27 Mixed States: General Case and Algebraic Structure
This chapter continues the mixed-state renormalization analysis of Chapter 26 with the general case of [ CPGSV16 , Section 4 ] . The source first places the tensor in vertical canonical form and then introduces its closed-chain operators. The algebraic characterization identifies the diagonal \(\chi \)-matrices governing the trace-power formula and ends in the fusion-isometry description.
27.1 Recall of the vertical canonical form and closed-chain operators
The general case begins with the vertical canonical form of [ CPGSV16 , Proposition 4.13 ] . In the notation of Definition 23.4.6, there is a coisometry \(U\) from the original physical space onto the retained nonzero sector space. Writing \(B=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \), it satisfies
where the matrices \(\mu _\alpha \) are positive and diagonal and the tensors \(\{ M_\alpha \} _\alpha \) form a basis of normal tensors. The matrix \(U^\dagger U\) is the support projection in the original physical space; the reconstruction identity retains the zero-sector complement allowed by the source. This result was established in Theorem 23.4.100.
The source next defines \(O_L(M_\alpha )\) by contracting cyclically the BNT bond indices of \(L\) copies in the vertical reading. Write the vertical physical label as a pair \((a,b)\) of horizontal bond indices and set \(\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}\). For \(\mathbf a=(a_1,\ldots ,a_L)\) and \(\mathbf b=(b_1,\ldots ,b_L)\), the contraction is
Thus \(O_L(M_\alpha )=\rho ^{(L)}(\widehat M_\alpha )\), where the trace is over the BNT bond space and the remaining row and column indices are the horizontal bond strings \(\mathbf a\) and \(\mathbf b\). These are the operators entering the algebra clause of [ CPGSV16 , Theorem 4.14 ] .
27.2 Algebra structure
A family of support \(*\)-algebras \(\{ \mathcal{A}_n\} _{n \ge 0}\), one at every blocking size, equipped with blocking maps \(m_n : \mathcal{A}_n \otimes \mathcal{A}_n \to \mathcal{A}_{2n}\) and inclusion maps \(\iota _n : \mathcal{A}_n \to \mathcal{A}_{n+1}\) realized, inside the ambient bond-space matrix algebra, by
An MPO tensor \(M\) satisfies the algebra-structure formulation used here when there exist algebra-structure data such that, for every positive blocking size \(n\),
where \(E_n\) is the blocked transfer map. This is a nontrivial algebraic condition, but it is still weaker than the full coefficient formulation of [ CPGSV16 , Theorem 4.14(ii) ] ; that formulation also includes the coefficient family \(c_{\alpha ,\beta ,\gamma }^{(L)}\).
Assume the doubled tensor is trace-preserving and the transfer map admits a positive-definite fixed point. If \(E_1^2 = E_1\), then there is a stationary family of support algebras
with multiplication \(m_n(x,y) = xy\) and inclusion \(\iota _n(x) = x\).
Under the trace-preserving normalization and the faithful fixed point, Wolf’s fixed-point-algebra theorem gives the \(*\)-subalgebra \(\operatorname{Fix}(E_1^{\dagger }) \subseteq M_{D}(\mathbb {C})\). Since \(E_1\) is idempotent, \(E_n = E_1^n = E_1\) for every \(n \ge 1\), so \(\operatorname{Fix}(E_n^{\dagger }) = \operatorname{Fix}(E_1^{\dagger })\) at every positive blocking size, giving the stationary tower \(\mathcal{A}_n = \operatorname{Fix}(E_1^{\dagger })\).
The two closure maps have the same local form:
![\begin{tenkz}[periodic, physical=updown, tensor style=box]
\tn[box, no legs]{X} & \tn[mpo]{M_\alpha}
\end{tenkz}](tenkz_svg/tenkz-be57bbdd2db015a7.svg)
\(A_\alpha (X)=\) ![\begin{tenkz}[periodic, physical=updown, tensor style=box]
\tn[box, no legs]{X} & \tn[mpo]{A_\alpha}
\end{tenkz}](tenkz_svg/tenkz-56224f9c9b02720a.svg)
In each cell, \(X\) is inserted on the virtual bond and the virtual indices are closed by \(\operatorname{tr}\); the physical legs remain open. Their spans are the fixed-point algebras compared in [ CPGSV16 , Appendix C.4, lines 1974–1980 ] .
Assume the doubled tensor is trace-preserving and the transfer map admits a positive-definite fixed point. If the transfer-map fusion criterion holds, then the MPO satisfies the algebra-structure formulation of the renormalization fixed-point condition.
Assume the doubled tensor is trace-preserving and the transfer map admits a positive-definite fixed point. If, for every positive blocked size \(n\),
then the algebra-structure formulation holds.
Take the constant tower \(\mathcal{A}_n := \operatorname{Fix}(E_1^{\dagger })\) for every \(n\). The stabilization hypothesis \(\operatorname{Fix}(E_n^{\dagger }) = \operatorname{Fix}(E_1^{\dagger })\) then gives \(\mathcal{A}_n = \operatorname{Fix}(E_n^{\dagger })\) at every positive blocking size \(n\), which is compatibility with \(M\).
This does not give the full converse direction of [ CPGSV16 , Theorem 4.14 ] : the compatibility condition above only sees stabilized adjoint fixed-point algebras, not the coefficient/BNT data. In particular, [ CPGSV16 , Theorem 4.14(ii) ] also requires the special diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). The coefficient data attached to the algebra tower are: a chosen basis of each support algebra, coordinate maps into a finite scalar family, and the corresponding multiplication / inclusion coefficient arrays.
For each blocked size \(n\), fix a basis \(\{ b_i\} _{i \in I_n}\) of \(\mathcal{A}_n\) over a finite index set \(I_n\). The blocked coefficients of \(x \in \mathcal{A}_n\) are the coordinates \((x_i)_{i \in I_n} \in \mathbb {C}^{I_n}\) with
Reconstructing a coefficient family gives the corresponding basis expansion \(\sum _i a_i b_i\) in \(\mathcal{A}_n\).
This is exactly the finite-basis reconstruction formula for the chosen basis of \(\mathcal{A}_n\).
For \(i,j \in I_n\), the blocked multiplication coefficients are the blocked coefficients \((c^{(n)}_{i,j,k})_{k \in I_{2n}}\) of \(m_n(b_i,b_j) \in \mathcal{A}_{2n}\):
In the ambient bond-space matrix algebra, \(b_i b_j = \sum _{k \in I_{2n}} c^{(n)}_{i,j,k} b_k\).
Apply the reconstruction formula of Theorem 27.2.7 to the element \(m_n(b_i,b_j) \in \mathcal{A}_{2n}\).
For \(i \in I_n\), the blocked inclusion coefficients are the blocked coefficients \((d_{ij})_{j \in I_{n+1}}\) of \(\iota _n(b_i) \in \mathcal{A}_{n+1}\):
Reconstructing the inclusion coefficients recovers the image of \(b_i\) under \(\iota _n\): \(\sum _{j \in I_{n+1}} d_{ij} b_j = \iota _n(b_i)\).
Coefficient extraction and reconstruction are mutually inverse coordinate maps for the chosen basis of \(\mathcal{A}_{n+1}\), so extracting the coefficients of \(\iota _n(b_i)\) and reconstructing returns the same element.
If the algebra tower is compatible with an MPO tensor \(M\) and \(E_n^\dagger X = X\) for a positive blocking size \(n\), then \(X \in \mathcal{A}_n\) and, writing \((X_i)_{i \in I_n}\) for its blocked coefficients, \(X = \sum _{i \in I_n} X_i b_i\).
Compatibility identifies \(\mathcal{A}_n\) with the adjoint fixed-point algebra of \(E_n\), so the fixed point first becomes an element of \(\mathcal{A}_n\) and then Theorem 27.2.7 reconstructs it.
