Dependencies

Under the BNT algebra conditions and blocked-basis comparison, for every positive blocking length \(n\), the blocked-basis coefficients are traces of powers of the matrices \(\chi _{\alpha ,\beta ,\gamma }\) selected by the source and target label maps. Equivalently, they are traces of powers of the pulled-back matrices \(\chi _{n,i,j,k}\):

Assume the conditions of Definition 21.2.3.35. For every positive blocked length \(n\), source indices \(i,j\) at length \(n\), and target index \(k\) at length \(2n\), the blocked-basis coefficients are traces of powers of the BNT matrices selected by the source and target comparison maps:

For any vertical canonical decomposition with fusion coisometries onto the nonzero product sectors and exact reconstruction, the recursive density operator satisfies, for every \(N{\gt}0\),

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

Let \(M\) generate MPDOs and be in normalized BNT-refined horizontal form. Suppose that \(M\) is an RFP. One choice of the vertical canonical decomposition and fusion coisometries gives the density decomposition and, for every \(N{\gt}0\), satisfies

Equivalently, \([A_N,T_N]=0\), as stated in [ 13 , lines 1000–1002 ] .

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

Equivalently,

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

A basis of normal tensors (BNT) for a total tensor \(A_{\mathrm{tot}}\) consists of \(g\) block tensors \((A_1, \ldots , A_g)\) with bond dimensions \((D_1, \ldots , D_g)\) such that each \(A_j\) is normal (Definition 2.5.5); at each system size \(N\geq 1\) the MPV of \(A_{\mathrm{tot}}\) lies in the span of the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\); and for sufficiently large \(N\) those block MPVs are linearly independent [ 17 , Definition IV.2 ] . In the canonical-form characterization of [ 13 , Section 2.3 ] , a BNT is a minimal family of representatives for \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) among the normal blocks.

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

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

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

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

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

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

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

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

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

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 the strengthened, normalized BNT-refined form of the canonical decomposition in [ 13 , Section 2.3 ] : unlike the literal predicate (Definition 7.6.7), the direct sum exhausts the full bond dimension \(D\), each copy weight is nonzero and normalized, and gauge-equivalent normal blocks are grouped over a minimal BNT representative with an independent gauge for each copy.

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 (RFP) if there exist 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 \(X\). In [ 13 , Definition 4.1 ] , this condition is formulated for tensors in canonical form; the zero-correlation-length consequence concerns the physical-trace transfer (Definition 20.6.1) rather than idempotence of the doubled-index transfer map. On the open coefficient tensors the two maps have types:

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

Equivalently, its Kraus map is trace-preserving.

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

This is the global positivity condition of [ 13 , Section 4 ] .

Let \(\{ \mathcal{A}_n\} _{n \ge 0}\) be a family of \(*\)-subalgebras of \(M_{D}(\mathbb {C})\), one at every blocking size, equipped with bilinear maps \(m_n : \mathcal{A}_n \times \mathcal{A}_n \to \mathcal{A}_{2n}\) and linear inclusion maps \(\iota _n : \mathcal{A}_n \to \mathcal{A}_{n+1}\). These maps are realized in the ambient bond-space matrix algebra by

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

This is the common block inverse used in [ 13 , Appendix C.2, lines 1666–1676 ] and [ 8 , lines 269–277 ] .

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

An MPO tensor \(M\) has a family of algebras of blocked adjoint fixed points if there exists a family of support algebras with blocking and inclusion maps as in Definition 21.2.1 such that, for every positive blocking size \(n\),

where \(E_n\) is the blocked transfer map. This equality is termed compatibility with \(M\). This condition concerns adjoint fixed points; it is not the algebra condition of [ 13 , Theorem 4.14(ii) ] , which requires a common BNT \(\{ M_\alpha \} _\alpha \), the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)}=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), and a vector \(m\), with \(m_\alpha =\operatorname{tr}(\mu _\alpha )\), that is an idempotent for the multiplication induced by \(c^{(1)}\).

For \(i \in I_n\), define the inclusion coefficients \((d^{(n)}_{ij})_{j \in I_{n+1}}\) by

For each positive blocking length \(n\), choose a family of diagonal matrices indexed by triples in the disjoint union of the chosen basis indices of \(\mathcal A_n\) and \(\mathcal A_{2n}\). For a multiplication triple \((i,j,k)\), where \(i,j\) label basis elements of \(\mathcal A_n\) and \(k\) labels a basis element of \(\mathcal A_{2n}\), write its size as \(r_{n,i,j,k}\in \mathbb {N}\) and its complex diagonal entries as \(\chi _{n,i,j,k,s}\), with \(0\le s{\lt}r_{n,i,j,k}\). The corresponding matrix is

Unlike the matrices indexed by fixed BNT labels in [ 13 , Theorem 4.14(ii) ] , the size and diagonal entries may depend on the blocking length and on the chosen basis indices.

For \(i,j \in I_n\), define the multiplication coefficients \((c^{(n)}_{i,j,k})_{k \in I_{2n}}\) by

These are coordinates in \(\mathcal{A}_{2n}\), not the same-length structure coefficients of the vertical operators \(O_L(M_\alpha )\).

Fix structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\), operators \(O_L(M_\alpha )\), and scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). The algebra conditions require a length-independent family of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) and

which reads, with each \(O_L(M_\alpha )\) drawn as a closed MPO chain,

 \(=\sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }\) 

together with

Thus the vector \(m=(m_\alpha )_\alpha \) is an idempotent for the multiplication induced by \(c^{(1)}\). These are the algebra conditions in [ 13 , Theorem 4.14(ii), labelled IV.13 in the source, lines 972–985 ] , without an additional comparison with a chosen blocked basis.

Choose a vertical canonical decomposition

Here the \(\mu _\alpha \) are positive diagonal matrices with nonempty multiplicity spaces, and \(U\) is a coisometry onto the retained sectors, so \(UU^\dagger =\mathbb {1}\). The decomposition also requires exact reconstruction:

These tensor identities hold for every vertical physical index. In particular, any complementary sector is zero. 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 algebra conditions when the tensors \(M_\alpha \) form a basis of normal tensors of the vertical tensor and the conditions of Definition 21.2.2.97 hold for these operators and scalars. These are the objects and equations in [ 13 , Theorem 4.14(ii), lines 972–993, and Appendix C.4, lines 2046–2064 ] ; a comparison with a two-site vertical canonical decomposition is an additional requirement.

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 [ 13 ] , lines 2067–2068 define \(R_1\) by (50), and lines 2078–2079 define \(R_2\) by (51). Lines 2081–2083 give the normalized map \(\widetilde R_1\) in (52). Lines 2058–2071 give \(\widetilde R_2\), including its common denominator, in (53). These four definitions use only the two vertical decompositions, their sector relabelling, and (49).

A comparison between the BNT structure coefficients and the chosen blocked-basis coefficients consists of an assignment of BNT labels together with the equality

for every \(n{\gt}0\) and all basis indices \(i,j\) of \(\mathcal A_n\) and \(k\) of \(\mathcal A_{2n}\). On the left, \(c^{(n)}_{i,j,k}\) is the coefficient of the \(k\)-th basis element of \(\mathcal A_{2n}\) in the blocked product. This comparison is an additional hypothesis, not the product identity for BNT operators at the same length.

An assignment of BNT labels consists, for every positive blocking length \(n\), of a map \(\sigma _n\) from the chosen basis indices of \(\mathcal A_n\) and a map \(\tau _n\) from the chosen basis indices of \(\mathcal A_{2n}\) to the fixed set of BNT labels. The combined map \(\sigma _n \sqcup \tau _n\) is defined on the disjoint union of these two index sets.

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 [ 13 , Appendix C.4, lines 2053–2080 ] . The existence of the unitaries in (42) is an assumption here. They are distinct from the transported direct-sum maps in [ 13 , Appendix C.4, lines 1955–1997 ] . No maps between the ambient full matrix algebras are included here.

Let \(\mu _\alpha \) and \(\nu _\gamma \) be the diagonal multiplicity matrices in the one-site and two-site vertical decompositions, respectively. Suppose that their diagonal entries and those of \(\chi _{\alpha ,\beta ,\gamma }\) are strictly positive, and that \(\sum _i\mu _{\alpha ,i}=m_\alpha \) for every \(\alpha \). If, for every \(\gamma \) and all sufficiently large \(L\),

then the multiset of entries \(\nu _{\gamma ,r}\) equals the multiset of products \(\mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k}\). These multiplicity matrices, together with this multiset equality and the one-site trace identity, give a comparison of the one-site and two-site multiplicity spectra.

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 \(\chi _{\alpha ,\beta ,\gamma }\). Set \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), and suppose that the vector \(m=(m_\alpha )_\alpha \) is an idempotent for the multiplication induced by \(c^{(1)}\). For \(L{\gt}0\), the operators \(O_L(M_\alpha )\) linearly span an algebra. The product law and the assumed idempotent condition are, respectively,

This construction realizes implication (iii)\(\Rightarrow \)(ii) of [ 13 , Theorem 4.14 ] .

For the same labelled tensors and positive diagonal matrices, put \(G_{\alpha ,\beta }^{ij} =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}\). Fusion coisometries onto the retained direct sum are 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, these maps are fusion isometries on the full product bond space.

The coisometry condition makes explicit the retained-row orientation of the source’s vertical CF decomposition. Exact reconstruction states that the discarded common corner is zero. This correction to the full-support interpretation is documented in [ 125 ] .

Let \(\{ M_\gamma \} \) be a finite family of tensors with a common physical dimension and bond dimensions depending on \(\gamma \). Fusion isometries on the full product bond space consist of diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) with positive entries and maps \(U_{\alpha ,\beta }\) satisfying \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) and, site by site,

This is the full-support specialization of the fusion statement in [ 13 , Theorem 4.14(iii) ] . The definition does not require \(\{ M_\gamma \} \) to be a BNT. In applications to an MPDO tensor, these tensors come from the vertical BNT of [ 13 , Proposition 4.13 ] ; their physical dimension in the vertical reading is the bond dimension of the original tensor. Statement (iii) of the source theorem also requires the vector \(m\), with \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) for the positive diagonal multiplicity matrices \(\mu _\alpha \) of the vertical decomposition, to be an idempotent for the multiplication induced by \(c_{\alpha ,\beta ,\gamma }^{(1)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })\). This condition is recorded separately in Definition 21.2.2.5.

The condition \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) on the whole product bond space is an additional restriction. The unrestricted source statement permits a common zero corner. Its support convention must therefore be made explicit: the map is a coisometry onto the retained direct sum, and exact reconstruction is required. This correction is documented in [ 125 ] .

The tensors \(M_\gamma \) generate the periodic operators \(O_L(M_\gamma )\). The positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) define the structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

Choose a vertical canonical decomposition, with nonempty positive diagonal matrices \(\mu _\alpha \) and a BNT \(\{ M_\alpha \} _\alpha \):

Here \(U\) is a coisometry onto the retained sectors, and its adjoint reconstructs \(\widetilde M\), including any common zero-sector complement. Active-sector fusion 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 vector with entries \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) is an idempotent for the multiplication induced by \(c^{(1)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })\). This is the active-support formulation of statement (iii) of [ 13 , Theorem 4.14, lines 986–993 ] , for the chosen decomposition of [ 13 , Proposition 4.13, lines 943–951 ] .

Let \(\Lambda \) be a fixed set of BNT labels. Consider complex coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\) for \(L\in \mathbb {N}\) and \(\alpha ,\beta ,\gamma \in \Lambda \). In [ 13 , Theorem 4.14(ii) ] , these are the structure coefficients for the products \(O_L(M_\alpha )O_L(M_\beta )\) at the same length \(L\), rather than for the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\). A family of diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) determines coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) = \sum _r\chi _{\alpha ,\beta ,\gamma ,r}^{\, L}\). The positive diagonal matrices constructed in [ 13 , Appendix C.4 ] give coefficients of this form; the definition for a general diagonal family does not require positivity.

For a finite set of BNT labels, the vector \(m=(m_\alpha )_\alpha \) is an idempotent for the multiplication induced by \(c^{(1)}\) when, for every label \(\gamma \),

This is the idempotent condition in [ 13 , Theorem 4.14(ii) ] , stated separately from the trace-power formula.

The structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\) are length independent when \(c^{(L)}_{\alpha ,\beta ,\gamma } = c^{(1)}_{\alpha ,\beta ,\gamma }\) for every positive length \(L\) and all labels; equivalently, the coefficients agree at any two positive lengths. This is the case singled out in the discussion following [ 13 , Theorem 4.14 ] .

The family \(\{ O_L(M_\alpha )\} _\alpha \) is linearly independent at length \(L\) when its operators are linearly independent, and eventually linearly independent when this holds for every length beyond some threshold. Eventual linear independence is the operator form of the third defining property of a basis of normal tensors in [ 13 , Section 2.3 ] , invoked for the vertical operators in [ 13 , Appendix C.4 ] .

