \) for the map in [ 194 , Equations (5.2)–(5.3) ] .

If \(A\) is injective, then \(\ker \Phi _A = \{ 0\} \).

Suppose the explicit uncle tensor \(U\) is injective. The actual canonical two-site parent interactions of \(C(\varepsilon )\) converge, in operator norm as \(\varepsilon \to 0\) through nonzero complex parameters, to \(h'_P=\Pi [U]\) on the original two physical sites. At every fixed periodic length \(N\ge 2\), the actual parent Hamiltonians converge in operator norm to \(H'_P=\sum _{i=1}^N(h'_P)_{i,i+1}\). The Hamiltonian limit also holds along real perturbation parameters.

Let \(\Lambda \in M_{d}(\mathbb {C})\) be invertible. If \(\dim \mathcal G_{N,L}(A)=1\), then \(\dim \mathcal G_{N,L}(\Lambda A)=1\).

If \(A\) becomes injective after blocking \(L_0 {\gt} 0\) sites and \(D \ge 1\), the parent Hamiltonian with interaction range \(2L_0\) has a unique ground state on every periodic chain with \(N \ge 2L_0\). In particular, this includes \(L_0=1\) and \(N=2\).

If \(A\) is normal, becomes injective after blocking \(L_0 {\gt} 0\) sites, and \(D \ge 1\), then the parent Hamiltonian with interaction range \(L_0 + 1\) has a unique ground state on every periodic chain with \(L_0+1\le N\).

Let \(\rho \) be positive definite with \(\mathcal{E}_A(\rho )=r\rho \) for some \(r{\gt}0\). For \(B^i:=r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}\),

If \(r=1\), then the unscaled gauge \(B^i:=\rho ^{-1/2}A^i\rho ^{1/2}\) is unital and generates the same MPV family as \(A\). This is the spectral-radius normalization and full-rank fixed-point gauge of [ 228 , Theorem 4 ] .

Let \(\rho \in M_D(\mathbb {C})\) be positive definite. The \(\rho \)-weighted inner product on \(M_D(\mathbb {C})\) is

This is the auxiliary inner product in [ 194 , Equation (5.6) ] .

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 \(b\in B\).

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 ] .

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(N\ge 2\), \(L_0+1{\lt}N\), and \(L_0{\lt}L\le N\), and let \(\psi \in \mathcal G_{N,L}(A)\) have an open-chain representation \(\psi =\Gamma _N(X)\). Suppose \(Y^{+}\) and \(Y^{-}\) are two families of matrices obtained from the two boundary-crossing cyclic-window constraints, and that they satisfy, for every physical letter \(j\) and every boundary condition \(\tau \),

For every physical letter \(\eta \in \{ 0,\ldots ,d{-}1\} \) and every word \(\mu \) on the complementary sites, define two boundary conditions by

Then the two families agree on these two boundary conditions: \(Y^{+}_{\tau ^{+}_{\eta }(\mu )} = Y^{-}_{\tau ^{-}_{\eta }(\mu )}\). This is the boundary-matrix comparison required for equality of the two restrictions crossing the periodic cut.