If an MPO tensor satisfies the algebra-structure formulation, then for every positive blocking size \(n\) and every \(X\), \(E_n^{\dagger }X=X\) implies \(E_1^{\dagger }X=X\).
Compatibility at size \(n\) places \(X\) in \(\mathcal{A}_n\). The inclusion map \(\iota _n\) lifts \(X\) into \(\mathcal{A}_{n+1}\), and since \(\iota _n\) is just an inclusion it does not change the underlying operator, so we continue to denote the lifted element by \(X\). Thus compatibility at size \(n + 1\) yields \(E_{n+1}^{\dagger } X = X\). Rewriting \(E_{n+1} = E_n \circ E_1\) and applying \((A \circ B)^{\dagger } = B^{\dagger } \circ A^{\dagger }\) together with the hypothesis \(E_n^{\dagger } X = X\) extracts \(E_1^{\dagger } X = X\).
For every \(n\), \(E_1^{\dagger }X=X\) implies \(E_n^{\dagger }X=X\).
Induction on \(n\) using \(E_{n+1} = E_n \circ E_1\) and \((A \circ B)^{\dagger } = B^{\dagger } \circ A^{\dagger }\); the base case is \(E_0 = \operatorname{id}\). This step does not require the algebra-structure data.
If \(E=E_1\) and \(E^\dagger X=\lambda X\), then \(E_n^\dagger X=\lambda ^n X\) for every \(n\).
The case \(n=0\) is \(E_0=\operatorname{id}\). For the induction step, \(E_{n+1}=E_n\circ E_1\), hence
Assume the algebra-structure formulation holds. If \(X\ne 0\), \(E_1^\dagger X=\lambda X\), and \(\lambda ^n=1\) for some \(n{\gt}0\), then \(\lambda =1\).
If an MPO tensor satisfies the algebra-structure formulation, then \(\operatorname{Fix}(E_n^{\dagger })=\operatorname{Fix}(E_1^{\dagger })\) for every positive blocking size \(n\).
Under the trace-preserving normalization and a positive-definite fixed point, if \(M\) satisfies the algebra-structure formulation, then the stationary tower \(\mathcal{A}_n = \operatorname{Fix}(E_1^\dagger )\) is itself compatible with \(M\): for every positive blocking size \(n\), \(\operatorname{Fix}(E_1^\dagger )=\operatorname{Fix}(E_n^\dagger )\).
There exists an MPO tensor of physical and bond dimension two such that the doubled tensor is trace-preserving, the transfer map has a positive-definite fixed point, and the support-algebra condition holds, but the fusion condition does not hold.
Take the phase-flip channel with Kraus operators \(K_0=\frac35 I\) and \(K_1=\frac45 Z\). Its transfer map satisfies
Hence, for every \(n{\gt}0\) and \(i\neq j\), \(E^n(X)_{ij}=(-7/25)^n X_{ij}\). Since \((-7/25)^n\neq 1\) for every \(n{\gt}0\) and \(E\) is self-adjoint, \(\operatorname{Fix}((E^n)^\dagger )=\operatorname{Fix}(E^\dagger )\). Thus the support-algebra condition holds. On the other hand, \(E^2(X)_{01}=\frac{49}{625}X_{01}\neq -\frac7{25}X_{01}=E(X)_{01}\). Therefore \(E^2\neq E\), so the fusion condition fails.
Write \(E=E_1\) for the one-site transfer map. The present algebra predicate gives \(\operatorname{Fix}((E^n)^\dagger )=\operatorname{Fix}(E^\dagger )\) for every \(n{\gt}0\). Hence a vector \(X\) with \(E^\dagger X=\lambda X\) and \(\lambda ^n=1\) lies in \(\operatorname{Fix}(E^\dagger )\), so \((\lambda -1)X=0\) and \(\lambda =1\) by Theorem 27.2.16. It does not imply \(E^2=E\), as Theorem 27.2.19 shows. More generally, if \(\Pi _{\mathrm{diag}}\) is the projection onto diagonal matrices and \(0{\lt}\varepsilon {\lt}1\), set \(E(X)=\varepsilon X+(1-\varepsilon )\Pi _{\mathrm{diag}}(X)\), so that
The fixed-point algebra is diagonal for every \(n{\gt}0\), but \(E^2\ne E\).
The missing source step is the coefficient argument in [ CPGSV16 , Appendix C.4 ] . One needs the same-length product law
and the length-one idempotent trace identity
The proof of [ CPGSV16 , Theorem 4.14 ] uses (??) and (??), positivity, and the simple-matrix lemma [ CPGSV16 , lines 1155–1163 ] to give
The reconstruction maps in [ CPGSV16 , lines 2065–2085 ] then use
Thus \(\operatorname{tr}(\nu _\gamma )=m_\gamma \) and the defining equality \(m_\gamma =\operatorname{tr}(\mu _\gamma )\) normalize the factors \(\nu _\gamma /m_\gamma \) and \(\mu _\gamma /m_\gamma \), respectively, in \(T=\widetilde R_2\widetilde T R_1\) and \(S=\widetilde R_1\widetilde S R_2\). The scalar Newton–Girard power-sum identity needed to obtain the trace-power form is already available as Theorem 10.5.1. To obtain the trace-power conclusion of [ CPGSV16 , Theorem 4.14(ii) ] , one must impose the BNT-label hypotheses as a separate condition, relate them to the one-site and two-site vertical canonical forms, and construct the maps in (??). Theorem 27.2.19 shows that the support-algebra condition alone does not imply the BNT-label hypotheses.
27.2.1 Diagonal \(\chi \)-matrices and the trace-power formula
The following statements isolate the form of [ CPGSV16 , Theorem 4.14(ii) ] : the structure coefficients of the length-\(L\) operator algebra are trace-powers of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). They state the coefficient identity and the elementary trace-power identity for diagonal matrices.
A family of diagonal complex matrices, indexed by ordered triples \((\alpha ,\beta ,\gamma )\) drawn from a common index type, consisting of a size \(r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\) and diagonal entries \(\chi _{\alpha ,\beta ,\gamma ,1}, \ldots , \chi _{\alpha ,\beta ,\gamma ,r_{\alpha ,\beta ,\gamma }} \in \mathbb {C}\) for each triple, giving the diagonal matrix
This is the matrix \(\chi _{\alpha ,\beta ,\gamma }\) of [ CPGSV16 , Theorem 4.14(ii) ] .
For every triple \((\alpha , \beta , \gamma )\), if \(\chi _{\alpha ,\beta ,\gamma }\) has size \(r_{\alpha ,\beta ,\gamma }\), then for every \(L \ge 0\),
where \(\chi _{\alpha ,\beta ,\gamma ,k}\) are the diagonal entries.
Diagonal matrices raise to powers entrywise and the trace of a diagonal matrix is the sum of its entries.
A binary compatibility predicate between an abstract structure-coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\) and a diagonal \(\chi \) family: the pair \((c, \chi )\) is said to be in trace-power form when
for every \(L \ge 0\) and every triple \((\alpha , \beta , \gamma )\), where \(r_{\alpha ,\beta ,\gamma }\) is the size of \(\chi _{\alpha ,\beta ,\gamma }\). This is the formal analog of the target identity of [ CPGSV16 , Theorem 4.14(ii) ] ; the existential version—“there exists \(\chi \) for which \((c,\chi )\) has trace-power form”—is obtained by quantifying over \(\chi \).
27.2.2 The BNT-label algebra law and idempotent trace vector
The same-length coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) from [ CPGSV16 , Theorem 4.14(ii) ] , indexed by fixed BNT labels \(\alpha ,\beta ,\gamma \). The coefficient is attached to the length-\(L\) product \(O_L(M_\alpha )O_L(M_\beta )\), not to the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\). A diagonal \(\chi \)-family canonically determines the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) = \sum _r\chi _{\alpha ,\beta ,\gamma ,r}^{\, L}\), as in [ CPGSV16 , Appendix C.4 ] .
