Dependencies

Under the hypotheses of Theorem 20.15.2.27, every absorbed BNT element \(\mathcal K_s\) generates positive semidefinite operators on all nonempty periodic chains. Indeed, \(P_s^{\otimes N}\sigma ^{(N)}(K)P_s^{\otimes N}\geq 0\) and \(n_s{\gt}0\), so (54) gives \(\sigma ^{(N)}(\mathcal K_s)\geq 0\). This is the first positivity conclusion in  [ CPGSV16 , Appendix C.2, lines 1753–1759 ] .

Let \(M\) be an MPO tensor and let its BNT canonical form have normal representatives \(A_x\), copy numbers \(n_x\), and copy-independent weights \(\mu _x\). Put \(\mathcal K_x=\mu _xA_x\), and assume that this canonical form generates the same matrix product vectors as \(M\) at every positive length. If the representatives \(\mathcal K_x\) have orthogonally supported commuting sector products, then

Hence the BNT copy numbers \(n_x\) are the natural multiplicities of the explicit GSNNCH form.

Let \(M\) have a nonempty basis-of-normal-tensors decomposition with positive copy numbers \(n_x\) and copy-independent weights \(\mu _x\). Put \(K_x=\mu _xA_x\), and suppose that the decomposition generates the same positive-length matrix product vectors as \(M\). If an independently supplied orthogonally supported commuting sector family realizes the tensors \(K_x\), and every \(K_x\) satisfies SAL, then \(M\) satisfies SAL.

Let \(M\) be an MPO tensor and let its BNT canonical form have normal representatives \(A_x\), copy numbers \(n_x\), and copy-independent weights \(\mu _x\). Put \(\mathcal K_x=\mu _xA_x\), and assume that this canonical form generates the same matrix product vectors as \(M\) at every positive length. Equivalently, for every \(N{\gt}0\),

Under the BNT projector-selection hypotheses, if every \(\mathcal K_x\) satisfies SAL, then \(M\) has the source GSNNCH form. Its outer sectors are the BNT sectors and its natural multiplicities are the copy numbers \(n_x\).

Under the one-letter product-algebra spanning hypothesis, there is a simultaneous left inverse for which every three-site family closure has the preceding diagonal-sector contraction and vanishing off-diagonal contractions.

Let a BNT decomposition have a common nonzero weight within each sector, and suppose that its one-site words span the product of the virtual matrix algebras. For every sector \(s\), assume a physical-sector factorization with positive neighboring operators. If distinct absorbed representatives obey the unstarred layer identities, then there are pairwise orthogonal projections \(P_s\) such that

for every \(s,\beta ,\alpha \).

Let \({\cal K}\) be injective. Let \(R\) and \(\rho \) give a three-site closure of \({\cal K}\), choose a Hayashi decomposition of \(\rho \), and fix virtual indices \(\alpha _1,\beta _3\) such that \(R_{\beta _3,\alpha _1}\ne 0\). For the corresponding closed sector tensors, \(\mathcal T_{\cal K}=\sum _k |l_k)(r_k|\).

For every MPO tensor \(K\) and every chain length \(N\),

If \(K\) generates MPDOs and \(N\geq 1\), then

Every injective MPO tensor satisfying SAL admits an \(\eta \)-local structure. Thus there is a single positive two-site bond whose cyclic translates commute pairwise and whose ordered product is the finite-chain density operator at every length \(N\geq 2\), with normalization scalar one. This is the conclusion of Proposition C.8.

For every ancillary dimension, the spin reduction obtained by tracing the ancillary index is trace-preserving and completely positive.

Let \(M\) be in normalized BNT-refined horizontal form, generate positive semidefinite ring operators, and satisfy the local renormalization fixed-point equations of Definition 4.1. Then \(M\) saturates the area law.

Let \(M\) be in normalized BNT-refined horizontal form, generate positive semidefinite ring operators, and satisfy the local renormalization fixed-point equations of Definition 4.1. Then \(\mathcal T_M^2=\mathcal T_M\) and \(M\) saturates the area law.

Under the hypotheses of Theorem 20.14.8, there exist simultaneously a one-site vertical decomposition of \(M\) and a two-site vertical decomposition of \(M^{[2]}\) of the respective forms stated in Theorems 20.14.8 and 20.14.9.

Let \(K\) be injective and satisfy SAL, and suppose that its doubled-index matrix-product tensor is normal. There is a positive physical-sector factorization whose selected two-site bond \(B\) has \(K\) itself as a normal fixed tensor representation. Its virtual dimension is the virtual dimension of \(K\), and for every \(N\geq 2\), \(\rho _N(K)=B_0B_1\cdots B_{N-1}\). In particular, the scalar is one. This is an existential statement about the bond selected in the proof of Proposition C.8; it does not assert uniqueness or normal rescalability for an arbitrary proportional commuting-bond presentation.

Let \(K_x\) be a nonempty finite family, let an orthogonally supported commuting sector family realize the tensors \(K_x\), and let \(n_x\) be positive natural numbers. Suppose that, for every \(N{\gt}0\),

If every \(K_x\) satisfies SAL, then \(M\) satisfies SAL.

Every injective MPO tensor satisfying SAL admits a physical-sector factorization such that, for every \(N\geq 1\) and every cyclic sector configuration \(k=(k_0,\ldots ,k_{N-1})\) with \(k_N=k_0\),

This is the projected-chain inequality in [ CPGSV16 , Appendix C.2, Lemma C.4, lines 1446–1450 ] . It does not assert positivity of the individual neighboring operators.

Every injective MPO tensor satisfying SAL admits a physical-sector factorization. This is only the raw algebraic factorization from Lemma C.4 and equations formK and etarl; coherent positivity of the neighboring operators, recurrence, and primitivity are not part of this conclusion.

Let \(\mathcal K\) be an injective normal matrix product density operator tensor satisfying the strong area law and literal zero correlation length,

Then there is a physical-sector factorization

with

positive semidefinite neighboring operators \(\eta _{k,h}\), and real families \((a_k)_k,(b_k)_k\) such that

and

The orientation of \(U\) is the physical-sector-factorization convention displayed above. No additional closure property of the sector tensors is asserted. This is the structural corollary at [ CPGSV16 , Appendix C.2, lines 1503–1506 ] .

Let \({\cal K}\) be an injective MPDO, let \(R\) give a three-site closure, and fix an \(\eta \)-structure and a nonzero tail entry. Let \(N\geq 1\). For a cyclic sector assignment \(k_0,\ldots ,k_{N-1}\), suppose that every concrete operator \(\eta _{k_n,k_{n+1}}\) obtained from these data is nonzero. Then there are nonzero scalars \(c_n\) such that

This is the specialization of Theorem 20.15.4.35 to the inverse-map sector operators; positivity of the cyclic product follows from the MPDO property.

A simple canonical-form tensor with scale-invariant source zero correlation length has non-nilpotent basis transfers and weights independent of the copy index.

A simple canonical-form tensor (Definition 20.11.20) with zero correlation length has non-nilpotent basis transfers and weights independent of the copy index.

Let \(D\geq 1\), and let \(M\) generate diagonal positive semidefinite operators at every positive chain length. If \(L+R{\gt}0\) and the trace at length \(L+R\) is positive, then the classical mutual information of its normalized diagonal coefficients satisfies \(I(X:Y)\leq 2\log D\).

Every nondegenerate purification renormalization fixed point satisfies the positive-length global purification predicate of Definition 4.3 in [ CPGSV16 , lines 744–758 ] .

For a block of \(L\) spin–ancilla pairs, regroup

and trace the ancillary factor. Denote this map by \(\operatorname{Tr}_{\mathrm{anc},L}\). It is the blockwise analog of the one-site ancillary trace in Definition 20.6.3.

For a fixed system size \(N\), a block length \(L\le N\), and an MPO tensor \(M\) such that \(\rho ^{(N)}(M)\) is positive semidefinite, let

denote the normalized state and the reduced state of the first \(L\) of \(N\) spins. The \(L\)-block entropy is \(S_L^{(N)}(M)=S(\rho _L^{(N)}(M))\).

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

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

Equivalently,

This is the fixed-length form of the block-injective canonical-form condition of [ CPGSV21 , lines 2113–2117 ] .

The coefficient form of a basis of normal tensors: each block \(A_j\) is a CPSV normal tensor (Definition 8.6.1), the MPV of \(A\) is given at every positive length by explicit 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. This is the literal coefficient-form definition of [ CPGSV16 , Section 2.3 ] . It differs from Definition 9.1.1 only in the chosen normality predicate. Theorem 9.2.5 proves the implication between the two forms.

A BNT sector presentation of a tensor \(A\) is a nonempty sector decomposition \(P\) such that every representative has a positive number of copies, every copy has nonzero weight, every representative \(C_j\) is normal, the representative matrix product vectors are eventually linearly independent, and, for every \(N{\gt}0\),

This records the nonzero canonical summands constructed at [ CPGSV16 , lines 217–246 ] : repeated copies and their weights remain explicit.

Let \(A\) have bond dimension \(D\). It is in the canonical form of [ CPGSV16 , 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 [ CPGSV16 ] is a separate normalization of these data, not part of canonical-form membership:

Every weight \(\mu _k\) is nonzero. This is the reading of [ CPGSV16 , 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 9.1.1; it is not part of canonical form itself.

Identify the local four-dimensional spin with two qubits and write \(U=\sigma _x\otimes \sigma _x\). Starting from one Bell pair on every bond, apply independently at each site the channel

Equivalently, use the local purification with Kraus amplitudes \(\sqrt3/2\) for the identity and \(1/2\) for the correlated flip. This is distinct from the repeated-left expression printed in the source.

Place one maximally entangled qubit pair on each bond of the four-spin ring and apply, independently at every spin, the corrected correlated flip with flip probability one quarter. A joint flip configuration \(s\in \{ 0,1\} ^4\) has probability

and its bond pattern \(t\in \{ 0,1\} ^4\) is \(t_n=s_n\oplus s_{n+1}\), with spin five identified with spin one. Write \(q\) for the pushforward of \(P\) to bond patterns. The weight family of a window of \(k\) consecutive spins, \(1\le k\le 3\), assigns to each configuration of its \(k-1\) interior bond labels and its two boundary qubits one quarter of the marginal probability of those interior labels; the four-spin window weight family is \(q\) itself.

Identify the ambient one-site basis with \(\{ |a,b\rangle :a,b\in \{ 0,1\} \} \). Define

Then \(\iota \) is injective and \(V^*V=I_2\). For a binary configuration \(\sigma \), write \(\iota \sigma \) for its sitewise image in the ambient basis.

For a binary word \(x=(x_1,\ldots ,x_L)\), let \(t(x)\) be the number of indices \(n\in \{ 1,\ldots ,L-1\} \) for which \(x_n\ne x_{n+1}\).

The coefficient matrix extracted from the four diagonal neighboring operators printed in [ CPGSV16 , Example 4.11, lines 907–924 ] is

In the effective binary encoding, the supported label \(k\in \{ 0,1\} \) selects the ambient two-qubit basis vector \(|k,k\rangle \). Let \(M\) be the cyclic-edge tensor for the ket–bra diagonal edge coefficient induced by \(W\). For a positive cycle length \(N\), define

This is a physical-dimension-\(2\) support model. The ambient physical-dimension-\(4\) realization is obtained by the support inclusion defined below; the two physical spaces are not identified by an equivalence.

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

This is the diagonal entry of \(\sigma _z^{\otimes N}\) at \(\boldsymbol \sigma \).

For the state in [ CPGSV16 , Example 4.12, lines 932–938 ] , at \(N=4\) regard the sites as \(A=\{ 0\} \), \(B=\{ 1,3\} \), and \(C=\{ 2\} \). Encode the two binary labels \((b_1,b_3)\) as one four-dimensional label \(b\), and write \(z_a\in \{ 1,-1\} \) for the eigenvalue of \(\sigma _z\) at \(a\). In the basis \((a,b,c)\leftrightarrow (a,b_1,c,b_3)\), let \(\rho _{ABC}\) be the normalized four-site state and set

Let \(M\) be the tensor with \(d=D=2\) whose only nonzero components are

and let \(\sigma _z=|0\rangle \langle 0|-|1\rangle \langle 1|\).

For a physical letter \(a\in \{ 0,1\} \), define its Pauli sign by

For \(n\in \mathbb {N}\) and \(\alpha ,\beta \in \{ 0,1\} \), define

These are the transition counts governed by the recurrence in [ CPGSV16 , Appendix D, lines 2131–2149 ] .

For \(n{\gt}0\) and \(\alpha ,\beta \in \{ 0,1\} \), define the matrix \(\rho ^{(n)}_{\alpha \beta }(A)\), indexed by physical words \(\sigma \) and \(\tau \), by

Closing the two boundary labels gives the transition count

This is the boundary-label closure used in [ CPGSV16 , Appendix D, line 2157 ] .

This is the concrete tensor in [ CPGSV16 , Appendix D, lines 2116–2121 ] . Its physical labels are triples \(i,j,k\in \{ 0,1\} \) and its virtual labels are \(\alpha ,\beta \in \{ 0,1\} \). The fusion weights satisfy

Every weight outside this zero set is positive, and

The canonical support tensor takes every nonzero \(N_{ijk}\) to be one.

The transition matrix from [ CPGSV16 , Appendix D, lines 2146–2149 ] is

For an MPS tensor \(A\) on bond space \(\mathbb {C}^D\), define the associated MPO tensor \(M\) on bond space \(\mathbb {C}^{D^2}\) by \(M^{ij}=A^i\otimes \overline{A^j}\). This is the single-ancilla purification tensor of [ CPGSV16 ] .

Two tensors \(A\) and \(B\) are eventually nonzero proportional if, for all sufficiently large \(N\), there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This auxiliary form is used with the eventual linear-independence statement Lemma E.1.10.

For square matrices \(A_n\) indexed by \(n=0,\ldots ,N-1\), possibly with different index types, their finite Kronecker product is the matrix with entries

Let \(R_1\) identify an \((N+1)\)-tuple with its first coordinate and the remaining \(N\)-tuple. Let \(R_2\) identify an \((N+2)\)-tuple with its first two coordinates and the remaining \(N\)-tuple.

