Dependencies

For every positive blocked length \(n\), the blocked-basis coefficients are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target label maps. Equivalently, they are traces of powers of the pulled-back blocked-basis matrices \(\chi _{n,i,j,k}\):

For every positive blocked length \(n\), the blocked-basis coefficients obtained from an existential BNT-label theorem witness are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target maps. Equivalently, they are traces of powers of the pulled-back blocked-basis \(\chi \)-matrices:

For any active-sector fusion clause and every \(N{\gt}0\), its all-label recursive density operator factors as

Thus the multiplicity-weight factor commutes with the reconstructed recursive factor. No length-independence or spectral hypothesis is used.

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form and suppose that it satisfies the renormalization fixed-point condition. One choice of the vertical canonical decomposition and fusion coisometries satisfies, for every \(N{\gt}0\),

Equivalently, \([A_N,T_N]=0\), as asserted in [ CPGSV16 , lines 1000–1002 ] . No additional compatibility hypothesis is imposed. This statement makes no length-independence, terminal spectral, projection, Hamiltonian, or Gibbs-state assertion.

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

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.

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.

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 \(M\) 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 is an MPDO if, for every \(N\geq 1\),

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

A family of support \(*\)-algebras \(\{ \mathcal{A}_n\} _{n \ge 0}\), one at every blocking size, equipped with blocking maps \(m_n : \mathcal{A}_n \otimes \mathcal{A}_n \to \mathcal{A}_{2n}\) and inclusion maps \(\iota _n : \mathcal{A}_n \to \mathcal{A}_{n+1}\) realized, inside the ambient bond-space matrix algebra, by

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 ] .

For each blocked size \(n\), fix a basis \(\{ b_i\} _{i \in I_n}\) of \(\mathcal{A}_n\) over a finite index set \(I_n\). The blocked coefficients of \(x \in \mathcal{A}_n\) are the coordinates \((x_i)_{i \in I_n} \in \mathbb {C}^{I_n}\) with

An MPO tensor \(M\) satisfies the blocked fixed-point-algebra tower used here when there exist blocked fixed-point-algebra data such that, for every positive blocking size \(n\),

where \(E_n\) is the blocked transfer map. This is a nontrivial algebraic condition, but it is still weaker than the full coefficient formulation of [ CPGSV16 , Theorem 4.14(ii) ] ; that formulation also includes the coefficient family \(c_{\alpha ,\beta ,\gamma }^{(L)}\).

For \(i \in I_n\), the blocked inclusion coefficients are the blocked coefficients \((d_{ij})_{j \in I_{n+1}}\) of \(\iota _n(b_i) \in \mathcal{A}_{n+1}\):

For each positive blocked size \(n\) and multiplication triple \((i,j,k)\), where \(i\) and \(j\) label basis elements of \(\mathcal{A}_n\) and \(k\) labels a basis element of \(\mathcal{A}_{2n}\), a size \(r_{n,i,j,k} \in \mathbb {N}\) and diagonal entries \(\chi _{n,i,j,k,1}, \ldots , \chi _{n,i,j,k,r_{n,i,j,k}} \in \mathbb {C}\), giving the diagonal matrix

Unlike the uniform BNT-label family in [ CPGSV16 , Theorem 4.14(ii) ] , the size and diagonal entries may depend on the blocked length and on the chosen basis indices.

For \(i,j \in I_n\), the blocked multiplication coefficients are the blocked coefficients \((c^{(n)}_{i,j,k})_{k \in I_{2n}}\) of \(m_n(b_i,b_j) \in \mathcal{A}_{2n}\):

Fix a coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\), BNT-label operators \(O_L(M_\alpha )\), and trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). They satisfy the BNT algebra clause when there is a length-independent family of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), the same-length product law holds,

and the trace scalars satisfy

This is the algebra statement in [ CPGSV16 , Theorem 4.14(ii), labelled IV.13 in the source, lines 972–985 ] , before comparison with a chosen blocked basis.

Choose a vertical canonical decomposition

Separating the two factors of the doubled physical index of \(M_\alpha \) gives an MPO tensor \(\widehat M_\alpha \). Define

The chosen decomposition satisfies the tensor-attached BNT algebra clause when the tensors \(M_\alpha \) form a basis of normal tensors and the BNT algebra clause holds for precisely these operators and these trace scalars. This is the data in [ CPGSV16 , Theorem 4.14(ii), lines 972–993, and Appendix C.4, lines 2046–2064 ] ; it does not include a comparison with a two-site vertical canonical decomposition.

Let

be the one-site sector algebra and the relabelled two-site sector algebra. Their retained physical spaces are

Let \(W_1\colon \mathbb {C}^d\to \mathcal K_1\) and \(W_2\colon \mathbb {C}^{d^2}\to \mathcal K_2^\sigma \) be the corresponding coisometries, so that \(W_i W_i^\dagger =I\). Write \(\Pi _{1,\gamma }\) for the projection onto the \(\gamma \)-summand of \(\mathcal K_1\) and \(\Pi _{2,\gamma }^\sigma \) for the projection onto the \(\sigma (\gamma )\)-summand of \(\mathcal K_2^\sigma \). Let

be the canonical relabelling of a pair of physical indices. For the positive diagonal multiplicity matrices \(\mu _\gamma \) and \(\nu _{\sigma (\gamma )}\), set

Define

In Appendix C.4 of [ CPGSV16 ] , lines 2067–2068 define \(R_1\) by (41), and lines 2078–2079 define \(R_2\) by (42). Lines 2081–2083 give the normalized map \(\widetilde R_1\) in (43). Lines 2058–2071 give \(\widetilde R_2\), including its common denominator, in (44). These four definitions use only the two vertical decompositions, their sector relabelling, and (40).

A comparison between the BNT-label coefficients and the chosen blocked-basis coefficients consists of a BNT-label assignment together with the equality

This is a blocked-basis comparison: the left hand side is expanded in the chosen basis of \(\mathcal A_{2n}\), so the statement is not itself the same-length BNT product law.

The BNT-label assignment consists, for every positive blocked length \(n\), of a source label map \(\sigma _n\) from the chosen basis of \(\mathcal A_n\) and a target label map \(\tau _n\) from the chosen basis of \(\mathcal A_{2n}\) to the fixed BNT labels. It also gives the combined label map \(\sigma _n \sqcup \tau _n\) on the disjoint union of these two blocked bases.

Work in the setting of Definition 21.2.3.19. Let

be the one-site sector algebra and the two-site sector algebra relabelled by the given sector matching. Suppose that \(d_\gamma =\widetilde d_{\sigma (\gamma )}\) and that there is a unitary \(U_\gamma \) on each matched sector such that

for every physical index \(i\). Denote by \(\iota _\gamma \) the coordinate identification induced by the equality of the two bond dimensions. Define mutually inverse linear maps

These are the converse-direction middle conjugations introduced in [ CPGSV16 , Appendix C.4, lines 2053–2080 ] . The existence of the unitaries in (34) is an assumption here. They are distinct from the transported direct-sum maps in [ CPGSV16 , Appendix C.4, lines 1955–1997 ] . No maps between the ambient full matrix algebras are included here.

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.

Suppose the fusion-isometry identity holds with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and suppose the trace scalars satisfy the length-one idempotent law for the corresponding trace-power coefficients. For \(L{\gt}0\), the resulting BNT algebra clause has coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) and records positivity of every diagonal entry. The product law and the assumed idempotent condition are, respectively,

This construction realizes implication (iii)\(\Rightarrow \)(ii) of [ CPGSV16 , Theorem 4.14 ] ; it does not assert either of the implications involving the renormalization fixed-point condition.

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 fusion-isometry family over a finite set of labels consists of a family of tensors \(M_\gamma \) with a common physical dimension and per-label bond dimensions, diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) with positive entries, and isometries \(U_{\alpha ,\beta }\) such that, site by site,

This is the full-support specialization of the fusion statement in [ CPGSV16 , Theorem 4.14(iii) ] ; the tensors are the vertically read basis of normal tensors from [ CPGSV16 , Proposition 4.13 ] , so the physical dimension is the bond dimension of the original tensor. Statement (iii) of the source theorem additionally asserts the idempotent identity for the trace scalars of the vertical decomposition, recorded separately by Definition 21.2.2.5.

Scope restriction (full-support fusion family): The displayed family assumes \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) on the whole product bond space. The unrestricted source statement permits a common zero corner and instead gives a coisometry onto the active direct sum, together with exact reconstruction. This distinction is documented in [ con26j ] .

The tensors of a fusion-isometry family generate the concrete BNT-label operator family \(O_L(M_\gamma )\), and the positive diagonal matrices give the positive trace-power witness for the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

Choose a vertical canonical decomposition

The fusion clause for this decomposition consists of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and coisometries \(U_{\alpha ,\beta }\) onto the active product sectors, satisfying

They obey \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), and the active direct sum reconstructs the product tensor:

The trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) satisfy the length-one idempotent identity for the coefficients determined by these same diagonal matrices. This is statement (iii) of [ CPGSV16 , Theorem 4.14, lines 986–993 ] , attached to the chosen decomposition of [ CPGSV16 , Proposition 4.13, lines 943–951 ] .

The same-length coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) from [ CPGSV16 , Theorem 4.14(ii) ] , indexed by fixed BNT labels \(\alpha ,\beta ,\gamma \). The coefficient is attached to the length-\(L\) product \(O_L(M_\alpha )O_L(M_\beta )\), not to the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\). A diagonal \(\chi \)-family canonically determines the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) = \sum _r\chi _{\alpha ,\beta ,\gamma ,r}^{\, L}\), as in [ CPGSV16 , Appendix C.4 ] .

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.

A BNT-label coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) is length independent when \(c^{(L)}_{\alpha ,\beta ,\gamma } = c^{(1)}_{\alpha ,\beta ,\gamma }\) for every positive chain length \(L\) and all labels; equivalently, the coefficients agree at any two positive lengths. This is the case singled out in the discussion following [ CPGSV16 , Theorem 4.14 ] .

A BNT-label operator family is linearly independent at length \(L\) when the operators \(O_L(M_\alpha )\) are linearly independent, and eventually linearly independent when this holds for every length beyond some threshold. Eventual linear independence is the operator-level form of the third defining property of a basis of normal tensors in [ CPGSV16 , Section 2.3 ] , invoked for the labelled operators in [ CPGSV16 , Appendix C.4 ] .

A BNT-label operator family records, for each chain length \(L\), the operators \(O_L(M_\alpha )\) indexed by the fixed BNT labels \(\alpha \). The ambient algebra may depend on \(L\) and is not identified with a chosen blocked support algebra. Write \(\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}\) for the vertical reading. The vertical indices of the \(L\) copies are contracted cyclically, while the horizontal legs remain open, as in [ CPGSV16 , lines 962–967 ] :

A BNT-label coefficient system and a diagonal \(\chi \)-family are compatible when, for every positive length \(L\),

The same matrix \(\chi _{\alpha ,\beta ,\gamma }\) is used for all positive \(L\).

A BNT-label operator family and a BNT-label coefficient system have same-length product form when, for every positive length \(L\),

This is the product identity appearing in [ CPGSV16 , Theorem 4.14(ii) ] , stated independently of the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\).

The BNT-label theorem data consist of a BNT algebra clause together with the comparison with the chosen blocked bases. Explicitly, for every \(L{\gt}0\) the product law is

while the idempotent scalar law is

For every positive blocked length \(n\) and blocked-basis indices \(i,j,k\), the source and target label maps \(\sigma _n,\tau _n\) give the comparison

They also determine the pulled-back blocked-basis \(\chi \)-family. These data are the source-side hypotheses from which the blocked-basis consequences below are derived. In the source-side special case where the coefficient family is canonically determined by the same \(\chi \)-family, theorem data are built from that \(\chi \)-family, its positivity, and the product, idempotent, and blocked-comparison statements for the resulting canonical coefficients. The same product and idempotent predicates may also be rephrased using the canonical coefficient family determined by the \(\chi \)-matrices carried by the data, with the corresponding displayed equations.

An existential BNT-label theorem witness consists of the finite BNT-label type, the same-length operator spaces, their algebraic structure, and the corresponding BNT-label theorem data, including the pulled-back blocked-basis \(\chi \)-family. The proposition-level form is the nonemptiness of this witness type. In the source-side special case where the coefficients are canonically determined by a \(\chi \)-family, the witness is obtained from that same \(\chi \)-family together with the remaining product, idempotent, and blocked-comparison statements. For any such witness, the product and idempotent predicates can be rephrased with the canonical coefficient family determined by its \(\chi \)-matrices, with the corresponding displayed equations.

A BNT-label trace-scalar family records the scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) indexed by the fixed BNT labels \(\alpha \), as in the idempotent condition of [ CPGSV16 , Theorem 4.14(ii) ] .

Fix the one-site diagonal multiplicity matrices \(\mu _\alpha \) and the two-site diagonal multiplicity matrices \(\nu _\gamma \) in the respective vertical canonical decompositions. A multiplicity-spectrum comparison records \(\operatorname{tr}(\mu _\alpha )=m_\alpha \) and, for every \(\gamma \), equality with multiplicity between the entries of \(\nu _\gamma \) and the products \(\mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k}\). This is the conclusion of the vertical BNT comparison and positive power-sum argument in [ CPGSV16 , Appendix C.4, lines 2048–2058 ] . The definition does not require this comparison to follow from a BNT algebra clause; constructing it from the MPDO tensor is a separate obligation.

The invertible-gauge sub-result for a source-derived two-site multiplicity spectrum additionally records, for every one-site sector \(\gamma \), equality of bond dimensions \(d_\gamma =d_{\sigma (\gamma )}\) and an invertible matrix \(Z_\gamma \in \mathrm{GL}(d_\gamma ,\mathbb {C})\) such that, for every physical index \(i\),

Thus the matched normal tensors have no residual scalar phase. This is the exact invertible conjugacy retained from [ CPGSV16 , Appendix C.4, lines 2053–2057 ] . It does not include the further conclusion on line 2057 that \(Z_\gamma \) may be chosen unitary.

For a tensor-attached BNT algebra clause, a source-derived two-site multiplicity spectrum consists of a vertical canonical decomposition

a bijection \(\sigma \) from the one-site sectors to its sectors such that \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length, and, for every \(\gamma \), the multiset equality

This is the decomposition and comparison retained in [ CPGSV16 , Appendix C.4, lines 2046–2058 ] .