A BNT-label operator family records, for each chain length \(L\), the operators \(O_L(M_\alpha )\) indexed by the fixed BNT labels \(\alpha \). The ambient algebra may depend on \(L\) and is not identified with a chosen blocked support algebra. Write \(\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}\) for the vertical reading. The vertical indices of the \(L\) copies are contracted cyclically, while the horizontal legs remain open, as in [ CPGSV16 , lines 962–967 ] :
A BNT-label operator family and a BNT-label coefficient system have same-length product form when, for every positive length \(L\),
This is the product identity appearing in [ CPGSV16 , Theorem 4.14(ii) ] , stated independently of the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\).
A BNT-label trace-scalar family records the scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) indexed by the fixed BNT labels \(\alpha \), as in the idempotent condition of [ CPGSV16 , Theorem 4.14(ii) ] .
A BNT-label trace-scalar family and a BNT-label coefficient system have idempotent coefficient form when, for every BNT label \(\gamma \),
This is the idempotent condition in [ CPGSV16 , Theorem 4.14(ii) ] , stated separately from the trace-power formula.
The BNT-label assignment consists, for every positive blocked length \(n\), of a source label map \(\sigma _n\) from the chosen basis of \(\mathcal A_n\) and a target label map \(\tau _n\) from the chosen basis of \(\mathcal A_{2n}\) to the fixed BNT labels. It also gives the combined label map \(\sigma _n \sqcup \tau _n\) on the disjoint union of these two blocked bases.
A comparison between the BNT-label coefficients and the chosen blocked-basis coefficients consists of a BNT-label assignment together with the equality
This is a blocked-basis comparison: the left hand side is expanded in the chosen basis of \(\mathcal A_{2n}\), so the statement is not itself the same-length BNT product law.
A BNT-label coefficient system and a diagonal \(\chi \)-family are compatible when, for every positive length \(L\),
The same matrix \(\chi _{\alpha ,\beta ,\gamma }\) is used for all positive \(L\).
Under positive-length BNT-label trace-power form, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).
Combine the defining positive-length trace-power identity with the trace formula for powers of a diagonal matrix.
The coefficient family canonically determined by a diagonal \(\chi \)-family satisfies the positive-length trace-power condition with respect to the same \(\chi \)-family.
This follows from the definition of the canonical coefficient family.
A positive BNT-label \(\chi \) witness consists of a length-independent diagonal \(\chi _{\alpha ,\beta ,\gamma }\)-family, positivity of every diagonal entry, and the positive-length trace-power identity for the BNT-label coefficients. This is the coefficient statement of [ CPGSV16 , Theorem 4.14(ii) ] . If the coefficient family is chosen canonically from the same \(\chi \)-family, positivity of the diagonal entries alone gives such a witness.
A positive BNT-label \(\chi \) witness gives, for every \(L{\gt}0\), the formula \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).
Apply the trace reformulation of the positive-length trace-power identity from the witness.
If the same-length BNT product identity holds and the BNT-label coefficients come from a positive length-independent \(\chi \)-witness, then for every \(L{\gt}0\),
Substitute the positive-length trace-power formula into the same-length BNT product expansion.
If the same-length BNT product identity holds with the coefficient family canonically determined by a diagonal \(\chi \)-family, then for every \(L{\gt}0\),
Substitute the canonical trace-power formula for the coefficients into the same-length BNT product expansion.
If the BNT trace scalars satisfy the idempotent coefficient condition and the BNT-label coefficients come from a positive length-independent \(\chi \)-witness, then
Substitute the length-one trace-power formula into the idempotent coefficient identity.
If the BNT trace scalars satisfy the idempotent coefficient condition with the coefficient family canonically determined by a diagonal \(\chi \)-family, then
Substitute the length-one canonical trace formula into the idempotent coefficient identity.
If the chosen blocked-basis coefficients are compared with the fixed BNT-label coefficient system, and that BNT-label system has a positive length-independent \(\chi \)-witness, then each positive-length blocked coefficient is the trace power of the corresponding BNT-label \(\chi \)-matrix:
Use the comparison equality to replace the blocked-basis coefficient by the corresponding BNT-label coefficient, and then apply the BNT-label trace-power formula. In the special case where the compared BNT-label coefficient family is already the canonical family determined by \(\chi \), the same conclusion follows directly from the canonical trace formula.
For each positive blocked size \(n\) and multiplication triple \((i,j,k)\), where \(i\) and \(j\) label basis elements of \(\mathcal{A}_n\) and \(k\) labels a basis element of \(\mathcal{A}_{2n}\), a size \(r_{n,i,j,k} \in \mathbb {N}\) and diagonal entries \(\chi _{n,i,j,k,1}, \ldots , \chi _{n,i,j,k,r_{n,i,j,k}} \in \mathbb {C}\), giving the diagonal matrix
Unlike the uniform BNT-label family in [ CPGSV16 , Theorem 4.14(ii) ] , the size and diagonal entries may depend on the blocked length and on the chosen basis indices.
For each positive blocked size \(n\), each multiplication triple \((i,j,k)\), and every exponent \(L\), \(\operatorname{tr}(\chi _{n,i,j,k}^{\, L}) = \sum _r \chi _{n,i,j,k,r}^{\, L}\).
Diagonal matrices raise to powers entrywise and the trace of a diagonal matrix is the sum of its entries.
The blocked multiplication coefficients are in trace-power form when, for every positive blocked size \(n\) and every multiplication triple \((i,j,k)\), the coefficient \(c^{(n)}_{i,j,k}\) equals \(\sum _r \chi _{n,i,j,k,r}^{\, n}\). Equivalently, by Lemma 27.2.2.19, it is the trace of the \(n\)-th power of the corresponding diagonal matrix. The exponent \(n\) is the source blocking length of the two factors in \(\mathcal{A}_n\); the product is expanded in the basis of \(\mathcal{A}_{2n}\).
Under trace-power form for the blocked multiplication coefficients, for every positive blocked size \(n\), \(c^{(n)}_{i,j,k}=\operatorname{tr}(\chi _{n,i,j,k}^{\, n})\).
Combine the defining trace-power identity with the trace formula for powers of a diagonal matrix.
A positive blocked \(\chi \) trace-power witness consists of a blocked \(\chi \) family, positivity of all its diagonal entries, and the positive-length trace-power identity for the blocked multiplication coefficients. This is the blocked-basis analog of the positive diagonal matrices in [ CPGSV16 , Theorem 4.14(ii) ] , without requiring a single family indexed only by the BNT labels.
Suppose that the chosen blocked-basis coefficients are compared with a fixed BNT-label coefficient system, and that this BNT-label system has a positive length-independent \(\chi \)-witness. Then the chosen blocked bases carry a positive blocked \(\chi \) trace-power witness. The construction is obtained by pulling back the BNT-label \(\chi _{\alpha ,\beta ,\gamma }\)-matrices along the source and target label maps.
Reindex the positive BNT-label \(\chi \)-family along the blocked-basis label map. Positivity is preserved by reindexing, and the trace-power identity follows by substituting the blocked-basis comparison into the BNT-label trace-power formula.
Let \(x_1,\ldots ,x_n\) be non-negative real numbers whose power sums \(\sum _k x_k^L\) take the same value at every exponent \(L \ge 1\). Then every \(x_k\) equals \(0\) or \(1\). The same holds for a finite family of complex numbers that are non-negative in the complex order. Nonnegativity cannot be dropped: the primitive cube roots of unity have power sums invariant under doubling the exponent, yet do not lie in \(\{ 0,1\} \).
If some entry exceeded one, its powers would grow without bound while all other summands stay non-negative, contradicting constancy of the power sums; hence every entry lies in \([0,1]\). Comparing the exponents one and two gives \(\sum _k x_k(1-x_k) = 0\) with non-negative summands, so each summand vanishes.