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

In tensor-network notation, (5) is represented by

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

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

Given per-block gauges \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\), form the block-diagonal element \(\bigoplus _k X_k\) and then reindex the dependent block-coordinate space with the standard bond index \(\{ 0,\ldots ,\sum _k D_k-1\} \) of the assembled tensor. The result is an element of \(\mathrm{GL}_{\sum _k D_k}(\mathbb {C})\).

An MPO tensor is in normalized BNT-refined horizontal form if its doubled-index MPS tensor is a BNT sector decomposition with an independent gauge on every repeated copy:

Here the \(A_j\) are the distinct minimal representatives, \(r_j{\gt}0\), and every weight \(\mu _{j,q}\) is nonzero, \(|\mu _{j,q}|\leq 1\), and at least one weight has modulus one. The BNT hypotheses include irreducibility, left-canonicality, normalized self-overlap, eventual linear independence of the representative MPV families, and the exclusion of similarity up to phase between distinct representatives. Repeated gauge-equivalent copies are retained over the same representative rather than separated as distinct blocks. The total bond dimension of the displayed direct sum is exactly \(D\). Equivalently, with the block-diagonal matrix \(X=\bigoplus _{j,q}X_{j,q}\),

This is a strengthened representative-grouped form of the canonical decomposition in [ CPGSV16 , Section 2.3 ] . In contrast with Definition 8.6.7, the retained direct sum has the full bond dimension, every copy weight is nonzero and normalized, and gauge-equivalent normal blocks are grouped over a BNT representative with a separate gauge for each copy. It is therefore not the literal CPSV canonical-form hypothesis.

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

An MPO tensor \(M\) satisfies the local purification RFP condition if it is an LPDO whose purifying tensor \(A\), viewed as a matrix product state tensor on the combined spin–ancilla index, is a pure-state renormalization fixed point. Thus

This local condition is motivated by the purification tensor formula [ CPGSV16 , lines 744–747 ] , but it is a one-site tensor-level condition rather than the global finite-chain definition in Definition 20.6.5.

An MPO tensor is a nondegenerate purification renormalization fixed point if it is the local ancillary contraction of a pure-state renormalization fixed point and \(\mathcal T_M=\sum _i M^{ii}\ne 0\). The nonzero condition records the nondegeneracy implicit in the density operators of [ CPGSV16 , lines 744–786 ] and excludes the zero purifying tensor.

An MPO tensor \(M\) is a renormalization fixed point if there exist two trace-preserving completely positive maps \(\mathcal{S}, \mathcal{T}\) on the physical indices such that \(\mathcal{S}[M_2(X)]=M_1(X)\) and \(\mathcal{T}[M_1(X)]=M_2(X)\) for every virtual operator \(X\), where \(M_1(X)_{ij}=\operatorname{tr}(M^{ij}X)\) and \(M_2(X)_{(i_1i_2)(j_1j_2)} =\operatorname{tr}(M^{i_1j_1}M^{i_2j_2}X)\) are the one- and two-site physical operators obtained by closing one or two tensors with \(X\). This condition is distinct from idempotence of the doubled-index transfer map for general mixed states. Its source zero-correlation-length consequence concerns the physical-trace transfer and is proved below. In [ CPGSV16 , Definition 4.1, lines 638–660 ] , this local condition is imposed on a tensor already in canonical form and generating matrix product density operators. The definition above isolates the two channel equations; results that use the source’s standing hypotheses state them separately. On the open coefficient tensors the two maps have types:

An MPO tensor \(M\) that generates an MPDO and whose positive-length finite-chain operators \(\rho ^{(N)}(M)\) all have nonzero trace (so the normalized states are well-defined) saturates the area law if \(I_L=I_{L+1}\) for all \(1\le L{\lt}\lfloor N/2\rfloor \) and all \(N\); that is, the mutual information is constant, \(I_1=I_2=\cdots =I_{\lfloor N/2\rfloor }\). This is [ CPGSV16 , Definition 4.6, line 811 ] .

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

Equivalently, its Kraus map is trace-preserving.

An MPO tensor is an LPDO if there exist a Kraus dimension \(d_K\), an inner bond dimension \(D'\), a bond-space identification \(e:\{ 0,\ldots ,D-1\} \simeq \{ 0,\ldots ,D'-1\} \times \{ 0,\ldots ,D'-1\} \) (equivalently, \(D\) and \((D')^2\) have the same cardinality), and matrices \(A^{(i,k)}\in M_{D'}(\mathbb {C})\) such that, for all physical indices \(i,j\),

where \(\otimes \) denotes the Kronecker product, \(\overline{(\cdot )}\) denotes entrywise complex conjugation, and \((\cdot )_{e,e}\) denotes reindexing along \(e\) back to \(\{ 0,\ldots ,D-1\} \). See [ CPGSV16 , Section 4.3 ] . The local contraction is

\( = \)

The auxiliary predicates state two matrix-theoretic ingredients used in the rank-one step of [ CPGSV16 , Appendix C.2, Lemma C.5 ] : constant traces of all positive powers and existence of a rank-one factorization \(T=ab^\top \).

The proposed implication from primitivity and constant trace powers to a rank-one factorization is not a valid general theorem. A concrete \(3\times 3\) counterexample appears in the archive. The corrected criterion proved here adds positive semidefiniteness and trace normalization, which provide the diagonalizability needed to turn the trace-power condition into a rank-one factorization.

An MPO tensor is an MPDO if, for every \(N\geq 1\),

This is the global positivity condition of [ CPGSV16 , Section 4 ] .

For sector weights \(p_k\), let \(I_*=\{ k:p_k\ne 0\} \). The one-site matrices and their spaces on \(I_*\) are

The real trace matrix on this index set is

Only the trace matrix is restricted: the physical direct sum still contains every zero-weight summand.

For a matrix \(X\) on the spin–ancilla space, the ancillary trace map is

This is the one-site trace over ancillary degrees of freedom appearing in [ CPGSV16 , lines 751 and 761–764 ] .

Let \(M_c^{il}\in M_{D_c}(\mathbb {C})\) be a finite labelled MPO tensor family. A simultaneous block left inverse consists of coefficients \(C_{(c,x,y),(i,l)}\) such that

This is the common block inverse used in [ CPGSV16 , Appendix C.2, lines 1666–1676 ] and [ BMW\(^{+}\)17 , lines 269–277 ] .

Suppose that every entry of the one-site multiplicity matrices, every entry of the two-site multiplicity matrices, and every entry of the matrices \(\chi _{\alpha ,\beta ,\gamma }\) is positive. Suppose also that \(\sum _i\mu _{\alpha ,i}=m_\alpha \) for every \(\alpha \). If, for every \(\gamma \) and all sufficiently large \(L\),

then the entries on the two sides agree as multisets. Together with the one-site trace identity \(\sum _i\mu _{\alpha ,i}=m_\alpha \), the one-site and two-site multiplicity entries and their dimensions define a BNT multiplicity-spectrum comparison.

Positivity makes all entries nonzero. Eventual equality of two finite sums of geometric sequences gives, for every \(L\geq 1\),

The finite power-sum identity determines both the cardinality and the multiset of entries.

An active-support fusion coisometry family consists of the same labelled tensors and positive diagonal matrices, together with maps \(U_{\alpha ,\beta }\) satisfying \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), the forward fusion identity

and exact reconstruction

Zero-dimensional retained sums are permitted. If in addition \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) for every pair, this family gives the preceding full-support fusion-isometry family.

Local fix (Figure-11 fusion coisometry): The source uses the retained-row orientation of Proposition 4.13. Thus its fusion maps are coisometries onto the active sectors; exact reconstruction records the discarded common zero corner. This convention is documented in [ con26j ] .

A BNT-label trace-scalar family and a BNT-label coefficient system have idempotent coefficient form when, for every BNT label \(\gamma \),

This is the idempotent condition in [ CPGSV16 , Theorem 4.14(ii) ] , stated separately from the trace-power formula.

For a family \(\{ \mathcal K_x\} _{x=0}^{g-1}\), write \(\mathcal K_x[\beta ,\alpha ]\in M_d(\mathbb {C})\) for the physical matrix obtained by fixing the two virtual indices. The layers are orthogonal when, for \(x\ne y\),

for every choice of virtual indices. Equivalently, for all physical indices \(i,j\),

The factor order is exactly \(\mathcal K_y\mathcal K_x\); no adjoint occurs. This is equation AppKxKy=0 in [ CPGSV16 , Appendix C.2, lines 1634–1639 ] .

For a normal sector \(s\) and a Markov sector \(k\), let

The assertion \(O_s^{(k)}\ne 0\) means that this block is nonzero for at least one pair of virtual indices \(\beta ,\alpha \).

Let \(C_x\) and \(C_y\) be two rows of a simultaneous left inverse, and choose a Hayashi decomposition of the three-site operator. In its \(k\)-th Markov sector, set

These are the two factors obtained in the projected contraction of  [ CPGSV16 , Appendix C.2, lines 1719–1725 ] .

Under the hypotheses of Theorem 20.15.2.10, define

This is equation Pis, restricted to the positive-weight Markov support retained here.

For sector tensors \((l_k)_\beta \) and \((r_k)_\alpha \), close their physical legs and set

This is the pairing operator displayed in [ CPGSV16 , Appendix C.2, lines 1473–1493 ] .

Tracing the two physical legs of a sector tensor leaves a vector over the virtual bond:

These are the closed tensors of [ CPGSV16 , Appendix C.2, lines 1473–1477 ] .

Suppose the weights over each representative do not depend on the copy index. Choose the common weight \(\mu _j\) from any copy over \(j\) and absorb it into the representative:

This is the convention of [ CPGSV16 , Appendix C.2, equation CFK, lines 1660–1665 ] .

For a fixed chain length \(N \ge 2\), this consists of one positive semidefinite two-site operator \(B\ge 0\) whose translated copies commute pairwise on the periodic \(N\)-site chain: \([B_{i,i+1},B_{j,j+1}]=0\).

Fix a BNT label \(s_0\). Each positive-weight Markov sector retains its unique BNT label, while every zero-weight Markov sector is assigned the label \(s_0\). If \(\widehat S_s\) is the set of Markov sectors with completed label \(s\), define

Let \(H=\bigoplus _{k,h}H_{k,h}\). Orthogonally controlling the completed preparations over the sector pairs gives a map

where \((A_{k,h,a})_a\) is a Kraus family for \(\overline{\mathcal S}_{k,h}\) and the tilde denotes its extension by zero outside \(H_{k,h}\). Thus the map discards inter-sector coherences and applies the completed preparation in each diagonal sector. On active sectors this is the sector-control formula for \(\mathcal S_1\) in [ CPGSV16 , Appendix C.2, lines 1548–1555 ] .

For every sector pair define

On an active pair this is precisely the preparation \(\mathcal S_{k,h}\) of [ CPGSV16 , Appendix C.2, lines 1551–1555 ] ; the second branch gives a normalized extension on a pair where the displayed source quotient is undefined.

Orthogonally controlling the completed direct-sum preparations gives

where \((B_{k,h,a})_a\) is a Kraus family for \(\overline{\mathcal T}_{k,h}\). This map discards inter-sector coherences and, on active sectors, is the sector-control formula for \(\mathcal T_1\) in [ CPGSV16 , Appendix C.2, lines 1523–1535 ] .

For every sector pair define

Thus \(\mathcal H^{\Omega }_{k,h}\) is the carrier space, whereas \(\Omega _{k,h}\), \(\widehat\Omega _{k,h}\), and \(\overline\Omega _{k,h}\) are operators on it: respectively the unnormalized neighboring operator, its normalization on an active pair, and its completed density on an arbitrary pair. On an active pair this is precisely the preparation \(\mathcal T_{k,h}\) of [ CPGSV16 , Appendix C.2, lines 1527–1535 ] ; the inactive branch is a normalized extension.

For the inverse-map sector tensors, define the complex matrix \(T_{k,h}:=(r_k|l_h)\). Equivalently, \(T_{k,h}=\operatorname{tr}(\eta _{k,h})\). This is the trace matrix in [ CPGSV16 , Appendix C.2, lines 1478–1481 ] .

Let \(U\) be a matrix on the physical space. The physical conjugation of \({\cal K}\) by \(U\) is the tensor \(\widetilde{\cal K}\) defined by

Its horizontal bond indices are unchanged. When \(U\) is unitary, this is the corresponding physical basis change.

For a physical-sector factorization with neighboring operators \(\eta _{k,h}\), retain a sector \(k\) when there is a sector \(h\) such that \(\eta _{k,h}\ne 0\) and \(h\leadsto k\), where \(\leadsto \) denotes reachability by nonzero neighboring operators. The first edge \(k\to h\) is explicit, so the resulting closed walk has positive length even when the return path is empty. The set \(C\) of retained sectors is therefore exactly the set of vertices occurring in positive-length directed cycles, in the sense of [ Bei12 , Section III ] . Define, for \(k,h\in C\),

For positive semidefinite neighboring operators the trace is real, so \(T_{k,h}=\operatorname{tr}(\eta _{k,h})\), and \(T_{k,h}{\gt}0\) is equivalent to \(\eta _{k,h}\ne 0\).

A scalar weight \(w(i,j;i',j')\) on two consecutive ket–bra pairs determines a translation-invariant matrix-product tensor with virtual dimension \(d^2\). Its incoming virtual pair is constrained to equal the current physical ket–bra pair, while its outgoing pair records the following physical pair.

Let \(N\geq 1\) and let \(k\) be a sector configuration. Define the matrix \(\Omega _k\) on \(I_k\) by

Thus \(\Omega _k=\eta _{k_0,k_1}\otimes \cdots \otimes \eta _{k_{N-1},k_0}\) after the neighboring tensor factors are ordered cyclically.

A periodic configuration is identified with the ordered pair consisting of its restriction to the cyclic window \(W_i^L\) and its restriction to the complementary cyclic interval.

For a doubled-index tensor \(B\) with physical dimension \(d^{2}\), the physical-trace transfer is the bond matrix obtained by closing the ket leg against the bra leg of one tensor:

This is the transfer object of the zero-correlation-length condition of [ CPGSV16 , Definition 4.2, lines 735–739 ] , applied to one element of a basis of normal tensors.