Let \(I=\{ 0,1\} \) and identify \(I\times I\) with four bond coordinates. Define the bond-four MPO tensor \(M\) by

This test model is motivated by [ CPGSV16 , Appendix C.4, lines 2048–2057 ] ; it is not a tensor stated there.

Let \(I=\{ 0,1\} \) and let \(E_{a,b}\) be the matrix units of \(M_{2}(\mathbb {C})\). Define a tensor \(A\) by

Its terminal matrix is

Independently, define the companion Bell projector by

and

The tensor \(A\) is the retained vertical tensor of the ambient model in Definition 21.2.3.1. Theorem 21.2.3.4 identifies \(P\) with its two-site operator after the canonical reindexing. For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\), set \(B^v=XA^vX^{-1}\) and define

Separately, define

For \(X\in \mathrm{GL}(2,\mathbb {C})\), define \(M_X\) by applying \(X\) to the ket index and \(X^{-1}\) to the bra index of \(M\):

This family is motivated by the sector comparison in [ CPGSV16 , Appendix C.4, lines 2048–2057 ] , with the physical-sector argument of [ CPGSV16 , Proposition 4.13, lines 1898–1921 ] as a template. Neither passage states the deformation or the results below.

Fix a final label \(d\). Summing the diagonal matrix-unit coefficients of the simultaneous block inverse defines

Here \(C_{(d,x,y),(i,j)}\) denotes the coefficient of the simultaneous block inverse corresponding to the matrix unit \(E_{xy}\) in block \(d\).

Fix labels \(a,b,c,d,e\). The five fusion trees in equation (33) of [ BMW\(^{+}\)17 ] determine five multiplicity spaces and five synthesis maps into the fourfold bond space. These spaces use the integer multiplicities \(N_{ab}^{c}\) of the complete zipper family, rather than the weighted positive-diagonal coordinates introduced earlier.

Each edge of the pentagon carries the printed \(F\)-matrix for the affected triple tensor factor and the identity on the untouched multiplicity factor. The three upper edges and the two lower edges define composites in the forward orientation. Replacing each edge by its inverse defines the opposite-oriented composites whose entries occur literally in equation (33).

Let \(\Lambda \) be a finite set. A complete zipper fusion family consists of positive bond dimensions \(D_a\), MPO blocks \(B_a^{ij}\in M_{D_a}(\mathbb {C})\), fusion multiplicities \(N_{ab}^{c}\), and matrices

The matrices \(X^{c+}_{ab,\mu }\) are independently chosen left inverses, not adjoints. For all labels and multiplicity indices,

The opposite product \(P_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }\) is the projector onto the product-MPO support; it need not be the identity on the ambient bond space. Every product-MPO letter is reconstructed on this support:

Writing \(B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}\), the two zipper identities are

Each block is injective, and the labelled block-letter matrix has a common left inverse: its contraction with \(B_d^{ij}\) is \(\delta _{cd}E_{xy}\) on the block selected by \((c,x,y)\). These are the fusion-tensor, reconstruction, zipper, and simultaneous inverse relations of [ BMW\(^{+}\)17 , lines 161, 181–200, and 269–277 ] .

A family of diagonal complex matrices, indexed by ordered triples \((\alpha ,\beta ,\gamma )\) drawn from a common index type, consisting of a size \(r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\) and diagonal entries \(\chi _{\alpha ,\beta ,\gamma ,1}, \ldots , \chi _{\alpha ,\beta ,\gamma ,r_{\alpha ,\beta ,\gamma }} \in \mathbb {C}\) for each triple, giving the diagonal matrix

This is the matrix \(\chi _{\alpha ,\beta ,\gamma }\) of [ CPGSV16 , Theorem 4.14(ii) ] .

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.

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

For some common length \(S\), suppose that for each final label \(\varepsilon \) there are coefficients \(c_\varepsilon (w)\), indexed by the words \(w\) of length \(S\) in the doubled physical alphabet, such that

This is the finite-word form of the simultaneous inverse relation \(B_d^+B_{d'}=\delta _{d,d'}1\) used in the fixed-channel argument of [ BMW\(^{+}\)17 ] .

The maps \(U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }\) and \(U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon }\) are obtained by restricting the two iterated fusion isometries to the summands with final label \(\varepsilon \). Their adjoints, after the canonical reassociation of the triple bond space, have the orientation and bracketing pattern of the weighted analogs of the two sides in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] .

Fix a final label \(\varepsilon \). The multiplicity spaces of the two bracketings are

These spaces have the bracketing pattern of the \((e,\mu ,\nu )\) and \((f,\lambda ,\sigma )\) index spaces in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] . Identifying the dimensions of the positive diagonal matrices with the fusion multiplicities of that equation requires the length-independent integer specialization and is not asserted here. The corresponding fixed-final row spaces are \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\).

For a full-support fusion family, let \(P_\gamma \) act on the bond space of \(M_\gamma \). Define

The terminal contraction is the same as in the source operator. The additional condition is that the active support is the whole product bond space.

Scope restriction (full-support fusion family): Full support is not an additional hypothesis of the arbitrary-chain source statement. This distinction is documented in [ con26f ] .

Let \(M\) generate matrix product density operators, and fix an active-sector fusion clause. For every terminal label \(\gamma \), close the horizontal operator leg of \(M_\gamma \) to obtain a positive semidefinite bond matrix \(K_\gamma \). Its spectral indices are pairs \(s=(\gamma ,k)\); write \(\lambda _s\) for the corresponding eigenvalue and \(E_s\) for the rank-one spectral projection. Recursively transport this family through every fusion history and the adjoint sequential coisometry, and denote the resulting all-label operator at positive length \(N\) by \(P_s^{(N)}\).

Scope restriction (active product BNT): only nonzero product corners occur in the fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction. See [ con26j ] .

A transfer-retract datum at blocked size \(n\) for an MPO tensor consists of a subspace \(\mathcal{A}_n \subseteq M_{D}(\mathbb {C})\) together with linear maps

where \(E_n\) denotes the blocked transfer map.

The blocked multiplication coefficients are in trace-power form when, for every positive blocked size \(n\) and every multiplication triple \((i,j,k)\), the coefficient \(c^{(n)}_{i,j,k}\) equals \(\sum _r \chi _{n,i,j,k,r}^{\, n}\). Equivalently, by Lemma 21.2.2.19, it is the trace of the \(n\)-th power of the corresponding diagonal matrix. The exponent \(n\) is the source blocking length of the two factors in \(\mathcal{A}_n\); the product is expanded in the basis of \(\mathcal{A}_{2n}\).

A binary compatibility predicate between an abstract structure-coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\) and a diagonal \(\chi \) family: the pair \((c, \chi )\) is said to be in trace-power form when

for every \(L \ge 0\) and every triple \((\alpha , \beta , \gamma )\), where \(r_{\alpha ,\beta ,\gamma }\) is the size of \(\chi _{\alpha ,\beta ,\gamma }\). This is the formal analog of the target identity of [ CPGSV16 , Theorem 4.14(ii) ] ; the existential version—“there exists \(\chi \) for which \((c,\chi )\) has trace-power form”—is obtained by quantifying over \(\chi \).

For labels \(\alpha , \beta , \gamma \), the left iterated fusion isometry \(U^{\mathrm L}_{\alpha ,\beta ,\gamma }\) is obtained by applying \(U_{\alpha ,\beta }\) (tensored with the identity on the bond space of \(M_\gamma \)) to fuse the first two tensors, then applying \(U_{\delta ,\gamma }\), for each label \(\delta \) appearing in the resulting direct sum, to fuse the outcome with the third tensor. Its codomain is indexed by pairs of labels \((\delta , \varepsilon )\) together with the multiplicity indices of \(\chi _{\alpha ,\beta ,\delta }\) and \(\chi _{\delta ,\gamma ,\varepsilon }\).

Let \(M\) generate matrix product density operators, suppose that its doubled-index MPS tensor is in literal CPSV canonical form, and suppose that \(M\) satisfies the renormalization fixed-point condition. Choose one active-sector BNT fusion tensor clause furnished by these hypotheses and denote it by \(\mathcal F_{\mathrm{CPSV}}(M)\).

Scope restriction (active product BNT): only nonzero product corners occur in the selected fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family, and no unsupported normal sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction of the product tensor. See [ con26j ] .

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 ] .

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) ] .

For three bond spaces, let

be the coordinate pullback induced by canonical reassociation, defined on elementary tensors by \(A_{D_1,D_2,D_3}(x_1\otimes (x_2\otimes x_3)) =(x_1\otimes x_2)\otimes x_3\).

A positive blocked \(\chi \) trace-power witness consists of a blocked \(\chi \) family, positivity of all its diagonal entries, and the positive-length trace-power identity for the blocked multiplication coefficients. This is the blocked-basis analog of the positive diagonal matrices in [ CPGSV16 , Theorem 4.14(ii) ] , without requiring a single family indexed only by the BNT labels.

A positive BNT-label \(\chi \) witness consists of a length-independent diagonal \(\chi _{\alpha ,\beta ,\gamma }\)-family, positivity of every diagonal entry, and the positive-length trace-power identity for the BNT-label coefficients. This is the coefficient statement of [ CPGSV16 , Theorem 4.14(ii) ] . If the coefficient family is chosen canonically from the same \(\chi \)-family, positivity of the diagonal entries alone gives such a witness.

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

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.

For labels \(\alpha , \beta , \gamma \), the right iterated fusion isometry \(U^{\mathrm R}_{\alpha ,\beta ,\gamma }\) is obtained by applying \(U_{\beta ,\gamma }\), tensored with the identity on the bond space of \(M_\alpha \), and then applying \(U_{\alpha ,\delta }\) for every intermediate label \(\delta \). Its codomain is indexed by the intermediate and final labels \((\delta ,\varepsilon )\) and by the multiplicity indices of \(\chi _{\beta ,\gamma ,\delta }\) and \(\chi _{\alpha ,\delta ,\varepsilon }\). This is the right-bracketed composite in the associativity equation of [ BMW\(^{+}\)17 , Section “Associativity and the pentagon equation” ] , before the change of basis by an \(F\)-matrix.

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.

For an MPDO in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition, fix the fusion clause selected by that construction. For every terminal label \(\gamma \), close the horizontal operator leg of \(M_\gamma \) to obtain a matrix \(K_\gamma \) on its bond space. Its spectral indices are pairs \(s=(\gamma ,k)\) with \(0\leq k{\lt}D_\gamma \). Write \(\lambda _s\) for the corresponding eigenvalue and \(E_s\) for the rank-one spectral projection, extended by zero on terminal labels different from \(\gamma \). Recursively transport this terminal family through every fusion history, then through the adjoint sequential fusion coisometry, and assemble over all sitewise label and multiplicity configurations. Denote the resulting all-label operator at positive chain length \(N\) by \(P_s^{(N)}\).

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

For a chain of positive length \(N\), choose at every site a BNT label \(\alpha _j\) and a diagonal coordinate \(q_j\) of \(\mu _{\alpha _j}\). Equivalently, putting \(L=N-1\), the chain has length \(L+1=N\) and contains \(L\) appended labels after the initial site. Define

The first label is the initial label of the recursive fusion, while the remaining labels are appended in the physical order \(\alpha _1,\ldots ,\alpha _{N-1}\). The recursive specification records this sequence in the reverse order \((\alpha _{N-1},\ldots ,\alpha _1)\), so removing its first entry removes the final physical site. This reversal affects only the recursive notation: the tensor product and the sequential fusion coisometry remain in the physical order \(\alpha _0,\ldots ,\alpha _{N-1}\). Let \(W_{N,c}\) be the resulting retained-row sequential coisometry for the configuration \(c\), and set \(Q_{N,c}=Q_N((\operatorname{tr}(M_\gamma ))_\gamma )\). The corresponding density block is \(m(\boldsymbol \alpha ,\boldsymbol q) W_{N,c}^\dagger Q_{N,c}W_{N,c}\). Thus the embedding \(\widetilde U\) in the source formula is \(W_{N,c}^\dagger \). The all-label recursive density operator is the direct sum of these blocks over every sitewise label and diagonal coordinate; denote it by \(R_N\). On the same direct-sum coordinates, define

Its scalar coefficients are precisely the selected diagonal coefficients of the factor \(\mu ^{\otimes N}\) in [ CPGSV16 , line 999 ] ; the identities mentioned there act on the complementary tensor factors of each block.

Let \(M\) generate matrix product density operators, fix an active-sector fusion clause, and let \(\mu \) be its strictly positive diagonal multiplicity matrix in the retained one-site coordinates. Define \(h=\operatorname{diag}(-\log \mu )\otimes I\) on two adjacent sites. For a chain of length \(L=N+2\), let \(H_L=\sum _{i=1}^{L}h_{i,i+1}\), with periodic indices.

The terminal spectral family has cardinality at most the physical one-site dimension \(d\). Fix an injection of its indices into \(\{ 1,\ldots ,d\} \) and extend the remaining eigenvalues and projectors by zero.

Scope restriction (retained vertical coordinates): the Hamiltonian and projectors act on the retained nonzero vertical sectors. Applying the adjoint retained-row map reconstructs the physical density operator, but does not provide a Hamiltonian or projectors on the original physical space. This restriction is recorded in [ con26r ] .

Scope restriction (positive chains of length at least two): the local interaction is defined on two spins, so these data are indexed by \(L=N+2\). The length-one boundary is recorded in [ con26i ] .

Let \(P_\gamma \) be operators on the bond space of the vertically read tensor \(M_\gamma \). Define one fusion layer of the recursive operator \(Q\) by

The source terminal matrices are \(P_\gamma =\operatorname{tr}(M_\gamma )\) after spectral refinement, where the trace closes the horizontal operator leg and leaves the bond indices. Before spectral refinement, the fusion identity implies

Indeed, summing the forward fusion identity over equal horizontal operator indices and distributing matrix multiplication over this finite sum gives (76). This is one pairwise step in the recursion of [ CPGSV16 , lines 999–1010 ] .

Scope restriction (one fusion layer): This entry does not assert the arbitrary-chain recursion described below. That recursion, the density decomposition, and the physical commuting Gibbs decomposition for \(L\geq 2\) are given in Theorems 21.3.46, 21.3.51, and 21.3.62, respectively. The source boundary at \(L=1\) is recorded in Remark 21.3.65.