For each length \(L\), let \(O_L(M_\alpha )\) be operators indexed by the fixed BNT labels \(\alpha \). Their ambient complex vector space, equipped with multiplication, 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 [ 13 , lines 962–967 ] . For \(\mathbf a=(a_1,\ldots ,a_L)\) and \(\mathbf b=(b_1,\ldots ,b_L)\),

Diagrammatically, with the cyclically contracted matrix index of each copy of \(\widehat M_\alpha \) drawn horizontally,

The coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\) satisfy the trace-power formula for a family of diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) when, for every positive length \(L\) and all labels,

The same matrix \(\chi _{\alpha ,\beta ,\gamma }\) is used for all positive \(L\). Its diagonal entries may be complex; positivity is not required here.

For a finite set of BNT labels, the operators \(O_L(M_\alpha )\) satisfy the product identity with coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\) when, for every positive length \(L\) and all labels \(\alpha ,\beta \),

In the diagram of Definition 21.2.2.2, the operator product joins the south legs of the first ring to the north legs of the second, and the identity reads

 \(=\displaystyle \sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }\)  

This is the product identity appearing in [ 13 , Theorem 4.14(ii) ] , stated independently of the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\).

Consider operators \(O_L(M_\alpha )\), structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\), and scalars \(m_\alpha \) indexed by a finite set of BNT labels. In the vertical MPDO decomposition, \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). Require the BNT algebra conditions and a comparison with chosen blocked bases. The algebra conditions require positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) for every \(L{\gt}0\), and, at every such length, the product law

The vector \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\), meaning that, for every \(\gamma \),

The blocked-basis comparison is an additional condition. For every positive blocking length \(n\), let \(\sigma _n\) assign BNT labels to the basis indices at length \(n\), and let \(\tau _n\) assign BNT labels to the basis indices at length \(2n\). For indices \(i,j\) at length \(n\) and \(k\) at length \(2n\), require

Here the left-hand side denotes the structure coefficient in the chosen blocked bases, while the right-hand side uses BNT labels. The same maps assign the diagonal matrix \(\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}\) to each triple of blocked-basis indices.

One may start with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and define the coefficients by their traces of powers. In that case, the product law, idempotent law, and blocked-basis comparison are required for those coefficients. Conversely, replacing coefficients satisfying the trace-power identity by \(\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) preserves these conditions at every positive length.

Fix the algebra-structure data. Consider a finite set of labels \(\alpha \), complex vector spaces equipped with a product at each length \(L\), and operators \(O_L(M_\alpha )\) in those spaces, together with scalars \(m_\alpha \), structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\), and length-independent diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) with positive diagonal entries. Require the product, idempotent, trace-power, and blocked-basis comparison conditions of Definition 21.2.3.31.

In a vertical basis-of-normal-tensors decomposition, \(O_L(M_\alpha )\) denotes the operator obtained by contracting \(L\) copies of \(M_\alpha \), and \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) for the positive diagonal multiplicity matrix \(\mu _\alpha \). The abstract conditions here do not themselves construct such a tensor decomposition. Existence means that these objects can be chosen simultaneously. The blocked-basis comparisons are additional conditions relating them to the fixed algebra-structure data, not clauses of the basis-of-normal-tensors algebra characterization.

One may instead start with the positive diagonal matrices and define \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\), provided the product, idempotent, and blocked-basis comparison conditions hold. For any choice satisfying the conditions above, replacing the coefficients by these traces at positive lengths preserves all three conditions.

Let \(m_\alpha \in \mathbb {C}\) be scalars indexed by the fixed BNT labels \(\alpha \). For the positive diagonal matrices \(\mu _\alpha \) in the vertical decomposition, these scalars are \(m_\alpha =\operatorname{tr}(\mu _\alpha )\), as in the idempotent condition of [ 13 , Theorem 4.14(ii) ] .

Let \(\mu _\alpha \) and \(\nu _\gamma \) be diagonal matrices with complex entries, representing one-site and two-site multiplicities, respectively. A multiplicity-spectrum comparison requires \(\operatorname{tr}(\mu _\alpha )=m_\alpha \) and, for every \(\gamma \), equality with multiplicity between the entries of \(\nu _\gamma \) and the products \(\mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k}\). For the positive diagonal matrices in the vertical canonical decompositions, this is the conclusion of the BNT comparison and positive power-sum argument in [ 13 , Appendix C.4, lines 2048–2058 ] . The definition itself does not require positivity or a realization by an MPDO tensor.

Exact invertible gauges for a two-site multiplicity spectrum require, for every one-site sector \(\gamma \), equality of the paired 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 \(i\) of the vertical tensor,

In the vertical direction the gauge acts on the virtual legs of the matched normal tensor:

\( = \)

Thus the matched normal tensors have no residual scalar phase. This is the exact invertible conjugacy in [ 13 , Appendix C.4, lines 2053–2057 ] . The definition does not require \(Z_\gamma \) to be unitary.

Let \(M\) have a vertical decomposition \(\widetilde M=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \) with a basis of normal tensors \(\{ M_\alpha \} _\alpha \) and positive diagonal matrices \(\mu _\alpha \), satisfying the BNT algebra condition. A two-site multiplicity spectrum consists of a vertical canonical decomposition

a bijection \(\sigma \) from the one-site sectors to the two-site 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 used in [ 13 , 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 model is motivated by [ 13 , 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 physical trace is \(J=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\). Define \(|\Phi \rangle =|0,0\rangle +|1,1\rangle \) and \(P=|\Phi \rangle \! \langle \Phi |\). Thus \(P\) is an unnormalized rank-one positive operator, not an orthogonal projection. The tensor \(A\) is the vertical tensor of the physical model in Definition 21.2.3.1. Theorem 21.2.3.4 identifies \(P\) with its two-site operator after reindexing. For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\), set \(B^v=XA^vX^{-1}\) and 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 [ 13 , Appendix C.4, lines 2048–2057 ] and the physical-sector argument of [ 13 , Proposition 4.13, lines 1898–1921 ] . Neither passage states this deformation or the results below.

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

Here \(C_{(d,x,y),(i,j)}\) denotes the coefficient of the simultaneous left 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 [ 8 ] 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 multiplicity coordinates weighted by the positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\).

Each edge of the pentagon carries the \(F\)-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 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 or an orthogonal projector. Write \(B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}\). Every product-MPO matrix is reconstructed on this support:

The two zipper identities are

Each block is injective, and the matrix of coefficients of all labelled blocks has a simultaneous left inverse. Contraction of its row indexed by \((c,x,y)\) with \(B_d^{ij}\) gives \(E_{xy}\) when \(d=c\) and zero otherwise. These are the fusion-tensor, reconstruction, zipper, and simultaneous inverse relations of [ 8 , lines 161, 181–200, and 269–277 ] .

Let \(I\) be an index set. A family of diagonal complex matrices indexed by \((\alpha ,\beta ,\gamma )\in I^3\) consists of a size \(r_{\alpha ,\beta ,\gamma }\in \mathbb {N}\) and entries \(\chi _{\alpha ,\beta ,\gamma ,k}\in \mathbb {C}\) for \(0\le k{\lt}r_{\alpha ,\beta ,\gamma }\), defining

A size of zero gives the empty diagonal matrix. The matrices \(\chi _{\alpha ,\beta ,\gamma }\) in [ 13 , Theorem 4.14(ii) ] additionally have positive diagonal entries.

For the mixed-Bell tensor \(Q\), put \(c=4\sqrt2/5\) and \(A=cQ\), viewed as a tensor over the sixteen-letter ket-bra alphabet.

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 [ 13 , Definition 4.2, lines 735–739 ] , applied to one element of a basis of normal tensors.

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

Fix a common length \(S\). The tensors admit common word selectors for the final labels at length \(S\) if, 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

for every \(\varepsilon '\ne \varepsilon \). 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 [ 8 ] .

The maps \(U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }\) and \(U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon }\) are obtained by restricting the rows of the two iterated fusion isometries to the summands with final label \(\varepsilon \). After the canonical reassociation of the triple bond space, their adjoints have the orientation and bracketing pattern of weighted versions of the two fusion maps in  [ 8 , 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 row and column convention for \(F\) in  [ 8 , Section 3.4 ] ; no equality with the fusion multiplicities of that source is asserted. The corresponding spaces for a fixed final label are \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\).

Suppose that the fusion isometries have full support, and let \(P_\gamma \) act on the bond space of \(M_\gamma \). Define

The terminal contraction is unchanged. Full support means that the active support is the whole product bond space.

Full support is an additional restriction here, not a hypothesis of the source statement for arbitrary chain lengths. This distinction is documented in [ 116 ] .

Let \(M\) generate MPDOs, and fix a fusion decomposition of its vertical BNT \(\{ M_\alpha \} _\alpha \) with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). For every terminal label \(\gamma \), close the horizontal operator leg of \(M_\gamma \) to obtain a positive semidefinite bond matrix \(K_\gamma \). Choose an orthonormal eigenbasis of each \(K_\gamma \). Its spectral indices are pairs \(s=(\gamma ,k)\); write \(\lambda _s\) for the corresponding eigenvalue and \(E_s\) for the rank-one orthogonal projection onto the chosen eigenvector. Recursively transport this family through every fusion history and the adjoint sequential coisometry. Taking the direct sum over all labels and multiplicities at positive length \(N\) defines the operator \(P_s^{(N)}\).

Only nonzero product sectors occur in the fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

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

where \(E_n\) denotes the blocked transfer map. No complete positivity or trace preservation is required of these linear maps.

Let \(M^{(2)}\) be the two-site blocking of \(M\) and write \(A=\widetilde{M^{(2)}}\). The grouped vertical product hypotheses require the phase-class grouping with physical reducing isometries and the pairwise grouped-corner Gram comparison for both \(M\) and \(M^{(2)}\). They require in addition the following three properties of \(A\):

1. Every orthogonal projection \(P\) satisfying \(PA^v=PA^vP\) for all \(v\) also satisfies \(A^vP=PA^vP\) for all \(v\).

2. Each irreducible isometric corner \(B\) with a positive definite eigenmatrix \(\rho \) satisfying \(\mathcal E_B(\rho )=r\rho \), \(r{\gt}0\), has no peripheral eigenvalue other than \(r\).

3. For every matrix \(P\), if \(PA^vP\ne 0\) for some letter \(v\), then its sector compression is nonzero at some finite chain length: \(\mathcal C_N(M^{(2)},P)\ne 0\) for some \(N\).

Both normalized BNT-refined horizontal form and literal CPSV canonical form, together with positivity of the generated operators, supply these hypotheses independently.

The blocked multiplication coefficients are in trace-power form when, for every positive blocking length \(n\) and every multiplication triple \((i,j,k)\), the coefficient \(c^{(n)}_{i,j,k}\) equals \(\sum _{s=0}^{r_{n,i,j,k}-1} \chi _{n,i,j,k,s}^{\, n}\). 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 blocking length of the two factors in \(\mathcal A_n\); the product is expanded in the basis of \(\mathcal A_{2n}\).

Let \(c^{(L)}_{\alpha ,\beta ,\gamma }\) be a family of complex coefficients and let \(\chi _{\alpha ,\beta ,\gamma }\) be diagonal matrices on the same index set. The coefficients have trace-power form with respect to \(\chi \) if, for every \(L\ge 0\) and every triple \((\alpha ,\beta ,\gamma )\),

Here \(r_{\alpha ,\beta ,\gamma }\) is the size of \(\chi _{\alpha ,\beta ,\gamma }\). This is the scalar identity appearing in [ 13 , Theorem 4.14(ii) ] . By itself, the definition neither requires positive diagonal entries nor asserts that the coefficients describe multiplication of vertical operators.

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 \((\delta , \varepsilon )\), the multiplicity indices of \(\chi _{\alpha ,\beta ,\delta }\) and \(\chi _{\delta ,\gamma ,\varepsilon }\), and the bond index of \(M_\varepsilon \).

Let \(M\) generate MPDOs, suppose that its MPS tensor with paired physical indices is in CF as defined by CPSV, and suppose that \(M\) satisfies the RFP condition. Choose one fusion decomposition of the vertical BNT on the nonzero product sectors furnished by these hypotheses and denote it by \(\mathcal F_{\mathrm{CPSV}}(M)\).

Only nonzero product sectors occur in the selected fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors, and no unsupported normal sector is inserted. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction of the product tensor. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

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 [ 13 , 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 [ 13 , Section 4.5 ] , this joins two vertically read tensors along an original horizontal bond. It is the tensor written \(M_\alpha M_\beta \) in [ 13 , Theorem 4.14(iii) ] .

For three bond spaces, let

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

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

Thus the ket and bra words are grouped separately. For \(L=0\), both words are empty and the unique local matrix is the identity. For \(L{\gt}0\), this is the MPO version of the physical blocking used for the basis of normal tensors in [ 13 , lines 317–345 ] . It is a local tensor, not the closed-chain operator \(O_L(M)\) of [ 13 , lines 962–967 ] .

Consider a family of blocked diagonal matrices with strictly positive diagonal entries satisfying \(c^{(n)}_{i,j,k}=\operatorname{tr}(\chi _{n,i,j,k}^{\, n})\) for every \(n{\gt}0\) and every multiplication triple \((i,j,k)\). This is a blocked-basis analogue of the positive diagonal matrices in [ 13 , Theorem 4.14(ii) ] , without requiring a single family indexed only by the BNT labels.