Let \(V:\mathbb {C}^d\to \mathbb {C}^e\) be a linear map. Its action on the doubled ket–bra index is the matrix \(W:\mathbb {C}^{d^2}\to \mathbb {C}^{e^2}\) with

For a cyclic sector assignment \((k_n)_{n\in \mathbb Z/N\mathbb Z}\), the horizontal bond fiber at \(n\) is

Shifting the left component by one site identifies \(\prod _n H_n\) with the vertex-indexed sector fiber used in the cyclic neighboring-operator product.

This defines the one-bond eta-local form constructed in [ CPGSV16 , Appendix C.2 ] : a two-site bond \(B\) as in Definition 20.16.1 together with, for every chain length \(N\ge 2\), a constant \(c{\gt}0\) such that

The bond acts on the full two-site space.

Every Hayashi decomposition witness \(h_\eta \) canonically produces an explicit neighboring \(\eta \)-family by Kronecker-multiplying the sector-reduced states on the neighboring bond spaces:

Positive semidefiniteness of each \(\eta _{k,h}\) follows from the fact that partial traces preserve positivity and that positive semidefiniteness is closed under Kronecker products. This is the sector-reduced extraction; the remaining connection to [ CPGSV16 , Appendix C.2 ] is the identification of this family with the inverse-map construction in the original simple-MPDO tensor coordinates.

Let \((l_k)_a\) and \((r_k)_a\) be sector tensors indexed by the horizontal virtual index \(a\). For neighboring sectors \(k,h\), define an operator on \(B_k^R\otimes B_h^L\) by

Thus \(\eta _{k,h}=r_k l_h\) is the contraction over the shared horizontal index in [ CPGSV16 , Appendix C.2, equation (etarl) ] . Positivity is not part of this definition.

For sectors \(k,h\), define the positive-operator candidate

The summand indexed by \(l\) acts on \((B_k^R\otimes B_l^L)\otimes (B_l^R\otimes B_h^L)\). This is the operator adjoined by \(\mathcal T_{k,h}\) in [ CPGSV16 , Appendix C.2, lines 1527–1535 ] .

Let \((\eta _{k,h})\) be a positive neighboring-operator family. A rank-one trace factorization consists of real families \((a_k)\) and \((b_h)\) such that

These identities occur in [ CPGSV16 , Appendix C.2, lines 1395–1402 ] .

The nonzero support graph of a neighboring-operator family has a directed edge \(k\to h\) precisely when \(\eta _{k,h}\ne 0\). Its support is recurrent when every such edge belongs to a directed cycle, equivalently when \(h\) can reach \(k\) whenever \(k\to h\) is an edge.

For a three-site density operator \(\rho _{ABC}\), the local \(\eta \)-structure is the quantum Markov decomposition on the middle subsystem \(B\) produced by equality in strong subadditivity. Concretely, after a unitary change of basis on \(B\), one has a finite direct-sum decomposition

If every neighboring operator \(\eta _{k,h}\) of Definition 20.15.3.1 is positive semidefinite, the family constitutes explicit neighboring \(\eta \)-data. The source obtains the positivity from the projected chain identity: writing \(\tilde\sigma = U^{\dagger \otimes N} \sigma ^{(N)}({\cal K})\, U^{\otimes N}\) for the \(N\)-site operator in the sector basis and \(Q_k\) for the projector onto the \(k\)-th sector of the physical splitting, every cyclic sector assignment \(k_1,\ldots ,k_N\) (with \(k_{N+1}=k_1\)) satisfies

and the sector tensors can be chosen so that each factor is positive semidefinite [ CPGSV16 , Appendix C.2, lines 1446–1450 ] . The all-length sector identity is Theorem 20.15.4.18. That algebraic identity does not by itself choose coherent positive representatives for all neighboring pairs, so positive semidefiniteness remains a hypothesis on the chosen representatives here. Theorem 20.15.4.39 provides the diagonal and paired positivity, and Theorem 20.15.4.42 a rescaled positive family under symmetric support.

Fix a Hayashi decomposition witness \(h_\eta \). An explicit neighboring \(\eta \)-family is a dependent family \((\eta _{k,h})\) indexed by sector pairs, where \(\eta _{k,h}\) acts on the neighboring bond space \(B_k^{R} \otimes B_h^{L}\). Requiring positivity for each pair gives the corresponding positive \(\eta \)-data.

Every explicit neighboring \(\eta \)-family determines a sector-by-sector trace matrix. We give both its complex-valued form and the real-part version, which is the direct data for the real Perron–Frobenius matrix \(T\) used in the rank-one step.

On a nonzero finite-dimensional space \(H\), define

For fixed sectors \((k,h)\), define the fiberwise form of \(\mathcal T_0\) by regrouping two sites and tracing \(R_k\otimes L_h\). For fixed \((k,l,h)\), define the fiberwise form of \(\mathcal S_0\) by regrouping three sites and tracing \((R_k\otimes L_l)\otimes (R_l\otimes L_h)\).

On fixed sector pairs, define \(\mathcal S_2\) and \(\mathcal T_2\) by the same inverse regroupings as the corresponding global shifts.

After \(R_1\) separates the first physical coordinate from the \(N\) spectator coordinates, define

where \(i,p\in \{ 0,\ldots ,d-1\} \) and \(u\in \{ 0,\ldots ,d-1\} ^N\). Thus, for a matrix \(Z\) indexed by \(((i,u),p)\),

A fixed tensor for a bond-product family consists of a positive virtual dimension and one matrix-product tensor \(C\), independent of \(N\), such that, for every \(N\ge 2\),

If \(d{\gt}0\), a cyclic scalar edge-weight expression for the entries of the product gives such a tensor with positive virtual dimension \(d^2\).

For a virtual matrix \(X\), the four-site physical closure of \(K\) is the operator \(K_4(X)\) whose matrix coefficients are

A rank-one trace factorization of neighboring matrices \(\eta _{q,h}\) consists of real numbers \(a_q,b_h\) such that

The factorization is an explicit assumption; it is not deduced from zero correlation length. It is the factorization invoked at [ CPGSV16 , Appendix C.2, line 1613 ] .

An MPO tensor \(M\) satisfies the global purification equation with a purifying tensor \(A\) if \(\rho ^{(N)}(M)=\rho ^{(N)}_p(A)\) for every positive length \(N\). Equivalently, for every positive length \(N\),

as in [ CPGSV16 , line 751 ] .

Let \(N\) count the sites that are left unchanged. For physical maps \(\mathcal T:M_d\to M_{d^2}\) and \(\mathcal S:M_{d^2}\to M_d\), define

using \(R_1\) and \(R_2\). Thus \(N=0\) leaves no spectator site, but the maps still act on a one-site or two-site operator; no empty ring occurs.

The two- and three-site physical spaces are reindexed by their canonical direct sums over sector pairs and triples, with each summand further regrouped into its boundary and neighboring factors. Write \(\mathfrak G_2\) and \(\mathfrak G_3\) for these reindexings of matrices.

This condition asks, for every chain length \(N\geq 2\), for one positive bond on the full two-site space and one positive normalization such that

This auxiliary condition contains no explicit sector labels or multiplicities.

This is the conjunction of the single-bond condition with the doubled-index transfer identity \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\). The source physical-trace condition is Definition 20.5.4.

An MPO tensor \(M\) has the single-bond commuting-form property if for every \(N \ge 2\) there is a two-site operator \(B\) as in Definition 20.13.2 and a constant \(c{\gt}0\) such that

The bond in this chainwise condition may depend on \(N\). It is therefore weaker than the single translation-invariant bond family used in the proposition at [ CPGSV16 , Appendix C.2, lines 1597–1619 ] .

The spin reduction obtained by tracing the ancillary index is trace-preserving completely positive: the ancillary trace map \(\operatorname{tr}_a\) satisfies the corresponding definition in the companion quantum-channel volume  [ LTC26 , “Trace-preserving completely positive map” ] . This is the structure described immediately after Definition 4.3 in [ CPGSV16 , lines 761–764 ] .

Fix virtual indices \(\beta ,\alpha \) and set \(\kappa _{\beta ,\alpha }(i,j)=\mathcal K^{ij}_{\beta ,\alpha }\). For a Hayashi sector \(k\), define the two inverse factors

These are the two factors in the sectorwise inverse-map calculation of [ CPGSV16 , Appendix C.2, lines 1407–1428 ] .

Let \(B=\bigoplus _k(b_1^{(k)}\otimes b_2^{(k)})\) be the middle-site decomposition, and let \(E_k\) be the coordinate projection onto its \(k\)-th summand. In the original physical basis, define \(Q_k=U_B^*E_kU_B\). These are the projectors introduced in [ CPGSV16 , Appendix C.2, lines 1364–1368 ] ; the unitary implements the basis change described in [ CPGSV16 , Appendix C.2, lines 1437–1440 ] .

A simple MPO tensor \(K\) is called injective if its doubled-index MPS tensor is injective. For such a tensor, the chosen decomposition map of the doubled-index MPS furnishes a concrete inverse tensor \(K^{-1}\) together with right- and left-physical realization maps that turn virtual bond insertions back into local physical operators.

Let \(\mathcal{K}\) be injective, let \(\rho \) be its three-site closure against a virtual tail \(R\), and choose a Hayashi decomposition of \(\rho \). Fix outer indices such that \(R_{\beta _3,\alpha _1}\ne 0\). The inverse-map tensors \(l_k,r_k\), together with the Hayashi physical decomposition and unitary, determine a physical-sector factorization of \(\mathcal{K}\).

The left and right factor spaces are nonzero because the corresponding Hayashi density matrices have trace one; their nonzero dimensions are conclusions rather than hypotheses. No positivity of the neighboring operators is asserted. This gives the factorization in [ CPGSV16 , Appendix C.2, equations AppUkU=rl and formK, lines 1383–1387 and 1434–1439 ] .

Let \(\mathcal K\) be an injective simple MPO tensor, and let \(R\in M_{D}(\mathbb {C})\) be the virtual contraction of the sites following a three-site word. For virtual indices \(\alpha _1,\beta _1,\alpha _3,\beta _3\), define

This is the contraction used in the proof of [ CPGSV16 , Appendix C.2, lines 1415–1438 ] .

An MPO tensor satisfies the bare global purification RFP predicate if it has a positive-length global purification RFP witness. Equivalently, it admits a purifying tensor \(A\) such that, for every positive length \(N\),

and \(\widehat A\) is a pure-state renormalization fixed point. These are the displayed conditions of Definition 4.3 in [ CPGSV16 , lines 756–764 ] ; the nondegenerate tensor-level predicate is Definition 20.6.13.

Reassociating four right-associated sites as \((123)4\), applying \(\widehat{\mathcal S}\otimes \operatorname{id}\) to the first three, and reassociating the result back to \(1(23)\) defines the localized coarse-graining channel \(\widehat{\mathcal S}_{(123)4}:M_{q^4}(\mathbb {C})\longrightarrow M_{q^3}(\mathbb {C})\) on right-associated physical coordinates.

Reassociating three right-associated sites as \((12)3\), applying \(\widehat{\mathcal T}\otimes \operatorname{id}\) to the first two, and reassociating the result back to \(1(2(34))\) defines the localized refinement channel \(\widehat{\mathcal T}_{(12)3}:M_{q^3}(\mathbb {C})\longrightarrow M_{q^4}(\mathbb {C})\) on right-associated physical coordinates.

For a virtual matrix \(P\), define

The ordinary neighboring contraction is the case \(P=I\).

For sectors \(k,h\), contract the common virtual index of the right and left sector tensors:

This is an operator on \(B_k^R\otimes B_h^L\). Positivity is not part of this definition; it is a separate hypothesis in Proposition C.7.

Suppose that every neighboring operator is positive semidefinite and that there are real numbers \(a_k,b_h\) such that

These are precisely the remaining hypotheses used in the construction of Proposition C.7 after the physical-sector factorization has been fixed [ CPGSV16 , Appendix C.2, lines 1389–1403 ] .

With the convention \(0^{-1}=0\), define for every pair \((k,h)\)

These operators are normalized when the pair is active, namely when \(a_kb_h\ne 0\).

Let \(\mathcal K\) be an MPO tensor and set \(Z_4=\operatorname{tr}[\rho ^{(4)}(\mathcal K)]\). Its normalized fourth-site virtual tail is

Thus the contraction of the fourth ket and bra indices is incorporated into the virtual closing matrix. This is the \(N=4\) specialization of the normalization convention and three-site marginal in [ CPGSV16 , lines 792–793 and 1343–1348 ] .

Let the vertical sectors be labelled by \(\alpha \), with simple matrix algebra \(M_{d_\alpha }\) and positive diagonal multiplicity matrix

Write \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). The normalized embedding and the left partial trace are

On a weighted sector, \(\widetilde R_\mu (\lambda _\alpha \mu _\alpha \otimes X_\alpha ) =\lambda _\alpha m_\alpha X_\alpha \). Thus the partial trace is the canonical extension to the full block matrix space of the inverse map displayed in [ CPGSV16 , Appendix C.4, lines 1957–1971 ] .

For a sector \(k\) and physical matrix indices \(x,y\) in its summand, let \(A_{k;x,y}\in M_{D}(\mathbb {C})\) be the corresponding entry of the sector-coordinate tensor. For a set \(S\) of sectors, define

These are the matrix spaces associated with the physical projections in [ CPGSV16 , Appendix C.2, lines 1463–1467 ] .

The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:

In the vertical reading of [ CPGSV16 , Section 4.5 ] this is the tensor obtained by joining two vertically read tensors along the original horizontal bond, the tensor written \(M_\alpha M_\beta \) in [ CPGSV16 , Theorem 4.14(iii) ] .

An original-label corner family for the retained product records, for every active corner of each copy pair, an original one-site BNT label, a positive coefficient, and an isometric inclusion into the raw product bond space. Distinct corners belonging to one copy pair have orthogonal ranges. Each inclusion intertwines the raw product tensor with the stated positive multiple of its original BNT representative, and the resulting corners reconstruct every tensor letter exactly.

Local fix (Figure-11 fixed-pair support): A copy pair with no active corner is represented by an empty family; no unsupported corner is inserted. See [ con26q ] .