Fix an initial label \(\alpha \) and append labels successively along a finite chain. For total chain length \(N\geq 1\), the appended-label list has length \(N-1\); the symbols \(W_N\) and \(Q_N\) below refer to this total chain length. A fusion history records, at every step, the preceding final label, the new final label, and an index in the corresponding multiplicity space. If \(h\) ends at \(\gamma (h)\), let

For terminal bond operators \(P_\gamma \), define

The associated sequential map is obtained by tensoring the preceding map with the identity on the newly appended bond and then applying the active pairwise fusion coisometry in every preceding-history block. This is the recursively typed operator and circuit in [ CPGSV16 , lines 999–1010 ] .

The canonical comparison from the right-associated full direct sum to the left-associated full direct sum is

Both intermediate labels and the final label remain in the direct sums. This definition does not select a fixed final label and makes no invertibility assertion. Its orientation is from the right-associated sum to the left-associated sum; consequently it is adjoint-oriented with respect to the printed-\(F\) row/column convention used here for [ BMW\(^{+}\)17 , Section 3.4 ] .

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 ] .

Let \((O_L(\alpha ))_{\alpha \in \Lambda }\) and \((O'_L(\alpha ))_{\alpha \in \Lambda '}\) be two labelled operator families. An explicit vertical transport consists of a bijection \(e:\Lambda '\to \Lambda \), nonzero scalars \(s_\alpha \), and algebra isomorphisms \(\Phi _L\) between the ambient operator algebras such that

for every positive length and every target label.

Given a label bijection \(e:\Lambda '\to \Lambda \) and scalars \(s_\alpha \), the transported coefficient family is

Here division is totalized by the convention \(z/0=0\).

Let

Define one-letter MPO tensors by \(M_P^{00}=P\) and \(M_Q^{00}=Q\). Equip them with one-label vertical BNT presentations whose normalized representatives and positive multiplicity weights are

The corresponding positive diagonal matrices in the algebra clause are \([1]\) and \([4/5]\). No horizontal canonical-form assumption is imposed on the sheared tensor \(M_Q\).

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, \(P_1Q_1=(PQ)_1\).

For injective families \(A_k\) and \(B_k\) with the same closed matrix product vectors, the per-block linear extension is the unique linear map \(T_k:M_{D_k}(\mathbb {C})\to M_{D_k}(\mathbb {C})\) satisfying \(T_k(A_k^i)=B_k^i\) for every letter \(i\).

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.

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 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,

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 \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

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

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

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.

Let \((A_i)_{i\in I}\) span \(M_{D}(\mathbb {C})\), and let \(C:\mathbb {C}^{m_2}\otimes \mathbb {C}^D\longrightarrow \mathbb {C}^{m_1}\otimes \mathbb {C}^D\) satisfy, for every \(i\in I\),

Then there is a matrix \(F:\mathbb {C}^{m_2}\to \mathbb {C}^{m_1}\) such that \(C=F\otimes \mathbb {1}_D\).

Let \(x_1,\ldots ,x_n\) be non-negative real numbers whose power sums \(\sum _k x_k^L\) take the same value at every exponent \(L \ge 1\). Then every \(x_k\) equals \(0\) or \(1\). The same holds for a finite family of complex numbers that are non-negative in the complex order. Nonnegativity cannot be dropped: the primitive cube roots of unity have power sums invariant under doubling the exponent, yet do not lie in \(\{ 0,1\} \).

Let \(a_1,\ldots ,a_m\) and \(b_1,\ldots ,b_n\) be nonzero complex numbers. If their power sums agree for every sufficiently large exponent, then \(\sum _i a_i^k=\sum _j b_j^k\) for every non-negative exponent \(k\).

Let \(F=\bigoplus _a F_a\) and \(G=\bigoplus _a G_a\). Then \(FGF^\dagger =\bigoplus _a F_aG_aF_a^\dagger \). Moreover, if \(r\) is a bijective reindexing of the row space of a matrix \(A\), then \(A_rXA_r^\dagger =(AXA^\dagger )_{r,r}\), where \(A_r\) denotes the corresponding row reindexing.

Let \(D{\gt}0\). The map \(A\mapsto A\otimes I_D\) is injective and preserves matrix products. Consequently, \((F\otimes I_D)(G\otimes I_D)=I\) implies \(FG=I\). In particular, orthonormal rows or columns of \(F\otimes I_D\) imply the corresponding orthonormality of \(F\).

If \(E=E_1\) and \(E^\dagger X=\lambda X\), then \(E_n^\dagger X=\lambda ^n X\) for every \(n\).

For every \(n\), \(E_1^{\dagger }X=X\) implies \(E_n^{\dagger }X=X\).

Under trace-power form for the blocked multiplication coefficients, for every positive blocked size \(n\), \(c^{(n)}_{i,j,k}=\operatorname{tr}(\chi _{n,i,j,k}^{\, n})\).

For each positive blocked size \(n\), each multiplication triple \((i,j,k)\), and every exponent \(L\), \(\operatorname{tr}(\chi _{n,i,j,k}^{\, L}) = \sum _r \chi _{n,i,j,k,r}^{\, L}\).

An active-support fusion coisometry satisfies

If the chosen blocked-basis coefficients are compared with the fixed BNT-label coefficient system, and that BNT-label system has a positive length-independent \(\chi \)-witness, then each positive-length blocked coefficient is the trace power of the corresponding BNT-label \(\chi \)-matrix:

The length-one idempotent law of a BNT algebra clause also holds for the coefficient family determined by its diagonal chi matrices:

Under positive-length BNT-label trace-power form, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

The coefficient family canonically determined by a diagonal \(\chi \)-family satisfies the positive-length trace-power condition with respect to the same \(\chi \)-family.

Identify each physical index \(p\in \{ 0,1,2,3\} \) with a pair \((p_1,p_2)\in \{ 0,1\} ^2\), and similarly write \(q=(q_1,q_2)\). The doubled-index letter at the paired index \((p,q)\) is the MPO letter \(R^{pq}\). Moreover, this bond matrix is the rank-one outer product

or equivalently

The \(16\) letters \(R^{pq}\) of the doubled-index tensor \(R^{\bullet \bullet }\) span the full bond matrix algebra \(M_4(\mathbb {C})\).

The transfer map of the dimer tensor \(R\) satisfies

Set

Then the transfer map of \(R\) has rank-one form

The tensor \(A\) is injective and normal. The matrix \(X\) is invertible but not unitary, and \(A\) and \(B\) generate the same matrix product vectors at every positive length. Nevertheless,

The matrix \(P\) is positive semidefinite while \(P_X\) is not Hermitian. Theorem 21.2.3.4 identifies \(P\) with the reindexed two-site operator of the ambient base model. It does not identify \(P_X\) with a two-site operator of a deformed matrix product density operator.

For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\) and every \(\omega \in \mathbb {R}\),

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 \(\operatorname{tr}(U^\dagger WU)=\operatorname{tr}(W)\).

The one-letter selector satisfies

Thus one physical letter selects the identity on the chosen final block and annihilates every other final block.

Let \(A\) have letters \(\mathbb {1},E_{00},E_{01},E_{10}\), let \(X=\operatorname{diag}(2,1)\), and set \(B^i=XA^iX^{-1}\). There are no \(c{\gt}0\) and \(V\in M_{2}(\mathbb {C})\) satisfying \(V^\dagger V=\mathbb {1}\) and \(V^\dagger B^iV=cA^i\) for every letter \(i\).

Neither \(T_A\) nor \(T_B\) in Lemma 21.2.2.67 is positive semidefinite.

Let \(A\) have letters \(\mathbb {1},E_{00},E_{01},E_{10}\), let \(X=\operatorname{diag}(2,1)\), and set \(B^v=XA^vX^{-1}\). After separating the doubled physical index, their terminal transfer matrices are

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.

The word evaluation of the product tensor along a configuration pair \((\sigma ,\tau )\) expands as a sum over the contracted middle configurations \(\rho \) of tensor products of word evaluations:

A positive BNT-label \(\chi \) witness gives, for every \(L{\gt}0\), the formula \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

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

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.

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 \(V\).

The product of the full comparison with its adjoint is the range projection of the left iterated fusion isometry:

The right side is not asserted to be the identity. Surjectivity of the left iterated fusion map requires completeness of the fusion decomposition, corresponding to the invertible total fusion matrix in [ BMW\(^{+}\)17 , lines 181–191 ] .

The product in the opposite order is the range projection of the right iterated fusion isometry:

The right side is not asserted to be the identity. Surjectivity of the right iterated fusion map likewise requires completeness of the fusion decomposition.

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 \(X\), \(M(X)=U^\dagger B(X)U\).

For block-diagonal letters whose block of index \(\gamma \) is the tensor product of a fixed matrix \(X_\gamma \) with a varying letter \(G_{\gamma ,l}\),

For an isometry \(U\) (a rectangular matrix with \(U^\dagger U=1\)) and a nonempty word of square matrices \(F_1,\ldots ,F_L\),

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.

Let \(D \ge 1\). If \(A\) is injective and \(A\), \(B\) generate the same matrix-product vectors, then the \(B^i\) are not all zero.

Let \(\alpha , \beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\). If \(\sum _i \alpha _i^k = \sum _i \beta _i^k\) for \(1 \le k \le n\), then the multisets \(\{ \alpha _i\} \) and \(\{ \beta _i\} \) are equal.

Two variants relax the equal-cardinality assumption. If \(\alpha : \{ 0, \ldots , m-1\} \to \mathbb {C}\) and \(\beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\) have no zero entries and their power sums agree for \(1 \le k \le \max (m,n)\), then \(m=n\) and the multisets are equal. Without a nonzero hypothesis, the list form \(\sum _{k=1}^{x_a} \lambda _{a,k}^N = \sum _{k=1}^{x_b} \lambda _{b,k}^N\) for \(1 \le N \le \max (x_a,x_b)\) gives equality of the nonzero multisets; if it also holds at \(N=0\) then \(x_a=x_b\) and the full lists agree. The nonzero variant is the one applied to the nonzero coefficients of a canonical form, which is how [ CPGSV16 , Lemma A.5 ] is used.

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 \(A\) be in literal CPSV canonical form. For canonical-form data and its BNT refinement, write

where \(B^i=GT_{\mathrm{grp}}^iG^{-1}\) and \(G\) is the listed block gauge. If marked matrices satisfy \(C=V^\dagger EV\), then for every word \(w\), including the empty word,

The marked matrix makes the closed chain nonempty; no equality of unmarked empty-word matrix product vectors is asserted.

For a linear physical mark \(f\), the linear marking of \(T_{\mathrm{grp}}\) is exactly its full grouped marked tensor. Each copy \((j,q)\) carries \(\lambda _{j,q}f(C_j)\), where \(\lambda _{j,q}=\nu _{k(j,q)}\zeta _{k(j,q)}\). For two linear marks \(f\) and \(g\), put \(F_A^u=\sum _i f_{ui}A^i\) and \(G_A^u=\sum _i g_{ui}A^i\). If

for every marked letter \(u\) and every word \(w\) of positive length, then \(F_A=G_A\) on the original bond space.

Thus the separation conclusion does not require the marked trace identity for the empty tail word. This is the positive-tail form of the argument in [ CPGSV16 , Appendix C.3, lines 1835–1858 ] : the source assumes first-site agreement at every positive system size, while its proof separates the representatives at a suitable nonzero tail length.

In the notation of Theorem 9.6.1.9, let \(k(j,q)\) be the listed block corresponding to the class-copy pair \((j,q)\). The representatives \(C_j\) define a sector decomposition \(P_{\mathrm{rep}}\) with copy weights

Every copy weight in \(P_{\mathrm{rep}}\) is nonzero. The representatives have simultaneous word-tuple separation at every sufficiently large length.

For every positive length \(N\), the grouped tensor \(T_{\mathrm{grp}}\) of Theorem 9.6.1.9 and the representative sector tensor generate the same matrix product vector. The original tensor has the same positive-length matrix product vectors as both:

Hence \(P_{\mathrm{rep}}\) is a BNT sector presentation of \(A\) whenever at least one listed block occurs.

For a canonical-form tensor generating matrix product density operators, [ CPGSV16 , Theorem 4.14 ] states that the fixed-point condition, the tensor-attached BNT algebra clause, and the displayed fusion isometry clause are equivalent.

The fixed-point condition is equivalent to the tensor-attached algebra clause. For the fusion clause, an absent product sector must have multiplicity zero, and the retained-row map is a coisometry with exact reconstruction. Under this active-support interpretation, the three conditions are equivalent by Theorem 21.3.29. Without these qualifications, the displayed fusion clause is not well-defined on empty product sectors and has the wrong isometry orientation.

Let \(M_\alpha \) be a normal tensor, and let \(X\) and \(Y\) be invertible gauges whose Gram matrices induce the same conjugation on each letter \(M_\alpha ^v\). Then there is a positive real number \(\omega \) such that

and \(\omega ^{-1/2}XY^{-1}\) is unitary. Theorem 17.4.70 gives the Figure 8 comparison in [ CPGSV16 , Proposition 4.13, lines 1909–1921 ] . This comparison supplies precisely the relative hypothesis for a sector and the distinguished sector. In the normalization \(Y=\mathbb {1}\), the resulting unitary is \(\omega ^{-1/2}X\). The common-target declarations attached to this entry are conditional algebraic corollaries, not the reflected marked-chain statement itself.

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

Under the hypotheses of Theorem 3.1.5, the map \(T\) satisfies \(T(MN) = T(M) T(N)\) for all \(M, N \in M_{D}(\mathbb {C})\).

Let \(P\in M_{m}(\mathbb {C})\) be an orthogonal projection, let \(\rho \in M_{n}(\mathbb {C})\) be positive semidefinite with \(\operatorname{tr}(\rho )=1\), and let \(\mathcal E\colon M_{m}(\mathbb {C})\to M_{n}(\mathbb {C})\) be completely positive. Define the trace-and-prepare map and its complementary restriction by

Both maps are completely positive, and \(\mathcal D_\rho \) is trace-preserving. If

for every \(X\in M_{m}(\mathbb {C})\), then the completion

is trace-preserving and completely positive. It agrees with \(\mathcal E\) on every matrix supported by \(P\):

If an MPO tensor satisfies the blocked fixed-point-algebra tower, then for every positive blocking size \(n\) and every \(X\), \(E_n^{\dagger }X=X\) implies \(E_1^{\dagger }X=X\).