A BNT-label coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) is length independent when \(c^{(L)}_{\alpha ,\beta ,\gamma } = c^{(1)}_{\alpha ,\beta ,\gamma }\) for every positive chain length \(L\) and all labels; equivalently, the coefficients agree at any two positive lengths. This is the case singled out in the discussion following [ CPGSV16 , Theorem 4.14 ] .
Suppose a BNT-label coefficient system is in trace-power form \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and is length independent. Then every diagonal entry \(\chi _{\alpha ,\beta ,\gamma ,r}\) lies in \(\{ 0,1\} \), as stated in [ CPGSV16 , Section 4.5 ] ; by positivity every entry equals one. Consequently, at every positive length, \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\).
For fixed labels, the trace-power form turns length independence into constancy of the power sums of the diagonal entries of \(\chi _{\alpha ,\beta ,\gamma }\), so by Lemma 27.2.2.24 every entry equals \(0\) or \(1\), and positivity excludes \(0\). The trace of \(\chi _{\alpha ,\beta ,\gamma }^{\, L}\) is then the number of diagonal entries.
If a BNT-label coefficient system is in trace-power form with respect to a diagonal \(\chi \)-family all of whose entries equal one, then for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } = r_{\alpha ,\beta ,\gamma }\), so the coefficient system is length independent.
The power sums of a family of ones do not depend on the exponent.
Starting from the same-length product law and the idempotent condition, in the particular case of natural-number structure coefficients independent of the length, one recovers all known non-chiral topologically ordered phases together with a description of their excitations [ BMW\(^{+}\)17 ] . The renormalization fixed-point density operators correspond to the boundary theories of the renormalization fixed-point representatives of those phases, which yields a large family of non-trivial examples. Whether there exist renormalization fixed points whose structure coefficients depend on the length is left open in [ CPGSV16 , Section 4.5 ] .
A BNT-label operator family is linearly independent at length \(L\) when the operators \(O_L(M_\alpha )\) are linearly independent, and eventually linearly independent when this holds for every length beyond some threshold. Eventual linear independence is the operator-level form of the third defining property of a basis of normal tensors in [ CPGSV16 , Section 2.3 ] , invoked for the labelled operators in [ CPGSV16 , Appendix C.4 ] .
Two coefficient systems satisfying the same-length product law for the same operator family agree at every positive length at which the operators are linearly independent.
Applying the same-length product law to both coefficient systems gives
Subtracting yields \(\sum _\delta (c^{(L)}_{\alpha ,\beta ,\delta } - c'^{(L)}_{\alpha ,\beta ,\delta })O_L(M_\delta ) = 0\), and linear independence forces every coefficient of the difference to zero.
At every positive length \(L\) at which the operators are linearly independent, the structure coefficients of the same-length product law satisfy
These are the restrictions that the associativity of the multiplication imposes on the structure coefficients, noted in [ CPGSV16 , Section 4.5 ] .
Applying the same-length product law twice in each order gives
Associativity of multiplication equates the two left-hand sides, so the right-hand sides agree; linear independence of the operators then forces the corresponding coefficient sums to agree for each \(\varepsilon '\).
For a coefficient system satisfying the same-length product law for a BNT-label operator family and in trace-power form, at every positive length \(L\) at which the labelled operators are linearly independent, the associativity constraint is a restriction on the family of diagonal matrices:
In [ CPGSV16 , Section 4.5 ] these restrictions, combined with the isometry statement, lead to a pentagon-like equation for the fusion isometries whose solutions come from unitary fusion categories.
Substitute the trace-power form of the coefficients into the associativity constraint.
Fix a coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\), BNT-label operators \(O_L(M_\alpha )\), and trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). They satisfy the BNT algebra clause when there is a length-independent family of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), the same-length product law holds,
and the trace scalars satisfy
This is the algebra statement in [ CPGSV16 , Theorem 4.14(ii), labelled IV.13 in the source, lines 972–985 ] , before comparison with a chosen blocked basis.
Choose a vertical canonical decomposition
Separating the two factors of the doubled physical index of \(M_\alpha \) gives an MPO tensor \(\widehat M_\alpha \). Define
The chosen decomposition satisfies the tensor-attached BNT algebra clause when the tensors \(M_\alpha \) form a basis of normal tensors and the BNT algebra clause holds for precisely these operators and these trace scalars. This is the data in [ CPGSV16 , Theorem 4.14(ii), lines 972–993, and Appendix C.4, lines 2046–2064 ] ; it does not include a comparison with a two-site vertical canonical decomposition.
Fix the one-site diagonal multiplicity matrices \(\mu _\alpha \) and the two-site diagonal multiplicity matrices \(\nu _\gamma \) in the respective vertical canonical decompositions. A multiplicity-spectrum comparison records \(\operatorname{tr}(\mu _\alpha )=m_\alpha \) and, for every \(\gamma \), equality with multiplicity between the entries of \(\nu _\gamma \) and the products \(\mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k}\). This is the conclusion of the vertical BNT comparison and positive power-sum argument in [ CPGSV16 , Appendix C.4, lines 2048–2058 ] . The definition does not require this comparison to follow from a BNT algebra clause; constructing it from the MPDO tensor is a separate obligation.
For a tensor-attached BNT algebra clause, a source-derived two-site multiplicity spectrum consists of a vertical canonical decomposition
a bijection \(\sigma \) from the one-site sectors to its sectors such that \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length, and, for every \(\gamma \), the multiset equality
This is the decomposition and comparison retained in [ CPGSV16 , Appendix C.4, lines 2046–2058 ] .
The invertible-gauge sub-result for a source-derived two-site multiplicity spectrum additionally records, for every one-site sector \(\gamma \), equality of bond dimensions \(d_\gamma =d_{\sigma (\gamma )}\) and an invertible matrix \(Z_\gamma \in \mathrm{GL}(d_\gamma ,\mathbb {C})\) such that, for every physical index \(i\),
Thus the matched normal tensors have no residual scalar phase. This is the exact invertible conjugacy retained from [ CPGSV16 , Appendix C.4, lines 2053–2057 ] . It does not include the further conclusion on line 2057 that \(Z_\gamma \) may be chosen unitary.
Let \(M\) be an MPDO in normalized BNT-refined horizontal form, equipped with a tensor-attached BNT algebra clause \(\widetilde M=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \). There is a vertical canonical decomposition of the two-site blocking,
Its normal-tensor sectors are in bijection, via \(\sigma \), with the sectors of the full-support one-site product expansion such that \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length and, for every \(\gamma \),
as multisets. Thus this two-site decomposition and relabelling realize the multiplicity-spectrum comparison of Appendix C.4, lines 2046–2058 of [ CPGSV16 ] . Moreover, the paired bond dimensions agree and there are invertible matrices \(Z_\gamma \) satisfying
exactly. This is only the invertible-gauge sub-result of lines 2053–2057; no unitary choice of \(Z_\gamma \) follows from this statement.
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
Multiplying the two one-site decompositions and using the BNT algebra law expresses the two-site closed chain in the basis \(M_\gamma \), with coefficient
All these coefficients are positive. Hence every basis sector occurs, and the characterization of bases of normal tensors matches these sectors bijectively with those of the two-site vertical canonical decomposition. Normalization makes each matching scalar have modulus one, while positivity of two consecutive coefficients makes it positive; it is therefore one. The normal-tensor matching theorem also supplies equality of the paired bond dimensions and an invertible gauge; after the scalar is one, its gauge-phase relation is precisely the exact conjugacy in (??). Eventual linear independence of the basis vectors gives the corresponding equality of power sums. The finite positive power-sum identity then gives the multiset equality (??).
The length-one idempotent law of a BNT algebra clause also holds for the coefficient family determined by its diagonal chi matrices:
The chi representation at length one gives \(c^{(1)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })\). Substitution into the idempotent law \(m_\gamma =\sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta \) gives the stated identity.