For given coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }\), consider diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), independent of \(L\), with strictly positive diagonal entries, such that \(c^{(L)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) for every \(L{\gt}0\) and all labels. This is the coefficient condition in [ 13 , Theorem 4.14(ii) ] . If the coefficients are defined by these traces, positivity of the diagonal entries suffices.

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 \((\delta ,\varepsilon )\), the multiplicity indices of \(\chi _{\beta ,\gamma ,\delta }\) and \(\chi _{\alpha ,\delta ,\varepsilon }\), and the bond index of \(M_\varepsilon \). This is the right-bracketed composite in the associativity equation of [ 8 , 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 [ 13 , lines 217–246 ] together with the simplicity condition of [ 13 , 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 20.12.5. The definition omits the global unit-weight normalization of [ 13 , 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 \(\mu _j\), set \(\mathcal K_j=\mu _j A_j\).

For an MPDO in normalized BNT-refined horizontal form satisfying the RFP condition, fix the fusion decomposition supplied 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. Choose an orthonormal eigenbasis of each \(K_\gamma \), with spectral indices \(s=(\gamma ,k)\) satisfying \(0\leq k{\lt}D_\gamma \). Write \(\lambda _s\) for the corresponding eigenvalue and \(E_s\) for the rank-one orthogonal projection onto the chosen eigenvector, 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 take the direct sum over all sitewise label and multiplicity configurations. Denote the resulting operator at positive chain length \(N\) by \(P_s^{(N)}\).

The normalized BNT-refined horizontal hypothesis is stronger than CF as defined by CPSV.

For a chain of positive length \(N\), choose at every site a BNT label \(\alpha _j\) and a diagonal coordinate \(q_j\) of \(\mu _{\alpha _j}\). There are \(N-1\) 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 \(c=(\boldsymbol \alpha ,\boldsymbol q)\) denote the configuration, let \(W_{N,c}\) be its sequential coisometry onto the retained direct sum, and set \(Q_{N,c}=Q_N((\operatorname{tr}(M_\gamma ))_\gamma )\). The corresponding density block is \(m(\boldsymbol \alpha ,\boldsymbol q) W_{N,c}^\dagger Q_{N,c}W_{N,c}\). The embedding \(\widetilde U\) in [ 13 , line 999 ] corresponds to \(W_{N,c}^\dagger \) on this block. The recursive density operator \(R_N\) is the direct sum of these blocks over all sitewise labels and diagonal coordinates. On the same direct-sum coordinates, define

Here \(\mathbb {1}_c\) is the identity on the product bond space for configuration \(c\). The scalar coefficients of \(A_N\) are the selected diagonal coefficients of the factor \(\mu ^{\otimes N}\) in [ 13 , line 999 ] ; the identities mentioned there act on the complementary tensor factors of each block.

Let \(M\) generate MPDOs, fix a fusion decomposition on the nonzero product sectors, and let \(\mu \) be its strictly positive diagonal multiplicity matrix on the retained one-site space. Define \(h=(-\log \mu )\otimes \mathbb {1}\) on two adjacent sites, where \(\log \mu \) is the diagonal matrix obtained by taking the logarithm of each diagonal entry. For a chain of length \(L=N+2\), with \(N\in \mathbb {N}\), let \(H_L=\sum _{i=0}^{L-1}h_{i,i+1}\), with periodic indices.

The terminal spectral family has cardinality at most the physical one-site dimension \(d\). Fix an injection of its indices into \(\{ 0,\ldots ,d-1\} \) and extend the eigenvalues and projectors by zero outside the image.

The Hamiltonian and projectors in this definition act on the retained nonzero vertical sectors, with reconstruction onto the physical space treated in [ 140 ] .

The local interaction is defined on two spins, so this definition applies to lengths \(L=N+2\). The length-one boundary is recorded in [ 122 ] .

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

In the source construction, the terminal matrices are \(P_\gamma =\operatorname{tr}(M_\gamma )\), with spectral refinement used to obtain projections. Here the trace closes the horizontal operator leg and leaves the bond indices of the vertically read tensor open. Before spectral refinement, the fusion identity implies

Summing the forward fusion identity over equal horizontal operator indices and distributing matrix multiplication gives (90). This is one pairwise step in the recursion of [ 13 , lines 999–1010 ] .

This definition describes one fusion layer. The arbitrary-chain recursion, the density decomposition, and the physical commuting Gibbs decomposition for \(N\geq 2\) are given in Theorems 21.3.47, 21.3.52, and 21.3.63, respectively. The source boundary at \(N=1\) is recorded in Remark 21.3.66.

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 operators \(W_N\) and \(Q_N\) are indexed by 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 sequential map \(W_N\) is obtained by tensoring the preceding map with the identity on the newly appended bond and then applying the pairwise fusion coisometry onto the nonzero sectors in each preceding-history block. This gives the recursive operator and sequential circuit in [ 13 , lines 999–1010 ] .

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

Here \(A_{D_\alpha ,D_\beta ,D_\gamma }\) is the permutation matrix identifying the two bracketings of the product bond space. Both intermediate labels and the final label remain in the direct sums. The comparison is oriented from the right-associated sum to the left-associated sum, matching the adjoint orientation of the \(F\)-matrix convention in [ 8 , Section 3.4 ] .

Let \(J\) be the basis permutation from an exterior site and an inserted bond to two sites:

Define the refinement and coarse-graining maps by

The partial trace is over the inserted bond, leaving the two exterior qubits as a single physical site.

For \(N\ge 0\), write each physical configuration as \(s=((l_m,r_m,f_m))_m\), with cyclic indices when \(N{\gt}0\). At \(N=0\) the configurations are empty. The incoming cells are \(e_N(s)_m=((r_{m-1},l_m),f_m)\). For cells \(a_m=(b_m,g_m)\), with \(b_m\in \{ 0,1\} ^2\), the inverse reconstructs site \(m\) as \(((b_m)_2,(b_{m+1})_1,g_m)\). Let \(E_N|s\rangle =|e_N(s)\rangle \), and let \(S_N\) separate a cell configuration into its bond string and flag string. Define

For \(N{\gt}0\), \(\omega _N\) is the uniform state on even-parity flag strings, and \(D_N\) is the product of independent bond states and that flag state, expressed in the physical site coordinates. At length zero, the empty-product convention gives \(\omega _0=D_0=2\mathbb {1}\) and \(U_0=\mathbb {1}\).

Let

A physical site consists of qubits \((l,r)\in \{ 0,1\} ^2\). With virtual indices in \(\{ 0,1\} ^2\), define the tensor \(Q\) by

where \(E_{a,b}\) is the matrix unit with row \(a\) and column \(b\). Both physical and virtual dimensions are four. For \(N\ge 0\), let \(B_N\) be the permutation of basis configurations given by

with cyclic indices when \(N{\gt}0\) and the unique empty configuration when \(N=0\). Its inverse sends a bond configuration \((b_m)_m\) to the site configuration \((((b_m)_2,(b_{m+1})_1))_m\).

On two bond qubits let

Define \(\sigma =\frac78\Pi _++\frac18\Pi _-\) and \(\sigma '=\frac78\Pi _+-\frac18\Pi _-\). On a flag qubit let \(X|f\rangle =|f+1\rangle \), with addition modulo two, and \(Z|f\rangle =(-1)^f|f\rangle \). The controlled Bell-sign flip is

The matrices of the twisted quantum dimer of Definition 21.2.2.33 are diagonal in the flag qubit. Viewed vertically, the tensor therefore splits along the flag value \(f\in \{ 0,1\} \) into two tensors \(M_f\) acting on the two-qubit space of \(L\) and \(R\), with matrices indexed by pairs of horizontal bond indices. Write \(\mu =\frac58\), so that \(\mu ^2=(x^2+y^2)/2\) at \(x=\frac78\), \(y=\frac18\). The rescaled sector \(\widehat M_f=M_f/\mu \) has, at the bond pair \(((p,p',k),(q,q',k'))\), the matrix

where \([k=k']\) is one when \(k=k'\) and zero otherwise. The matrix unit has row \((p,q)\) and column \((p',q')\). The entry of this matrix between the qubit configurations \((l,r)\) and \((l',r')\) is \(\mu ^{-1}\) times the bond-matrix entry of the twisted dimer at the physical indices \(((l,r,f),(l',r',f))\) carrying the common flag value \(f\). The horizontal blocks of Definition 21.2.2.33 and the two vertical flag sectors are all matrix-unit tensors of this form, differing only in their one-site coefficients. The closed operators \(O_L(\widehat M_f)\) on a ring of \(L\) sites give the operators indexed by these two labels in Definition 21.2.2.2. Their basis-of-normal-tensors properties are proved below, rather than assumed in this definition. This tensor is motivated by the question of [ 13 , lines 995–1010 ] .

For \(N\) sites, let \(H_N\) be the basis permutation separating each site’s two bond qubits from its flag:

Unlike the incoming-bond regrouping, this separation does not move any qubit between physical sites.

Each site carries two qubits \(L\), \(R\) and a flag qubit \(f\), so the physical index set is \(\{ 0,1\} ^3\), read as \((l,r,f)\). The bond index set is also \(\{ 0,1\} ^3\), read as \((p,p',k)\), where \(k\) labels one of two horizontal blocks. Let \(X\) and \(Z\) denote the Pauli matrices. With \(x=\frac78\), \(y=\frac18\), \(P_\pm =\frac12(\mathbb {1}\pm X)\), and the \(2\times 2\) matrices \(C_0=xP_++yP_-\) and \(C_1=xP_+-yP_-\), the bond matrix for the pair of physical indices \(((l,r,f),(l',r',f'))\) is

Here \(\tau _0=\mathbb {1}\) and \(\tau _1=Z\). Thus each bond matrix is a sum of two matrix units, one in each block. The tensor is the direct sum along \(k\) of the two bond-dimension-four blocks obtained by keeping one summand. This example is motivated by the question of [ 13 , lines 995–1010 ] ; it is not stated there.

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 [ 13 , Theorem 4.9, lines 851–856 ] .

Let \(\{ O_L(M_\alpha )\} _{\alpha \in \Lambda }\) and \(\{ O_L(M'_\alpha )\} _{\alpha \in \Lambda '}\) be two operator families. A change of vertical representatives consists of a bijection \(e:\Lambda '\to \Lambda \), nonzero complex scalars \(s_\alpha \), and complex algebra isomorphisms \(\Phi _L\) between the ambient operator algebras. For every positive length \(L\) and every \(\alpha \in \Lambda '\), these satisfy

Given a label bijection \(e:\Lambda '\to \Lambda \) and complex scalars \(s_\alpha \), define

For this definition, use the convention \(z/0=0\).

Let

Define MPO tensors with one-dimensional physical space by \(M_P^{00}=P\) and \(M_Q^{00}=Q\). Thus \(M_Q\) is the gauge transform of \(M_P\) by \(X\):

\( = \)

Equip them with single-element vertical basis-of-normal-tensors decompositions whose normalized representatives and positive multiplicity weights are

The corresponding positive diagonal matrices \(\chi \) in the algebra condition 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 16.4.4, 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 linear extension on block \(k\) 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 20.6.2.

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 [ 13 , 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  [ 13 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

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

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 [ 13 , 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 [ 13 , 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 [ 13 , 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 \(x_0,\ldots ,x_{n-1}\) be non-negative real numbers whose power sums \(\sum _k x_k^L\) take the same value at every integer exponent \(L \ge 1\). Then every \(x_k\) equals \(0\) or \(1\). The same holds for a finite family of complex numbers whose imaginary parts vanish and whose real parts are non-negative.

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

If \(r\) is a bijective reindexing of the row space of a matrix \(A\), then

where \(A_r\) denotes the corresponding row reindexing.

Let \(M_0,\ldots ,M_{L-1}\) be square matrices of the same finite size over a commutative semiring, with \(L\ge 1\). Applying the ordered product \(M_0\cdots M_{L-1}\) to a vector \(v\) and reading the entry at \(a\) sums, over all paths \((t_0,\ldots ,t_L)\) with \(t_0=a\), the product of the entries \((M_n)_{t_n t_{n+1}}\) along the path times \(v_{t_L}\). Consequently,

with the cyclic convention \(t_L=t_0\).

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 blocking length \(n\), \(c^{(n)}_{i,j,k}=\operatorname{tr}(\chi _{n,i,j,k}^{\, n})\).

For each positive blocking length \(n\), each multiplication triple \((i,j,k)\), and every exponent \(L\in \mathbb {N}\), \(\operatorname{tr}(\chi _{n,i,j,k}^{\, L}) = \sum _{s=0}^{r_{n,i,j,k}-1} \chi _{n,i,j,k,s}^{\, L}\).

A fusion coisometry onto the retained direct sum satisfies

Suppose the chosen blocked-basis coefficients satisfy the comparison equality with the BNT structure coefficients, and the latter have a trace-power representation by positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) independent of length. Then, for every \(n{\gt}0\) and every multiplication triple \((i,j,k)\),

Suppose that the BNT algebra conditions hold. The vector \(m\) is then an idempotent for the multiplication induced by the length-one coefficients \(\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })\): for every \(\gamma \),

Under the trace-power condition at positive lengths, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

The coefficients defined by \(c^{(L)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) for a family of diagonal matrices satisfy the trace-power condition at positive lengths for those matrices.