Let \(\{ K_x\} _{x=0}^{g-1}\) be MPO tensors on a common one-site space. An orthogonally supported family of commuting sector products consists of pairwise orthogonal one-site projections \(P_x\) and positive two-site operators \(B^{(x)}\) such that

and, for every \(N\geq 2\),

Together with natural multiplicities \(n_x\), these objects determine the corresponding finite-chain sector decomposition in Definition 20.13.1.

Every exactly realized orthogonal commuting sector family determines a proportionally realized orthogonal commuting sector family with the same one-site projections and two-site bonds.

For an MPO tensor \(K\), define \(K^\sharp \) by

Thus \(K^\sharp [\beta ,\alpha ]=K[\beta ,\alpha ]^*\), while the virtual indices retain their order. For every periodic chain,

For an integer \(L\geq 0\), group \(L\) adjacent physical sites and denote the resulting local tensor by \(M^{[L]}\). It has a length-\(L\) ket word \(I=(i_1,\ldots ,i_L)\) and a length-\(L\) bra word \(J=(j_1,\ldots ,j_L)\) as its physical indices, and

Thus the ket and bra words are grouped separately. For \(L=0\), both words are empty and the unique local matrix is the identity. For \(L{\gt}0\), this is the MPO version of the physical blocking used for the basis of normal tensors in [ CPGSV16 , lines 317–345 ] . It is a local tensor, not the closed-chain operator \(O_L(M)\) of [ CPGSV16 , lines 962–967 ] .

Transporting \(\mathcal S\) through the canonical three-site and two-site coordinate identifications gives a channel \(\widehat{\mathcal S}:M_{q^3}(\mathbb {C})\longrightarrow M_{q^2}(\mathbb {C})\).

Define

Here the first application acts on the first three sites and leaves the fourth site unchanged. The superscript again denotes successive overlapping applications.

Let \(N\geq 1\). For a cyclic sector configuration \(k=(k_0,\ldots ,k_{N-1})\), write

Applying the one-site sector decomposition at every site gives a bijection

On \(I_k\), define the cyclic neighboring product by

where site labels are read modulo \(N\).

The coordinate equivalences used by the coarse-graining channel expose the direct sums over site-sector pairs and triples. Write \(\mathfrak R_2\) and \(\mathfrak R_3\) for the corresponding matrix reindexings. Their inverses send every sector block to the corresponding physical matrix entries. These reindexings preserve the left-right order within every physical site. They differ from the regrouped coordinates \(\mathfrak G_2\) and \(\mathfrak G_3\) of Definition 20.15.3.39, which place the two outer subspins before the neighboring subspins. If \(R_{\Phi _2}\) and \(R_{\Phi _3}\) denote the two regrouping maps, then

A physical-sector factorization of an MPO tensor \(\mathcal{K}\) consists of a finite decomposition

whose left and right factors are nonzero, an isometry \(U\) on the physical space, and matrix families \((l_k)_\beta \) and \((r_k)_\alpha \) such that every physical slice satisfies

No positivity, trace factorization, or quantum-Markov decomposition is included in this datum. This is the factorization displayed in [ CPGSV16 , Appendix C.2, lines 1381–1388 ] .

Let \(K\) and \(L\) have virtual dimensions \(D\) and \(E\), respectively, and suppose that a rectangular matrix \(V\in M_{D\times E}(\mathbb {C})\) satisfies the following identity for every physical index \(i\):

Every physical-sector factorization of \(K\) induces one of \(L\) with the same physical decomposition. This construction combines the virtual corner in [ CPGSV16 , Section 2.3, lines 195–219 ] with the physical-sector factorization in [ CPGSV16 , Appendix C.2, lines 1381–1439 ] .

Let \(K\) and \(L\) have virtual dimensions \(D\) and \(E\), respectively. Suppose that matrices \(X\in M_{E\times D}(\mathbb {C})\) and \(Y\in M_{D\times E}(\mathbb {C})\) satisfy the following identity for every physical index \(i\):

Every physical-sector factorization of \(K\) induces one of \(L\) with the same physical decomposition. Its virtual tensor families are

This construction abstracts the virtual corner in [ CPGSV16 , Section 2.3, lines 195–219 ] and the physical-sector factorization in [ CPGSV16 , Appendix C.2, lines 1381–1439 ] .

Let \(S\) be a set of active sectors in a physical-sector factorization, and suppose that \((l_k)_\beta =0\) for every \(k\notin S\) and every \(\beta \). Retain every physical summand and define

Then \(\widehat l_k\otimes \widehat r_k=l_k\otimes r_k\) for every \(k\), so the physical-slice factorization is unchanged. The neighboring operators become

This is the reparameterization freedom in the factorization and neighboring contraction of [ CPGSV16 , Appendix C.2, lines 1434–1445 ] .

Let \(z_k\in \mathbb {C}\) satisfy \(|z_k|=1\) for every sector. Replacing

leaves every physical-slice factorization unchanged.

On a sector pair with \(a_kb_h\ne 0\), let \(\mathcal S_{k,h}(X)=X\otimes \frac{\eta _{k,h}}{a_kb_h}\). If \(a_kb_h=0\), positivity and vanishing trace imply \(\eta _{k,h}=0\); choose any density operator on that unused summand. Taking the orthogonal direct sum over \((k,h)\) defines \(\mathcal S_1\).

Define the channel from three physical sites to two physical sites by \(\mathcal S=\mathcal S_2\mathcal S_1\mathcal S_0\). This is the coarse-graining map in Proposition C.7 [ CPGSV16 , Appendix C.2, lines 1547–1563 ] .

On a sector pair with \(a_kb_h\ne 0\), set \(\widehat\Omega _{k,h}=(a_kb_h)^{-1}\Omega _{k,h}\). If \(a_kb_h=0\), positivity and vanishing trace imply \(\Omega _{k,h}=0\); choose any density operator on that summand and denote the resulting completed density by \(\overline\Omega _{k,h}\). The sectorwise preparation is \(X\longmapsto X\otimes \overline\Omega _{k,h}\). Taking the orthogonal direct sum over the outer sectors \((k,h)\) gives the preparation stage \(\mathcal T_1\) of [ CPGSV16 , Appendix C.2, lines 1523–1535 ] . The zero-weight choice makes the source map total without imposing a nonzero-weight hypothesis.

Define the channel from two physical sites to three physical sites by \(\mathcal T=\mathcal T_2\mathcal T_1\mathcal T_0\). Thus \(\mathcal T_0\) traces the neighboring subspins, \(\mathcal T_1\) adjoins the completed three-site neighboring density, and \(\mathcal T_2\) restores the three complete physical sectors. This is the refinement map in [ CPGSV16 , Appendix C.2, lines 1522–1545 ] .

For every outer-sector pair \((k,h)\), define

This is the direct sum of the two neighboring contractions appearing in the three-site closure factorization [ CPGSV16 , Appendix C.2, lines 1510–1516 ] .

In the two-site sector coordinates, define

Thus the outer factors carry the identity, while the neighboring factors carry the operator \(\eta _{k,h}\). This is the local bond in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1581–1593 ] .

Let \(P\) be an orthogonal one-site projection and let \(K\) be an MPO tensor fixed by compression with \(P\) on its two physical indices. A physical support restriction consists of a dimension \(e\) and an isometry \(V:\mathbb {C}^e\longrightarrow \mathbb {C}^d\) such that

for which the restricted tensor \(K_P=V^*KV\) is injective and its inclusion into the ambient physical space recovers \(K\) exactly.

Transporting \(\mathcal T\) through the canonical two-site and three-site coordinate identifications gives a channel \(\widehat{\mathcal T}:M_{q^2}(\mathbb {C})\longrightarrow M_{q^3}(\mathbb {C})\), where all multiple physical indices are associated to the right.

Let \(F\) be a physical-sector factorization whose neighboring operators satisfy Definition 20.15.3.32, and let \(\widehat{\mathcal K}\) be its sector-coordinate tensor. Define

Here the second application acts on the first two sites and leaves the third site unchanged. Thus the superscript denotes two successive overlapping applications, as in [ CPGSV16 , Appendix C.2, lines 1821–1825 ] .

Let \(V_2\) be the two-site tensor product of the project’s coordinate map, so that \(V_2XV_2^\dagger \) expresses a physical operator \(X\) in sector coordinates. Thus \(V_2=U_{\mathrm{paper}}^\dagger \) in the convention of [ CPGSV16 , Appendix C.2, lines 1581–1583 ] . Define

and identify pairs of physical indices with functions \(\{ 0,1\} \to \{ 0,\ldots ,d-1\} \). This is the physical two-site bond in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1581–1593 ] .

A vertical-sector family \(X=(X_\alpha )_\alpha \) is positive when every \(X_\alpha \) is positive semidefinite. Its total sector trace is

A purification RFP is viewed together with trace-preserving spin reduction by applying Corollary 20.6.7 to the ancillary trace map appearing in its purification equation. This records the structure obtained after tracing the ancillary indices in [ CPGSV16 , lines 761–764 ] , without adding a separate hypothesis to Definition 4.3.

Let \(\{ K_x\} _{x=0}^{g-1}\) be MPO tensors on a common one-site space. A proportionally realized orthogonal commuting sector family consists of pairwise orthogonal one-site projections \(P_x\) and positive two-site operators \(B^{(x)}\) such that

For every sector \(x\) and every \(N\geq 2\), there is a number \(a_{x,N}{\gt}0\) such that

For a purifying spin–ancilla tensor \(A\), the reduced spin density matrix is defined by

This is the coefficient form of the ancillary trace in [ CPGSV16 , eq. (4.7), line 751 ] .

An MPO tensor \(M\) has a bare global purification RFP witness if it satisfies the global purification equation for some purifying spin–ancilla tensor \(A\), and \(\widehat A\) is a pure-state renormalization fixed point: \(\mathcal{E}_{\widehat A}\circ \mathcal{E}_{\widehat A}=\mathcal{E}_{\widehat A}\). This records the displayed conditions of Definition 4.3 in [ CPGSV16 , lines 756–764 ] . It does not include the nondegeneracy implicit when the subsequent theorem speaks of density operators.

Given matrices \(A^{(i,k)}\) with a spin index \(i\) and an ancillary index \(k\), the associated MPS tensor on the product physical space is \(\widehat A^{(i,k)}=A^{(i,k)}\). This is the tensor whose finite-chain vector appears in the purification formula of [ CPGSV16 , lines 744–751 ] .

A retained-product spectral family records, simultaneously for every ordered pair of retained BNT copies, all nonzero normal corners, their positive spectral coefficients, and their local isometries. Its active labels form the dependent union of the local corner families; this union is enumerated by a single finite type. The local isometries and their inclusions remain part of the data, and their range projections can be formed when needed.

Retain the coordinates \((\alpha ,a,i)\), where \(\alpha \) labels a simple sector, \(a\) labels a diagonal entry of \(\mu _\alpha \), and \(0\leq i{\lt}d_\alpha \). The family \((Y_\alpha )_\alpha \) is represented on this space by \(\bigoplus _\alpha Y_\alpha \). Write \(\iota \) for this inclusion, \(\pi \) for extraction of the diagonal summands, and \(P=\iota \pi \). Composing \(\iota \) and \(\pi \) with \(R_\mu \) and \(\widetilde R_\mu \) defines their retained-coordinate forms.

Define \(\mathcal S_0\) by the three-site regrouping followed, for every outer pair \((k,h)\), by the partial trace over

Thus, writing \(R_{\Phi _3}\) for the regrouping reindexing,

The map \(\mathcal S_2\) is matrix reindexing along the inverse two-site equivalence. Thus

For an arbitrary virtual matrix \(X\), define the outer boundary operator on \(B_k^L\otimes B_h^R\) by

The map \(X\mapsto B_{k,h}(X)\) is complex-linear. No positivity of \(X\) or \(B_{k,h}(X)\) is assumed.

For a sector configuration \(k=(k_0,\ldots ,k_{N-1})\), let

Applying the one-site Hayashi decomposition at every site gives a bijection

We write \(\operatorname{reindex}_{\Phi _N}(M)\) for the matrix obtained by transporting both indices of \(M\) along this bijection.

Fix outer virtual indices \(\alpha _1,\beta _3\) with the chosen Hayashi decomposition, and let \(l_k,r_k\) be the sector tensors of Definition 20.15.2.50. For each pair of sectors \((k,h)\), the neighboring operator contracts the right sector tensor of one site with the left sector tensor of the following site over the virtual bond they share:

It acts on the neighboring bond space \(B_k^{R}\otimes B_h^{L}\). This is the operator displayed in [ CPGSV16 , Appendix C.2, lines 1441–1445 ] .

For a real \(n\times n\) matrix \(T\) and a real vector space \(V\), this structure consists of vectors \(|l_0),\ldots ,|l_{n-1})\in V\) and linear functionals \((r_0|,\ldots ,(r_{n-1}|\) on \(V\) satisfying the pairing identity \(T_{k,h}=(r_k|l_h)\) of [ CPGSV16 , Appendix C.2 ] together with the zero-correlation-length identity

displayed in the proof of [ CPGSV16 , Lemma C.5 ] . The vectors are the closed sector tensors obtained there from the sector splitting of an injective simple tensor.

Fix outer virtual indices \(\alpha _1,\beta _3\) and a Hayashi sector \(k\). The sector tensors are

These are the tensors of the factorization displayed in [ CPGSV16 , Appendix C.2, lines 1435–1448 ] .

In sector coordinates, let \(A_{k;x,y}\) be the virtual matrix obtained from the \((x,y)\) entry within sector \(k\). The sector-coordinate family contains all transformed physical matrices, while the virtual-matrix family consists of all matrices \(A_{k;x,y}\).

A tensor generating MPDOs satisfies this predicate if it has a normalized horizontal basis-of-normal-tensors decomposition whose basis elements all have non-nilpotent physical-trace transfer. This records the horizontal-canonical-form data of the already-blocked tensor \(K\) fixed in  [ CPGSV16 , Appendix C.2, line 1628 ] .