Suppose that every entry of the one-site multiplicity matrices, every entry of the two-site multiplicity matrices, and every entry of the matrices \(\chi _{\alpha ,\beta ,\gamma }\) is positive. Suppose also that \(\sum _i\mu _{\alpha ,i}=m_\alpha \) for every \(\alpha \). If, for every \(\gamma \) and all sufficiently large \(L\),
then the entries on the two sides agree as multisets. Together with the one-site trace identity \(\sum _i\mu _{\alpha ,i}=m_\alpha \), the one-site and two-site multiplicity entries and their dimensions define a BNT multiplicity-spectrum comparison.
Positivity makes all entries nonzero. Eventual equality of two finite sums of geometric sequences gives, for every \(L\geq 1\),
The finite power-sum identity determines both the cardinality and the multiset of entries.
Suppose that the BNT algebra clause holds and that the one-site and two-site vertical multiplicity spectra have been compared as above. Then every two-site multiplicity matrix has the prescribed trace:
This is the normalization used for the preparation matrix \(\nu _\gamma /m_\gamma \) in the algebra-to-RFP implication of [ CPGSV16 , Appendix C.4, lines 2058–2064 ] .
Sum the multiset equality of the two spectra. For each \(\alpha ,\beta ,\gamma \), the resulting product sum satisfies
The identities
therefore give
where the final equality is the length-one idempotent law.
The BNT-label theorem data consist of a BNT algebra clause together with the comparison with the chosen blocked bases. Explicitly, for every \(L{\gt}0\) the product law is
while the idempotent scalar law is
For every positive blocked length \(n\) and blocked-basis indices \(i,j,k\), the source and target label maps \(\sigma _n,\tau _n\) give the comparison
They also determine the pulled-back blocked-basis \(\chi \)-family. These data are the source-side hypotheses from which the blocked-basis consequences below are derived. In the source-side special case where the coefficient family is canonically determined by the same \(\chi \)-family, theorem data are built from that \(\chi \)-family, its positivity, and the product, idempotent, and blocked-comparison statements for the resulting canonical coefficients. The same product and idempotent predicates may also be rephrased using the canonical coefficient family determined by the \(\chi \)-matrices carried by the data, with the corresponding displayed equations.
Let \(c^{(L)}_{\alpha ,\beta ,\gamma }\) be the coefficient family and let \(\chi _{\alpha ,\beta ,\gamma }\) be the positive diagonal matrices carried by the theorem data. For every \(L{\gt}0\),
where \(c^{\chi }\) is the canonical coefficient family determined by \(\chi \), and every diagonal entry of every \(\chi _{\alpha ,\beta ,\gamma }\) is strictly positive. Consequently, for every \(L{\gt}0\), the same-length product identity reads
and, at \(L=1\), the idempotent identity reads
The positive trace-power witness gives \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\). Substitution into the product law and its length-one idempotent specialization gives the stated formulas.
BNT-label theorem data give a positive blocked-basis \(\chi \) trace-power witness by pulling the length-independent BNT-label \(\chi \)-family back along the blocked-basis comparison maps: writing \(\chi _{n,i,j,k}\) for the resulting blocked-basis diagonal matrix at a positive blocked length \(n\), \(\chi _{n,i,j,k} =\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}\). Thus the two matrices have equal size, diagonal entries, and traces of all powers.
Apply the BNT-to-blocked witness construction to the comparison and the positive BNT-label \(\chi \)-witness belonging to the theorem data.
For every positive blocked length \(n\), the blocked-basis coefficients are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target label maps. Equivalently, they are traces of powers of the pulled-back blocked-basis matrices \(\chi _{n,i,j,k}\):
The blocked-basis comparison identifies the coefficient with the trace power of the BNT-label matrix selected by the source and target maps. By definition, the pulled-back matrix \(\chi _{n,i,j,k}\) is this same matrix.
An existential BNT-label theorem witness consists of the finite BNT-label type, the same-length operator spaces, their algebraic structure, and the corresponding BNT-label theorem data, including the pulled-back blocked-basis \(\chi \)-family. The construction of such a witness from an MPDO tensor is the outstanding step toward [ CPGSV16 , Theorem 4.14(ii) ] . The proposition-level form is the nonemptiness of this witness type. In the source-side special case where the coefficients are canonically determined by a \(\chi \)-family, the witness is obtained from that same \(\chi \)-family together with the remaining product, idempotent, and blocked-comparison statements. For any such witness, the product and idempotent predicates can be rephrased with the canonical coefficient family determined by its \(\chi \)-matrices, with the corresponding displayed equations.
A proposition-level BNT-label theorem witness contains the source-side same-length product law, which holds for every positive length \(L\), and the idempotent scalar law,
A witness also carries the positive-length trace-power law \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})\) and positivity of the diagonal \(\chi _{\alpha ,\beta ,\gamma }\)-entries. Substituting the trace-power law into (??) and (??) rewrites them with the coefficients as \(\chi \)-traces,
At every positive length, the BNT-label coefficient family also agrees with the canonical coefficient family determined by these \(\chi \)-matrices. Hence both families are represented by the \(\chi \)-trace formulas (??) and (??). If \(\sigma _n\) and \(\tau _n\) are the source and target label maps for a positive blocked length \(n\), then the blocked-basis coefficients satisfy
Unpack the existential witness and apply the corresponding witness-level equations.
An existential BNT-label theorem witness gives a positive blocked-basis \(\chi \) trace-power witness for the chosen algebra-structure data: a blocked \(\chi \)-family with positive diagonal entries such that, for every positive blocked length \(n\),
The blocked-basis \(\chi \)-family is the pullback of the BNT-label \(\chi \)-family along the source and target comparison maps \(\sigma _n,\tau _n\). The proposition-level nonempty form of the witness gives the same existential blocked-basis family, stated either through this trace-power predicate or through the coefficient equation it unfolds to.
Apply the corresponding theorem-data consequence carried by the witness.
For every positive blocked length \(n\), the blocked-basis coefficients obtained from an existential BNT-label theorem witness are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target maps. Equivalently, they are traces of powers of the pulled-back blocked-basis \(\chi \)-matrices:
Apply the trace-power identity of the positive blocked-basis \(\chi \)-witness obtained from the existential witness.
27.3 Fusion isometries
The next definitions isolate the transfer-map content underlying the fusion-isometry picture of [ CPGSV16 , Section 4.5 ] : they record when the blocked transfer map factors through its support algebra as a retract, which is the idempotence criterion, not the physical fusion isometry of two tensors into one. For each blocked size \(n \ge 1\), the doubled-index blocked tensor of an MPO has a blocked transfer map \(E_n\) acting on bond-space matrices, and the support algebra is modeled by a subspace through which \(E_n\) factors as a retract. The physical fusion isometries of [ CPGSV16 , Theorem 4.14(iii) ] are recorded at the end of this section, together with the derivation of the same-length product law from them.
A transfer-map fusion-isometry datum at blocked size \(n\) for an MPO tensor consists of a subspace \(\mathcal{A}_n \subseteq M_{D}(\mathbb {C})\) together with linear maps
where \(E_n\) denotes the blocked transfer map.
An MPO tensor \(M\) satisfies the fusion-isometry formulation of the renormalization fixed-point condition when for every blocked size \(n \ge 1\) there exists transfer-map fusion-isometry data for the blocked transfer map \(E_n\).
A one-site fusion-isometry datum implies transfer-map idempotence. Hence the fusion-isometry formulation, which supplies such data at every positive blocked size, implies the same condition.
Apply the definition at blocked size \(n=1\). By (??),
Since \(E_1\) is the original transfer map, this is exactly transfer-map idempotence.
A one-site transfer-map fusion-isometry datum exists if and only if the MPO transfer map is idempotent.
The forward implication is the retract calculation \(E_1^2=S_1T_1S_1T_1=S_1T_1\). Conversely, if \(E_1\) is idempotent, then it factors through its range, giving the required one-site datum.
The fusion-isometry formulation is equivalent to transfer-map idempotence.