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 matrix at the paired index \((p,q)\) is the MPO matrix \(R^{pq}\). This bond matrix has the outer-product factorization

or equivalently \(R^{pq}=\frac{25}{32}u_{pq}v_{pq}^{\mathsf T}\), where \(u_{pq}(a)=A^a_{p_1q_1}\) and \(v_{pq}(b)=A^b_{p_2q_2}\).

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 \(\mathcal{E}_R^2=\frac{337}{512}\mathcal{E}_R\).

Set \(s=(\frac45,\frac45,\frac35,\frac35)\), \(F=\operatorname{diag}(s_0^2,s_1^2,s_2^2,s_3^2)\), and

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

The tensor \(A\) is injective and proportional to a normal tensor. The matrix \(X\) is invertible but not unitary, and \(A\) and \(B\) generate the same matrix product vectors at every positive length. Their physical traces satisfy

Nevertheless, \(P\) is positive semidefinite while \(P_X\) is not Hermitian. Theorem 21.2.3.4 identifies \(P\) with the reindexed two-site operator of the physical model. Equality of the vertical matrix product vectors does not imply that the corresponding physical deformation generates MPDOs.

For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\) and every \(\omega \in \mathbb {R}\), we have \(X^\dagger X\ne \omega \mathbb {1}\).

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 coefficients \(s_d\) satisfy

Thus contraction over one physical index pair selects the identity on the chosen final block and annihilates every other final block.

The representative \(A\) is left-canonical and normal. Its physical-trace transfer \(\mathcal B_A=\sum _i A^{ii}\) satisfies \(\operatorname{tr}\mathcal B_A=c\ne 0\).

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

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 21.2.2.100 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, closing the physical legs gives

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

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

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:

For every tensor \(M\) and every blocking length \(L\), the physical-trace transfer of the \(L\)-site blocking of \(M\) is the \(L\)-th power of the physical-trace transfer of \(M\).

If the coefficients have a trace-power representation by positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) independent of \(L\), then for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

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. The operator \(V^{\otimes N}\rho ^{(N)}(M)(V^{\otimes N})^\dagger \) is generated by the tensor obtained by applying \(X\mapsto VXV^\dagger \) to the one-site physical matrices of \(M\).

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

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

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

Surjectivity of the left iterated fusion map corresponds to completeness of the fusion decomposition in [ 8 , lines 181–191 ] .

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

Surjectivity of the right iterated fusion map likewise corresponds to completeness of the fusion decomposition.

For \(k\in \{ 0,1\} \), put \(\sigma _k=\frac78\Pi _++(-1)^k\frac18\Pi _-\), where \(\Pi _\pm \) are the Bell projections of Definition 21.2.2.36. Let \(C_k\) be the matrix of Definition 21.2.2.33. For \(r,l,r',l'\in \{ 0,1\} \), in right-then-left bond order,

The operator \(V\) of Definition 21.2.2.36 is self-adjoint and satisfies \(V^2=\mathbb {1}\) on the full bond–flag space. It obeys

For a matrix-unit tensor with one-site coefficients \(c\), the ordered product along a nonempty pair of words vanishes unless both words satisfy the open bond-matching condition. When they do, the product is the matrix unit whose row is given by the left bits of the first indices and whose column is given by the right bits of the last indices, multiplied by the product of the coefficients. On a ring of positive length the closed operator therefore has entries \(\mathbb {1}[\text{cyclic bond match}](\sigma ,\tau ) \prod _n c(\sigma _n,\tau _n)\), where the first factor is the indicator that both words satisfy the cyclic bond-matching condition. In particular, for a positive length \(L\) and words \(\sigma _n=(p_n,p'_n,k_n)\), \(\tau _n=(q_n,q'_n,k'_n)\) of horizontal bond indices,

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 [ 13 ] ; it lets the MPDO block-entropy theory apply to pure-state block entropies.

Each \(T_k\) is bijective.

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

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

Let \(A=\bigoplus _{s\in S}C_s\) be a finite direct sum of nonzero normal tensors \(C_s=(C_s^i)_i\) of bond dimension \(D_s\), with repetitions allowed, and let \(B=(B^i)_i\) be an arbitrary tensor of bond dimension \(D_B\). Assume that the periodic vectors of \(A\) and \(B\) agree at every positive length, \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N{\gt}0\); equivalently, \(\operatorname{tr}(B^w)=\operatorname{tr}(A^w)\) for every nonempty word \(w\). Then there are an integer \(z\geq 0\), an integer \(r=|S|+z\), an invertible matrix \(G\in \mathrm{GL}_{D_B}(\mathbb {C})\), and maps \(V_s:\mathbb {C}^{D_s}\to \mathbb {C}^{D_B}\), \(W_s:\mathbb {C}^{D_B}\to \mathbb {C}^{D_s}\), with the following properties.

1. Every \(GB^iG^{-1}\) is block upper triangular for one common decomposition into \(r\) blocks.

2. Exactly \(|S|\) of the diagonal slots are matched with the target slots: writing \(j(s)\) for the position matched with \(s\), there is \(X_s\in \mathrm{GL}_{D_s}(\mathbb {C})\) with \(\bigl(GB^iG^{-1}\bigr)_{j(s),j(s)}=X_sC_s^iX_s^{-1}\).

3. The remaining \(z\) diagonal slots are one-dimensional zero tensors.

4. The compression pairs are mutually biorthogonal: \(W_sV_t= \operatorname{id}_{D_s}\) if \(s=t\) and \(0\) otherwise.

5. For every nonempty word \(w\), \(W_sB^wV_s=C_s^w\).

6. Writing \(\widehat C^i=\sum _{s\in S}V_sC_s^iW_s\) and \(R^i=B^i-\widehat C^i\), every product \(R^{i_1}R^{i_2}\cdots R^{i_r}=0\): the non-unital algebra generated by the \(R^i\) is nilpotent.

7. \(D_B=\sum _{s\in S}D_s+z\).

In the setting of Theorem 29.5.1, suppose that the unital algebra generated by the matrices \(B^i\) is closed under conjugate transposition, that is, \((B^i)^\dagger \) lies in the unital algebra generated by the \(B^j\) for every \(i\). Then the word-algebra module of \(B\) is semisimple, and \(B\) admits a block triangular form as in Theorem 29.5.1 whose remainder vanishes: the block triangular form is block diagonal.

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

Let

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

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

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

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

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

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

and

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

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

Let

For every \(N{\gt}0\), its periodic operator is the normalized periodic operator of the printed tensor and has trace one:

It generates positive semidefinite operators. For \(p=(a,b)\in \{ 0,1\} ^2\), set \(\pi (p)=a\mathbin \oplus b\) and

The Kraus families \((C_p)_p\) and \((R_p)_p\) define trace-preserving completely positive maps \(\mathcal S\) and \(\mathcal T\), respectively, and for every virtual matrix \(X\) they satisfy

This proves the two channel equations in Definition 4.1 without further assumptions. Compatibility with the line-246 global unit-weight canonical convention is outside the conclusion of this theorem.

Let \(A\) have canonical-form data

For every \(L\geq 1\), \(A^{[L]}\) has canonical-form data with the same coisometry \(U\), the blocked normal tensors \(A_k^{[L]}\), and the weights \(\mu _k^L\):

Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension normal tensors, no two of which are gauge-phase equivalent after identifying equal bond dimensions. Then the matrix product vectors \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\), as in [ 13 , Appendix A ] .

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 inserted marked matrix ensures that the closed chain is nonempty.

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.

This positive-tail formulation corresponds to the argument in [ 13 , Appendix C.3, lines 1835–1858 ] : first-site agreement at positive system sizes suffices to separate representatives at a nonzero tail length without requiring empty-word data.

Let \(e:\{ 0,\ldots ,d'-1\} \to \{ 0,\ldots ,d-1\} \) be a bijection and set

The transfer maps of \(A\) and \(e^*A\) agree. In particular, physical relabeling preserves left-canonical normalization and CPSV normality. If \(A\) has canonical-form data

then \(e^*A\) has the same weights and coisometry and the relabeled normal blocks:

In the notation of Theorem 8.6.1.9, 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 8.6.1.9 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 tensor in CF generating MPDOs, [ 13 , Theorem 4.14 ] states that the RFP condition, the algebra and idempotent relations for a vertical BNT, and the fusion-isometry and idempotent relations are equivalent.

The RFP condition is equivalent to the algebra and idempotent relations. For the fusion statement, an absent product sector must have multiplicity zero, and the map onto the retained sectors is a coisometry with exact reconstruction. Under this active-support interpretation, the three conditions are equivalent by Theorem 21.3.30. Without these qualifications, the displayed fusion statement is not well-defined on empty product sectors and has the wrong isometry orientation.

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 has a compatible family of support algebras, 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 has a compatible family of support algebras, then \(\operatorname{Fix}(E_n^{\dagger })=\operatorname{Fix}(E_1^{\dagger })\) for every positive blocking size \(n\).

Let \(M\) generate a family of MPDOs with physical dimension \(d\) and bond dimension \(D\), and choose a vertical canonical decomposition satisfying the algebra conditions of Definition 21.2.2.98. Let \(M_\gamma \) be any retained normal tensor. Separating its doubled physical index gives two physical legs of dimension \(D\), corresponding to the original auxiliary indices of \(M\). Closing these legs gives the bond matrix

\( = \)

Then \(T_\gamma \succeq 0\). This follows from the chosen decomposition and positivity of the one-site operator \(\rho ^{(1)}(M)\); it does not require fusion maps or a renormalization fixed point hypothesis.

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

then there is a compatible family of support algebras.

Assume there is a compatible family of support algebras. 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 whose doubled-tensor transfer map is trace-preserving and has a positive-definite fixed point, and which has a compatible family of support algebras, although its doubled-tensor transfer map is not idempotent.

Suppose that the labelled tensors of a fusion-isometry family with length-independent structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) form a BNT. There is a common positive blocking length 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 \(\mathbb {C}^{D_\alpha D_\beta } \cong \mathbb {C}^{D_\alpha }\otimes \mathbb {C}^{D_\beta }\). The two zipper identities, exact reconstruction, injectivity, and the simultaneous left inverse of the block coefficients follow as conclusions rather than additional hypotheses.

Assume the transfer map of the doubled tensor is trace-preserving and 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 20.15.82, then

This is the idempotent coefficient identity in [ 13 , Appendix C.4, lines 2030–2042 ] .

Under the hypotheses and with the sector correspondence of Theorem  20.15.81, 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 [ 13 , Appendix C.4, lines 2011–2018 ] .

In [ 13 , Appendix C.4, line 2013 ] , the exponent is printed as \(m_\gamma ^L/n_\gamma \); the tensor scaling requires \((m_\gamma /n_\gamma )^L\) (see [ 153 ] ).

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 \(M\) generate MPDOs, and suppose that its MPS tensor obtained by pairing the physical indices is in canonical form. Suppose also that \(M\) has a vertical decomposition \(\widetilde M=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \) satisfying the BNT algebra condition. 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 one-site product expansion, in which every sector occurs. The tensors \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length and, for every \(\gamma \),

as multisets. This is the multiplicity-spectrum comparison of Appendix C.4, lines 2046–2058 of  [ 13 ] . The paired bond dimensions agree, and there are invertible matrices \(Z_\gamma \) satisfying, for every physical index \(i\) of the vertical tensor,

This statement asserts exact invertible conjugacy, without requiring the matrices \(Z_\gamma \) to be unitary.

Let \(M\) be an MPDO whose doubled-index tensor is in literal CPSV canonical form. If \(M\) has a tensor-attached basis-of-normal-tensors 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 [ 13 , Theorem 4.14, lines 972–993 ] .

Here the common marked comparison needed to normalize the exact sector gauges is supplied by Theorem 21.2.3.27 [ 28 ] , with the physical maps completed on the discarded complements as in [ 159 ] .

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:

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

The relabelled two-site maps satisfy

Identities (60) and (61) combine the one-site weighted sector form of [ 13 , Appendix C.4, lines 2048–2051 ] with, respectively, [ 13 , Appendix C.4, lines 2067–2068 ] and [ 13 , Appendix C.4, lines 2081–2083 ] . Identity (62) combines the matched weighted sector form of [ 13 , Appendix C.4, lines 2048–2064 ] with the map \(R_2\) of [ 13 , Appendix C.4, lines 2078–2079 ] . Identity (63) combines the same sector form with the normalized map \(\widetilde R_2\) of [ 13 , 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 (56) may be strict: \(W_i^\dagger W_i\) projects onto the retained nonzero sectors, so positive elements supported on the discarded orthogonal complement lose trace under the compression.

Suppose the chosen blocked-basis coefficients satisfy the comparison equality with the BNT structure coefficients, and the latter have a trace-power representation by positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) independent of length. Then there are positive blocked diagonal matrices giving the blocked coefficients by the trace-power formula. They are obtained by reindexing the BNT diagonal matrices along \(\sigma _n\sqcup \tau _n\); on multiplication triples this gives \(\chi _{n,i,j,k}=\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}\).