If an MPO tensor satisfies the blocked fixed-point-algebra tower, then \(\operatorname{Fix}(E_n^{\dagger })=\operatorname{Fix}(E_1^{\dagger })\) for every positive blocking size \(n\).

Let \(M\) be a matrix product density operator with physical dimension \(d\) and bond dimension \(D\), equipped with a tensor-attached BNT algebra clause, and let \(M_\gamma \) be any retained normal tensor. Separating its doubled physical index gives two physical legs of dimension \(D\). Closing these legs gives the bond matrix

Then \(T_\gamma \succeq 0\). This is a necessary consequence of the tensor-attached clause and positivity of the one-site operator; it does not require fusion maps or a renormalization fixed-point hypothesis.

Assume the doubled tensor is trace-preserving and the transfer map admits a positive-definite fixed point. If, for every positive blocked size \(n\),

then the blocked fixed-point-algebra tower holds.

Assume the blocked fixed-point-algebra tower holds. If \(X\ne 0\), \(E_1^\dagger X=\lambda X\), and \(\lambda ^n=1\) for some \(n{\gt}0\), then \(\lambda =1\).

There exists an MPO tensor of physical and bond dimension two such that the doubled tensor is trace-preserving, the transfer map has a positive-definite fixed point, and a blocked fixed-point-algebra tower, while the doubled-index transfer map is not idempotent.

Suppose that the labelled tensors of a length-independent fusion family form a source basis of normal tensors. There is one common positive blocking at which they determine a complete zipper fusion family. Its bond dimensions are the original labelled bond dimensions, its fusion multiplicities are \(N_{\alpha \beta }^{\gamma }=\dim \chi _{\alpha ,\beta ,\gamma }\), its synthesis map is the adjoint of \(U_{\alpha ,\beta }\), and its analysis map is \(U_{\alpha ,\beta }\), after the canonical identification \(\mathbb {C}^{D_\alpha D_\beta } \cong \mathbb {C}^{D_\alpha }\otimes \mathbb {C}^{D_\beta }\). In particular, the two zipper identities, literal reconstruction, injectivity, and the simultaneous block-letter inverse are conclusions; none is an additional hypothesis.

Assume the doubled tensor is trace-preserving and the transfer map admits a positive-definite fixed point. If \(E_1^2 = E_1\), then there is a stationary family of support algebras

with multiplication \(m_n(x,y) = xy\) and inclusion \(\iota _n(x) = x\).

Assume that every blocked BNT label has positive multiplicity and that the blocked vertical tensor has the exact coisometric reconstruction \(T=U_2^\dagger C_2U_2\). Let \(E_{\gamma ,0}\) be the canonical inclusion of the first copy of label \(\gamma \), and put \(F_\gamma =U_2^\dagger E_{\gamma ,0}\). Then \(F_\gamma \) is an isometry, intertwines \(T\) with the distinguished weighted BNT copy in both directions, and compresses \(T\) to that copy exactly. For an active product corner covered by \(\gamma \), transport \(F_\gamma \) along the stated equality of bond dimensions. The transported map remains an isometry and obeys the corresponding transported compression identity.

Fix one-site and two-site vertical canonical decompositions whose sectors are identified by the transported unitary relabelling. Suppose that the product tensors have positive diagonal fusion matrices \(\chi _{\alpha ,\beta ,\gamma }\) and exact forward and reverse coisometric fusion identities. If the blocked operator has the two simultaneous representations of Theorem 20.14.82, then

This is the idempotent coefficient identity in [ CPGSV16 , Appendix C.4, lines 2030–2042 ] .

Under the hypotheses and with the sector correspondence of Theorem  20.14.81, let \(B\) be the vertical reading of the two-site blocking, and write \(O_L(A)=\operatorname{Tr}_{\mathrm{bond}}(A_{i_1}\cdots A_{i_L})\) for the length-\(L\) closed-chain operator. If \(L{\gt}0\), then the same equivalence \(\sigma \), dimension identifications, and unitaries appearing in the tensor-letter identity satisfy

These are the two representations in [ CPGSV16 , Appendix C.4, lines 2011–2018 ] .

Local fix (blocked coefficient exponent): CPSV16 Appendix C.4, line 2013 prints \(m_\gamma ^L/n_\gamma \). The line-2008 tensor scaling and line 2040 give \((m_\gamma /n_\gamma )^L\). This is documented in [ con26x ] .

For every integer \(L\geq 0\), pairing the ket and bra letters site by site gives the canonical identification

Under this identification, the doubled-index MPS tensor of \(M^{[L]}\) is the physical reindexing of the \(L\)-blocked doubled-index MPS tensor of \(M\). In particular, one is injective if and only if the other is.

For every integer \(L\geq 0\) and two MPO tensors \(M,N\) with the same physical dimension, let juxtaposition denote the MPO tensor product obtained by contracting the intermediate physical index. Then \((MN)^{[L]}=M^{[L]}N^{[L]}\). For \(L=0\), each blocked tensor is the unique empty-word local matrix, namely the identity, so the equality reduces to the product of two identity local tensors.

Let \(M\) be an MPDO whose doubled-index MPS tensor is in literal CPSV canonical form, equipped with a tensor-attached BNT algebra clause \(\widetilde M=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \). There is a vertical canonical decomposition of the two-site blocking,

Its normal-tensor sectors are in bijection, via \(\sigma \), with the sectors of the full-support one-site product expansion such that \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length and, for every \(\gamma \),

as multisets. Thus this two-site decomposition and relabelling realize the multiplicity-spectrum comparison of Appendix C.4, lines 2046–2058 of [ CPGSV16 ] . Moreover, the paired bond dimensions agree and there are invertible matrices \(Z_\gamma \) satisfying

exactly. This is only the invertible-gauge sub-result of lines 2053–2057; no unitary choice of \(Z_\gamma \) follows from this statement.

Let \(M\) be an MPDO whose doubled-index tensor is in literal CPSV canonical form. If \(M\) has a tensor-attached BNT algebra clause, then there are trace-preserving completely positive maps

such that, for every virtual matrix \(X\),

Thus \(M\) is a renormalization fixed point in the sense of Definition 20.2.1. This is implication (ii)\(\Rightarrow \)(i) of [ CPGSV16 , Theorem 4.14, lines 972–993 ] .

Local fix (mixed one-site/two-site prefix): The common marked comparison needed to normalize the exact sector gauges is supplied by Theorem 21.2.3.28, as recorded in [ con26n ] .

Local fix (zero-sector complement): The physical maps are completed on the discarded complements as recorded in [ con26z ] .

Represent a sector family by its block-diagonal matrix. For a map whose domain is a sector algebra, first extract the diagonal sector blocks from an arbitrary matrix. With these canonical full-matrix representatives, each of \(R_1\), \(R_2\), \(\widetilde R_1\), and \(\widetilde R_2\) is completely positive.

The raw retractions are left inverses of their normalized embeddings:

No identity is asserted for either reverse composite on the full ambient matrix algebra.

For every vertical letter \(v\), the one-site maps satisfy

The relabelled two-site maps satisfy

The first and second one-site identities combine the one-site weighted sector form of [ CPGSV16 , Appendix C.4, lines 2048–2051 ] with, respectively, [ CPGSV16 , Appendix C.4, lines 2067–2068 ] and [ CPGSV16 , Appendix C.4, lines 2081–2083 ] . The first two-site identity combines the matched weighted sector form of [ CPGSV16 , Appendix C.4, lines 2048–2064 ] with the map \(R_2\) of [ CPGSV16 , Appendix C.4, lines 2078–2079 ] . The second combines the same sector form with the normalized map \(\widetilde R_2\) of [ CPGSV16 , Appendix C.4, lines 2070–2071 ] .

For arbitrary ambient matrices \(X\in M_{d}(\mathbb {C})\) and \(Y\in M_{d}(\mathbb {C})\otimes M_{d}(\mathbb {C})\),

If \(X\) and \(Y\) are positive semidefinite, then

In the other direction, for \(Z\in \mathcal A_1\) and \(Z'\in \mathcal A_2^\sigma \),

Hence the canonical full-matrix representatives of \(\widetilde R_1\) and \(\widetilde R_2\) are trace-preserving completely positive maps.

The inequalities in (47) may be strict. Indeed, \(W_i^\dagger W_i\) is the projection onto the retained nonzero sectors and need not be the identity on the ambient space. A positive matrix supported on the discarded orthogonal complement is annihilated by the corresponding compression. Thus \(R_1\) and \(R_2\) are not asserted to preserve the trace on all ambient positive matrices, and no term acting on the discarded complement is included.

Suppose that the chosen blocked-basis coefficients are compared with a fixed BNT-label coefficient system, and that this BNT-label system has a positive length-independent \(\chi \)-witness. Then the chosen blocked bases carry a positive blocked \(\chi \) trace-power witness. The construction is obtained by pulling back the BNT-label \(\chi _{\alpha ,\beta ,\gamma }\)-matrices along the source and target label maps.

Suppose that the labelled doubled-index tensors form the basis of normal tensors, and that their fusion-isometry family has positive trace-power coefficients independent of the positive chain length. Then there is one integer \(L{\gt}0\) such that the blocked tensors \(M_\gamma ^{[L]}\) span their full product matrix algebra in one letter. Hence every \(M_\gamma ^{[L]}\) is injective.

For this same \(L\), write

where \(r_{\alpha ,\beta ,\gamma }\) is the dimension of the specialized chi space. The original fusion isometry satisfies

the two blocked zipper identities

and the literal reconstruction

In the setting of the preceding theorem, suppose in addition that the doubled-index tensor of \(M\) is in literal CPSV canonical form. Assume that, for every positive tail length, the identity-dressed marked chain equals the reflected-adjoint marked chain of \(M_\gamma \) in the adjoint two-site blocking. No equality for the empty tail is assumed. Then, for every vertical letter \(v\),

Consequently, there is a real number \(\omega _\gamma {\gt}0\) such that

Scope restriction (conditional reflected target): This theorem assumes that the identity-dressed physical-letter family has the same reflected-adjoint target as the gauge-dressed marked chains at every positive tail length. The oblique construction alone cannot provide this identity: adjoint reflection exchanges \(L_\gamma \) and \(R_\gamma \) and is governed by the Gram-dressed compression (19). The mixed-prefix comparison below derives the target under the standing canonical-form and positivity assumptions. This distinction is recorded in [ con26n ] .

Under the hypotheses of the preceding conditional theorem, define

Then \(U_\gamma \) is unitary and, for every physical index \(i\),

More generally, the same conclusion follows from any exact invertible sector gauge whose Gram matrix is a positive scalar multiple of the identity. This is the inverse-square-root normalization used in [ CPGSV16 , Proposition 4.13, lines 1903–1908 ] , applied at Appendix C.4, lines 2053–2057.

Scope restriction (conditional reflected target): The physical-letter coefficients are supplied by Theorem 21.2.3.24, while their common positive-tail reflected target is an assumption of this theorem. Appendix C.4, lines 2048–2057, does not itself spell out the star-compatible comparison for the raw representative. The mixed-prefix theorem below supplies it under the standing assumptions, as recorded in [ con26n ] .

Suppose that the one-site and two-site sector decompositions are paired by a relabelling \(\sigma \) and unitaries \(U_\gamma \) satisfying

for every sector \(\gamma \) and every vertical letter \(i\). With the maps of the preceding two subsections, set

Let

The matrices \(P_1\) and \(P_2\) are orthogonal projections. If \(d{\gt}0\), put \(\tau _1=d^{-1}I_d\) and \(\tau _2=d^{-2}I_{d^2}\), and define

If \(d=0\), take both maps to be zero; their domain and codomain matrix algebras then consist only of zero. Then \(\mathcal T\) and \(\mathcal S\) are trace-preserving completely positive maps on the full physical matrix algebras. For every virtual matrix \(Z\), they satisfy

Hence \(M\) is a renormalization fixed point in the sense of Definition 20.2.1.

The compositions in (57) are the formulas of [ CPGSV16 , Appendix C.4, lines 2065–2085 ] . Those formulas determine the action after compression to the retained physical sectors. With rectangular coisometries, the literal composites annihilate the discarded zero-sector complements and fail to preserve trace there. Definition 4.1 nevertheless requires maps on the full physical algebras [ CPGSV16 , Definition 4.1, lines 638–660 ] . The source does not specify a trace-restoring modification on the discarded complements. Formula (59) makes one such choice when the physical space is nonzero. Replacing \(\tau _1,\tau _2\) by any density matrices gives the same action on the physical operators generated by \(M\). The conclusion is conditional on the unitaries in (56); their existence is not asserted here.

Let the one-site and blocked two-site BNT decompositions have the source-derived exact sector pairing, including the matched multiplicity spectrum, and suppose that \(M\) generates matrix product density operators and that its doubled-index tensor is in literal CPSV canonical form. Assume, for every sector \(\gamma \), every positive tail length, every pair of open bond indices, and every pair of two-site tail words, that the identity-dressed marked-chain coefficient equals the corresponding reflected-adjoint marked-chain coefficient of \(M_\gamma \) in the adjoint two-site blocking, with both tail words reversed on the reflected-adjoint side. This is the additional comparison needed to reproduce the Figure 8 argument in [ CPGSV16 , Proposition 4.13, lines 1903–1921 ] at its invocation in Appendix C.4, line 2057. Then there are trace-preserving completely positive maps \(\mathcal T\) and \(\mathcal S\) satisfying the identities in (60) for every virtual matrix \(X\). Hence \(M\) is a renormalization fixed point in the sense of Definition 20.2.1. The target is an explicit hypothesis of this theorem. It is derived from the BNT algebra clause by the mixed-prefix comparison in Theorem 21.2.3.28.

The maps \(\widetilde T\) and \(\widetilde S\) are completely positive and preserve the total trace on the direct sums. They are mutually inverse:

They also carry the matched tensor letters in both directions:

Suppose that the one-site and two-site sectors are related by the exact invertible gauges above. Let \(\gamma \) be a one-site sector matched with a two-site sector by the bijection \(\sigma \), and identify their bond dimensions. There are an isometry \(V_\gamma \) from the sector bond space into the blocked physical-index space and a scalar \(c_\gamma {\gt}0\) such that, for every vertical letter \(v\),

If, in addition, \(M\) generates matrix product density operators, then for every pair of open bond indices and every pair of two-site tail words, the marked-chain coefficient of

equals the corresponding reflected-adjoint marked-chain coefficient in the adjoint two-site blocking. This is one of the two marked comparisons in Figures 7–8 of [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] , applied to the sector matching of Appendix C.4, lines 2048–2057.