The forward implication is Theorem 27.3.3. Conversely, if the transfer map \(E\) is idempotent, then every positive blocked transfer map equals \(E\) and therefore factors through its range. This range-factorization gives the required fusion-isometry data.
For the diagonal MPO associated to a pure MPS tensor, the fusion-isometry formulation reduces to the usual pure-state RFP condition.
Combine Theorem 27.3.5 with the diagonal pure-state recovery theorem for the MPO predicate.
The fusion-isometry formulation for all positive blocked sizes is equivalent to the existence of a one-site transfer-map fusion-isometry datum.
Combine the one-site criterion with the equivalence between the all-blocked fusion formulation and transfer-map idempotence.
The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:
In the vertical reading of [ CPGSV16 , Section 4.5 ] this is the tensor obtained by joining two vertically read tensors along the original horizontal bond, the tensor written \(M_\alpha M_\beta \) in [ CPGSV16 , Theorem 4.14(iii) ] .
For three bond spaces, let
be the coordinate pullback induced by canonical reassociation, defined on elementary tensors by \(A_{D_1,D_2,D_3}(x_1\otimes (x_2\otimes x_3)) =(x_1\otimes x_2)\otimes x_3\).
The letters of the two parenthesizations of a triple product are related by the canonical bond reassociation. Equivalently, for every pair of physical indices \(i,k\),
Expand both product tensors. Reassociation preserves every elementary tensor factor, while exchanging the order of the two finite sums gives the same contraction over the two intermediate physical indices.
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:
Induction on the length, using the mixed-product property of the tensor product to merge one letter at a time.
For every chain length \(L\), the operator generated by the product tensor is the product of the generated operators: \(O_L(M\mathbin {\cdot }N)=O_L(M)O_L(N)\).
Take traces in (??): the trace of a tensor product is the product of the traces, and the sum over middle configurations is the entrywise form of the operator product.
For an isometry \(U\) (a rectangular matrix with \(U^\dagger U=1\)) and a nonempty word of square matrices \(F_1,\ldots ,F_L\),
The inner factors \(U^\dagger U\) telescope to the identity.
For block-diagonal letters whose block of index \(\gamma \) is the tensor product of a fixed matrix \(X_\gamma \) with a varying letter \(G_{\gamma ,l}\),
Blocks multiply blockwise and tensor products multiply factorwise, so the fixed factors accumulate into the matrix power \(X_\gamma ^L\).
Let \(F=\bigoplus _a F_a\) and \(G=\bigoplus _a G_a\). Then \(FGF^\dagger =\bigoplus _a F_aG_aF_a^\dagger \). Moreover, if \(r\) is a bijective reindexing of the row space of a matrix \(A\), then \(A_rXA_r^\dagger =(AXA^\dagger )_{r,r}\), where \(A_r\) denotes the corresponding row reindexing.
The first identity follows because block-diagonal matrices multiply blockwise. The second follows by commuting bijective row reindexing with multiplication and adjunction.
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 \(F\).
Evaluate the amplified identity on one basis vector of the nonzero factor. Multiplicativity of the Kronecker product gives the product and adjoint specializations.
A fusion-isometry family over a finite set of labels consists of a family of tensors \(M_\gamma \) with a common physical dimension and per-label bond dimensions, diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) with positive entries, and isometries \(U_{\alpha ,\beta }\) such that, site by site,
This is the isometry statement of [ CPGSV16 , Theorem 4.14(iii) ] ; the tensors are the vertically read basis of normal tensors of [ CPGSV16 , Proposition 4.13 ] , so the physical dimension is the bond dimension of the original tensor. Statement (iii) of the source theorem additionally asserts the idempotent identity for the trace scalars of the vertical decomposition, recorded separately by Definition 27.2.2.5.
Scope restriction (full-support fusion family): The displayed family assumes \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) on the whole product bond space. The unrestricted source statement permits a common zero corner and instead gives a coisometry onto the active direct sum, together with exact reconstruction. This distinction is documented in docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
An active-support fusion coisometry family consists of the same labelled tensors and positive diagonal matrices, together with maps \(U_{\alpha ,\beta }\) satisfying \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), the forward fusion identity
and exact reconstruction
Zero-dimensional retained sums are permitted. If in addition \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) for every pair, this family gives the preceding full-support fusion-isometry family.
Local fix (Figure-11 fusion coisometry): The source uses the retained-row orientation of Proposition 4.13. Thus its fusion maps are coisometries onto the active sectors; exact reconstruction records the discarded common zero corner. This convention is documented in docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
For fixed labels \(\alpha ,\beta \), put \(G_{\alpha ,\beta }^{ij}=\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}\). Then [ CPGSV16 , Theorem 4.14(iii), equation (89), lines 986–991 ] gives
![\tnpic[rows={op,op}]{
\tnfuse[combined=west]{U_{\alpha,\beta}} &
\tn[mpo, up=$i$]{M_\alpha} &
\tnfuse[combined=east]{U_{\alpha,\beta}^{\dagger}} \\
& \tn[mpo, down=$j$]{M_\beta} &
}](tenkz_svg/tenkz-58117702415fe904.svg)
\(=\) ![\tnpic[physical=updown]{
\tn[mpo, up=$i$, down=$j$]{G_{\alpha,\beta}}
}](tenkz_svg/tenkz-069fb9d80e0952f5.svg)
An active-support fusion coisometry satisfies
Substitute exact reconstruction (??) into the left-hand sides and use \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\).
For every pair of labels and physical indices, write \(G_{\alpha ,\beta }^{ij} =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma } \otimes M_\gamma ^{ij}\). The fusion isometry satisfies the two chi-weighted zipper identities
The corresponding identities in [ BMW\(^{+}\)17 , Equation (21) ] have identity matrices on the multiplicity spaces in place of the positive diagonal matrices \(\chi \). Their unweighted form requires the length-independent integer specialization and is not a consequence of the present hypotheses. Even after that specialization, comparison of the two triple-fusion orders and final-label separation are still needed to construct an invertible \(F\)-move.
Multiply the fusion identity (??) on the right by \(U_{\alpha ,\beta }\) and on the left by \(U_{\alpha ,\beta }^\dagger \), respectively, and use \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\).
Suppose the structure coefficients have positive trace-power form and are independent of the positive chain length. Then \(\chi _{\alpha ,\beta ,\gamma }=1_{r_{\alpha ,\beta ,\gamma }}\) for every triple of labels. If
the fusion isometries satisfy
These are the identity-weight specialization of Equations (19)–(21) in [ BMW\(^{+}\)17 , Section 3 ] . The MPDO fusion identity appears in [ CPGSV16 , Theorem 4.14(iii), lines 986–991 ] ; its identity-weight specialization follows from the length-independent case at [ CPGSV16 , line 1010 ] .
Length independence makes every positive diagonal entry of \(\chi _{\alpha ,\beta ,\gamma }\) equal to one. Substitute the resulting identity matrices into Theorem 27.3.20. For completeness, insert \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) on both sides of the product letter and apply the fusion identity between the two inserted factors, yielding (??).
Suppose that the labelled doubled-index tensors form the basis of normal tensors, and that their fusion-isometry family has positive trace-power coefficients independent of the positive chain length. Then there is one integer \(L{\gt}0\) such that the blocked tensors \(M_\gamma ^{[L]}\) span their full product matrix algebra in one letter. Hence every \(M_\gamma ^{[L]}\) is injective.
For this same \(L\), write
where \(r_{\alpha ,\beta ,\gamma }\) is the dimension of the specialized chi space. The original fusion isometry satisfies
the two blocked zipper identities
and the literal reconstruction
Choose the positive block length supplied by simultaneous BNT blocking. The canonical pairing of the blocked ket and bra words transfers the one-letter product-algebra span to the blocked MPO tensors, and projection to each labelled factor gives injectivity. Along a nonempty block, the products of the conjugated fusion letters telescope using only \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\). Length independence replaces every chi matrix by the identity on its own chi space. The two zipper identities and reconstruction then follow by multiplying the blocked fusion identity (??) by \(U_{\alpha ,\beta }\) or \(U_{\alpha ,\beta }^\dagger \) on the appropriate side.