Suppose that the labelled doubled-index tensors form a BNT and admit fusion isometries on the full product bond spaces. Suppose also that \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), is independent of \(L{\gt}0\). Then there is one integer \(L{\gt}0\) such that the tuples of letters of the blocked tensors \(M_\gamma ^{[L]}\) span the full product of their matrix algebras. Hence every \(M_\gamma ^{[L]}\) is injective.

For this same \(L\), write

where \(r_{\alpha ,\beta ,\gamma }\) is the dimension of the multiplicity space on which \(\chi _{\alpha ,\beta ,\gamma }\) acts. The original fusion isometry satisfies

the two blocked zipper identities

and exact reconstruction

Let \(Z_\gamma \) be an exact invertible sector gauge and suppose \(Z_\gamma ^\dagger Z_\gamma =\omega _\gamma \mathbb {1}\) with \(\omega _\gamma {\gt}0\). Define \(U_\gamma =\omega _\gamma ^{-1/2}Z_\gamma \). Then \(U_\gamma \) is unitary and, for every vertical letter \(v\),

This is the inverse-square-root normalization used in [ 13 , Proposition 4.13, lines 1903–1908 ] , applied at Appendix C.4, lines 2053–2057.

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 ambient sector maps and unitary conjugations defined above, 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 (69) are the formulas of [ 13 , 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; Formula (71) provides a trace-preserving completion on those complements when the physical space is nonzero, yielding the same action on the physical operators generated by \(M\) for any choice of reference states \(\tau _1,\tau _2\).

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 space and a scalar \(c_\gamma {\gt}0\) such that, for every vertical letter \(v\),

\( = c_\gamma \)

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 [ 13 , Proposition 4.13, lines 1909–1919 ] , applied to the sector matching of Appendix C.4, lines 2048–2057.

The compression in (24) selects a conjugate of \(M_\gamma \), not \(M_\gamma \) itself. It does not provide an isometric compression onto a positive multiple of \(M_\gamma \) or establish the Gram identity

This distinction is documented in [ 28 ] . The comparison with a common unblocked tail below proves the Gram identity without requiring such a compression of the two-site blocking.

For a matched two-site sector, choose \(V_\gamma \) and \(c_\gamma {\gt}0\) as in (24), and set

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

The first identity replaces the isometry on one side by the oblique pair \((L_\gamma ,R_\gamma )\):

\( = c_\gamma \)

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 express every horizontal slice of \(M_\gamma \) as a linear combination of the physical letters of the two-site blocking:

This gives the inclusion in the physical-letter span used to compare the two vertical forms in [ 13 , Appendix C.4, lines 2048–2057 ] .

In general \(L_\gamma \ne R_\gamma \), so (28) is not an isometric compression onto a positive multiple of \(M_\gamma \). 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 (29), rather than by a second copy of \(M_\gamma \). The linear-span identity alone gives neither a common reflected tensor nor the Gram identity

This prevents the direct two-site comparison from proving the Gram identity. The theorem below instead compares the one-site compression onto \(M_\gamma \) with the two-site compression onto \(Z_\gamma M_\gamma Z_\gamma ^{-1}\), keeping the same unblocked 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 compression selects a positive-weight copy in the one-site vertical decomposition of [ 13 , Proposition 4.13, statement lines 943–951, and Appendix C.4, lines 2046–2057 ] .

Suppose a chosen vertical canonical decomposition satisfies the active-sector fusion and idempotent relations. Then its operators \(O_L(M_\alpha )\) satisfy the algebra relations. For every positive chain length \(L\), the structure 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 [ 13 , Theorem 4.14, lines 972–993 ] , with the active-support interpretation of (iii).

Suppose that the operators \(\{ O_L(M_\alpha )\} _\alpha \) satisfy the product law with structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) for diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). At every positive length \(L\) at which the operators are linearly independent, the associativity constraint becomes

In [ 13 , Section 4.5 ] , these restrictions in the positive diagonal setting, together with the isometry statement, lead to a pentagon-like equation for the fusion isometries. Solutions constructed from unitary fusion categories give the family of renormalization fixed point MPDOs described there.

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

These are the restrictions that associativity of multiplication imposes on the structure coefficients, noted in [ 13 , Section 4.5 ] .

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

If \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\) and the coefficients have a trace-power representation by positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) independent of \(L\), then

If \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\), where the coefficients are defined by traces of powers of diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), then

Suppose the BNT structure coefficients have the trace-power form \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and are length independent. Then every diagonal entry \(\chi _{\alpha ,\beta ,\gamma ,r}\) lies in \(\{ 0,1\} \), as stated in [ 13 , Section 4.5 ] ; by positivity every entry equals one. Consequently, at every positive length, \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\), where \(r_{\alpha ,\beta ,\gamma }\) is the size of the diagonal matrix. The structure coefficient is this multiplicity, not necessarily one.

If the BNT structure coefficients have a trace-power representation by diagonal matrices all of whose diagonal entries equal one, then for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma }\), so the coefficients are length independent.

The tensor \(R\) has a vertical basis of normal tensors with one element \(M_0\). This component has \(16\) physical letters, bond dimension \(4\), and

Each letter of \(M_0\) is a ket copy of \(A\) over a bra copy of \(\overline A\) that share no index:

\( = \)

The positive diagonal multiplicity and coefficient matrices are

The vertical tensor is reconstructed exactly as

For every \(L{\gt}0\), the vertical operator satisfies

At length one, \(m_0=\operatorname{tr}(\mu _0)=25/32\) and \(c^{(1)}_{0,0,0}=32/25\). Thus the vector \(m=(m_0)\) is an idempotent for the multiplication induced by \(c^{(1)}\), since \(m_0=c^{(1)}_{0,0,0}m_0^2\):

The doubled-index tensor \(R^{\bullet \bullet }\), with bond dimension \(4\) and \(16\) letters \(R^{pq}\) for \(p,q\in \{ 0,\ldots ,3\} \), is in canonical form as in Definition 7.6.7 (28). It consists of a single normal block occupying the full bond space (\(r=1\), \(D_1=4\), \(U=\mathbb {1}\)), with weight \(\mu =\sqrt{337/512}\). This is the unique positive weight for which the normalized block \(\mu ^{-1}R^{\bullet \bullet }\) has an idempotent transfer map, and hence spectral radius one.

For every positive chain length \(N\), the horizontal operator factors as

where \(B\) sends each bit-string basis vector to the basis vector of the unique cyclic-bond-matching physical-index string with those first bits, and \(B^{\mathsf T}B=\mathbb {1}\). Here \(W_N(a,b)=\prod _n W(a_n,b_n)\) is the \(N\)-fold Kronecker power of \(W\). For the Walsh–Hadamard matrix \(H=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\) and \(\chi =\operatorname{diag}(1,\tfrac {7}{25})\), one has \(H\chi H=2W\). Thus the entries of \(\chi \) are the eigenvalues of the local factor \(W\). Theorem 21.2.2.78 identifies this matrix as the coefficient matrix \(\chi _{0,0,0}\) of a vertical basis of normal tensors with one element.

The coefficients \(c^{(L)}_{0,0,0}\) have a trace-power representation with the fixed positive diagonal matrix \(\chi =\operatorname{diag}(1,\tfrac {7}{25})\), as in Definition 21.2.2.11. For every \(L{\gt}0\),

The equality of traces follows from the diagonalization in Theorem 21.2.2.80. Here \(W^L\) is the ordinary matrix power of the fixed \(2\times 2\) matrix \(W\), not the Kronecker power \(W_L\) in \(\rho ^{(L)}(R)\). Theorem 21.2.2.78 identifies these coefficients with the structure coefficients of the vertical basis of normal tensors \(\{ M_0\} \).

The tensor \(R\) of Theorem 21.2.2.32 is a renormalization fixed point in the sense of Definition 20.2.1. The refinement map \(T\) from one site to two sites is completely positive and trace preserving, with 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 \(S\) from two sites to one site is completely positive and trace preserving, with four Kraus operators. Two use the amplitude \(1/\sqrt2\) and the eigenvectors of \(W\) on physical-index pairs satisfying the bond-matching condition. The other two assign the pairs that fail this condition deterministically, so that the four Kraus operators resolve the identity. These maps satisfy \(T[M_1(X)]=M_2(X)\) and \(S[M_2(X)]=M_1(X)\) for every boundary matrix \(X\), with \(M=R\).

The operator \(\rho ^{(N)}(R)\) is positive semidefinite for every positive chain length \(N\), so \(R\) generates MPDOs. Writing each bond matrix \(R^{pq}\) as a rank-one product scaled by \(25/32\), the closed-chain trace of a length-\(N\) configuration \((\sigma ,\tau )\) gives

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. In the product, \(W\) is evaluated on the corresponding bond bits. It is the local factor from Theorem 21.2.2.32, with eigenvalues \(\{ 1,7/25\} \). Thus \(\rho ^{(N)}(R)\) 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 \(W\).

Theorem 21.2.2.31 gives length-dependent structure coefficients for a specified BNT presentation, as asked in [ 13 , lines 995–999 ] . Its diagonal matrix \(\chi \) has the single entry \(\sqrt{1/2}\), and rescaling that entry to \(1\) makes the coefficient constant. The diagonal matrix constructed here has two distinct positive entries, and its trace powers remain length dependent under every uniform positive rescaling. Define four matrices in \(M_{2}(\mathbb {C})\) by

Identify the physical index set \(\{ 0,1,2,3\} \) with \(\{ 0,1\} \times \{ 0,1\} \) and let \(R\) be the bond-dimension-four MPO tensor obtained by exchanging physical and virtual indices in the matrices \(B^{ab}=A^a\otimes \overline{A^b}\) and applying the indicated normalization:

Here \((p,q)\) are the physical indices and \((a,b)\) are 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 zero-correlation-length condition of [ 13 , Definition 4.2, lines 736–741 ] . Define the positive diagonal matrix \(\chi =\operatorname{diag}(1,\frac{7}{25})\) and its trace powers by

Hence \(c^{(1)}\ne c^{(2)}\), so these coefficients are not length independent. For every real number \(s{\gt}0\), the rescaled matrix \(s\chi =\operatorname{diag}(s,\frac{7s}{25})\) has trace powers

which are again not length independent. Unlike the single-entry matrix of Theorem 21.2.2.31, no uniform positive rescaling makes both diagonal entries equal to one.

The unscaled coefficients are the structure coefficients of an explicit one-element vertical basis-of-normal-tensors presentation of \(R\), motivated by the discussion in [ 13 , lines 995–1010 ] .

Let \(\widehat R=\mu ^{-1}R^{\bullet \bullet }\) be the single normal block in this 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 condition in the definition of simplicity.

There is a tensor \(M\) generating MPDOs that has the direct-sum form \(M^i=\bigoplus _k\mu _k M_k^i\) with normalized normal tensor blocks \(M_k\), is a renormalization fixed point, and has a vertical basis-of-normal-tensors decomposition satisfying the algebra condition whose structure coefficients depend on the positive length \(L\). One may take \(M=R\) and the single-element basis-of-normal-tensors decomposition already constructed, for which

This exhibits length-dependent structure coefficients for the chosen vertical decomposition of \(R\), addressing the question following Theorem 4.14 of [ 13 ] without requiring the unit-weight normalization of [ 23 ] .

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

The two diagonal entries of \(\chi \) are eigenvalues of the local factor \(W\): the uniform vector has eigenvalue \(1\), and the alternating vector has eigenvalue \(7/25\). The transfer map \(\mathcal{E}_A(Y)=\sum _a A^aY(A^a)^\dagger \) has eigenvalues \(1,7/25,0,0\), counted with multiplicity. The identity is fixed, the traceless diagonal matrix \(\operatorname{diag}(1,-1)\) has eigenvalue \(7/25\), and the off-diagonal matrix units are annihilated. Thus the entries of \(\chi \) are also the nonzero eigenvalues of \(\mathcal{E}_A\).

Suppose the product identity for BNT operators at the same length holds and its coefficients have a trace-power representation by positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) independent of \(L\). Then, for every \(L{\gt}0\),

If the product identity for BNT operators at the same length holds with coefficients defined by traces of powers of diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), then for every \(L{\gt}0\),

Suppose that the BNT algebra conditions and blocked-basis comparison hold. Let \(\chi _{\alpha ,\beta ,\gamma }\) be the associated positive diagonal matrices. For every \(L{\gt}0\),

where \(c^{\chi ,(L)}_{\alpha ,\beta ,\gamma }\) denotes the trace of the \(L\)th power of \(\chi _{\alpha ,\beta ,\gamma }\). Drawn as a closed chain of \(L\) copies of \(\chi _{\alpha ,\beta ,\gamma }\), the coefficient is

\(c^{(L)}_{\alpha ,\beta ,\gamma }={}\)

Every diagonal entry of every \(\chi _{\alpha ,\beta ,\gamma }\) is strictly positive. Consequently, for every \(L{\gt}0\), the product identity reads

and the idempotent identity reads

Suppose that the BNT algebra conditions and blocked-basis comparison hold. Positive diagonal matrices representing the blocked coefficients as traces of powers are obtained by pulling back the length-independent matrices \(\chi _{\alpha ,\beta ,\gamma }\) along the comparison maps. At every positive blocking length \(n\), define \(\chi _{n,i,j,k} =\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}\). Thus the two matrices have equal size, diagonal entries, and traces of all powers.