The definition concerns this fixed representative. Its witness includes the global unit-weight convention of  [ CPGSV16 , line 246 ] , whereas the non-nilpotency condition itself is unchanged by nonzero scalar rescaling. It supplies the Appendix C.2 hypothesis on the already-blocked tensor and is not the simplicity predicate of Definition 20.11.6, which quantifies over blocking lengths and presentations. No invariance under rescaling or replacement of the canonical presentation is asserted.

A tensor generating MPDOs is simple if there is a positive physical blocking length \(L\) such that the doubled-index tensor of the \(L\)-site blocking has a BNT sector presentation \(P\) for which every representative \(B_j\) has non-nilpotent physical-trace transfer \(\mathcal T_{B_j}\). This records the standing canonical-block convention of [ CPGSV16 , lines 217–246 ] together with the simplicity condition of [ CPGSV16 , lines 815–822 ] .

Every representative in \(P\) has a positive number of copies, and every copy has nonzero weight. The blocking length is existential, and presentation independence follows from Theorem 20.11.5. The definition omits the global unit-weight normalization of [ CPGSV16 , line 246 ] and does not require a normalized horizontal canonical-form witness.

For a one-site matrix \(V\), write \(V^{\otimes N}\) in the configuration basis as

If the copies of the \(j\)-th normal tensor have common coefficient \(\mu _j\), let \(\mathcal K_j=\mu _j A_j\) denote the representative with this coefficient absorbed into its local tensor.

A density operator on a periodic chain is a Gibbs state of a nearest-neighbor commuting Hamiltonian if

where the one-site spaces belonging to distinct labels \(x\) are orthogonal and \([h^{(x)},\tau _1(h^{(x)})]=0\). With the projector-limit convention of [ CPGSV16 , Definition 4.8, lines 829–850 ] , this is equivalently

with commuting neighboring translates. The orthogonal sum and the multiplicities are part of the definition. Definition 4.8 decomposes the global Hilbert space into the uniform product sectors and requires the associated one-site spaces to be mutually orthogonal; it does not impose a resolution of a larger one-site identity by the sector projections.

For \(N=L+K\), split a physical configuration into consecutive intervals of lengths \(L\) and \(K\), flatten each interval to a finite index, and denote the resulting bipartite form of the normalized periodic MPO by \(\sigma ^{(N)}_{L:K}\). The flattened local maps on the two factors are denoted by \(\widehat{\mathcal T}_L\) and \(\widehat{\mathcal S}_K\).

An MPO tensor \(M\) satisfies the strong renormalization fixed-point relation if it admits a matrix \(P\geq 0\) on the physical space and a unitary \(U\) on the two-site physical space such that, for all two-site physical indices \(i=(i_1,i_2)\) and \(j=(j_1,j_2)\),

Equivalently, the literal orientation of the source diagram is \(M_2=U(M_1\otimes P)U^\dagger \), with \(M\) on the left physical site and \(P\) on the right. There is no virtual change of basis and no scalar proportionality. This is the relation labelled Strong-RFP in [ CPGSV16 , Appendix D, lines 2109–2113 ] .

Let \(S\) be the finite set of physical sectors and let \(k=(k_0,\ldots ,k_{L-1})\in S^L\) be a retained sector word. For \(R\geq 1\) and \(x,y\in I_k\), define

where \(kt\) and \(xz\) denote concatenation of sector words and their corresponding fiber coordinates. The cases \(R=1,2,3\) are denoted by \(\Gamma _k^{(1)}\), \(\Gamma _k^{(2)}\), and \(\Gamma _k^{(3)}\).

Define \(\mathcal T_0\) by the two-site regrouping followed, in each pair \((k,h)\), by the partial trace over \(R_k\otimes L_h\). Its output is indexed by the retained outer factors \(L_k\otimes R_h\). Thus, writing \(R_{\Phi _2}\) for the regrouping reindexing,

The map \(\mathcal T_2\) is matrix reindexing along the inverse three-site equivalence. On each summand it sends

Let \(\mathcal K\) be an MPO tensor and let \(R\in M_{D}(\mathbb {C})\). A tripartite operator \(\rho \) is the three-site closure of \(\mathcal K\) against \(R\) if

This is the algebraic form of the three-site reduced operator in [ CPGSV16 , Appendix C.2, lines 1422–1433 ] .

Let \((\mathcal K_s)_s\) be a finite family of MPO tensors, with \(\mathcal K_s^{ij}\in M_{D_s}(\mathbb {C})\), and let \(R_s\in M_{D_s}(\mathbb {C})\). A three-site operator is a family closure when

Given a simultaneous left inverse \(C\) as in Theorem 21.3.83, contracting \(C\) against the first and third physical sites of \(\rho \) defines the outer inverse contraction

The closure is the sector-sum form of the displayed equation for \(\sigma _3^{(N)}\) in [ CPGSV16 , Appendix C.2, lines 1343–1348 ] , used with the block-canonical tensor in the Case II calculation at [ CPGSV16 , Appendix C.2, lines 1726–1732 ] . The outer contraction is the sector-family form of the Case I contraction in [ CPGSV16 , Appendix C.2, lines 1415–1438 ] ; the same construction is used with sector projectors at lines 1719–1725.

Write the two-site bond on the ordered pair of physical indices, and identify a three-site configuration \((x_0,x_1,x_2)\) with the left-associated triple \(((x_0,x_1),x_2)\). These are common coordinates for the two bonds on sites \((0,1)\) and \((1,2)\).

Let \(B\) be an operator on two copies of a finite-dimensional space \(H\). On the right-associated space \(H\otimes (H\otimes H)\), define

where \(\alpha :(H\otimes H)\otimes H\to H\otimes (H\otimes H)\) is the canonical associator.

First pass to the physical basis selected by \(U\) and form the three-site closure of the resulting sector-coordinate tensor. Restrict this closure to the sectors \((k,l,h)\) and reindex its physical factors by

Denote the resulting matrix by \(\mathcal{K}_{3;k,l,h}(X)\).

Regroup three sites by

Equivalently,

The forward equivalence prepares the four middle subspins for \(\mathcal S_0\); its inverse is the shift \(\mathcal T_2\) in [ CPGSV16 , Appendix C.2, lines 1535–1540 and 1547 ] .

This records one positive semidefinite two-site bond \(B\ge 0\) whose translated copies commute pairwise on the periodic \(N\)-site chain at every chain length \(N\ge 2\): \([B_{i,i+1},B_{j,j+1}]=0\).

A translation-invariant positive two-site bond determines finite-chain commuting-form data at every length \(N\ge 2\).

Define maps between the one-site and two-site simple-sector algebras by

The partial trace and diagonal-summand extraction define these maps on the full matrix algebras. Their agreement with the normalized maps in [ CPGSV16 , Appendix C.4, lines 1972–1979 ] is asserted on the weighted block-diagonal image. At this stage there is no assertion that the image is preserved, that the two maps are inverse, completely positive, or trace preserving, or that the sector labels have been matched. The final renormalization fixed-point conclusion is separate.

For an MPO tensor \(K\), its two-site blocking \(K^{[2]}\) has physical indices \((i_0,i_1)\) and \((j_0,j_1)\) and matrices

This is the blocking used in [ CPGSV16 , Theorem 4.9, lines 851–856 ] .

First pass to the physical basis selected by \(U\), so that each slice is written in the sector coordinates of Definition 20.15.3.3. Form the two-site closure of this sector-coordinate tensor, restrict it to the physical sectors \((k,h)\), and reindex its physical factors by

Denote the resulting matrix by \(\mathcal{K}_{2;k,h}(X)\).

In sector-adapted coordinates, regroup two sites by

Globally this gives the dependent-sum equivalence

The forward equivalence prepares the neighboring factors \(R_k\otimes L_h\) for the partial trace in \(\mathcal T_0\); its inverse is the shift \(\mathcal S_2\). These are the reorderings in [ CPGSV16 , Appendix C.2, lines 1521–1522 and 1555–1559 ] .

Let \(A\) be a tensor whose physical label is a pair \((a,b)\) of horizontal bond indices. For a bond matrix \(X\), define

The order of the entries of \(X\) is chosen so that \(A(X)_{ij}=\operatorname{tr}(M^{ij}X)\) when \(A_{ab,ij}=M^{ij}_{ab}\).

Let \(U_1\) and \(U_2\) be the physical coordinate changes in vertical canonical forms of \(M\) and its two-site blocking, respectively. Let \(J\) be the canonical relabelling from matrices indexed by pairs of one-site physical indices to matrices indexed by one blocked physical index. For the refinement and coarse-graining maps \(T\) and \(S\), define

These are the physical maps after absorbing the two vertical coordinate changes, as in [ CPGSV16 , Appendix C.4, lines 1955–1979 ] . They are intermediate maps on the retained physical coordinate spaces; the normalized sector maps \(R_1,R_2,\widetilde R_1,\widetilde R_2\) have not yet been composed with them.

Let \(F\) be a linear endomorphism of \(\mathcal A=\prod _\alpha M_{d_\alpha }\), and write \(\mathcal F=\{ X\in \mathcal A:F(X)=X\} \). For \(L\geq 0\), define

This is the product space used in the dimension argument of [ CPGSV16 , Appendix C.4, lines 1980–1993 ] .

View the MPO tensor \(M\) vertically as the family of physical-space operators \(M_{ab}\) of Definition 17.4.4, indexed by the virtual indices, with matrix elements \((M_{ab})_{ij}=M^{ij}_{ab}\). A matrix \(P\in M_{d}(\mathbb {C})\) then acts on the tensor by

and on an \((N+1)\)-site chain by the first-spin operator \(P\otimes \mathbb {1}^{\otimes N}\). On the letters of the vertically viewed tensor these actions are the left and right matrix products, \((PM)_{ab}=PM_{ab}\) and \((MP)_{ab}=M_{ab}P\). The first-spin operators compose site by site, \(P_1Q_1=(PQ)_1\).

Let \(Z_T(X)\) be the matrix obtained by applying the normalized one-site embedding, expanding by \(U_1^\dagger \), applying \(T\), and relabelling the two-site indices, before compression by \(U_2\). Define \(Z_S(Y)\) in the reverse direction, before compression by \(U_1\).

For scalars \(c_\alpha \ne 0\), the map

is a linear equivalence.

For vertical BNT tensors \(M_\alpha \) with simple matrix algebras \(M_{d_\alpha }\) and a family of complex scalars \((c_\alpha )_\alpha \), define

For each \(L\geq 0\), let

Write \(V_m\) for the specialization obtained by taking \(c_\alpha =m_\alpha =\operatorname{tr}(\mu _\alpha )\).

A physical index for an \(L\)-site block determines a word by decoding the index into its \(L\) original physical letters and listing them in site order.

In the inverse-map factorization, retain every physical sector and its factor spaces, and set

The physical-slice factorization is unchanged. This treats the zero-weight freedom left by the Hayashi decomposition used in [ CPGSV16 , Appendix C.2, lines 1415–1445 ] .

For words \(u=(i_1,\ldots ,i_N)\) and \(v=(j_1,\ldots ,j_N)\) in \(\{ 0,\ldots ,d{-}1\} ^N\), the word evaluation is

The empty pair of words evaluates to the identity, \(M^{\varnothing ,\varnothing }=\mathbb {1}_D\), and word pairs of different lengths evaluate to \(0\).

The operator generated by \(M\) on \(N\) sites is the matrix \(\rho ^{(N)}(M)\) indexed by configurations \(\sigma ,\tau \in \{ 0,\ldots ,d{-}1\} ^N\) and given by

An MPO tensor \(M\) satisfies the doubled-index transfer condition when its completely positive transfer map is idempotent:

This is the condition obtained from the doubled-index MPS view. It is not the physical-trace condition of Definition 20.5.2.

For a chain of length \(N=L+R\), call \(\rho ^{(N)}(M)\) diagonal if \(\rho ^{(N)}(M)_{u,v}=0\) whenever \(u\ne v\). Call \(M\) globally diagonal if \(\rho ^{(N)}(M)\) is diagonal for every \(N{\gt}0\). For configurations \(x\) and \(y\) on the two consecutive blocks, define

An MPO tensor \(M\) has zero correlation length when its physical-trace transfer is idempotent:

This is the literal fixed-tensor condition of [ CPGSV16 , Definition 4.2, lines 735–739 ] .

The physical-trace transfer of an MPO tensor \(M\) is the virtual matrix obtained by contracting the ket and bra physical legs of one tensor:

This is the transfer object appearing in the zero-correlation-length condition of [ CPGSV16 , Definition 4.2, lines 735–739 ] .

An MPO tensor satisfies the nonzero up-to-scalar physical-trace relation when, for some real number \(\lambda {\gt}0\), the following conditions hold:

The nonzero condition excludes the degenerate zero transfer, and the positive scalar makes the condition invariant under positive real rescaling of \(M\). This is a scale-invariant relation at the level of the generated state family. It is broader than the paper’s literal diagram, which fixes \(\lambda =1\) for the displayed tensor.

An MPO tensor is a family of matrices \(\{ M^{ij}\} _{i,j=0}^{d-1}\) with \(M^{ij} \in M_{D}(\mathbb {C})\), indexed by a ket index \(i\) and a bra index \(j\). Diagrammatically,

By combining the pair \((i,j)\) into a single physical index in \(\{ 0,\ldots ,d^2-1\} \), one obtains an MPS tensor with physical dimension \(d^2\) and bond dimension \(D\), with \(M^{\mathrm{MPS}}_{(i,j)}=M^{ij}\), identifying \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \) with \(\{ 0,\ldots ,d^2{-}1\} \) via the product encoding.

The transfer map associated to \(M\) is the linear map \(\mathcal{E}_M:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined 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 transfer map of \(A\) satisfies the idempotence criterion when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

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

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

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

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

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

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

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

For the same fixed system size \(N\), block length \(L\le N\), and positive semidefiniteness of \(\rho ^{(N)}(M)\), the mutual information between a block of \(L\) spins and the rest is \(I_L=S_L^{(N)}(M)+S_{N-L}^{(N)}(M)-S_N^{(N)}(M)\).