The three rows below are, respectively, the two unweighted zipper identities and their local reconstruction consequence. The symbol \(H\) denotes \(H_{\alpha ,\beta }^{ij}\), and the open physical legs carry the indices \(i,j\). The reconstruction uses only \(U_{\alpha ,\beta }^{\dagger }U_{\alpha ,\beta }=1\); it does not assert \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\), surjectivity, or completeness of the total fusion space. The zipper orientation is that of [ BMW\(^{+}\)17 , Section 3, Equations (19)–(21) ] ; the MPDO identity is [ CPGSV16 , Theorem 4.14(iii), lines 986–991 ] , with identity weights in the length-independent case at [ CPGSV16 , line 1010 ] .
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\(=\) ![\tnpic[rows={op:fused,wire}]{
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{U_{\alpha,\beta}^{\dagger}} &
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The tensors of a fusion-isometry family generate the concrete BNT-label operator family \(O_L(M_\gamma )\), and the positive diagonal matrices give the positive trace-power witness for the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).
![\tnpic[compact, rows={op,op}, periodic]{
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\tn[mpo, down=$j_L$]{M_\beta}
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An active-support fusion coisometry family, and hence also a full-support fusion-isometry family, satisfies the same-length product law with the trace-power coefficients of its diagonal matrices. For every chain length \(L\ge 1\),
This derives the product law of [ CPGSV16 , Theorem 4.14(ii) ] from the isometry statement of [ CPGSV16 , Theorem 4.14(iii) ] . The active-support formulation is the one used in [ CPGSV16 , Appendix C.4, lines 2020–2029 ] .
The product of the operators is generated by the product tensor. For a nonempty word \(W\) of letters of the product tensor, the conjugated letters multiply to the conjugated word, and the closed trace is unchanged. For an active-support family, exact reconstruction writes the word as \(U_{\alpha ,\beta }^\dagger W U_{\alpha ,\beta }\), and cyclicity of trace together with \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\) removes the outer maps. In the full-support case one may equivalently use the isometric orientation: \(\operatorname{tr}(U_{\alpha ,\beta }WU_{\alpha ,\beta }^\dagger ) =\operatorname{tr}(WU_{\alpha ,\beta }^\dagger U_{\alpha ,\beta })=\operatorname{tr}(W)\). By the fusion identity the conjugated letters are the block-diagonal tensor-product letters of the weighted direct sum. Their word product splits block by block with the diagonal factors accumulating into \(\chi _{\alpha ,\beta ,\gamma }^{L}\); taking the trace block by block and factor by factor yields the coefficients \(\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) in (??).
Suppose the fusion-isometry identity holds with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and suppose the trace scalars satisfy the length-one idempotent law for the corresponding trace-power coefficients. For \(L{\gt}0\), the resulting BNT algebra clause has coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) and records positivity of every diagonal entry. The product law and the assumed idempotent condition are, respectively,
This construction realizes implication (iii)\(\Rightarrow \)(ii) of [ CPGSV16 , Theorem 4.14 ] ; it does not assert either of the implications involving the renormalization fixed-point condition.
Choose a vertical canonical decomposition
The fusion clause for this decomposition consists of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and coisometries \(U_{\alpha ,\beta }\) onto the active product sectors, satisfying
They obey \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), and the active direct sum reconstructs the product tensor:
The trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) satisfy the length-one idempotent identity for the coefficients determined by these same diagonal matrices. This is statement (iii) of [ CPGSV16 , Theorem 4.14, lines 986–993 ] , attached to the chosen decomposition of [ CPGSV16 , Proposition 4.13, lines 943–951 ] .
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form. If the one-site and two-site physical closures are related in both directions by trace-preserving completely positive maps as in Definition 4.1, then a vertical canonical decomposition of \(M\) has the active-sector fusion clause above. In particular, its positive diagonal fusion matrices satisfy the length-one idempotent trace-scalar identity.
Choose vertical canonical decompositions of the one-site tensor and its two-site blocking. Transporting the two physical maps through these decompositions identifies their normal sectors and yields the positive active-sector fusion of each product \(M_\alpha M_\beta \). For every positive length, the blocked operator is therefore both a sum over the two-site multiplicity matrices and the product of the two one-site sums. At all sufficiently large lengths, the normal-sector operators are linearly independent, so their coefficients agree. Equality of the resulting positive power sums identifies the two multisets of multiplicity entries. Summing this multiset identity gives \(m_\gamma =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta \), which completes the fusion clause.
Suppose the active-sector fusion clause holds for a chosen vertical canonical decomposition. Then the algebra clause holds for the same decomposition. For every positive chain length \(L\), its coefficients are \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), determined by the same diagonal matrices. Thus statement (iii) implies statement (ii) of [ CPGSV16 , Theorem 4.14, lines 972–993 ] . No assertion concerning the renormalization fixed-point condition is made here.
Iterating the fusion identity along \(L\) sites and taking the closed trace gives \(O_L(M_\alpha )O_L(M_\beta ) =\sum _\gamma \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})O_L(M_\gamma )\). Writing \(m_\alpha =\operatorname{tr}(\mu _\alpha )\), the assumed length-one idempotent identity is \(m_\gamma =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta \). These are the product and idempotent identities of the algebra clause. Retaining the chosen vertical decomposition and the same diagonal matrices gives the tensor-attached algebra clause.
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The associativity of the algebra product in [ CPGSV16 , Equation (87) ] constrains the diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and, through the isometry statement, induces a compatibility equation between the fusion isometries of the two ways of bracketing a triple product, \((M_\alpha M_\beta ) M_\gamma \) and \(M_\alpha (M_\beta M_\gamma )\) [ CPGSV16 , lines 995–999 ] . The source attributes the construction of that equation, and the classification of its solutions, to [ BMW\(^{+}\)17 ] . The next definitions and theorems first record the tensor-level content of the two bracketings and then compare the resulting full direct sums. Restriction of that comparison to a fixed final label is a separate step.
For labels \(\alpha , \beta , \gamma \), the left iterated fusion isometry \(U^{\mathrm L}_{\alpha ,\beta ,\gamma }\) is obtained by applying \(U_{\alpha ,\beta }\) (tensored with the identity on the bond space of \(M_\gamma \)) to fuse the first two tensors, then applying \(U_{\delta ,\gamma }\), for each label \(\delta \) appearing in the resulting direct sum, to fuse the outcome with the third tensor. Its codomain is indexed by pairs of labels \((\delta , \varepsilon )\) together with the multiplicity indices of \(\chi _{\alpha ,\beta ,\delta }\) and \(\chi _{\delta ,\gamma ,\varepsilon }\).
The left iterated fusion isometry satisfies \((U^{\mathrm L}_{\alpha ,\beta ,\gamma })^\dagger U^{\mathrm L}_{\alpha ,\beta ,\gamma }=1\).
The composite of two isometries is an isometry.
Site by site,
Apply the fusion identity for \(U_{\alpha ,\beta }\) to the first two tensors, summed against the physical index contracting with the third tensor. Regroup the result over \(\delta \) and use blockwise conjugation to obtain \(\chi _{\alpha ,\beta ,\delta }\) tensored with a letter of \(M_\delta M_\gamma \). The fusion identity for \(U_{\delta ,\gamma }\) gives the second direct sum.
For labels \(\alpha , \beta , \gamma \), the right iterated fusion isometry \(U^{\mathrm R}_{\alpha ,\beta ,\gamma }\) is obtained by applying \(U_{\beta ,\gamma }\), tensored with the identity on the bond space of \(M_\alpha \), and then applying \(U_{\alpha ,\delta }\) for every intermediate label \(\delta \). Its codomain is indexed by the intermediate and final labels \((\delta ,\varepsilon )\) and by the multiplicity indices of \(\chi _{\beta ,\gamma ,\delta }\) and \(\chi _{\alpha ,\delta ,\varepsilon }\). This is the right-bracketed composite in the associativity equation of [ BMW\(^{+}\)17 , Section “Associativity and the pentagon equation” ] , before the change of basis by an \(F\)-matrix.