Suppose the objects in Definition 21.2.3.35 exist for the chosen algebra-structure data. Then there are diagonal matrices \(\chi _{n,i,j,k}\) with positive diagonal entries such that, for every positive blocked length \(n\), source indices \(i,j\) at length \(n\), and target index \(k\) at length \(2n\),

These matrices are obtained by selecting the BNT matrices \(\chi _{\alpha ,\beta ,\gamma }\) along the source and target comparison maps \(\sigma _n,\tau _n\). Their dimensions, diagonal entries, and traces of powers agree with those of the selected matrices. The same construction proves existence when the BNT operators, scalars, and matrices are assumed to exist without a fixed choice.

Suppose the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\). Then \(\chi _{\alpha ,\beta ,\gamma }=1_{r_{\alpha ,\beta ,\gamma }}\) for every triple of labels, where \(r_{\alpha ,\beta ,\gamma }\) is the dimension of the corresponding multiplicity space. If

the fusion isometries satisfy

These are the identity-weight specialization of the fusion and zipper relations in [ 8 , Section 3 ] . The MPDO fusion identity appears in [ 13 , Theorem 4.14(iii), lines 986–991 ] ; its identity-weight specialization follows from the length-independent case discussed in [ 13 , line 1010 ] .

Let \(M\) generate MPDOs, and suppose that its MPS tensor obtained by pairing the physical indices is in canonical form. 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 proof compares a one-site compression onto \(M_\gamma \) with a two-site compression onto \(Z_\gamma M_\gamma Z_\gamma ^{-1}\), keeping the same unblocked tail. This supplies the common comparison missing from the direct application of [ 13 , Proposition 4.13, lines 1909–1919 ] in  [ 13 , Appendix C.4, lines 2048–2057 ] . The correction to the direct two-site argument is recorded in [ 28 ] .

Suppose that the BNT algebra conditions hold and that the one-site and two-site vertical multiplicity spectra satisfy the multiset comparison just defined. Then every two-site multiplicity matrix has trace \(m_\gamma \):

This is the normalization used for the preparation matrix \(\nu _\gamma /m_\gamma \) in the implication from the algebra condition to the renormalization fixed point property in [ 13 , Appendix C.4, lines 2058–2064 ] .

Let \(P\) and \(Q\) be BNT sector presentations of the same doubled-index tensor. Every representative of \(P\) has non-nilpotent physical-trace transfer if and only if every representative of \(Q\) does.

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 weighted zipper identities

The corresponding identities in [ 8 , 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 separation of the final labels 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 \(M\) generates MPDOs. These are unnormalized rank-one positive operators, not orthogonal projections. At \(N=0\), the closed MPO contraction instead gives \(4\mathbb {1}\), whereas the empty-word outer product is defined to be \(\mathbb {1}\). Thus the positive-length restriction is necessary.

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 \(P\) is exactly the two-site physical operator in ordered-pair coordinates.

The unique label and copy identify the vertical sector coordinate with \(I\). The corresponding permutation matrix \(W\) satisfies \(WW^\dagger =\mathbb {1}\). For every \(Z\in M_{2}(\mathbb {C})\), conjugation by \(W\) and reconstruction by its adjoint give

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 \(M\) itself is in canonical form, without a change of basis or blocking.

Put \(s=2^{-1/2}\) and \(\widehat A=sA\). The tensor \(\widehat A\) is left-canonical and is a normal tensor with transfer spectral radius one. A vertical basis of normal tensors consists of the single tensor \(M_0=\widehat A\). With one copy, take the positive diagonal matrices \(\mu _0=(s^{-1})\) and \(\chi _{0,0,0}=(s)\). The structure coefficient is \(c_{0,0,0}^{(L)}=\operatorname{tr}(\chi _{0,0,0}^L)=s^L\). For \(L\geq 1\),

 \(=s^L\) 

Each site carries \(M_0\) with its doubled physical index separated into an upper and a lower leg, so each periodic row is \(O_L(M_0)\). The weighted tensor \(\mu _0\otimes M_0\) reconstructs \(A\) exactly. For \(m_0=\operatorname{tr}(\mu _0)=s^{-1}\), the identity \(m_0=c_{0,0,0}^{(1)}m_0^2\) says that the vector \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\). At \(L=0\), however, \(O_0(M_0)=2\mathbb {1}\), so \(O_0(M_0)^2=4\mathbb {1}\ne 2\mathbb {1}\); the algebra identity therefore requires positive length.

For \(X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}\), the physical deformation \(M_X\) does not generate MPDOs.

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

If \(M_X\) generates MPDOs, then there is a real number \(\omega {\gt}0\) such that \(X^\dagger X=\omega \mathbb {1}\). Thus a physical similarity with nonscalar \(X^\dagger X\) does not preserve MPDO positivity in this family.

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

For every vertical letter \(v\), the vertical tensor of \(M_X\) satisfies \(\widetilde M_X^v=XA^vX^{-1}\). Drawing the vertical letter \(v\) as the upper leg, the physical similarity becomes a gauge on the virtual legs of \(A\):

\( = \)

Encode the operator indices of the vertical tensor \(A\) as directed edges on \(I\). For \(L\geq 1\), let \(q_L(w)\) indicate that consecutive edges in \(w\) meet cyclically. The closed operator in this direction is

Thus \(O_L(A)\) is an orthogonal projection and satisfies an algebra law with one label and structure coefficient one. It is distinct from the unnormalized GHZ rank-one operator in the original physical direction in Theorem 21.2.3.3. At \(L=0\), the vertical contraction is \(O_0(A)=2\mathbb {1}\), whereas the empty cyclic indicator gives \(\mathbb {1}\). In particular, this contraction is not idempotent at length zero, so neither the projection identity nor the coefficient-one algebra law extends to \(L=0\).

Let \(Y\in \mathrm{GL}(2,\mathbb {C})\). If \(YJY^{-1}\) and \((Y\otimes Y)P(Y^{-1}\otimes Y^{-1})\) are Hermitian, then there is a real number \(\omega {\gt}0\) such that \(Y^\dagger Y=\omega \mathbb {1}\). The comparisons in [ 13 , Appendix C.4, lines 2048–2057 ] and [ 13 , Proposition 4.13 ] motivate this finite-dimensional criterion; neither source states this model-specific theorem. Theorem 21.2.3.4 identifies \(P\) with the two-site operator of the undeformed physical model.

The physical tensor \(M\) has a vertical basis of normal tensors with one label, one copy, and bond dimension two. Its normalized representative is \(M_0=\widehat A\), its multiplicity matrix is \(\mu _0=(s^{-1})\), and \(\chi _{0,0,0}=(s)\) gives the structure coefficient \(c_{0,0,0}^{(L)}=\operatorname{tr}(\chi _{0,0,0}^L)=s^L\) at every positive length. The coisometry \(W\) identifies the vertical sector coordinates with the original physical coordinates. Both the decomposition and its reconstruction hold exactly for every vertical letter.

The vertical tensor of \(M\) is exactly the matrix-unit tensor \(A\) from Definition 21.2.2.109.

Regard the tensors \(A\) and \(B\) of Lemma 21.2.2.103 as one-block families. They are injective and have equal closed matrix product vectors at every length, but the linear extension on their virtual block is not positive.

For every triple of labels let \(Y^c_{ab}\) be an invertible \(N_{ab}^{c}\times N_{ab}^{c}\) matrix, write \(\Pi _{ab}=\bigoplus _c (Y^c_{ab})^{\mathsf T}\otimes \mathbb {1}_{D_c}\), and let \(X_{ab}\) and \(X^+_{ab}\) be the synthesis and analysis matrices whose blocks are the \(X^c_{ab,\mu }\) and the \(X^{c+}_{ab,\mu }\). Replace them by \(X_{ab}\Pi _{ab}\) and \(\Pi _{ab}^{-1}X^+_{ab}\). The transposes occur because the fusion tensors are columns of \(X_{ab}\). The new fusion tensors are

The new matrices, with the same blocks, multiplicities and simultaneous left inverse, again form a complete zipper fusion family. The matrix \(F^{abc}_d\) is the only matrix satisfying (120), and the matrix \(F'^{abc}_d\) of the new family is

Its inverse transforms as \((F'^{abc}_d)^{-1}=G_{\mathrm L}^{-1}(F^{abc}_d)^{-1}G_{\mathrm R}\). This is the gauge freedom of the fusion tensors in [ 8 , after eq. gauge ] and the induced transformation of the \(F\)-symbols in [ 199 , lines 415–422 ] .

The three-edge and two-edge composites of the forward \(F\)-matrices are equal. The corresponding composites of the inverse \(F\)-matrices are also equal. In the upper/lower convention of [ 8 , equation (33) ] , the inverse composite equality is

Every entry in (124) is an entry of the inverse of the forward matrix in (120), not the reciprocal of an individual entry.

Let \(B_a^{ij}\), \(a\in \Lambda \), be injective MPO blocks of positive bond dimension whose labelled letters have a simultaneous left inverse as in Definition 21.3.85. Suppose that for every pair \(a,b\) the product tensor \(B_{ab}\) admits a multi-block compression, as in Theorem 29.5.1, onto \(N_{ab}^{c}\) copies of each block \(B_c\), with vanishing remainder. Then the compression maps \(V_{(c,\mu )}\) and \(W_{(c,\mu )}\) are the fusion tensors \(X^c_{ab,\mu }\) and their left inverses \(X^{c+}_{ab,\mu }\) of a complete zipper fusion family. The vanishing remainder is the absence of nonzero blocks above the diagonal assumed in [ 8 , lines 181–191 ] . In particular, if for every nonempty word the word trace of every product \(B_{ab}\) is the sum of the word traces of \(N_{ab}^{c}\) copies of each \(B_c\), and \((B_{ab}^{ik})^\dagger \) lies in the unital algebra generated by the \(B_{ab}^{jl}\) for every \(i,k\), then such compressions exist and the conclusion holds.

Let \(B_1,\ldots ,B_m\) be MPO blocks of positive bond dimension which, read as tensors on the doubled physical alphabet, are normal, and no two of which are gauge equivalent up to a nonzero scalar. Then there is \(L{\gt}0\) such that the tuples \((B_1^w,\ldots ,B_m^w)\) of the words \(w\) of length \(L\) span \(\bigoplus _cM_{D_c}\), as in Definition 8.3.1, so the blocked tensors \(B_c^{[L]}\) are injective and have a simultaneous left inverse at one blocked site, as in Definition 21.3.83: there are coefficients \(C_{(c,x,y),(I,J)}\) with \(\sum _{I,J}C_{(c,x,y),(I,J)}\bigl((B_d^{[L]})^{IJ}\bigr)_{x'y'} =\delta _{cd}\delta _{xx'}\delta _{yy'}\). If every product \(B_aB_b\) admits a multi-block compression onto \(N_{ab}^c\) copies of each \(B_c\) with vanishing remainder, as in Theorem 21.3.87, then the same compression maps are the fusion tensors of a complete zipper fusion family of the blocked tensors \(B_c^{[L]}\), with no simultaneous left inverse among the hypotheses. Under the same hypotheses on the blocks, this applies in the star-closed case of Theorem 21.3.87. The simultaneous left inverse at one unblocked site, as printed in  [ 8 , line 269 ] , can fail for injective, inequivalent blocks, and the scalar in the non-equivalence cannot be dropped  [ 109 ] .

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 \(F\)-move of [ 8 , Section 3.4 ] .

Let \(M\) generate MPDOs, and suppose its doubled-physical-index tensor is in CF with normalized NT blocks. Consider the following conditions:

1. \(M\) is an RFP: there are tpCPM \(S,T\) with \(S[M_2(X)]=M_1(X)\) and \(T[M_1(X)]=M_2(X)\) for every \(X\).

2. A vertical canonical decomposition of \(M\) has positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that the \(O_L(M_\alpha )\) satisfy the product law with \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) for every \(L{\gt}0\), and the vector with entries \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) is an idempotent for the multiplication induced by \(c^{(1)}\).

3. A vertical canonical decomposition of \(M\) has active-sector fusion coisometries with exact reconstruction and positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and the vector with entries \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) is an idempotent for the multiplication induced by \(c^{(1)}\).

Then

The first equivalence matches (i)\(\iff \)(ii) of [ 13 , Theorem 4.14 ] , while the second gives the active-support form (iii\(_{\mathrm{act}}\)).

Only nonzero product corners occur in (iii\(_{\mathrm{act}}\)); a sector absent from a fixed product pair has zero fusion multiplicity. This restriction is recorded in [ 21 ] .

A fixed pair may have empty active support, and the fusion map onto the retained sectors is a coisometry. These corrections are recorded in [ 84 ] and [ 125 ] .

The implication (ii)\(\Rightarrow \)(i) uses the comparison of mixed one-site and two-site prefixes recorded in [ 28 ] .

The physical maps in (ii)\(\Rightarrow \)(i) are completed on the discarded complements as recorded in [ 159 ] .