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ CPGSV16 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

Let \(M\) have physical dimension two and bond dimension three, with

For an MPO tensor \(M\), a length \(N\), and a virtual operator \(X\), define the physical operator \(M_N(X)\) by

This is linear in \(X\). The displayed order chooses the cut of the cyclic virtual contraction immediately before the first site, so that \(X\) follows the last site. The one- and two-site operators are constructed in [ CPGSV16 , lines 638–654 and Definition 4.1 ] . When \(M=\mathcal K\), write these operators as \(\mathcal K_N(X)\); the cases \(N=2,3\) occur in [ CPGSV16 , Proposition C.7, lines 1510–1516 ] .

An MPS tensor \(A\) is a renormalization fixed point if there is an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), with coefficients \(V_{(i_1,i_2),j}\) and \(V^\dagger V=\mathbb {1}\), such that, for all physical indices \(i_1,i_2\),

This is the relation \(AA=A\) in [ CPGSV16 , Definition 3.2 ] .

For an MPS tensor \(A\), a system size \(N\), and a block length \(L\le N\), let

denote the normalized state and the reduced state of the first \(L\) spins. The \(L\)-block entropy is \(S_L^{(N)}(A)=S(\rho _L^{(N)}(A))\). The tensor saturates the area law if \(S_L^{(N)}(A)=S_{L+1}^{(N)}(A)\) for all \(1\le L{\lt}\lfloor N/2\rfloor \) and all \(N\). This is [ CPGSV16 , Definition 3.13, line 600 ] .

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

Given a sector decomposition \(P\), these conditions define the normalized, left-canonical, aperiodic BNT surface used below. This is stronger than the abstract BNT of Definition 9.1.1 and the literal coefficient-form BNT of Definition 9.1.2: besides spanning and eventual linear independence, it fixes irreducible left-canonical representatives, their self-overlap normalization, gauge-phase separation, and the canonical-form weight normalization. It imposes no equal-modulus factorization or strict ordering on the raw weights. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ CPGSV16 , Section 2.3 ] canonical-form normalization holds,

The MPV expansion takes the raw two-layer form

matching [ CPGSV16 , Section 2.3 ] and [ CPGSV21 , Definition IV.2 ] . The canonical-form normalization admits every example of the source paper (in particular \(C \oplus D\), \(C \oplus (-C)\), \(C \oplus (1/2) C\), and \(C \oplus (-C) \oplus (1/2) C\)) and is not derivable from the per-block normality hypotheses alone.

A sector-weight tuple over \(g\) basis blocks consists of multiplicities \(r_j {\gt} 0\), nonzero weights \(\mu _{j,q}\) for \(q \in \{ 1,\ldots ,r_j\} \), and the sector coefficients

A sector decomposition consists of the basis blocks \((A_j)_{j=1}^g\) together with these multiplicities and weights. If \(A_j\) has bond dimension \(D_j\), then the flattened copy family has total bond dimension

For every type \(\alpha \), evaluation at the three coordinates gives the canonical equivalence

Its forward map is

An MPS tensor \(A=\{ A^i\} _{i=0}^{d-1}\) determines an MPO tensor by placing \(A^i\) on the diagonal in the bra and ket indices:

Equivalently, \(M^{ii}=A^i\) and \(M^{ij}=0\) for \(i\ne j\).

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.

An MPO tensor is in vertical canonical form if there are a basis of normal tensors \(\{ M_\alpha \} _\alpha \) for its vertically viewed tensor \(\widetilde M\), positive diagonal matrices \(\mu _\alpha =\operatorname{diag}(\omega _{\alpha ,0},\ldots ,\omega _{\alpha ,r_\alpha -1})\) with \(r_\alpha \geq 1\), and a matrix \(U\) from the physical space onto the retained nonzero sector space such that

This is the conclusion of [ CPGSV16 , Proposition 4.13, lines 1863–1870 ] . In this orientation \(UU^\dagger =I\) on the retained space, while \(U^\dagger U\) is its support projection in the original physical space. The reconstruction identity says that every letter of \(\widetilde M\) is supported on this projection, so the discarded orthogonal complement and the off-diagonal corners vanish. The orthogonal complement may be a zero sector: the source’s general canonical-form construction explicitly permits the sum of the nonzero block dimensions to be strictly smaller than the original dimension [ CPGSV16 , lines 214–225 ] . Since each \(\mu _\alpha \) is diagonal, the summand \(\mu _\alpha \otimes M_\alpha \) is the direct sum \(\bigoplus _{k=0}^{r_\alpha -1}\omega _{\alpha ,k}M_\alpha \) of weighted copies of \(M_\alpha \), so the right-hand side is a block-diagonal tensor over the pairs \((\alpha ,k)\) in which the multiplicities \(r_\alpha \) and the diagonal entries \(\omega _{\alpha ,k}\) stay explicit; the coefficients \(m_\alpha =\operatorname{tr}(\mu _\alpha )=\sum _{k=0}^{r_\alpha -1}\omega _{\alpha ,k}\) appearing in the renormalization fixed-point characterization are recovered from them.

The vertical direction of an MPO tensor exchanges the notion of physical and virtual indices [ CPGSV16 , line 943 ] : the vertically viewed tensor \(\widetilde M\) has its letters indexed by the horizontal bond pair \((a,b)\), and each letter is the operator on the physical space with matrix elements \((\widetilde M_{ab})_{ij}=M^{ij}_{ab}\). Concatenating \(\widetilde M\) vertically generates the matrix product vectors of the vertical direction.

For a normalized positive semidefinite finite-chain operator, every block entropy is non-negative: \(0\le S_m\).

For a normalized positive semidefinite finite-chain operator, the block entropy of the empty block vanishes: \(S_0=0\).

For an MPS tensor \(A\), system size \(N\), and block length \(L\le N\), the MPDO block entropy of the tensor \(M^{ij}=A^i\otimes \overline{A^j}\) agrees with the pure-state block entropy: \(S_L^{(N)}(M)=S_L^{(N)}(A)\).

A tensor with one-dimensional bond space and identity transfer map is a normal tensor.

Suppose that, after a change of coordinates, \(A\) is the dependent direct sum of tensors \(B_k\). If \(A\) has invariant-projector closure and no nontrivial periodic vectors, then every \(B_k\) has both properties. The assertion also applies to summands of dimension zero.

If \(x\geq 0\), then the complex coercion of its real square root satisfies \(((\sqrt{x}:\mathbb R):\mathbb C)^2=x\).

Every block in a coefficient-form basis of normal tensors has positive bond dimension.

If \(L\le N\), then a site \(k\) belongs to the cyclic support \(S_i\) exactly when its cyclic offset from \(i\) is below \(L\): \(k\in S_i \quad \Longleftrightarrow \quad (k-i)\bmod N {\lt} L\).

For length-\(L\) cyclic windows, \(i\sim _Lj\) if and only if \(j\sim _Li\).

Every nonempty closed directed walk of length \(N\) determines a cyclic family of vertices \(v_0,\ldots ,v_{N-1}\). Each \(v_n\to v_{n+1}\) is an edge, where \(v_N=v_0\), and for any edge weights in a commutative monoid, \(\operatorname {wt}(w)=\prod _{n=0}^{N-1}\kappa _{v_n,v_{n+1}}\).

The canonical four-coordinate regrouping and its inverse satisfy

Let \(a,n\in \mathbb N\), and let \(f\) be defined on the integer interval \([0,n]\). If \(f(m)=f(m+1)\) for every \(a\le m{\lt}n\), then \(f(i)=f(j)\) for all \(i,j\in [a,n]\).

A finite Kronecker product of nonzero matrices is nonzero. Replacing one factor \(A_i\) by \(cA_i\) multiplies the whole finite product by \(c\).

Let \(I\) be finite, let \(p_i\) be weights in a type with zero, and let \(f_i\) take values in a commutative additive monoid. If \(f_i=0\) whenever \(p_i=0\), then

Let \(\mathcal A_1,\mathcal A_2\subseteq M_{D}(\mathbb {C})\) be nonzero linear subspaces. If \(XY=0\) for every \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), then \(\mathcal A_1+\mathcal A_2\ne M_{D}(\mathbb {C})\). No vanishing condition on products in the reverse order is required. This is the matrix-algebra obstruction used in [ CPGSV16 , Appendix C.2, lines 1465–1470 ] .

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

Let \(E_k:H_k\to \bigoplus _lH_l\) be the canonical inclusion and let \(B=\bigoplus _lB_l\). For distinct \(k\) and \(\ell \), the corresponding inclusions have orthogonal ranges. The identities are

In particular, \(B\) commutes with the range projection \(E_kE_k^\dagger \).

Let \({\cal K}\) be an MPO tensor with a physical-sector factorization and sector weights \(p_k\), and suppose that every neighboring operator \(\eta _{k,h}\) is positive semidefinite. On the active sector set \(I_*\), define the trace-squared matrix

Then, entrywise,

This matrix is a project derivation preparing the Case-I route toward [ CPGSV16 , Appendix C.2, Lemma C.5, lines 1473–1499 ] ; it is not a matrix defined in that source.

For a weighted block-diagonal tensor \(A=\bigoplus _\alpha \mu _\alpha A_\alpha \), one has \(A(X)=\bigoplus _\alpha \mu _\alpha A_\alpha (X)\).

Let \(L\) and \(K\) be fixed matrices of compatible sizes. Then \(\left((a,b)\mapsto L A_{ab}K\right)(X)=L A(X)K\).

Across the consecutive cut after the first \(L\) sites, the two marginals of \(\sigma ^{(N)}_{L:N-L}\) are the reduced states on \(L\) and \(N-L\) sites, up to the standard finite-index identifications. Consequently, \(I_L(M)\) is the ordinary bipartite mutual information of \(\sigma ^{(N)}_{L:N-L}\).

Let \(U:\mathbb {C}^n\to \mathbb {C}^r\) satisfy \(UU^\dagger =1\). If \(Z\geq 0\), then

For every \(W\in \mathbb {C}^{r\times r}\), one also has \(\operatorname{tr}(U^\dagger WU)=\operatorname{tr}(W)\).

Let \((P_i)_i\) be finitely many operators on \(H\) and let each \(\Phi _i:\operatorname{End}_{\mathbb {C}}(H)\to \operatorname{End}_{\mathbb {C}}(K)\) be completely positive. Then

is completely positive. No completeness or orthogonality condition on the \(P_i\) is required.

For every copy \(q\) over \(j\), one has \(\mu _{j,q}=\mu _j\).

For a sector decomposition with basis elements \(A_{j}\) and weights \(\mu _{j,q}\), the physical-trace transfer of the assembled tensor is the block direct sum of the weighted transfers of the copies:

The physical-trace transfer of the doubled-index view of an MPO tensor is the physical-trace transfer of the tensor itself.

Reindexing a cyclic neighboring-operator block along its horizontal bond fiber gives the finite Kronecker product \(\bigotimes _{n=0}^{N-1}\eta _{k_n,k_{n+1}}\).

For every pair of sectors and every pair of matrix indices, the entry of \(\eta _{k,h}\) is the contraction displayed in (82).

A vertex \(h\) is reachable from \(k\) in the nonzero sector support if and only if there is a finite directed walk from \(k\) to \(h\). Consequently, the support is recurrent if and only if every directed edge admits a directed return walk.

Suppose \(\eta _{q,h}\) has a rank-one trace factorization. For fixed boundary sectors \(u,v\), set

Then

This is the corrected edge contraction at [ CPGSV16 , Appendix C.2, lines 1613–1617 ] .

Let \(M\) be an MPO tensor and let \(c\in \mathbb C\). If two physical words \(u,v\) have the same length, then

If \(\mathcal S\) and \(\mathcal T\) are trace-preserving completely positive, then so are \(\widehat{\mathcal S}_N\) and \(\widehat{\mathcal T}_N\) for every \(N\).

For a physical word \(\sigma \) of length \(N+1\),

For a physical word \(\sigma \) of length \(N+2\),

For every sector \(x\) of an explicit GSNNCH decomposition and every finite chain length \(N\geq 2\), all cyclic translates of the sector bond commute: \([\tau _i(B^{(x)}),\tau _j(B^{(x)})]=0\).

If a finite-chain operator of an MPO tensor has the explicit sector form of Definition 20.13.1 and nonzero trace, then its trace normalization has the sector form as well.

Each sector product \(\prod _{j=1}^N\tau _j(B^{(x)})\) of an explicit GSNNCH decomposition is positive semidefinite; hence so are the represented operator

and every chain operator equal to a positive multiple of it.

Each sector product of an explicit GSNNCH decomposition is invariant under the cyclic shift of the periodic chain, where spin \(N+1\) is identified with the first; hence so are the represented operator and every chain operator equal to a positive multiple of it [ CPGSV16 , lines 838–842 ] .

Let \(B\) act on \(L\) consecutive spins of the periodic \(N\)-site chain, embedded at base site \(i\), and let \(r\) denote the cyclic shift of sites. Reindexing both chain configurations by \(r\) carries the embedded operator to its translate at the shifted base site:

An operator embedded into the periodic chain from a positive semidefinite local factor is positive semidefinite; in particular every translate \(\tau _i(B^{(x)})\) of a sector bond is positive semidefinite.

For every pair of sectors \((k,h)\), the spaces carrying \(\eta _{k,h}\) and \(\Omega _{k,h}\) are nonzero-dimensional.

Every left and right factor in a Hayashi sector is nonzero-dimensional.

The set of Hayashi sectors is nonempty.

If an MPO tensor \(K\) is in normalized BNT-refined horizontal form, then its two-site blocking \(K^{[2]}\) is again in that form.

The word encoded by an \(L\)-site physical-block index has length \(L\).

Localizing the preparation \(Y\mapsto Y\otimes P\) at the first site gives

Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. For \(m_\alpha =\operatorname{tr}(\mu _\alpha )\),

Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. The normalized embedding \(R_\mu \) sends a positive sector family to a positive retained matrix and satisfies \(\operatorname{tr}(R_\mu (X))=\operatorname {Tr}_{\mathrm{sec}}(X)\). Conversely, the sectorwise partial trace sends a positive retained matrix to a positive sector family and satisfies \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde R_\mu (Z))=\operatorname{tr}(Z)\). These trace identities follow by summing the traces of the diagonal sectors; deleting the off-diagonal sector entries does not change the trace.