Scope restriction (gauge-dressed corner): This theorem by itself constructs only the physical corner in (14). It does not construct a same-sided identity-dressed physical corner for \(M_\gamma \) and does not prove the common-target Gram identity

This restriction is documented in [ con26n ] . The mixed-prefix comparison below proves the Gram identity without constructing a same-sided raw corner for the two-site blocking.

In the setting of the preceding theorem, choose \(V_\gamma \) and \(c_\gamma {\gt}0\) as in (14), and set

Then, for every vertical letter \(v\),

Write \(\mathcal{R}(M_\gamma )^{rs}\) for the horizontal slice \(v\mapsto (M_\gamma ^v)_{rs}\). For open bond indices \(r,s\) and blocked physical indices \(i,j\), define

These coefficients realize every identity-dressed horizontal slice in the physical-letter span of the two-site blocking:

This supplies the physical-letter span needed for the direct comparison of the two vertical forms at [ CPGSV16 , Appendix C.4, lines 2048–2057 ] .

Scope restriction (oblique realization): In general \(L_\gamma \ne R_\gamma \). Thus (18) is not a positive same-sided corner. At the marked-chain level, adjoint reflection exchanges the two maps together with the bond-pair and tail reversal. The reflected orientation is therefore governed by the Gram-dressed compression in (19), rather than by a second copy of the raw tensor. The theorem supplies the physical-letter realization needed for the identity-dressed mark, but it proves neither a common reflected target nor the Gram identity

This is the obstruction in the direct two-site argument. The mixed-prefix theorem below instead compares the one-site raw corner with the two-site gauge corner while retaining the same one-site tail.

In the setting above, each sector \(\gamma \) admits an isometry \(W_\gamma \colon \mathbb {C}^{d_\gamma }\to \mathbb {C}^d\) and a scalar \(a_\gamma {\gt}0\) such that

for every vertical letter \(v\). Here the bond space of \(M_\gamma \) is identified with that of its matched two-site sector. This is the distinguished positive-weight copy in the one-site vertical decomposition of [ CPGSV16 , Proposition 4.13, statement lines 943–951, and Appendix C.4, lines 2046–2057 ] .

Suppose the active-sector fusion clause holds for a chosen vertical canonical decomposition. Then the algebra clause holds for the same decomposition. For every positive chain length \(L\), its coefficients are \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), determined by the same diagonal matrices. Thus statement (iii) implies statement (ii) of [ CPGSV16 , Theorem 4.14, lines 972–993 ] . No assertion concerning the renormalization fixed-point condition is made here.

For a coefficient system satisfying the same-length product law for a BNT-label operator family and in trace-power form, at every positive length \(L\) at which the labelled operators are linearly independent, the associativity constraint is a restriction on the family of diagonal matrices:

In [ CPGSV16 , Section 4.5 ] these restrictions, combined with the isometry statement, lead to a pentagon-like equation for the fusion isometries whose solutions come from unitary fusion categories.

At every positive length \(L\) at which the operators are linearly independent, the structure coefficients of the same-length product law satisfy

These are the restrictions that the associativity of the multiplication imposes on the structure coefficients, noted in [ CPGSV16 , Section 4.5 ] .

Two coefficient systems satisfying the same-length product law for the same operator family agree at every positive length at which the operators are linearly independent.

If the BNT trace scalars satisfy the idempotent coefficient condition and the BNT-label coefficients come from a positive length-independent \(\chi \)-witness, then

If the BNT trace scalars satisfy the idempotent coefficient condition with the coefficient family canonically determined by a diagonal \(\chi \)-family, then

Suppose a BNT-label coefficient system is in trace-power form \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and is length independent. Then every diagonal entry \(\chi _{\alpha ,\beta ,\gamma ,r}\) lies in \(\{ 0,1\} \), as stated in [ CPGSV16 , Section 4.5 ] ; by positivity every entry equals one. Consequently, at every positive length, \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\).

If a BNT-label coefficient system is in trace-power form with respect to a diagonal \(\chi \)-family all of whose entries equal one, then for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma }\), so the coefficient system is length independent.

The tensor \(R\) has a one-label vertical BNT presentation. Its component has \(16\) physical letters, bond dimension \(4\), and

The \(1\times 1\) multiplicity matrix and the \(2\times 2\) positive coefficient matrix are

The vertical tensor is reconstructed exactly by

For every \(L{\gt}0\), its closed component operator satisfies

At length one, \(m_0=\operatorname{tr}(\mu _0)=25/32\) and \(c^{(1)}_{0,0,0}=32/25\), so the idempotent identity is

The doubled-index tensor \(A=R^{\bullet \bullet }\) (bond dimension \(4\), \(16\) letters \(R^{pq}\), \(p,q\in \{ 0,\ldots ,3\} \)) is in the canonical form of Definition 8.6.7 (25): a single retained normal block occupying the full ambient bond dimension \(4\) (\(r=1\), \(D_1=4\), \(U=\mathbb {1}\)), with weight \(\mu =\sqrt{337/512}\), the unique positive scalar making the retained block’s transfer map exactly idempotent (hence spectral-radius one).

For every positive chain length \(N\), the closed operator of \(R\) factors literally as

where \(B\) is the boundary partial isometry sending a chain of first bits \(a\) to the unique cyclic-bond-matching physical-index string reading \(a\) back through its first bond bit (\(B^{\mathsf T}B=1\)), and \(W_N(a,b)=\prod _nW(a_n,b_n)\) is the \(N\)-fold Kronecker power of \(W\). The Walsh–Hadamard change of basis \(H=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\) diagonalizes \(W\) exactly:

Thus the matrix \(\chi \) diagonalizes the local factor \(W\) in the horizontal closed-operator formula. Theorem 21.2.2.45 identifies the same matrix with the coefficient matrix of an explicit one-label vertical BNT presentation. The factorization here still concerns the original horizontal operator \(\rho ^{(N)}(R)\); it is not the scalar reconstruction formula for the rotated vertical contraction.

The hand-defined family \(c^{(L)}_{0,0,0}\) is a positive BNT-label \(\chi \) trace-power model (Definition 21.2.2.11) with \(\chi =\operatorname {diag}(1,\tfrac {7}{25})\). For every \(L{\gt}0\),

The matrix \(\chi \) diagonalizes the local factor \(W\) in Theorem 21.2.2.47, so

Hence \(c^{(L)}=\operatorname{tr}(W^L)=1+(7/25)^L\). Here \(W^L\) is the ordinary matrix power of the fixed \(2\times 2\) matrix \(W\), not the Kronecker power \(W_L\) inside \(\rho ^{(L)}(R)\). Theorem 21.2.2.45 identifies this coefficient model with the explicit one-label vertical component. It does not identify \(O_L(M_0)\) with the original horizontal operator \(\rho ^{(L)}(R)\).

The tensor \(R\) of Theorem 21.2.2.30 is a renormalization fixed point in the sense of Definition 20.2.1. The refinement map \(\mathcal T\) has two Kraus operators built from the eigenvectors of \(W\), with amplitude \(5/8\) on the uniform eigenvector and \(\sqrt7/8\) on the alternating eigenvector. The coarse-graining map \(\mathcal S\) has four Kraus operators: two share the amplitude \(1/\sqrt2\) and the eigenvectors of \(W\) on the physical-index pairs satisfying the bond-matching condition, and two are a deterministic assignment on the pairs that fail it, chosen so the family resolves the identity. This proves the fixed-point condition directly. The following theorem constructs the corresponding one-label tensor-attached BNT algebra clause explicitly.

The closed operator \(\rho ^{(N)}(R)\) is positive semidefinite for every positive chain length \(N\): \(R\) is an MPDO. Writing each bond matrix \(R^{pq}\) as a rank-one product of letter-column vectors scaled by \(25/32\), the closed-chain trace of a length-\(N\) configuration \((\sigma ,\tau )\) collapses entrywise to

where the indicator is one exactly when both \(\sigma \) and \(\tau \) satisfy the cyclic bond-matching condition (consecutive letters share their bond bit around the ring), and \(W\) is the local factor from Theorem 21.2.2.30 with eigenvalues \(\{ 1,7/25\} \). So \(\rho ^{(N)}(R)\) is \((25/32)^N\) times the Hadamard product of the rank-one bond-matching indicator with the pullback of the \(N\)-fold Kronecker power of \(W\); the Schur product theorem then gives positivity.

Theorem 21.2.2.29 answers the question of [ CPGSV16 , lines 995–999 ] affirmatively, but its one-label block has a single diagonal entry \(\sqrt{1/2}\), and rescaling that entry to \(1\) makes the displayed coefficient constant. The family constructed here has a block with two distinct positive diagonal entries, and its length dependence survives every uniform positive rescaling of the block. The four letter matrices are the scaled matrix units of \(M_{2}(\mathbb {C})\)

Identify the physical index set \(\{ 0,1,2,3\} \) with \(\{ 0,1\} \times \{ 0,1\} \) and let \(R\) be the bond-four MPO tensor obtained by undoing the vertical reading of the letters \(B^{ab}=A^a\otimes \overline{A^b}\):

with \((p,q)\) the physical indices and \((a,b)\) the bond indices. Its physical-trace transfer is the rank-one projection

where \(t_a=\operatorname{tr}(A^a)\); it satisfies \(\mathcal T_R^2=\mathcal T_R\), the literal zero-correlation-length condition of [ CPGSV16 , Definition 4.2, lines 736–741 ] . The hand-defined one-label coefficient model is in trace-power form with \(\chi =\operatorname {diag}(1,\frac{7}{25})\):

Hence \(c^{(1)}\ne c^{(2)}\), so the family is not length independent. For every real number \(s{\gt}0\), the rescaled block \(s\chi =\operatorname {diag}(s,\frac{7s}{25})\) has coefficient

which is again not length independent: if \(c_s\) were constant in \(L\), comparing the lengths \(1\), \(2\), and \(3\) would force \((1+\lambda )(1+\lambda ^3)=(1+\lambda ^2)^2\) with \(\lambda =\frac{7}{25}\), which is false. Unlike the single-entry block of Theorem 21.2.2.29, no uniform rescaling makes both diagonal entries equal to one.

The theorem establishes the obstruction for the displayed coefficient family and for every uniform positive rescaling of its diagonal matrix. The unscaled family is attached below to an explicit one-label vertical BNT presentation of \(R\). No such tensor attachment is asserted here for the arbitrary rescaled families. Positivity, literal CPSV canonical form, and the renormalization fixed-point property of \(R\) are proved separately. The tensor is a project example motivated by the discussion in [ CPGSV16 , lines 995–1010 ] ; it is not a tensor stated there.

Let \(\widehat R=\mu ^{-1}R^{\bullet \bullet }\) be the single retained normal block in the displayed canonical form, where \(\mu =\sqrt{337/512}\). Then

Consequently neither \(\mathcal T_R\) nor \(\mathcal T_{\widehat R}\) is nilpotent. Thus the retained basis element satisfies the non-nilpotency clause in simplicity.

This conclusion concerns only that clause for the displayed one-block presentation, and it makes no claim about other presentations or blockings.

There is an MPDO \(M\) in literal CPSV canonical form that is a renormalization fixed point and has a tensor-attached BNT presentation whose structure coefficients depend on the positive chain length. One may take \(M=R\) and the one-label presentation above, for which

This gives a normalization-free variant of the question after Theorem 4.14 for the displayed vertical presentation. The statement does not assert that length dependence is invariant under a change of presentation. It also does not supply the source’s additional canonical normalization: every weight has modulus at most one and at least one has modulus one.

The dimer tensor \(R\) is simple. The positive blocking length is \(L=1\). After the canonical relabeling of the doubled physical alphabet, the basis of normal tensors has one member, namely the retained block \(\widehat R\), and its physical-trace transfer is non-nilpotent.

The two diagonal entries of \(\chi \) are eigenvalues of the local factor \(W\): the uniform vector is fixed (eigenvalue \(1\)) and the alternating vector is scaled by \(7/25\). The per-site transfer map \(\varphi (Y)=\sum _aA^aY(A^a)^\dagger \) has eigenvalues \(\{ 1,7/25,0,0\} \) with explicit eigenvectors: the identity is fixed, the traceless diagonal \(\operatorname {diag}(1,-1)\) is scaled by \(7/25\), and the off-diagonal matrix units are annihilated. This identifies the spectral data of \(W\) and of \(\varphi \) with \(\chi \).

If the same-length BNT product identity holds and the BNT-label coefficients come from a positive length-independent \(\chi \)-witness, then for every \(L{\gt}0\),

If the same-length BNT product identity holds with the coefficient family canonically determined by a diagonal \(\chi \)-family, then for every \(L{\gt}0\),

Let \(c^{(L)}_{\alpha ,\beta ,\gamma }\) be the coefficient family and let \(\chi _{\alpha ,\beta ,\gamma }\) be the positive diagonal matrices carried by the theorem data. For every \(L{\gt}0\),

where \(c^{\chi }\) is the canonical coefficient family determined by \(\chi \), and every diagonal entry of every \(\chi _{\alpha ,\beta ,\gamma }\) is strictly positive. Consequently, for every \(L{\gt}0\), the same-length product identity reads

and, at \(L=1\), the idempotent identity reads

BNT-label theorem data give a positive blocked-basis \(\chi \) trace-power witness by pulling the length-independent BNT-label \(\chi \)-family back along the blocked-basis comparison maps: writing \(\chi _{n,i,j,k}\) for the resulting blocked-basis diagonal matrix at a positive blocked length \(n\), \(\chi _{n,i,j,k} =\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}\). Thus the two matrices have equal size, diagonal entries, and traces of all powers.

An existential BNT-label theorem witness gives a positive blocked-basis \(\chi \) trace-power witness for the chosen algebra-structure data: a blocked \(\chi \)-family with positive diagonal entries such that, for every positive blocked length \(n\),

The blocked-basis \(\chi \)-family is the pullback of the BNT-label \(\chi \)-family along the source and target comparison maps \(\sigma _n,\tau _n\). The proposition-level nonempty form of the witness gives the same existential blocked-basis family, stated either through this trace-power predicate or through the coefficient equation it unfolds to.