The sixteen matrices \(Q^{ij}\) span \(M_{4}(\mathbb {C})\) and satisfy \(\sum _{i,j}(Q^{ij})^\dagger Q^{ij}=(25/32)\mathbb {1}\).

The tensor \(Q\) is simple in the sense of [ 13 , Definition 4.7 ] .

If two doubled-index tensors are related by an invertible gauge and a nonzero phase, then the physical-trace transfer of one is nilpotent if and only if that of the other is nilpotent. Identifying equal bond dimensions does not alter this condition.

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

Suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\) and that the operators \(O_L(M_\alpha )\) are linearly independent for all sufficiently large \(L\). Then, for each final label \(\varepsilon \),

Consequently a matrix between these two multiplicity spaces is square, establishing the numerical associativity statement of [ 8 , lines 237–241 ] .

Suppose the structure coefficients admit a trace-power presentation by positive diagonal matrices. They are length independent if and only if, for every triple of labels, \(c^{(2)}_{\alpha ,\beta ,\gamma }=c^{(1)}_{\alpha ,\beta ,\gamma }\) and \(c^{(3)}_{\alpha ,\beta ,\gamma }=c^{(1)}_{\alpha ,\beta ,\gamma }\).

If the support algebras are compatible with an MPO tensor \(M\) and \(E_n^\dagger X = X\) for a positive blocking size \(n\), then \(X \in \mathcal{A}_n\). Writing \((X_i)_{i \in I_n}\) for its coordinates gives \(X = \sum _{i \in I_n} X_i b_i^{(n)}\).

Fix four labels \(\alpha ,\beta ,\gamma ,\delta \) in a BNT fusion-isometry family, and write \(A=\{ 0,\ldots ,D_\alpha -1\} \), \(B=\{ 0,\ldots ,D_\beta -1\} \), \(C=\{ 0,\ldots ,D_\gamma -1\} \), and \(D=\{ 0,\ldots ,D_\delta -1\} \) 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 21.3.94.

Suppose that \(\{ M_\alpha \} _\alpha \) is a BNT and that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\). 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 MPDOs, and fix a fusion decomposition on the nonzero product sectors of a vertical canonical decomposition of \(M\). Suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L\) for all positive lengths. There are non-negative numbers \(\lambda _0,\ldots ,\lambda _{d-1}\) and one fixed Hermitian two-site matrix \(\widehat h\), all independent of the chain length. For every \(L\geq 2\), let \(\widehat H_L=\sum _{j=0}^{L-1}\widehat h_{j,j+1}\), with periodic indices. The translates of \(\widehat h\) commute pairwise, and there are orthogonal projections \(\widehat P_0^{(L)},\ldots ,\widehat P_{d-1}^{(L)}\) on the physical chain such that

This is the implication from the fusion decomposition in the proof of [ 13 , lines 999–1016 ] .

Only nonzero product sectors occur in the fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors, and no unsupported normal sector is inserted. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction of the product tensor. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

To define the operators on the full physical space, transport the retained one-site energy through the adjoint coisometry and extend it by zero energy on the orthogonal complement. Transport the retained projectors through the adjoint sitewise coisometry. This supplies the physical complement correction described in [ 140 ] .

The conclusion holds for \(L=N+2\), with \(N\in \mathbb {N}\). The definition of the local interaction in [ 13 ] uses two spins but gives no length-one convention. See [ 122 ] .

Let \(M\) generate MPDOs, fix a fusion decomposition on the nonzero product sectors, and suppose that its structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) are independent of \(L\) for all positive lengths, with \(\chi _{\alpha ,\beta ,\gamma }\) positive diagonal. 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 establishes the terminal spectral decomposition of [ 13 , lines 1003–1012 ] .

Only nonzero product sectors occur in the fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

Fix a vertical canonical decomposition of \(M\) with fusion coisometries onto the nonzero product sectors and exact reconstruction. For every \(N{\gt}0\), its vertical coisometry \(V\) and recursive density operator \(R_N\) satisfy

Only nonzero product sectors enter the fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The source fixed-pair decomposition requires a support correction: a fixed product pair may have no nonzero sectors. See [ 84 ] .

The fusion map is correspondingly a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction. See [ 125 ] .

The vertical map may also be rectangular. The reverse equality uses exact reconstruction from the retained sectors rather than a square-unitary identity. See [ 159 ] .

Let \(M\) generate MPDOs, and fix a fusion decomposition on the nonzero product sectors of a vertical canonical decomposition of \(M\). Suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L\) for all positive lengths. There are non-negative numbers \(\lambda _0,\ldots ,\lambda _{d-1}\), independent of the chain length. The two-site matrix \(h\) is Hermitian, and for every \(L\geq 2\) its periodic translates commute pairwise and satisfy

There are orthogonal projections \(P_0^{(L)},\ldots ,P_{d-1}^{(L)}\) such that

Applying the adjoint of the coisometry restricted to the retained rows at every site reconstructs \(\rho ^{(L)}(M)\) from the right-hand side of (109).

Only nonzero product sectors occur in the fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors, and no unsupported normal sector is inserted. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction of the product tensor. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

The formula holds for \(R_L\) on the retained vertical space; the extension to the physical space is analyzed in [ 140 ] .

The conclusion holds for \(L=N+2\), with \(N\in \mathbb {N}\). The definition of the local interaction in [ 13 ] uses two spins but does not state a length-one convention; see [ 122 ] .

Fusion coisometries onto the active sectors with exact reconstruction, and also fusion isometries on the full product bond space, give the operator product law with structure coefficients determined by the diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). For every chain length \(L\ge 1\),

This derives the product law of [ 13 , Theorem 4.14(ii) ] from the isometry statement of [ 13 , Theorem 4.14(iii) ] . The active-support formulation is the one used in [ 13 , 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 [ 13 , Proposition 4.13, lines 1909–1919 ] .

Let \(M\) generate matrix product density operators, have the grouped vertical product hypotheses, and satisfy the renormalization fixed-point condition. Then \(M\) has an active-sector BNT fusion tensor clause. In particular, its positive diagonal fusion matrices obey the length-one multiplicity relation \(m_\gamma =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta \).

Let \(M\) generate matrix product density operators and have grouped vertical product inputs. Suppose that its one-site and two-site vertical tensors have exact coisometric weighted reconstructions with positive multiplicities and weights, and that the two-site representatives form a basis of normal tensors. Suppose also that a sector bijection \(\sigma \) identifies the bond dimensions and that, after this identification, the unitary matrices \(V_\alpha \) satisfy \(A_{2,\sigma (\alpha )}^v= \frac{\operatorname{tr}(\mu _{1,\alpha })}{\operatorname{tr}(\mu _{2,\sigma (\alpha )})} V_\alpha A_{1,\alpha }^vV_\alpha ^\dagger \) for every vertical letter \(v\). Then 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 establishes the positive fusion decomposition of [ 13 , Appendix C.4, lines 2020–2029 ] , with active corners and coisometric fusion following [ 21 , 84 , 125 ] .

Suppose the objects in Definition 21.2.3.35 exist. For any such choice, the product law holds for every positive length \(L\) and all labels \(\alpha ,\beta \), and the scalar identity holds for every label \(\gamma \):

Thus the vector \(m=(m_\alpha )_\alpha \) is an idempotent for the multiplication induced by \(c^{(1)}\). The diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) have positive diagonal entries and satisfy \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) for every \(L{\gt}0\). Substituting this identity into (37) and (38) gives

At positive lengths, the coefficients therefore agree with those defined directly by the traces of powers of \(\chi \).

Let \(\sigma _n\) and \(\tau _n\) be the source and target label maps in the additional blocked-basis comparison at a positive length \(n\). For source indices \(i,j\) at length \(n\) and a target index \(k\) at length \(2n\), the blocked-basis coefficients satisfy

Let \(M\) generate MPDOs and satisfy the normalized BNT-refined horizontal-form hypothesis. Suppose there are tpCPM \(S,T\) on the original physical indices with \(S[M_2(X)]=M_1(X)\) and \(T[M_1(X)]=M_2(X)\) for every \(X\). Then a vertical canonical decomposition of \(M\) has active-sector fusion. Its positive diagonal fusion matrices make the vector \(m=(\operatorname{tr}(\mu _\alpha ))_\alpha \) an idempotent for the multiplication induced by \(c^{(1)}\).

The normalized BNT-refined horizontal hypothesis is stronger than CF with normalized NT blocks. See [ 177 ] .

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

Suppose that \(M_\varepsilon \) is injective. Let \(C\) be a matrix from the right final-sector space to the left 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 \(C=F\otimes 1_{D_\varepsilon }\). Here the Kronecker decomposition holds for any given intertwiner \(C\) between the fixed-final-label representations.

For every final label \(\varepsilon \), the left-associated restriction satisfies

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 21.3.42, \(U^{\mathrm L}_{\alpha ,\beta ,\gamma } (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^{\dagger }=1\).

There is a tensor \(M\) in canonical form that generates matrix product density operators, is a renormalization fixed point, and has a vertical basis-of-normal-tensors presentation with one element and structure coefficient \(c^{(L)}=(\sqrt{1/2})^L\). In particular, \(c^{(1)}\ne c^{(2)}\), so these structure coefficients for the exhibited presentation depend on the length.

Let \(M\) generate MPDOs, suppose that its MPS tensor with paired physical indices is in CF as defined by CPSV, and suppose that \(M\) satisfies the RFP condition. Select the fusion decomposition on the nonzero product sectors supplied by these three hypotheses, and assume that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) of this same decomposition are independent of \(L\) for all positive lengths.

There are non-negative numbers \(\lambda _0,\ldots ,\lambda _{d-1}\) and one fixed Hermitian two-site matrix \(\widehat h\), all independent of the chain length. For every \(L=N+2\), with \(N\in \mathbb {N}\), let \(\widehat H_L=\sum _{j=0}^{L-1}\widehat h_{j,j+1}\), with periodic indices. The translates of \(\widehat h\) commute pairwise, and there are pairwise orthogonal projections \(\widehat P_0^{(L)},\ldots ,\widehat P_{d-1}^{(L)}\) on the physical chain such that

This is the physical-space conclusion of [ 13 , lines 1013–1016 ] , under the CF and RFP hypotheses of [ 13 , Theorem 4.14, lines 972–993 ] .

Only nonzero product sectors occur in the selected fusion decomposition. A normal label absent from a fixed product pair has zero fusion multiplicity. See [ 21 ] .

The fixed-pair decomposition in Figure 11 requires a support correction: a fixed product pair may have no nonzero sectors, and no unsupported normal sector is inserted. See [ 84 ] .

The fusion map restricted to the retained rows is a coisometry onto the direct sum of nonzero sectors, and its adjoint gives exact reconstruction of the product tensor. This corrects the fusion-map convention in Figure 11. See [ 125 ] .

The retained one-site energy is transported through the adjoint coisometry and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry. This supplies the physical complement correction described in [ 140 ] .

The conclusion holds for \(L=N+2\). The definition of the local interaction in [ 13 ] uses two spins but gives no length-one convention. See [ 122 ] .

Let \(M\) generate MPDOs \(\rho ^{(N)}(M)\), and suppose that its doubled-physical-index MPS tensor is in CF, with normalized NT blocks. Suppose there are trace-preserving completely positive maps (tpCPM) \(S,T\) on the original physical indices such that \(S[M_2(X)]=M_1(X)\) and \(T[M_1(X)]=M_2(X)\) for every \(X\). Then a vertical canonical decomposition of \(M\) has active-sector fusion. Its positive diagonal fusion matrices satisfy

This is the corrected active-support formulation of implication (i)\(\Rightarrow \)(iii) in [ 13 , Theorem 4.14, lines 972–993 ] , rather than the unqualified printed statement.

For each fixed product pair, only its active normal corners occur; an absent label has zero fusion multiplicity. See [ 21 ] .

A fixed product pair may have an empty active family, so no unsupported normal sector is added. See [ 84 ] .

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. This corrects the orientation of the printed map. See [ 125 ] .

Suppose that \(M\) admits the phase-class grouping with physical isometries of Definition 16.4.92 and the grouped-corner Gram comparison of Definition 16.4.69. 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

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

Neither \(A\) nor \(B\) can occur, up to the bond-dimension identification, as a retained sector of bond dimension two in a vertical canonical decomposition of an MPDO tensor satisfying Definition 21.2.2.98.

Let \(A_c^q\in M_{D_c}(\mathbb {C})\) be a finite family of tensors. If the one-letter tuples \((A_c^q)_c\), indexed by \(q\), 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 simultaneous block left inverse used in [ 8 , lines 269–277 ] . For a BNT, the full product-algebra span is obtained after the common blocking described in [ 8 , 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 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 tensors \(\{ M_\alpha \} _\alpha \) with the given fusion isometries form a BNT and that the structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of the positive chain length. Then every pair fusion isometry is surjective: \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\). In particular, common final-label selectors and the coisometry identity follow from these hypotheses. This is a consequence of the common BNT word span in [ 13 , lines 317–345 ] , the fusion identity at lines 986–993, and length independence at lines 995–1010.