Suppose that \(K_x\) is the periodic product of translates of a positive two-site operator \(B_x\), and that \(B_x\) is supported on \(P_x\otimes P_x\). If \(N\geq 2\) and \(1\leq L\leq N\), then the normalized marginal of \(K_x\) on the first \(L\) sites satisfies

Let \(S\) be a square matrix with entries in a commutative semiring, indexed by a finite set \(I\), and let \(N\geq 0\). Then, for \(a,b\in I\),

the sum, over length-\(N\) walks \(v\) in \(I\) from \(a\) to \(b\), of the product of matrix entries along the walk. This is an elementary combinatorial identity used to derive the cyclic-product form of \(\operatorname{tr}(S^N)\) in the Case-I route.

Recording the trace-preserving ancillary reduction does not change the purification RFP condition: \(\mathrm{PRFP}_{\mathrm{tp}}(M)\Longleftrightarrow \mathrm{PRFP}(M)\).

Let \((P_i)_i\) be finitely many orthogonal projections on \(H\), not necessarily resolving the identity, and let each \(\Phi _i:\operatorname{End}_{\mathbb {C}}(H)\to \operatorname{End}_{\mathbb {C}}(K)\) be trace-preserving and completely positive. Then, for every \(X\),

Let \(a,b,c,N\in \mathbb N\) satisfy \(a+b+c\leq N\). For configurations \(u,v\) on the first \(a+b\) sites,

where the sum ranges over configurations on the last \(c\) sites. The same identity holds after replacing an arithmetically equal block length by its canonical identification.

The diagonal-summand extraction is a left inverse of the inclusion: \(\pi \iota =\operatorname{id}\).

For \(m_\alpha =\operatorname{tr}(\mu _\alpha )\),

If \(\alpha \neq \beta \), then \(P(Z)_{(\alpha ,a,i),(\beta ,b,j)}=0\).

The projection onto the vertical-sector diagonal summands is idempotent: \(P^2=P\).

For every retained-coordinate matrix \(Z\), \(P(Z)_{(\alpha ,a,i),(\alpha ,b,j)} =Z_{(\alpha ,a,i),(\alpha ,b,j)}\).

Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. Then the retained-coordinate partial trace is a left inverse of the normalized retained-coordinate embedding.

For every horizontal bond matrix \(X\), contraction of \(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \) gives \(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha (X)\).

Let \(M\) be in normalized BNT-refined horizontal form, generate positive semidefinite ring operators, and satisfy the local renormalization fixed-point equations. This horizontal hypothesis is stronger than literal CPSV canonical form. Then \(\operatorname{tr}\rho ^{(N)}(M)\neq 0\) for every \(N\geq 1\).

Suppose that \(N=(a+1)+(b+2)=(a+2)+(b+1)\). The local closure equations imply

If an MPO tensor \({\cal K}\) satisfies SAL, then its physical-trace transfer is nonzero:

Suppose a tensor with scale-invariant source zero correlation length is gauge-equivalent, through block-diagonal copy gauges, to an assembled sector decomposition. There is a scalar \(\lambda {\gt}0\) such that every copy satisfies

If \(p_k=0\), then \((l_k)_\beta =0\) for every virtual index \(\beta \). Consequently, \(\eta _{h,k}=0\) for every sector \(h\).

Suppose a tensor satisfying the literal zero-correlation-length equation is gauge-equivalent, through block-diagonal copy gauges, to an assembled sector decomposition. Every copy satisfies

This is the block equation in [ CPGSV16 , Appendix C.2, lines 1652–1657 ] .

A tensor satisfying normalized simplicity in the chosen blocked canonical-form setting is in horizontal canonical form.

Let \(V\) be a one-site matrix, not necessarily square. Applying \(V^{\otimes N}\) to both physical legs of the length-\(N\) operator generated by \(M\) is the operator generated by the tensor whose one-site physical matrices are conjugated by \(V\).

Let \(M\) have source zero correlation length. For every \(N\geq 0\), the normalized \(N\)-site marginal obtained by tracing two sites from \(\sigma ^{(N+2)}(M)\) equals the normalized \(N\)-site marginal obtained by tracing one site from \(\sigma ^{(N+1)}(M)\).

This is the first replacement at [ CPGSV16 , Appendix C.2, line 1613 ] . It does not imply the rank-one trace factorization asserted later on that line.

Let \({\cal K}\) be an MPO tensor with a physical-sector factorization and sector weights \(p_k\), such that the left tensor of every zero-weight sector vanishes. Then, for \(N\geq 1\), the sum over all sector labelings of the cyclic trace-square product reduces to the sum over active-sector labelings:

This is the reconciliation, flagged as an open reconciliation gap when Lemma 20.15.6.27 and Theorem 20.15.6.28 were added, between the full sector index set \(I\) and the active sector set \(I_*\).

For ket and bra configurations \(\sigma ,\tau \), \(V^{(N)}(\widehat M)_{(\sigma ,\tau )} =H^{(N)}(M)_{\sigma ,\tau }\). If \(p_0=(a,b)\) is the first doubled index and \(\sigma ',\tau '\) are the ket and bra configurations on the remaining \(N\) sites, contraction with \(Y_L\), \(Y_R\), and \(Y_C\) gives, respectively,

Let \(K\) be an MPO tensor and \(U\) an isometry on the physical space, \(U^\dagger U=1\). Then

where \(\widetilde K\) is \(K\) conjugated by \(U\) at every physical site. This is a project derivation preparing the Case-I route of [ CPGSV16 , Appendix C.2, lines 1434–1448 ] (the basis change defining \(\sigma _{\mathrm{tilde}}\)); it is not a claim from that source.

Summing the diagonal word evaluations of an MPO tensor over all length-\(N\) configurations closes the vertical loops site by site,

and hence \(\operatorname{tr}H^{(N)}=\operatorname{tr}(E^N)\).

If a matrix indexed by a finite type has trace one, then its index type is nonempty.

If the physical coarse-graining map \(S\) is completely positive, then the transported map \(S_{\mathrm v}\) in vertical canonical coordinates is completely positive.

If the physical refinement map \(T\) is completely positive, then the transported map \(T_{\mathrm v}\) in vertical canonical coordinates is completely positive.

For every bond matrix \(X\), set

Under the hypotheses of the preceding theorem, the transported composites satisfy

Thus the one-site and two-site contraction families lie in the respective fixed-point spaces, as in Appendix C.4, lines 1974–1980. These equations do not assert that either composite is the identity on the whole sector algebra; that conclusion requires the fixed-point structure argument of Appendix C.4, lines 1980–1993.

Write \(\widehat R_2\) for normalized state preparation, \(\widehat{\widetilde R}_1\) for controlled partial trace, and \(W_i\) for retained-coordinate reindexing. The full-matrix extension of the transported coarse-graining map is

Write \(\widehat R_1\) for normalized state preparation, \(\widehat{\widetilde R}_2\) for controlled partial trace, and \(W_i\) for retained-coordinate reindexing. The full-matrix extension of the transported refinement map is

The contraction of the assembled vertical tensor is the weighted block diagonal of the sector contractions:

If \(r\in \{ 0,\ldots ,D^2-1\} \) encodes the ordered pair \((\lfloor r/D\rfloor ,r\bmod D)\), then \(M\! \left(E_{(r\bmod D),\lfloor r/D\rfloor }\right)=M_r\).

Let \(B_{ab}\) be an arbitrary vertical tensor. If \(U\widetilde M_{ab}U^\dagger =B_{ab}\) for every pair \((a,b)\), then, for every bond matrix \(X\), \(U M(X)U^\dagger =B(X)\).

Let \(M=(M_{ab})_{a,b=0}^{D-1}\) be a matrix-product tensor whose physical index is identified with the ordered pair \((a,b)\). Then \(M(E_{ba})=M_{ab}\).

For the vertically viewed tensor \(\widetilde M_{ab,ij}=M^{ij}_{ab}\), bond-matrix contraction is the one-site physical closure: \(\widetilde M(X)=M(X)\).

Let \(B_{ab}\) be an arbitrary vertical tensor. If \(\widetilde M_{ab}=U^\dagger B_{ab}U\) for every pair \((a,b)\), then, for every bond matrix \(X\), \(M(X)=U^\dagger B(X)U\).

Suppose that

is a one-site vertical canonical-form identity. Contracting against an arbitrary bond matrix \(X\) gives

If the reverse letterwise identity is also given, then

These are the contracted canonical-form identities used in [ CPGSV16 , Appendix C.4, lines 1955–1979 ] .

Let \(\iota \) be block-diagonal inclusion of the weighted vertical-sector block family into its full matrix algebra, and let \(W\) be the canonical reindexing to retained physical coordinates. Then the retained-coordinate inclusion is \(\iota _{\mathrm v}=W\circ \iota \).

Let \(\pi \) be diagonal-block extraction to the weighted vertical-sector block family, and let \(W\) be the canonical reindexing to retained physical coordinates. Then the retained-coordinate extraction is \(\pi _{\mathrm v}=\pi \circ W^{-1}\).

Suppose that the physical maps are completely positive and trace preserving and \(U_iU_i^\dagger =1\). For positive sector families \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), the matrices \(Z_T(X)\) and \(Z_S(Y)\) are positive and

After the final compression and sectorwise partial trace,

Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. Then \(m_\alpha \ne 0\) for every \(\alpha \), and \(\widetilde R_\mu \circ R_\mu =\operatorname{id}\).

For a bond matrix \(X\), set \(V_2(X)=\bigoplus _\beta n_\beta A_\beta (X)\). Suppose that both vertical canonical forms satisfy their exact reconstruction identities, the two-site multiplicity spaces are nonzero, their diagonal weights are positive, \(S(\mathcal M^{(2)}(X))=\mathcal M^{(1)}(X)\), and \(U_1U_1^\dagger =1\). Then \(U_1^\dagger U_1 Z_S(V_2(X))=Z_S(V_2(X))\). This is the coarse-graining support identity used in [ CPGSV16 , Appendix C.4, lines 1955–1980 ] .

Under the hypotheses of Lemma 20.14.61, suppose in addition that \(U_2U_2^\dagger =1\) and that \(S\) is completely positive and trace preserving. Then \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde S(V_2(X)))=\operatorname {Tr}_{\mathrm{sec}}(V_2(X))\).

For a bond matrix \(X\), set \(V_1(X)=\bigoplus _\alpha m_\alpha M_\alpha (X)\). Suppose that both vertical canonical forms satisfy their exact reconstruction identities, the one-site multiplicity spaces are nonzero, their diagonal weights are positive, \(T(\mathcal M^{(1)}(X))=\mathcal M^{(2)}(X)\), and \(U_2U_2^\dagger =1\). Then \(U_2^\dagger U_2 Z_T(V_1(X))=Z_T(V_1(X))\). This is the refinement support identity used in [ CPGSV16 , Appendix C.4, lines 1955–1980 ] .

Under the hypotheses of Lemma 20.14.60, suppose in addition that \(U_1U_1^\dagger =1\) and that \(T\) is completely positive and trace preserving. Then \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde T(V_1(X)))=\operatorname {Tr}_{\mathrm{sec}}(V_1(X))\).

For a word \(w=((a_1,b_1),\ldots ,(a_L,b_L))\),

Fix \(L\geq 0\) and suppose that every \(c_\alpha \) is nonzero. If the simultaneous word tuples \((M_\alpha ^w)_\alpha \) of length \(L\) span \(\prod _\alpha M_{d_\alpha }\), then \(C_L(V_c)=\prod _\alpha M_{d_\alpha }\).

For every MPO tensor \(M\) and cut lengths \(L,R\), the total diagonal coefficient mass satisfies \(Z=\operatorname{Re}\operatorname{tr}\rho ^{(L+R)}(M)\).

For an MPS tensor \(A\), the pure-state operator \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\) equals the MPDO generated by the tensor \(M^{ij}=A^i\otimes \overline{A^j}\), acting on a bond space of dimension \(D^2\):

This is the single-ancilla case of the purification picture of [ CPGSV16 ] ; it lets the MPDO block-entropy theory apply to pure-state block entropies.

For words \(u_1,v_1\) of equal length and arbitrary words \(u_2,v_2\), word evaluation factors as \(M^{u_1u_2,\, v_1v_2}=M^{u_1,v_1}M^{u_2,v_2}\).

If \(M\) satisfies the nonzero up-to-scalar physical-trace relation, then there exists \(\lambda {\gt}0\) such that \(\lambda ^{-1}\mathcal T_M\) is idempotent.

The transfer map of an MPO tensor agrees with the transfer map of its doubled-index MPS view: \(\mathcal{E}_M=\mathcal{E}_{M^{\mathrm{MPS}}}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for every \(N\) and every configuration \(\sigma = (i_1, \ldots , i_N)\), \(V^{(N)}(\zeta A)_\sigma = \zeta ^{N} V^{(N)}(A)_\sigma \).

Multiplying every block weight in a block-diagonal MPS tensor by a scalar \(c\) multiplies every length-\(N\) MPV coefficient by \(c^N\).

Let \(g\) be a block count, \(\dim ,\operatorname{mult}:\{ 0,\ldots ,g-1\} \to \mathbb {N}\), weights \(\omega _{\alpha j}\in \mathbb {C}\) for \(0\leq j{\lt}\operatorname{mult}_\alpha \), and blocks \(A_\alpha \) of bond dimension \(\dim _\alpha \). The vertical direct-sum tensor is the block-diagonal direct sum over the flattened index \((\alpha ,j)\) that repeats \(A_\alpha \) with weight \(\omega _{\alpha j}\), written \(\bigoplus _\alpha \bigoplus _j\omega _{\alpha j}A_\alpha \). Its matrix product vector decomposes as

Let \(M\) generate an MPDO whose finite-chain operator \(\rho ^{(N)}(M)\) is positive semidefinite with nonzero trace. For every block length \(L\le N\), one has \(0\le I_L\).

For every block length \(L\le N\), the mutual information of the first \(L\) spins equals that of their complement: \(I_L=I_{N-L}\).

For a normalized positive semidefinite finite-chain operator, the mutual information of the empty block vanishes: \(I_0=0\).

For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the mutual information of the purification MPDO equals twice the pure-state block entropy: \(I_L=2S_L^{(N)}(A)\).