Suppose the structure coefficients have positive trace-power form and are independent of the positive chain length. Then \(\chi _{\alpha ,\beta ,\gamma }=1_{r_{\alpha ,\beta ,\gamma }}\) for every triple of labels. If

the fusion isometries satisfy

These are the identity-weight specialization of Equations (19)–(21) in [ BMW\(^{+}\)17 , Section 3 ] . The MPDO fusion identity appears in [ CPGSV16 , Theorem 4.14(iii), lines 986–991 ] ; its identity-weight specialization follows from the length-independent case at [ CPGSV16 , line 1010 ] .

Let \(M\) be an MPDO whose doubled-index tensor is in literal CPSV canonical form, and let the one-site and two-site sectors be paired by the exact gauges \(Z_\gamma \) above. Then, for every sector \(\gamma \) and every vertical letter \(v\),

Consequently there is a real number \(\omega _\gamma {\gt}0\) such that

and there is a unitary \(U_\gamma \) satisfying

The identity-dressed marked chains therefore also have the positive-tail reflected target used in the conditional statements above.

Local fix (mixed one-site/two-site prefix): The proof compares a one-site raw corner with a two-site gauge corner while retaining the same unblocked tail. This supplies the common comparison omitted when [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] is invoked in [ CPGSV16 , Appendix C.4, lines 2048–2057 ] . The distinction from the unsuccessful direct two-site comparison is recorded in [ con26n ] .

Suppose that the BNT algebra clause holds and that the one-site and two-site vertical multiplicity spectra have been compared as above. Then every two-site multiplicity matrix has the prescribed trace:

This is the normalization used for the preparation matrix \(\nu _\gamma /m_\gamma \) in the algebra-to-RFP implication of [ CPGSV16 , Appendix C.4, lines 2058–2064 ] .

For every pair of labels and physical indices, write \(G_{\alpha ,\beta }^{ij} =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma } \otimes M_\gamma ^{ij}\). The fusion isometry satisfies the two chi-weighted zipper identities

The corresponding identities in [ BMW\(^{+}\)17 , Equation (21) ] have identity matrices on the multiplicity spaces in place of the positive diagonal matrices \(\chi \). Their unweighted form requires the length-independent integer specialization and is not a consequence of the present hypotheses. Even after that specialization, comparison of the two triple-fusion orders and final-label separation are still needed to construct an invertible \(F\)-move.

For every \(N\geq 1\), let \(|\mathrm{GHZ}_N\rangle =|0\rangle ^{\otimes N}+|1\rangle ^{\otimes N}\). Then

Hence \(M\) is an MPDO tensor. The displayed operator is an unnormalized rank-one positive operator; it is not asserted to be an orthogonal projection. At \(N=0\), the closed MPO word instead gives \(4\mathbb {1}\), whereas the empty GHZ outer product is \(\mathbb {1}\); hence the positive-length restriction is sharp.

Let \(\varepsilon \) identify a binary word of length two with its ordered pair of letters, and put \(|\Phi \rangle =|0,0\rangle +|1,1\rangle \). Then

Thus the companion Bell matrix is exactly the two-site ambient operator in ordered-pair coordinates.

The unique label and copy identify the retained coordinate with \(I\). The corresponding permutation matrix \(W\) satisfies, for every \(Z\in M_{2}(\mathbb {C})\),

Conjugation by \(W\) gives

and the adjoint reconstructs the original matrix:

As an MPS with four doubled physical symbols, \(M\) is the direct sum of four bond-one normal tensors, one supported on each symbol, all with weight one. Hence \(M\) is in literal CPSV canonical form.

Put \(s=2^{-1/2}\) and \(\widehat A=sA\). The tensor \(\widehat A\) is left-canonical and normal. With one label, one copy, multiplicity weight \(s^{-1}\), and \(\chi =(s)\), the length-\(L\) coefficient is \(s^L\). For \(L\geq 1\),

Moreover,

These data reconstruct \(A\) and satisfy the positive-\(\chi \), product, and idempotent algebra conditions. At \(L=0\), however, \(\widehat Q_0=2\mathbb {1}\), so \(\widehat Q_0^2=4\mathbb {1}\ne 2\mathbb {1}\); the geometric algebra identity therefore also requires positive length.

For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\), the physical deformation \(M_X\) is not a matrix product density operator.

After identifying one-site words with their single letters in \(I\), the one-site operator of \(M_X\) is

If \(M_X\) is a matrix product density operator, then there is a real number \(\omega {\gt}0\) such that

Thus this natural physical-similarity family is excluded whenever \(X^\dagger X\) is nonscalar.

Under the ordered-pair identification \(\varepsilon \) from Theorem 21.2.3.4,

For every vertical letter \(v\),

Encode the retained symbols as directed edges on \(I\). For \(N\geq 1\), let \(q_N(w)\) indicate that consecutive edges in \(w\) meet cyclically. The raw retained closed operator is

It satisfies

Thus \(Q_N\) is a \(*\)-projection and obeys the one-label coefficient-one algebra law. It is distinct from the ambient unnormalized GHZ rank-one operator in Theorem 21.2.3.3. At \(N=0\), the raw retained contraction is \(2\mathbb {1}\), whereas the empty cyclic indicator gives \(\mathbb {1}\). In particular, the raw contraction is not idempotent at length zero, so neither the projection identity nor the coefficient-one algebra law extends to \(N=0\).

Let \(Y\in \mathrm{GL}(2,\mathbb {C})\). If the terminal-matrix similarity \(YJY^{-1}\) and the independent Bell-projector similarity \((Y\otimes Y)P(Y^{-1}\otimes Y^{-1})\) are Hermitian, then there is a real number \(\omega {\gt}0\) such that

The comparisons in [ CPGSV16 , Appendix C.4, lines 2048–2057 ] and [ CPGSV16 , Proposition 4.13 ] motivate this finite-dimensional criterion; neither source states this model-specific theorem. Theorem 21.2.3.4 supplies the exact base-model interpretation of \(P\), but the premise does not construct a corresponding deformed tensor-attached model.

The ambient tensor \(M\) has a tensor-attached BNT algebra clause with one label, one copy, bond dimension two, normalized representative \(\widehat A\), multiplicity weight \(s^{-1}\), coefficient \(s^L\), and retained-coordinate coisometry \(W\). Its forward and reconstruction identities hold exactly for every vertical letter.

The vertical tensor of \(M\) is exactly the matrix-unit singleton \(A\) from Definition 21.2.2.76.

Regard the tensors \(A\) and \(B\) of Lemma 21.2.2.70 as one-block families. They are injective and have equal closed matrix product vectors at every length, but their per-block linear extension is not positive. This concerns the virtual extension determined by the closed vectors; it does not negate a comparison of positive physical sector realizations, nor does it decide whether the tensor-attached marked comparison admits a positive-coefficient identity-dressed realization.

The three-edge and two-edge composites of the forward printed \(F\)-matrices are equal. The corresponding composites of the inverse \(F\)-matrices are also equal. In the upper/lower convention of [ BMW\(^{+}\)17 , equation (33) ] , the latter equality is

Every displayed entry denotes the inverse of the forward matrix in the \(F\)-move convention.

Fix \(a,b,c,d\in \Lambda \). There is an invertible matrix

whose rows are indexed by \((f,\lambda ,\sigma )\) and whose columns are indexed by \((e,\mu ,\nu )\). Its entries satisfy

The matrix in the opposite orientation is its two-sided inverse. This is the printed \(F\)-move of [ BMW\(^{+}\)17 , Section 3.4 ] .

Let \(M\) be an MPDO whose doubled-index tensor is in literal CPSV canonical form. Consider the following conditions:

1. \(M\) is a renormalization fixed point;

2. \(M\) has a tensor-attached BNT algebra clause;

3. \(M\) has the active-support BNT fusion clause.

Then

The first equivalence is the equivalence of (i) and (ii) in [ CPGSV16 , Theorem 4.14, lines 972–985 ] . The second is the corrected active-support form corresponding to the printed conditions (i) and (iii), rather than an assertion of the unrestricted printed condition (iii).

Scope restriction (active product BNT): Only nonzero product corners occur in (iii\(_{\mathrm{act}}\)); a sector absent from a fixed product pair has zero fusion multiplicity. This is recorded in [ con26c ] .

Local fix (fixed-pair support and coisometry): A fixed pair may have empty active support, and the retained-row fusion map is a coisometry. These points are recorded in [ con26q ] and [ con26j ] .

Local fix (mixed one-site/two-site prefix): The implication (ii)\(\Rightarrow \)(i) uses the marked comparison recorded in [ con26n ] .

Local fix (zero-sector complement): The physical maps in (ii)\(\Rightarrow \)(i) are completed on the discarded complements as recorded in [ con26z ] .

The matrix \(\tau _H\) is positive definite and has trace one.

Suppose that the positive trace-power coefficients are independent of the positive chain length and that the labelled operators are eventually linearly independent. Then, for each final label \(\varepsilon \),

Consequently a matrix between the two fixed-final multiplicity spaces is square. This is the numerical associativity statement of [ BMW\(^{+}\)17 , lines 237–241 ] ; it does not assert completeness of the fusion maps or invertibility of the fixed-final comparison.

If the algebra tower is compatible with an MPO tensor \(M\) and \(E_n^\dagger X = X\) for a positive blocking size \(n\), then \(X \in \mathcal{A}_n\) and, writing \((X_i)_{i \in I_n}\) for its blocked coefficients, \(X = \sum _{i \in I_n} X_i b_i\).

Fix four labels \(\alpha ,\beta ,\gamma ,\delta \) in a BNT fusion-isometry family, and write \(A=\{ 1,\ldots ,D_\alpha \} \), \(B=\{ 1,\ldots ,D_\beta \} \), \(C=\{ 1,\ldots ,D_\gamma \} \), and \(D=\{ 1,\ldots ,D_\delta \} \) for their bond-coordinate sets. Let

denote the canonical reassociation. Define

The three-edge and two-edge reassociations from \((((A\times B)\times C)\times D)\) to \(A\times (B\times (C\times D))\) coincide:

If \(P(r)_{x,y}=1\) when \(r(x)=y\) and is zero otherwise, then the corresponding coordinate-permutation matrices satisfy

These are equalities of Cartesian bond-coordinate identifications only; they are distinct from the fusion-multiplicity identity in Theorem 21.3.90.

Suppose that the labelled tensors form a basis of normal tensors and the positive trace-power coefficients are independent of the positive chain length. If each \(P_\gamma \) is a self-adjoint idempotent on the bond space, then \(Q^{\mathrm{fs}}_{\alpha ,\beta }\) is a self-adjoint idempotent and \(\widehat Q^{\mathrm{fs}}_{\alpha ,\beta }\) is an orthogonal projection.

Let \(M\) generate matrix product density operators, and fix an active-sector fusion clause for a vertical canonical decomposition of \(M\). Suppose that the trace-power coefficients of its positive diagonal fusion matrices are independent of the positive chain length. There are non-negative numbers \(\lambda _1,\ldots ,\lambda _d\) and one fixed Hermitian two-site matrix \(\widehat h\), all independent of the chain length. For every \(L\geq 2\), let \(\widehat H_L=\sum _{j=1}^{L}\widehat h_{j,j+1}\). The translates of \(\widehat h\) commute pairwise, and there are orthogonal projections \(\widehat P_1^{(L)},\ldots ,\widehat P_d^{(L)}\) on the physical chain such that

This is the fusion-clause implication in the proof of  [ CPGSV16 , lines 999–1016 ] .

Scope restriction (active product BNT): only nonzero product corners occur in the fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family, and no unsupported normal sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction of the product tensor. See [ con26j ] .

Local fix (physical complement): the retained one-site energy is compressed to physical coordinates and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry. See [ con26r ] .

Scope restriction (positive chains of length at least two): the conclusion is proved for \(L=N+2\). Definition 4.8 of [ CPGSV16 ] defines the local interaction on two spins but gives no length-one convention. See [ con26i ] .

Let \(M\) generate matrix product density operators, fix an active-sector fusion clause, and suppose that its positive trace-power coefficients are independent of the positive chain length. Then every terminal eigenvalue \(\lambda _s\) is non-negative. For every \(N{\gt}0\), the operators \(P_s^{(N)}\) are pairwise orthogonal projections and

This is the terminal spectral decomposition of [ CPGSV16 , lines 1003–1012 ] ; it makes no Hamiltonian or Gibbs-state assertion.

Scope restriction (active product BNT): only nonzero product corners occur in the fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction. See [ con26j ] .

Fix an active-sector fusion clause for a vertical canonical decomposition of \(M\). For every \(N{\gt}0\), its retained-row vertical coisometry \(V\) and all-label recursive density operator \(R_N\) satisfy

Scope restriction (active product BNT): only nonzero product corners occur in the fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction. See [ con26j ] .

Local fix (rectangular vertical map): the reverse equality uses exact retained-sector reconstruction rather than a square-unitary identity. See [ con26z ] .

Let \(M\) generate matrix product density operators, and fix an active-sector fusion clause for a vertical canonical decomposition of \(M\). Suppose that the trace-power coefficients of its positive diagonal fusion matrices are independent of the positive chain length. There are non-negative numbers \(\lambda _1,\ldots ,\lambda _d\), independent of the chain length. For every \(L\geq 2\), the same two-site matrix \(h\) is Hermitian, its periodic translates commute pairwise, and

There are orthogonal projections \(P_1^{(L)},\ldots ,P_d^{(L)}\) such that

Applying the adjoint retained-row map at every site reconstructs \(\rho ^{(L)}(M)\) from the right-hand side of (95).

Scope restriction (active product BNT): only nonzero product corners occur in the fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family, and no unsupported normal sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction of the product tensor. See [ con26j ] .

Scope restriction (retained vertical coordinates): the formula is an equality for \(R_L\) on the retained vertical space, not the literal physical-space equality asserted in [ CPGSV16 , lines 1013–1016 ] . Exact reconstruction by the adjoint retained-row map is weaker than the existence of physical projectors and a finite physical Hamiltonian. See [ con26r ] .

Scope restriction (positive chains of length at least two): the conclusion is proved for \(L=N+2\). Definition 4.8 of  [ CPGSV16 ] defines \(h\) on two spins but does not state a length-one convention; see [ con26i ] .