Suppose the structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), 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 [ 8 , 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 given, with \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\), and satisfy both

Then the simultaneous blocked representations imply that the vector with entries \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) is an idempotent for the multiplication induced by \(c^{(1)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })\). Thus the chosen decomposition satisfies the active-sector fusion and idempotent relations. With the stated comparison and fusion hypotheses, this is the final step of the construction in [ 13 , Appendix C.4, lines 2001–2046 ] .

Let \(M\) generate MPDOs, be in normalized BNT-refined horizontal form, and satisfy the RFP condition. Select the fusion decomposition on the nonzero product sectors supplied by that condition, and suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) of this same decomposition are independent of \(L\) for all positive lengths. Then the physical commuting Gibbs decomposition of Theorem 21.3.62 holds for every \(L=N+2\), with \(N\in \mathbb {N}\).

The retained one-site energy is transported through the adjoint coisometry and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry, supplying the physical complement correction described in [ 140 ] . The conclusion holds for \(L=N+2\) (see [ 122 ] regarding length one).

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 coordinate family \((a_i)_{i\in I_n}\) gives \(\sum _i a_i b_i^{(n)}\) 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 \(\sigma ,\tau \),

Adjunction exchanges the oriented horizontal bond pair and reverses the unmarked tail, without identifying \(A\) with \(A^\sharp \) letterwise.

For \(Z=\operatorname{diag}(1,-1)\), the transfer map satisfies \(\mathcal{E}_A(Z)=\frac{7}{25}Z\).

For the matrix unit \(E_{10}\), the transfer map satisfies \(\mathcal{E}_A(E_{10})=0\).

For the matrix unit \(E_{01}\), the transfer map satisfies \(\mathcal{E}_A(E_{01})=0\).

For the explicit four-letter tensor \(A\), the transfer map \(\mathcal{E}_A(Y)=\sum _a A^aY(A^a)^\dagger \) satisfies \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\).

Suppose that the one-site vertical tensor has the stated exact coisometric reconstruction, and that \(M\) has the grouped vertical product hypotheses. 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 \(J_j^\dagger (\widetilde{M^{(2)}})^iJ_j=\lambda _jA_j^i\). These explicit composite inclusions retain the range projections required for the sector-weight comparison.

For every final label \(\varepsilon \), the right-associated restriction satisfies

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 \(M\) has a compatible family of support algebras, then the stationary family \(\mathcal{A}_n = \operatorname{Fix}(E_1^\dagger )\) is itself compatible with \(M\): for every positive blocking size \(n\), \(\operatorname{Fix}(E_1^\dagger )=\operatorname{Fix}(E_n^\dagger )\).

Let \(M\) generate MPDOs, be in normalized BNT-refined horizontal form, and satisfy the RFP condition. Fix the fusion decomposition supplied by that construction, and suppose that its structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L\) for all positive lengths. Then \(K_\gamma \) is positive semidefinite for every terminal label, so \(\lambda _s\geq 0\). Write the positive chain length as \(N=n+1\), where \(n\in \mathbb {N}\). For every \(N{\gt}0\),

For distinct spectral indices \(s\) and \(t\), \(P_s^{(N)}P_t^{(N)}=0\). The recursive factor and multiplicity weights satisfy

Consequently,

Length independence is assumed for the coefficients of the fixed fusion decomposition supplied by the RFP construction, not for an arbitrary fusion decomposition.

The normalized BNT-refined horizontal hypothesis is stronger than CF as defined by CPSV.

This establishes the spectral decomposition of [ 13 , lines 1003–1012 ] .

Let \(M\) generate MPDOs, and choose a vertical canonical decomposition with BNT \(\{ M_\gamma \} _\gamma \) 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 [ 13 , lines 1010–1016 ] .

The trace closes the horizontal operator leg and leaves the two bond indices of the vertically read tensor \(M_\gamma \) open. See [ 116 ] .

Let \(M\) generate MPDOs and be in normalized BNT-refined horizontal form. Suppose that \(M\) is an RFP. 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 statement applies to positive lengths under the normalized BNT-refined horizontal hypothesis.

For a rectangular vertical map, (100) uses exact reconstruction from the retained nonzero sectors rather than a square-unitary identity. See [ 159 ] .

Let \(M\) generate MPDOs, be in normalized BNT-refined horizontal form, and satisfy the RFP condition. Select the fusion decomposition on the nonzero product sectors supplied by that condition, and suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) of this same decomposition are independent of \(L\) for all positive lengths. Then the commuting Gibbs decomposition on the retained vertical space in Theorem 21.3.60 holds for every \(L=N+2\), with \(N\in \mathbb {N}\).

The Hamiltonian and projectors act on the retained nonzero vertical sectors, with the physical-space extension treated in [ 140 ] . The conclusion holds for \(L=N+2\) (see [ 122 ] regarding length one).

Let \(P_\gamma \) be self-adjoint idempotents on the terminal bond space of \(M_\gamma \). Suppose that the matrices \(\chi _{\alpha ,\beta ,\gamma }\) are diagonal with strictly positive entries, and that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) are independent of \(L\) for every positive length \(L\). Then

is self-adjoint and idempotent, where \(r_{\alpha ,\beta }^{\gamma }\) is the dimension of \(\chi _{\alpha ,\beta ,\gamma }\). This is the single recursive fusion step in [ 13 , lines 999–1010 ] . Since the fusion map onto the nonzero sectors satisfies \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=\mathbb {1}\), the 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, and let \(M_{\alpha ,\boldsymbol \beta }\) denote the left-associated product tensor. The sequential fusion map \(W_N\) is a coisometry onto the retained direct sum, so \(W_NW_N^\dagger =\mathbb {1}\). For every pair of horizontal operator indices \(i,j\), it carries the product tensor to the recursive direct sum:

Consequently, after closing the horizontal operator index,

Exact reconstruction gives the reverse identities

If the diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) have strictly positive entries and give structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) independent of every positive length \(L\), then \(Q_N(P)\) is self-adjoint and idempotent whenever every terminal matrix \(P_\gamma \) is self-adjoint and idempotent.

The source embedding \(\widetilde U_N\) is \(W_N^\dagger \). The correction to the source fusion map is to use a coisometry onto the nonzero sectors; exact reverse fusion also requires reconstruction from those sectors, not merely the coisometry identity. See [ 125 ] .

This theorem fixes the initial label and the list of appended labels. In Theorem 21.3.52, the direct sum is taken over all labels and multiplicity coordinates. The commutator is a separate assertion; see [ 116 ] .

For every triple \((\alpha , \beta , \gamma )\) and every \(L \ge 0\),

where \(r_{\alpha ,\beta ,\gamma }\) is the size of \(\chi _{\alpha ,\beta ,\gamma }\) and \(\chi _{\alpha ,\beta ,\gamma ,k}\) are its 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 hypotheses and with the notation of Theorem  20.15.80, 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 \(X\), every sector \(\alpha \), and every tensor letter \(a\),

This is the coefficient identity in [ 13 , 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 that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\) and that 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 block 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 \(\varepsilon '\ne \varepsilon \), \(C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma }=0\).

Suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\) and that 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 simultaneous selector identities are additional hypotheses. The conclusion gives invertibility at a fixed final label, as obtained from the simultaneous inverse in  [ 8 , lines 247–277 ] .

Suppose that the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of \(L{\gt}0\) and that common word selectors exist. If \(\varepsilon \ne \varepsilon '\), then the block of the full comparison from the right final sector \(\varepsilon '\) to the left final sector \(\varepsilon \) vanishes: \(P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}=0\).

Suppose the structure coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), with positive diagonal \(\chi _{\alpha ,\beta ,\gamma }\), are independent of the positive chain length. Write \(r_{\alpha ,\beta }^{\gamma }\) for the dimension of \(\chi _{\alpha ,\beta ,\gamma }\). 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 an \(F\)-transformation for a fixed final label 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, but not yet invertibility of the multiplicity matrix for a fixed final label.

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 21.3.81. 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 (with opposite orientation to the row and column convention for \(F\) in [ 8 , Section 3.4 ] ).

The vertical sectors of the twisted quantum dimer of Definition 21.2.2.33 satisfy the algebra and idempotent conditions of Definition 21.2.2.98, using the decomposition of Theorem 21.2.2.62 and the structure coefficients of Theorem 21.2.2.54. The two flag sectors are normal tensors in the sense of [ 13 , lines 231–235 ] : each is injective at one site and left-canonical,

so its transfer map has spectral radius one. Together with the spanning and linear independence already proved, this gives a basis of normal tensors of the vertically viewed tensor, as in [ 13 , Proposition 4.13, lines 945–959 ] . The positive diagonal matrices \(\chi _{ff',f+f'}=\alpha \), \(\chi _{ff',f+f'+1}=\beta \) give \(c^{(L)}_{ff'g}=\operatorname{tr}(\chi _{ff'g}^L)\), and the closed operators satisfy the product law of Theorem 21.2.2.58. The multiplicity matrices are \(\mu _0=\mu _1=[\mu ]\), so \(m_0=m_1=\operatorname{tr}([\mu ])=\frac58\). The vector \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\), since

No positive rescaling of these two sectors makes the structure coefficients length independent.

For every \(N{\gt}0\), the tensor \(Q\) of Definition 21.2.2.43 satisfies

Diagrammatically, the bond state \(\sigma \) joins the right qubit of each site to the left qubit of the next one:

\(=\) 

This operator is positive semidefinite and has trace one. In particular, \(Q\) generates MPDOs. The identity excludes \(N=0\), where the empty closed operator is \(4\) and the empty bond product is \(1\).

The maps \(\mathcal T_Q\) and \(\mathcal S_Q\) of Definition 21.2.2.49 are trace-preserving completely positive maps. For every one-site operator \(Y\), \(\mathcal S_Q(\mathcal T_Q(Y))=Y\). For every virtual boundary matrix \(X\in M_{4}(\mathbb {C})\), the one-site and two-site physical closures satisfy

Thus \(Q\) is a renormalization fixed point in the sense of Definition 20.2.1.

For a bond matrix \(X\), the one-site physical closure \(M_1(X)\) of the twisted dimer has entries

The two-site closure \(M_2(X)\) vanishes unless the consecutive physical indices satisfy \(r_1=l_2\) and \(r'_1=l'_2\). When these conditions hold, its entry is the sum over a common block label \(k\) of the product of the two coefficients times \(X_{(r_2,r'_2,k),(l_1,l'_1,k)}\). The physical-trace transfer of the block \(k=0\) is \(\sum _{l,r}C_0[l,l]\, E_{(l,l),(r,r)}\), and that of the block \(k=1\) vanishes because \(\operatorname{tr}\tau _1=0\). Thus the latter block has nilpotent physical-trace transfer in the sense used by [ 13 , Definition 4.7, lines 815–822 ] .

The bond operator \(\sigma \) is positive semidefinite and has trace one. Every tensor power \(\sigma ^{\otimes N}\) has trace one, including \(N=0\). For every \(N{\gt}0\), \(D_N\) is positive semidefinite and \(\operatorname{tr}D_N=\operatorname{tr}\rho ^{(N)}(M)=1\).

Let \(\widehat M\) be the density-normalized tensor of Theorem 20.16.8.61. For every \(N\geq 0\), the flag operator satisfies \(\omega _N=\rho ^{(N)}(\widehat M)\). In particular, the flag family has an MPDO representation satisfying both RFP channel equations. For \(N{\gt}0\), \(\omega _N\) is positive semidefinite and has trace one. At \(N=0\), both \(\omega _0\) and \(\rho ^{(0)}(\widehat M)\) are the scalar \(2\).

For every positive length \(L\) and all flag values \(f,f'\in \{ 0,1\} \), the closed operators of the two rescaled flag sectors satisfy

Drawing each sector with its matrix index horizontal, as in the diagram for Definition 21.2.2.3, the product law reads, with \(g=f+f'\) and \(\bar g=f+f'+1\),

\(=\alpha ^L\)   \(+\, \beta ^L\) 

where \(\alpha =\frac{7}{10}\), \(\beta =\frac1{10}\), and the flag values are added modulo two. Thus the operators of Definition 21.2.2.55 satisfy the product law of Definition 21.2.2.3, in the form of [ 13 , Theorem 4.14(ii), lines 972–985 ] , with the structure coefficients of Theorem 21.2.2.54. These coefficients depend on \(L\), and no positive rescaling of the two sectors makes them length independent. This product law alone does not identify the sectors as a basis of normal tensors; that property is established below.

Let \(B_0\), \(B_1\) denote the two horizontal blocks of the twisted quantum dimer \(M\) of Definition 21.2.2.33, read as tensors with the doubled physical index \(((l,r,f),(l',r',f'))\) and bond dimension four, and let \(\widehat B_k=\frac85\, B_k\). The matrices of each \(\widehat B_k\) span the full algebra of \(4\times 4\) matrices, and \(\sum _{i,j}(\widehat B_k^{ij})^\dagger \widehat B_k^{ij}=\mathbb {1}\). Hence each \(\widehat B_k\) is a normal tensor with spectral radius one. The doubled-index tensor is in canonical form in the sense of Definition 7.6.7: with the permutation \(U\) of the bond index \((p,p',k)\) into the block label \(k\) followed by the block bond index \((p,p')\),

The physical-trace transfer of \(\widehat B_1\) vanishes.