For an MPS tensor \(A\) of bond dimension \(D\) and a block length \(L\le N\), the reduced state \(\rho _L^{(N)}(A)\) of the first \(L\) spins has rank at most \(D^2\).

For a sector decomposition \(S\) with associated tensor \(A_{\mathrm{tot}}\), the MPV satisfies

Suppose that \(C\simeq C'\otimes E\). The partial trace over \(E\) is given entrywise by

It preserves positivity and trace.

Closing the virtual legs with the identity gives the periodic MPO operator: \(M_N(1)=\rho ^{(N)}(M)\).

The general three-site physical closure has coefficients

For \(M=\mathcal K\), this is the operator \(\mathcal K_3(X)\) in [ CPGSV16 , Proposition C.7, lines 1510–1516 ] .

For the maximally mixed witness \(M_{\mathrm w}\), closing the ket and bra physical legs gives \(\mathcal T_{M_{\mathrm w}} =M_{\mathrm w}^{00}+M_{\mathrm w}^{11} =\tfrac 12+\tfrac 12=1\).

If \(X\) and \(-X\) are positive semidefinite complex matrices, then \(X=0\).

For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the pure-state block entropy of the empty block vanishes: \(S_0^{(N)}(A)=0\).

Let \(A\) be an MPS tensor, let \(L\le N\), put \(m=N-L\), and write \(c=(\operatorname{tr}\rho ^{(N)})^{-1}\). Then the pure block entropy is the charpoly-root entropy sum of the \(D^2\times D^2\) bond environment:

The roots are counted with algebraic multiplicity.

For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\) and every block length \(L\le N\), the pure-state block entropy is non-negative: \(0\le S_L^{(N)}(A)\).

For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the entropy of the whole chain vanishes: \(S_N^{(N)}(A)=0\).

If an MPS tensor \(A\) saturates the area law, then the block entropies coincide for all \(1\le L,L'\le \lfloor N/2\rfloor \): \(S_L^{(N)}(A)=S_{L'}^{(N)}(A)\).

A positive semidefinite matrix of trace one and rank at most one has zero von Neumann entropy.

For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the reduced state \(\rho _L^{(N)}(A)\) of the first \(L\) spins is a density matrix: it is positive semidefinite and has unit trace.

Let \(A\) be a renormalization fixed point. For every \(L\ge 1\) and \(m\ge 1\), the operator-Schmidt Grams of the \(L\)-block and of the \(m\)-site complement are the one-site Grams:

If an MPO tensor \(M\) saturates the area law, then the mutual informations coincide, \(I_L=I_{L'}\), for all \(1\le L,L'\le \lfloor N/2\rfloor \).

Suppose that isometries \(V_k\) satisfy

Then \(A\) and \(\bigoplus _k\mu _kB_k\) have the same matrix product vectors at every positive length.

Let \(A\) be an MPS tensor of bond dimension \(D\). For a length-\(\ell \) configuration \(\sigma \) and a length-\(m\) configuration \(\tau \), write the operator-Schmidt factors as

The \(D^2\times D^2\) Gram matrix of the left factor equals the \(\ell \)-fold transfer map evaluated on a matrix unit:

where the sum runs over length-\(\ell \) configurations \(\sigma \). The Gram of the right factor \(R_m\) of an \(m\)-spin complement satisfies the bond-swapped identity

where the sum runs over length-\(m\) configurations \(\tau \).

For an MPS tensor \(A\) and a block length \(L\le N\), the block entropies of a block and its complement coincide: \(S_L^{(N)}(A)=S_{N-L}^{(N)}(A)\).

A positive semidefinite matrix of trace one has positive rank.

For \(i,j\in \mathbb {Z}/N\mathbb {Z}\), if the two length-two cyclic supports starting at \(i\) and \(j\) overlap, then

Let \(M\) generate an MPDO whose finite-chain operator \(\rho ^{(N)}(M)\) is positive semidefinite with nonzero trace. For every block length \(L\) with \(2L+1\le N\), one has \(I_L\le I_{L+1}\). The implication \(1\le L{\lt}\lfloor N/2\rfloor \Longrightarrow 2L+1\le N\) shows that this contains the range of [ CPGSV16 , Proposition 4.5, line 801 ] .

Let \(A\) be an MPS tensor with \(V^{(N)}(A)\neq 0\). For every block length \(L\) with \(2L+1\le N\), covering \(1\le L{\lt}\lfloor N/2\rfloor \), \(S_L^{(N)}(A)\le S_{L+1}^{(N)}(A)\). This is the pure-state area law of [ CPGSV16 , Section 3, line 599 ] .

Let \(A\) be an MPS tensor of bond dimension \(D\geq 1\) with \(V^{(N)}(A)\neq 0\). For every block length \(L\le N\), \(S_L^{(N)}(A)\leq 2\log D\), uniformly in \(L\) and \(N\). This is the area law of [ CPGSV16 , Section 3, line 599 ] : the block entropy is bounded by a constant independent of the block size.

Let \(A\) be an MPS tensor of bond dimension \(D\geq 1\) with \(V^{(N)}(A)\neq 0\). For the purification MPDO of \(A\), whose generated operator is the pure state \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\), the mutual information between a block of \(L\) spins and the rest satisfies \(I_L\leq 4\log D\), uniformly in \(L\) and \(N\).

Let \((A_k)_{k}\) be a finite family of block tensors of physical dimension \(d\), and fix a length \(L\); write \(A_k^w\) for the length-\(L\) word evaluation of \(A_k\) along a word \(w \in \{ 0,\ldots ,d{-}1\} ^L\). The finite-length spanning hypothesis is that the tuple-valued evaluations span the full product algebra of block matrices,

The corresponding finite-length separation property is that some length \(L\) makes the trace pairing against block words nondegenerate: there is a length \(L\) such that, for every tuple \((\Delta _k)_k\) of block matrices,

Spanning at length \(L\) gives separation at the same length: a trace functional that vanishes on every tuple \((A_k^w)_k\) vanishes on their span, hence on the whole product algebra, so nondegeneracy of the product trace pairing forces every \(\Delta _k=0\). Combined with injectivity, left-canonicality, and nonzero block weights, this finite-length separation supplies the older per-copy trace-separation hypotheses for the block-diagonal tensor.

The spanning hypothesis has an equivalent entrywise form. For a block index \(k\) and matrix entry \((a,b)\), let \(w\longmapsto (A_k^w)_{a,b}\) be the corresponding scalar word family, indexed jointly by the triple \((k,a,b)\). Linear independence of this scalar family already implies the spanning condition, hence the separation property, and hence the older per-copy trace-separation hypotheses directly.

The block-injectivity proposition of [ CPGSV16 , lines 340–345 ] gives a concrete sufficient criterion for the spanning condition: a common length \(L\) with every block \(L\)-block-injective, together with a length \(S\) at which, for every \(k\), there is a coefficient family \((c_w)_{w\in \{ 0,\ldots ,d{-}1\} ^S}\) with

Concatenating the injective prefix with this finite selector family produces the spanning condition at length \(L+S\), hence the separation property and the older per-copy trace-separation hypotheses.

The map \(\operatorname{Tr}_{\mathrm{anc},L}\) is trace-preserving and completely positive.

If \(X\) is an operator on a spin–ancilla block, then for \(i,j\in [d]^L\),

For \(i,j\in [d]^L\) and \(x,y\in [d]^K\), applying the ancillary traces to both blocks across the cut \(L\mid K\) gives

If the characteristic roots of \(\rho _L^{(N)}(M)\) are the real multiset \(\{ r_j\} _j\), then

For the canonical inclusion \(V_s\) of a matrix block, \(V_sV_s^\dagger \) is the projection onto that block.

Let \(A\) and \(B\) be MPS tensors, possibly with different bond dimensions. Suppose that \(U\) satisfies \(UU^\dagger =\mathbb {1}\) and \(A^i=U^\dagger B^iU\) for every physical index \(i\). Then \(A\) and \(B\) generate the same matrix product vectors at every positive length.

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 is the existence form of simultaneous block injectivity associated with [ CPGSV16 , lines 317–345 ] ; no numerical bound on \(L\) or \(L_0\) is asserted here.

Let \(A\) have a canonical-form MPV expansion

at every positive length, where the tensors \(\widetilde A_k\) are normal tensors of positive bond dimension and every coefficient \(\mu _k\) is nonzero. Let \((A_j)_{j=1}^{g}\) be tensors of positive bond dimension. Then \((A_j)_{j=1}^{g}\) is a basis of normal tensors for \(A\) if and only if:

1. every \(A_j\) is normal;

2. for every \(k\) there are an index \(j\), an invertible matrix \(X_k\), and a phase \(e^{i\phi _k}\) such that \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) for every physical index \(i\);

3. no two distinct tensors \(A_j\) and \(A_{j'}\) are related in this way.

This is [ CPGSV16 , Proposition 2.7 ] , read with the nonzero-coefficient convention of Definition 8.6.7.

Two BNT presentations of the same positive-length matrix product vector family have equally many representatives. After a permutation, corresponding representatives have equal bond dimension and are related by an invertible gauge and a unit-modulus phase.

The flip-configuration distribution is a probability distribution on \(\{ 0,1\} ^4\). The bond-pattern distribution takes the value \(41/128\) on the unflipped pattern, \(15/128\) on each of the four patterns flipping two cyclically adjacent bonds, \(9/128\) on the two patterns flipping two opposite bonds and on the fully flipped pattern, and zero on the eight patterns with an odd number of flipped bonds. Its one-bond marginal is \((5/8,3/8)\), and its two-adjacent-bond marginal assigns \(7/16\) to the doubly unflipped pair and \(3/16\) to each of the three remaining pairs. All the window weight families are probability distributions.

The normalized reduced states on one, two, three, and four consecutive sites have characteristic-polynomial root multisets

These spectra are consequences of the corrected \(p=1/4\) tensor and are not formulas stated in the source.

The exact scalar ratio from the four-site calculation satisfies

For \(\tilde I_1=\tilde S_1+\tilde S_3-\tilde S_4\) and \(\tilde I_2=2\tilde S_2-\tilde S_4\),

The corrected tensor is locally purified and hence positive at every chain length. It has source zero correlation length.

For the corrected tensor at \(p=1/4\) on the four-site ring, let \(S_L\) be the entropy of \(L\) consecutive spins, with natural logarithms. Then

Consequently,

The tensor has source zero correlation length and does not saturate the area law. These identities concern only the corrected fixed value \(p=1/4\).

Let \(\tilde S_k\) denote the entropy \(\sum _x h(w(x))\) of the window weight family for \(k\) consecutive spins, with \(h(x)=-x\log x\) and \(\log \) the natural logarithm. Then

Divided by \(\log 2\) these are the printed decimals \(2\), \(2.9544\), \(3.8802\), and \(2.7839\).

For every positive \(N\), the ambient trace equals the binary partition function:

The isometric inclusion of the effective binary tensor into the two-qubit one-site space does not satisfy saturation of the area law.

The supported one-site slices satisfy \(\widetilde M^{\iota (k),\iota (h)}=M^{k,h}\), while a one-site slice vanishes if either physical leg lies outside the image of \(\iota \). For every positive \(N\),

Let \(1\leq L\leq N\) and let \(x\in \{ 0,1\} ^L\). The effective-binary block state is diagonal, and its entry at \(x\) is

This includes \(L=1\) and \(L=N\). It is a project-derived consequence of the finite periodic model, not a formula stated in CPSV16.

For every positive \(N\) and binary words \(\sigma ,\tau \),

These identities first concern the effective binary support model. The ambient definition below transports the operator and trace identities, but no spectrum, entropy, saturation of the area law, or zero correlation length assertion is made here.

For every \(L\leq 4\), let \(\rho _L:=\rho _L^{(4)}(M)\) be the effective reduced state and \(\widetilde\rho _L:=\rho _L^{(4)}(\widetilde M)\) the ambient reduced state. Let \(\chi _A\) denote the characteristic polynomial of \(A\). Then

For the positive block lengths \(L=1\), \(2\), \(3\), and \(4\), the full ambient root multiset is the effective multiset together with zero of multiplicity \(2\), \(12\), \(56\), and \(240\), respectively.

The effective binary tensor generates positive semidefinite periodic operators. On the ring of length four, write \(S_L=S(\rho _L)\). With \(h(x)=-x\log x\), where \(\log \) is the natural logarithm,

On the periodic ring of length four, let \(\rho _L:=\rho _L^{(4)}(M)\) be the effective-binary reduced state, and let \(\chi _A\) denote the characteristic polynomial of a matrix \(A\). Then

Here \(a^{[m]}\) denotes a root \(a\) of multiplicity \(m\). These are the effective nonzero spectra, not the full ambient root multisets.

For \(I_L=S_L+S_{4-L}-S_4\),

Consequently, the effective binary tensor does not satisfy saturation of the area law.

The exact scalar ratio from the four-site calculation satisfies

Let \(\rho ^{(N)}_2\) denote the normalized two-site marginal of the effective binary model on the periodic ring of length \(N\). For the constant-zero word,

The same values occur at the corresponding supported entries of the ambient two-qubit-site marginals. Hence neither the effective tensor nor its ambient realization has source zero correlation length. Moreover, their one-site traces give

Thus both displayed tensors fail the literal diagram in [ CPGSV16 , Definition 4.2, lines 735–739 ] .

The first obstruction uses the nonzero up-to-scalar physical-trace relation. The final inequalities concern its literal scalar-one specialization, not the distinct legacy convention based on idempotence of the doubled-index transfer map.

The normalized four-site state is diagonal with probability \(1/8\) on the eight even-parity configurations and zero on the eight odd-parity configurations. Its relevant marginals are

Consequently,

so \(S(ABC)+S(B)=5\log 2{\lt}6\log 2=S(AB)+S(BC)\). Thus equality in strong subadditivity fails for this tripartition.

For every \(N{\gt}0\),

and \(\rho ^{(N)}(M)\) is positive semidefinite.

Tracing one site removes the \(\sigma _z^{\otimes N}\) term. More generally, every nonempty proper normalized \(L\)-site reduced state is \(2^{-L}I^{\otimes L}\). Therefore its entropy is \(L\log 2\), the mutual information is independent of \(L\), and \(M\) satisfies saturation of the area law.