An active-support fusion coisometry family, and hence also a full-support fusion-isometry family, satisfies the same-length product law with the trace-power coefficients of its diagonal matrices. For every chain length \(L\ge 1\),

This derives the product law of [ CPGSV16 , Theorem 4.14(ii) ] from the isometry statement of [ CPGSV16 , Theorem 4.14(iii) ] . The active-support formulation is the one used in [ CPGSV16 , Appendix C.4, lines 2020–2029 ] .

Let \(M\) generate positive semidefinite operators on every nonempty chain, and suppose that one vertical corner is a positive scalar multiple of a similarity of a tensor \(A\),

Then the marked-chain coefficient of the Gram-dressed tensor \((X^\dagger X)A^v(X^\dagger X)^{-1}\) equals the reflected-adjoint marked-chain coefficient of \(A\) in the adjoint tensor, with the two tail words reversed. This is the single-corner, open-index conjugation step from Figures 7–8 of [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] .

A proposition-level BNT-label theorem witness contains the source-side same-length product law, which holds for every positive length \(L\), and the idempotent scalar law,

A witness also carries the positive-length trace-power law \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) and positivity of the diagonal \(\chi _{\alpha ,\beta ,\gamma }\)-entries. Substituting the trace-power law into (29) and (30) rewrites them with the coefficients as \(\chi \)-traces,

At every positive length, the BNT-label coefficient family also agrees with the canonical coefficient family determined by these \(\chi \)-matrices. Hence both families are represented by the \(\chi \)-trace formulas (31) and (32). If \(\sigma _n\) and \(\tau _n\) are the source and target label maps for a positive blocked length \(n\), then the blocked-basis coefficients satisfy

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form. If the one-site and two-site physical closures are related in both directions by trace-preserving completely positive maps as in Definition 4.1, then a vertical canonical decomposition of \(M\) has the active-sector fusion clause above. In particular, its positive diagonal fusion matrices satisfy the length-one idempotent trace-scalar identity.

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis. See [ con26y ] .

Reconstructing the inclusion coefficients recovers the image of \(b_i\) under \(\iota _n\): \(\sum _{j \in I_{n+1}} d_{ij} b_j = \iota _n(b_i)\).

Suppose that \(M_\varepsilon \) is injective. Let \(C\) be a matrix from the right fixed-final-sector space to the left fixed-final-sector space. Under the canonical identifications with \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\), assume that, for every pair of physical indices \(i,j\),

Then there is a rectangular matrix \(F:\mathcal H^{\mathrm R}_{\varepsilon }\to \mathcal H^{\mathrm L}_{\varepsilon }\) such that \(C=F\otimes 1_{D_\varepsilon }\). This conclusion is conditional on the supplied matrix \(C\). It does not assert the existence or invertibility of \(C\) or \(F\), and it does not identify the dimensions of the positive diagonal matrices \(\chi \) with the fusion multiplicities in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] .

For every final label \(\varepsilon \), the left-associated restriction satisfies the positive-diagonal weighted identity

Site by site,

The left iterated fusion isometry satisfies \((U^{\mathrm L}_{\alpha ,\beta ,\gamma })^\dagger U^{\mathrm L}_{\alpha ,\beta ,\gamma }=1\).

Under the hypotheses of Theorem 21.3.41, \(U^{\mathrm L}_{\alpha ,\beta ,\gamma } (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^{\dagger }=1\).

There is an MPDO \(M\) in literal CPSV canonical form that is a renormalization fixed point and has a tensor-attached BNT presentation with one label and

In particular, \(c^{(1)}\ne c^{(2)}\), so the displayed coefficient system depends on the chain length. This is an existential statement about the exhibited BNT presentation; it does not assert that length dependence is invariant under every possible presentation of \(M\).

Let \(M\) generate matrix product density operators, suppose that its doubled-index MPS tensor is in literal CPSV canonical form, and suppose that \(M\) satisfies the renormalization fixed-point condition of Definition 4.1. Select the active-sector fusion clause supplied by these three hypotheses, and assume that the trace-power coefficients of this same clause are independent of the positive chain length.

There are non-negative numbers \(\lambda _1,\ldots ,\lambda _d\) and one fixed Hermitian two-site matrix \(\widehat h\), all independent of the chain length. For every \(L=N+2\), let \(\widehat H_L=\sum _{j=1}^{L}\widehat h_{j,j+1}\). The translates of \(\widehat h\) commute pairwise, and there are pairwise orthogonal projections \(\widehat P_1^{(L)},\ldots ,\widehat P_d^{(L)}\) on the physical chain such that

This is the physical-coordinate conclusion of  [ CPGSV16 , lines 1013–1016 ] , under the literal canonical form and renormalization fixed-point hypotheses of  [ CPGSV16 , Theorem 4.14, lines 972–993 ] .

Scope restriction (active product BNT): only nonzero product corners occur in the selected fusion clause. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): a fixed product pair may have an empty active family, and no unsupported normal sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): the retained-row fusion map is a coisometry onto the active direct sum, and its adjoint gives exact reconstruction of the product tensor. See [ con26j ] .

Local fix (physical complement): the retained one-site energy is compressed to physical coordinates and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry. See [ con26r ] .

Scope restriction (positive chains of length at least two): the conclusion is proved exactly for \(L=N+2\). Definition 4.8 of  [ CPGSV16 ] defines the local interaction on two spins but gives no length-one convention. See [ con26i ] .

Let \(A_M\) be the doubled-index MPS tensor associated to an MPO tensor \(M\). If \(A_M\) is in literal CPSV canonical form and \(L\geq 1\), then the doubled-index MPS tensor associated to \(M^{[L]}\) is again in literal CPSV canonical form. The same conclusion holds for the concrete two-site blocking \(M^{[2]}\).

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. If the one-site and two-site physical closures are related in both directions by trace-preserving completely positive maps as in Definition 4.1, then a vertical canonical decomposition of \(M\) has the active-sector fusion clause. In particular, its positive diagonal fusion matrices satisfy

This is the corrected active-support formulation of implication (i)\(\Rightarrow \)(iii) in [ CPGSV16 , Theorem 4.14, lines 972–993 ] , rather than the unqualified printed statement.

Scope restriction (active product BNT): For each fixed product pair, only its active normal corners occur; an absent label has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): A fixed product pair may have an empty active family, so no unsupported normal sector is added. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): Each fusion map is a coisometry onto the active direct sum, and its adjoint reconstructs the whole product tensor, including any discarded common zero corner. See [ con26j ] .

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. Then there are normal tensors \(A_\alpha \), positive multiplicities \(r_\alpha \), positive numbers \(c_{\alpha ,q}\), and a coisometry \(U\) such that the \(A_\alpha \) form a CPSV basis of normal tensors for \(\widetilde M\) and, writing

one has

Under the hypotheses of Theorem 20.14.8, the concrete two-site block \(M^{[2]}\) has a decomposition of the same form: there are normal tensors \(\widehat A_\alpha \), \(r^{[2]}_\alpha {\gt}0\), \(c^{[2]}_{\alpha ,q}{\gt}0\), and a coisometry \(U^{[2]}\) such that

The tensors \(\widehat A_\alpha \) form a CPSV basis of normal tensors for \(\widetilde{M^{[2]}}\).

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. Given one-site and two-site vertical canonical decompositions and their trace-ratio unitary sector correspondence, there are positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and matrices \(U_{\alpha ,\beta }\) such that

Every diagonal entry of \(\chi _{\alpha ,\beta ,\gamma }\) is positive. This is the positive fusion decomposition in [ CPGSV16 , Appendix C.4, lines 2020–2029 ] .

Scope restriction (active product BNT): Only active product corners occur. A label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): A fixed product pair may have no active corner, and no unsupported sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): The retained-row map is a coisometry onto the active direct sum, and its adjoint gives the exact reconstruction. See [ con26j ] .

In the ambient bond-space matrix algebra, \(b_i b_j = \sum _{k \in I_{2n}} c^{(n)}_{i,j,k} b_k\).

Neither \(A\) nor \(B\) can occur, up to the bond-dimension identification, as a retained bond-two sector of a tensor-attached BNT algebra clause for a matrix product density operator.

Let \(A_c^q\in M_{D_c}(\mathbb {C})\) be a finite labelled tensor family. If the one-letter tuples \(q\longmapsto (A_c^q)_c\) span the full product algebra \(\prod _c M_{D_c}(\mathbb {C})\), then there are coefficients \(C_{(c,x,y),q}\) such that

For an MPO tensor the letter is the ket–bra pair \(q=(i,k)\), so the same coefficients form the common block-letter left inverse used in [ BMW\(^{+}\)17 , lines 269–277 ] . The full product-algebra span is obtained after the common blocking described in [ BMW\(^{+}\)17 , lines 427–431 ] .

After identifying one blocked physical index with two original indices, \((K^{[2]})_1(X)=K_2(X)\).

A one-site transfer-retract datum exists if and only if the doubled-index MPO transfer map is idempotent.

The letters of the two parenthesizations of a triple product are related by the canonical bond reassociation. Equivalently, for every pair of physical indices \(i,k\),

For every chain length \(L\), the operator generated by the product tensor is the product of the generated operators: \(O_L(M\mathbin {\cdot }N)=O_L(M)O_L(N)\).

Suppose that the labelled tensors in the fusion family form a basis of normal tensors and that the positive trace-power coefficients are independent of the positive chain length. Then every pair fusion isometry is surjective: \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\). In particular, final-label selectors and coisometry are conclusions, not additional hypotheses. This is a derived consequence of the common BNT word span in [ CPGSV16 , lines 317–345 ] , the fusion identity at lines 986–993, and length independence at lines 995–1010.

Suppose the positive trace-power coefficients are independent of the positive chain length, and suppose that common selectors exist at a positive length. Then every pair fusion isometry is surjective: \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\). The common selectors are the finite-word form of the simultaneous inverse used in [ BMW\(^{+}\)17 , line 269 ] .

Let \(\mathcal D_1\) and \(\mathcal D_2\) be positive vertical canonical decompositions of an MPO tensor and its two-site blocking. Suppose their normal sectors are identified by a bijection \(\sigma \) and unitary conjugacies with the corresponding multiplicity-trace ratios. Suppose also that positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and coisometries \(U_{\alpha ,\beta }\) are supplied, with \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\), and satisfy both

Then the simultaneous blocked representations yield the missing length-one idempotent trace-scalar law and complete the tensor-attached fusion clause. This completes the fusion-clause construction extracted from [ CPGSV16 , Appendix C.4, lines 2001–2046 ] .

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition. Select the active-sector fusion clause supplied by that condition, and suppose that the trace-power coefficients of this same clause are independent of the positive chain length. Then the physical-coordinate commuting Gibbs decomposition of Theorem 21.3.61 holds for every \(L=N+2\).

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

Local fix (physical complement): the retained one-site energy is compressed to physical coordinates and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry. See [ con26r ] .

Scope restriction (positive chains of length at least two): the conclusion is proved for \(L=N+2\); no length-one two-site convention is asserted. See [ con26i ] .

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

Reconstructing a coefficient family gives the corresponding basis expansion \(\sum _i a_i b_i\) in \(\mathcal{A}_n\).

Let \(M\) generate positive semidefinite operators on every nonempty chain, let \(A\) be a vertical tensor, and suppose that a matrix \(V\) selects the vertical corner \(V^\dagger \widetilde M_vV=cA^v\), where \(c{\gt}0\). For vertical indices \(r,s\), write \((\mathcal{R}(A)^{rs})_{ab}=(A^{(a,b)})_{rs}\), and set \((A^\sharp )^v=(A^{v^{\mathrm{op}}})^\dagger \). Then, for every pair of equal-length tail words \(\sigma ,\tau \),

Thus the adjoint exchanges the oriented horizontal bond pair and reverses the unmarked tail. In particular, this is not a same-tail identity and does not identify \(A\) with \(A^\sharp \) letterwise.

For \(Z=\operatorname {diag}(1,-1)\), the same transfer map satisfies

For the matrix unit \(E_{10}\), the same transfer map satisfies

For the matrix unit \(E_{01}\), the same transfer map satisfies

For the explicit four-letter tensor \(A\), its per-site transfer map \(\varphi (Y)=\sum _aA^aY(A^a)^\dagger \) satisfies

Suppose that the one-site vertical tensor has the stated exact coisometric reconstruction, that \(M\) is in normalized BNT-refined horizontal form, and that \(M\) generates positive operators. Then simultaneous retained-product spectral data exist. In copy coordinates, the assembled vertical tensor is the direct sum of the weighted simple blocks. The canonical inclusion of each copy is an isometry, intertwines in both directions, and its compression selects exactly that weighted block. Likewise, the canonical inclusion of a copy pair, followed by its local corner isometry, is an isometry into the retained product bond space and intertwines in both directions. Composing once more with the adjoint of the squared vertical coisometry gives an isometry into the bond space of the blocked vertical tensor. Under the exact coisometric reconstruction, these ambient inclusions intertwine each active weighted corner with the blocked vertical tensor in both directions. In particular, if \(J_j\) is the ambient inclusion of an active corner, then \(J_j^\dagger \mathcal V(M^{(2)})^iJ_j=\lambda _jA_j^i\). These explicit composite inclusions retain the range projections required for the sector-weight comparison.

For every final label \(\varepsilon \), the right-associated restriction satisfies the positive-diagonal weighted identity

Site by site,

The right iterated fusion isometry satisfies \((U^{\mathrm R}_{\alpha ,\beta ,\gamma })^\dagger U^{\mathrm R}_{\alpha ,\beta ,\gamma }=1\).

Under the same hypotheses, \(U^{\mathrm R}_{\alpha ,\beta ,\gamma } (U^{\mathrm R}_{\alpha ,\beta ,\gamma })^{\dagger }=1\).

Under the trace-preserving normalization and a positive-definite fixed point, if \(M\) satisfies the blocked fixed-point-algebra tower, then the stationary tower \(\mathcal{A}_n = \operatorname{Fix}(E_1^\dagger )\) is itself compatible with \(M\): for every positive blocking size \(n\), \(\operatorname{Fix}(E_1^\dagger )=\operatorname{Fix}(E_n^\dagger )\).

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition. Fix internally the fusion clause supplied by that construction, and suppose that its positive trace-power coefficients are independent of the positive chain length. Then \(K_\gamma \) is positive semidefinite for every terminal label, so \(\lambda _s\geq 0\). Write the total positive chain length as \(N=n+1\), where \(n\in \mathbb {N}\) is its predecessor. For every \(N{\gt}0\),

For distinct spectral indices \(s\) and \(t\),

The recursive factor and multiplicity weights satisfy

Consequently,

The length-independence assumption here is clause-relative: it is stated on the coefficients of the fixed clause selected from the renormalization fixed-point construction, not on an arbitrary fusion clause.

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

This theorem stops at [ CPGSV16 , lines 1003–1012 ] ; it does not assert the commuting nearest-neighbor Hamiltonian or Gibbs-state conclusion beginning at line 1013.

Let \(M\) be a matrix product density operator, and choose a vertical canonical decomposition with basis tensors \(M_\gamma \) and positive diagonal multiplicity matrices \(\mu _\gamma \). Then every terminal transfer matrix \(T_\gamma =\operatorname{tr}(M_\gamma )\) is positive semidefinite. This supplies the positivity needed for the spectral decomposition invoked in [ CPGSV16 , lines 1010–1016 ] ; it asserts neither that \(T_\gamma \) is a projection nor the subsequent spectral, Hamiltonian, or Gibbs decomposition.

Local fix (terminal trace orientation): The trace closes the horizontal operator leg and leaves the two bond indices of \(M_\gamma \) open. See [ con26f ] .

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form and suppose that it satisfies the renormalization fixed-point condition. There is one vertical canonical decomposition and one compatible family of fusion coisometries such that, for every \(N{\gt}0\), the rectangular vertical coisometry \(V\) satisfies the two exact equations

No additional compatibility hypothesis is imposed. The statement is restricted to positive lengths and does not assert the commutator at line 1001, length independence, or the subsequent spectral and Gibbs decomposition.

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

Local fix (rectangular vertical map): The second equality uses exact reconstruction from the retained nonzero sectors of Proposition 4.13 and Appendix C.4, lines 2020–2029, not a square-unitary identity. See [ con26z ] .

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition. Select the active-sector fusion clause supplied by that condition, and suppose that the trace-power coefficients of this same clause are independent of the positive chain length. Then the retained-coordinate commuting Gibbs decomposition of Theorem 21.3.59 holds for every \(L=N+2\).

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

Scope restriction (retained vertical coordinates): the Hamiltonian and projectors act on the retained nonzero vertical sectors. Applying the adjoint retained-row map reconstructs the physical density operator, but does not provide a Hamiltonian or projectors on the original physical space. See [ con26r ] .

Scope restriction (positive chains of length at least two): the conclusion is proved for \(L=N+2\); no length-one two-site convention is asserted. See [ con26i ] .

Let \(P_\gamma \) be self-adjoint idempotents on the terminal bond space of \(M_\gamma \). Suppose that the matrices \(\chi _{\alpha ,\beta ,\gamma }\) have the positive-length trace-power form of the label-coefficient family and that this family is independent of the positive chain length. Then

is self-adjoint and idempotent. This is the single recursive fusion step in [ CPGSV16 , lines 999–1010 ] . Since the active fusion map satisfies \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=I\), the transported operator \(U_{\alpha ,\beta }^{\dagger }Q_{\alpha ,\beta }U_{\alpha ,\beta }\) is also an orthogonal projection on the product bond space.

For every initial label and every finite list of appended labels, let \(N\geq 1\) be one plus the length of that list. The sequential fusion map \(W_N\) is a retained-row coisometry. Letter by letter, it carries the left-associated product tensor to the recursive direct sum:

Consequently, after closing the horizontal operator index,

Exact reconstruction gives the reverse identities

If the positive trace-power coefficients are independent of the positive chain length and the terminal matrices are self-adjoint idempotents, then this recursive operator is self-adjoint and idempotent.

Local fix (Figure-11 fusion coisometry): The source embedding \(\widetilde U_N\) is \(W_N^\dagger \), the adjoint of the retained-row coisometry used above. Exact reverse fusion uses the active-support reconstruction of Appendix C.4, lines 2020–2029. See [ con26j ] .

Scope restriction (fixed initial label): This theorem concerns one chosen initial label and list of appended labels. The following theorem performs the vertical-canonical assembly over all labels and multiplicity coordinates. The commutator remains outside the present result; see [ con26f ] .

For every triple \((\alpha , \beta , \gamma )\), if \(\chi _{\alpha ,\beta ,\gamma }\) has size \(r_{\alpha ,\beta ,\gamma }\), then for every \(L \ge 0\),

where \(\chi _{\alpha ,\beta ,\gamma ,k}\) are the diagonal entries.

A one-site transfer-retract datum implies doubled-index transfer-map idempotence.

If the doubled-index transfer map is idempotent, then transfer-retract data exist at every positive blocked size.

Under the one-site and two-site vertical basis-of-normal-tensors decompositions and the two completely positive, trace-preserving physical-closure transport identities, assume that the one-site tensor is in normalized BNT-refined horizontal form and defines a matrix product density operator. Then there are positive diagonal multiplicity matrices \(\chi _{\alpha ,\beta ,\gamma }\) and matrices \(U_{\alpha ,\beta }\) such that every diagonal entry is positive and

while the exact reconstruction is

This is the positive fusion decomposition of CPSV16, Appendix C.4, lines 2020–2029.

Scope restriction (active product BNT): Only active product corners occur. A label absent from a fixed product pair has zero fusion multiplicity. See [ con26c ] .

Local fix (Figure-11 fixed-pair support): A fixed product pair may have no active corner, and no unsupported sector is inserted. See [ con26q ] .

Local fix (Figure-11 fusion coisometry): The retained-row map is a coisometry onto the active direct sum, and its adjoint gives the exact reconstruction. See [ con26j ] .

Under the hypotheses and with the notation of Theorem  20.14.80, let \(\mu _\alpha \) and \(\nu _\beta \) be the positive diagonal multiplicity matrices in the one-site and two-site vertical canonical forms, and set \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) and \(n_\beta =\operatorname{tr}(\nu _\beta )\). For every horizontal bond matrix \(X\) and every sector \(\alpha \),

Consequently, for every tensor letter \(a\),

This is the coefficient identity in [ CPGSV16 , Appendix C.4, lines 2001–2008 ] . It compares the traces of the multiplicity matrices; it does not assert equality of their dimensions or of their individual diagonal entries.

Suppose the positive trace-power coefficients are independent of the positive chain length and common word selectors separate the final labels. Fix labels \(\alpha ,\beta ,\gamma ,\varepsilon \), and suppose that \(M_\varepsilon \) is injective at the present blocking. Write \(C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma } =P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}\) for the corner from the right final sector \(\varepsilon '\) to the left final sector \(\varepsilon \). There is a rectangular matrix \(F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }\) such that, after the canonical index identifications, \(C^{\varepsilon ,\varepsilon }_{\alpha ,\beta ,\gamma } =F_\varepsilon \otimes 1_{D_\varepsilon }\). For every \(\varepsilon '\ne \varepsilon \), \(C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma }=0\). The selector identities and injectivity at the present blocking are additional hypotheses here. The theorem neither asserts that \(F_\varepsilon \) is square or invertible nor identifies it with the printed \(F\)-matrix of [ BMW\(^{+}\)17 , Section 3.4 ] ; no pentagon identity is asserted.

Suppose the positive trace-power coefficients are independent of the positive chain length and common selectors exist at a positive length. For every final label \(\varepsilon \), write \(C_\varepsilon =P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon }\). Then

The conclusion is conditional on the simultaneous selector identities. It is the fixed-output-channel invertibility consequence of the simultaneous inverse used in [ BMW\(^{+}\)17 , lines 247–277 ] .

Suppose the positive trace-power coefficients are independent of the positive chain length and common word selectors exist. If \(\varepsilon \ne \varepsilon '\), then the corner of the full comparison from the right final sector \(\varepsilon '\) to the left final sector \(\varepsilon \) vanishes: \(P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}=0\). The assertion is conditional on the displayed word-selector identities. It neither states that a diagonal corner is invertible nor identifies one with an \(F\)-matrix, and it does not assert the pentagon identity.

Suppose the positive trace-power coefficients are independent of the positive chain length. For every pair of physical indices \(i,k\), let

and define \(D^{\mathrm R}_{ik}\) by the right-associated multiplicities \(r_{\beta ,\gamma }^{\delta }\) and \(r_{\alpha ,\delta }^{\varepsilon }\). Then

This is an identity on the full direct sums. It neither extracts a fixed-final \(F\)-transformation nor proves invertibility or the pentagon identity.

Under the same hypotheses, \(C_{\alpha ,\beta ,\gamma }^{\dagger }C_{\alpha ,\beta ,\gamma }=1\). This proves invertibility on the full direct sums. It does not yet prove invertibility of a fixed-final multiplicity matrix.

Under the same hypotheses, the comparison of the full direct sums satisfies \(C_{\alpha ,\beta ,\gamma }C_{\alpha ,\beta ,\gamma }^{\dagger }=1\).

Assume the hypotheses of Theorem 21.3.80. Suppose in addition that \(M_\varepsilon \) is injective at the present blocking and that \(D_\varepsilon {\gt}0\). There is a matrix \(F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }\) such that

Thus \(F_\varepsilon \) is invertible. Its orientation is opposite to the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] ; no pentagon identity is asserted.

Let \(M\) generate matrix product density operators, let \(A\) be a vertical tensor, and suppose that an isometry \(V\) selects from the two-site blocking a positive multiple of an invertible conjugate of \(A\):

For every pair of open bond indices and every pair of equal-length words in the original physical alphabet, the marked-chain coefficient of

followed by the corresponding unblocked tail of \(M\) equals the reflected-adjoint marked-chain coefficient of \(A\) followed by the reversed tail of \(M^\dagger \). Thus the first two sites are compressed jointly, while all later sites remain unblocked.

Local fix (mixed one-site/two-site prefix): This is the common unblocked-tail comparison needed when [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] is invoked in [ CPGSV16 , Appendix C.4, lines 2048–2057 ] . It is recorded in [ con26n ] .

Suppose that two tensor-attached vertical BNT presentations are related by an explicit vertical transport. At every positive length \(L\) at which the target operators are linearly independent, their structure coefficients satisfy

Theorem 21.2.2.59 shows why a bare equality of horizontal periodic families cannot replace the explicit transport hypothesis.

Suppose that source and target operator families satisfy same-length product laws and are related by an explicit vertical transport. At a positive length where the target operators are linearly independent, the target coefficient family equals the transported source coefficient family.

Under the hypotheses of the preceding theorem, if \(s_\alpha s_\beta =s_\gamma \), then

Under the hypotheses of the abstract covariance theorem, if \(s_\alpha s_\beta =s_\gamma \), then

An explicit vertical transport carries a same-length product law to the transported coefficient family.

The tensors \(M_P\) and \(M_Q\) are related by the nonunitary gauge \(X\). For every positive length \(N\), their unnormalized periodic operators are equal:

In particular, both tensors generate matrix product density operators. Their chosen vertical presentations nevertheless have structure coefficients

Hence there is no relabelling of the singleton label sets for which, at every positive length,

Indeed, the two sides at length one are \(4/5\) and \(1\).

Proposition 4.13 and Theorem 4.14 of [ CPGSV16 ] instead start with one horizontally canonical matrix product density operator and choose its vertical decomposition. The tensor \(M_Q\) above is not asserted to be a horizontal canonical representative. Thus the theorem excludes an additional cross-presentation conclusion under bare periodic equality; it does not alter the source’s fixed-presentation characterization. It also leaves Theorem 21.2.2.50 unchanged, since that theorem compares two coefficient systems for the same vertical operator family.

The coefficients in the \(P\) presentation are length independent, whereas those in the \(Q\) presentation are not. More precisely,

Let \((d_\alpha )_\alpha \) and \((r_\alpha )_\alpha \) be finite families of dimensions with \(r_\alpha {\gt}0\), and suppose that every diagonal entry of \(\mu _\alpha \) is positive. Then the normalized embedding

is a direct-sum Kraus map. Without any condition on the multiplicities, the map

is a direct-sum Kraus map. Its restriction to the weighted sectors is the retraction in [ CPGSV16 , Appendix C.4, lines 1961–1970 ] .

The per-block extension \(T_k\) is bijective.

The per-block extension satisfies \(T_k(MN)=T_k(M)T_k(N)\) for all \(M,N\in M_{D_k}(\mathbb {C})\).

Let \(M_\alpha \) be a normal tensor. Any matrix commuting with every matrix of \(M_\alpha \) is a scalar multiple of the identity. In particular, if \(X\ne 0\) and \(X^\dagger X\) commutes with every matrix of \(M_\alpha \), then \(X^\dagger X=\omega \mathbb {1}\) for a necessarily positive constant \(\omega \), and \(\omega ^{-1/2}X\) is unitary. This is the normalized form \(X_{\alpha ,k}^\dagger X_{\alpha ,k}=\omega _{\alpha ,k}\mathbb {1}\) of the Gram-matrix proportionality in the proof of [ CPGSV16 , Proposition 4.13 ] , with \(X_{\alpha ,0}=\mathbb {1}\) and \(U_{\alpha ,k}=\omega _{\alpha ,k}^{-1/2}X_{\alpha ,k}\). Deriving the commutation from self-adjointness of the sector compressions is a separate step.

If \(\omega {\gt}0\) and \(X^\dagger X=\omega \mathbb {1}\), then \(\omega ^{-1/2}X\) is unitary.

If an invertible matrix \(Z\) satisfies \(Z^\dagger Z=\omega \mathbb {1}\) with \(\omega {\gt}0\), then conjugation by \(\omega ^{-1/2}Z\) equals conjugation by \(Z\).

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

A tensor in normalized BNT-refined horizontal form that generates matrix product density operators is also in canonical form in the vertical direction: there exist a basis of normal tensors \(\{ M_\alpha \} _\alpha \) for the vertically viewed tensor \(\widetilde M\), positive diagonal matrices \(\mu _\alpha \), and a coisometry \(U\) from the physical space onto the retained nonzero sectors. Writing \(B=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \), one has

The conclusion includes the basis-of-normal-tensors condition in [ CPGSV16 , Proposition 4.13 ] , together with the positive weights and the coisometric realization. Its hypothesis is the normalized BNT-refined form of Definition 17.4.1, rather than the literal CPSV canonical form assumed in Theorem 17.4.111.