The right iterated fusion isometry satisfies \((U^{\mathrm R}_{\alpha ,\beta ,\gamma })^\dagger U^{\mathrm R}_{\alpha ,\beta ,\gamma }=1\).
The two successive fusion maps are isometries on each intermediate sector, so their block-diagonal composite is an isometry.
Site by site,
Apply the fusion identity for \(U_{\beta ,\gamma }\) to \(M_\beta M_\gamma \). In the summand indexed by \(\delta \), exchange the first multiplicity factor with the bond space of \(M_\alpha \). Blockwise conjugation regroups the contraction as \(\chi _{\beta ,\gamma ,\delta }\otimes (M_\alpha M_\delta )^{ij}\). The fusion identity for \(U_{\alpha ,\delta }\) gives the sum over \(\varepsilon \).
The canonical comparison from the right-associated full direct sum to the left-associated full direct sum is
Both intermediate labels and the final label remain in the direct sums. This definition does not select a fixed final label and makes no invertibility assertion. Its orientation is from the right-associated sum to the left-associated sum; consequently it is adjoint-oriented with respect to the printed-\(F\) row/column convention used here for [ BMW\(^{+}\)17 , Section 3.4 ] .
The product of the full comparison with its adjoint is the range projection of the left iterated fusion isometry:
The right side is not asserted to be the identity. Surjectivity of the left iterated fusion map requires completeness of the fusion decomposition, corresponding to the invertible total fusion matrix in [ BMW\(^{+}\)17 , lines 181–191 ] .
Substitute (??). The isometry identity \((U^{\mathrm R})^\dagger U^{\mathrm R}=1\) and the two-sided inverse identities for the permutation matrix \(A\) leave \(U^{\mathrm L}(U^{\mathrm L})^\dagger \).
The product in the opposite order is the range projection of the right iterated fusion isometry:
The right side is not asserted to be the identity. Surjectivity of the right iterated fusion map likewise requires completeness of the fusion decomposition.
Substitute (??). The isometry identity \((U^{\mathrm L})^\dagger U^{\mathrm L}=1\) and the two-sided inverse identities for \(A\) leave \(U^{\mathrm R}(U^{\mathrm R})^\dagger \).
Suppose the positive trace-power coefficients are independent of the positive chain length. For every pair of physical indices \(i,k\), let
and define \(D^{\mathrm R}_{ik}\) by the right-associated multiplicities \(r_{\beta ,\gamma }^{\delta }\) and \(r_{\alpha ,\delta }^{\varepsilon }\). Then
This is an identity on the full direct sums. It neither extracts a fixed-final \(F\)-transformation nor proves invertibility or the pentagon identity.
Length independence makes every positive diagonal factor equal to the identity. Substitute the left and right triple-fusion identities into (??), cancel the two isometries, and apply Theorem 27.3.10.
For some common length \(S\), suppose that for each final label \(\varepsilon \) there are coefficients \(c_\varepsilon (w)\), indexed by the words \(w\) of length \(S\) in the doubled physical alphabet, such that
This is the finite-word form of the simultaneous inverse relation \(B_d^+B_{d'}=\delta _{d,d'}1\) used in the fixed-channel argument of [ BMW\(^{+}\)17 ] .
Suppose the positive trace-power coefficients are independent of the positive chain length, and suppose that common selectors exist at a positive length. Then every pair fusion isometry is surjective: \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\). The common selectors are the finite-word form of the simultaneous inverse used in [ BMW\(^{+}\)17 , line 269 ] .
For a word \(w\) of the positive selector length, the identity \(U_{\alpha ,\beta }^{\dagger }U_{\alpha ,\beta }=1\) gives \(D^w=U_{\alpha ,\beta }(M_\alpha M_\beta )^w U_{\alpha ,\beta }^{\dagger }\), where \(D=\bigoplus _\gamma 1_{r_{\alpha ,\beta }^{\gamma }}\otimes M_\gamma \). Hence the range projection \(P=U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }\) satisfies \(PD^w=D^w\). Summing the selector polynomials over the final label gives the identity on the full direct sum, and therefore \(P=1\).
Suppose that the labelled tensors in the fusion family form a basis of normal tensors and that the positive trace-power coefficients are independent of the positive chain length. Then every pair fusion isometry is surjective: \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\). In particular, final-label selectors and coisometry are conclusions, not additional hypotheses. This is a derived consequence of the common BNT word span in [ CPGSV16 , lines 317–345 ] , the fusion identity at lines 986–993, and length independence at lines 995–1010.
The basis-of-normal-tensors argument gives a positive common word length at which the labelled word evaluations span the full product matrix algebra. Choosing the tuple which is the identity in one labelled block and zero in every other block gives common final-label selectors. Apply Theorem 27.3.40.
Let \(P_\gamma \) be operators on the bond space of the vertically read tensor \(M_\gamma \). Define one fusion layer of the recursive operator \(Q\) by
The source terminal matrices are \(P_\gamma =\operatorname{tr}(M_\gamma )\) after spectral refinement, where the trace closes the horizontal operator leg and leaves the bond indices. Before spectral refinement, the fusion identity implies
Indeed, summing the forward fusion identity over equal horizontal operator indices and distributing matrix multiplication over this finite sum gives (??). This is one pairwise step in the recursion of [ CPGSV16 , lines 999–1010 ] .
Scope restriction (one fusion layer): This entry does not assert the arbitrary-chain recursion described below, nor the vertical-canonical assembly and commuting Gibbs decomposition. The remaining statements are recorded in docs/paper-gaps/cpsv16_topological_projector_recursion.tex.
Let \(P_\gamma \) be self-adjoint idempotents on the terminal bond space of \(M_\gamma \). Suppose that the matrices \(\chi _{\alpha ,\beta ,\gamma }\) have the positive-length trace-power form of the label-coefficient family and that this family is independent of the positive chain length. Then
is self-adjoint and idempotent. This is the single recursive fusion step in [ CPGSV16 , lines 999–1010 ] . Since the active fusion map satisfies \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=I\), the transported operator \(U_{\alpha ,\beta }^{\dagger }Q_{\alpha ,\beta }U_{\alpha ,\beta }\) is also an orthogonal projection on the product bond space.
The trace-power hypothesis and length independence give \(\chi _{\alpha ,\beta ,\gamma }=1\). Hence
Finally, the coisometry identity \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=I\) and Theorem 4.6.4 show that \(U_{\alpha ,\beta }^{\dagger }Q_{\alpha ,\beta }U_{\alpha ,\beta }\) is an orthogonal projection.
Fix an initial label \(\alpha \) and append labels successively along a finite chain. For total chain length \(N\geq 1\), the appended-label list has length \(N-1\); the symbols \(W_N\) and \(Q_N\) below refer to this total chain length. A fusion history records, at every step, the preceding final label, the new final label, and an index in the corresponding multiplicity space. If \(h\) ends at \(\gamma (h)\), let
For terminal bond operators \(P_\gamma \), define
The associated sequential map is obtained by tensoring the preceding map with the identity on the newly appended bond and then applying the active pairwise fusion coisometry in every preceding-history block. This is the recursively typed operator and circuit in [ CPGSV16 , lines 999–1010 ] .
For every initial label and every finite list of appended labels, let \(N\geq 1\) be one plus the length of that list. The sequential fusion map \(W_N\) is a retained-row coisometry. Letter by letter, it carries the left-associated product tensor to the recursive direct sum:
Consequently, after closing the horizontal operator index,
Exact reconstruction gives the reverse identities
If the positive trace-power coefficients are independent of the positive chain length and the terminal matrices are self-adjoint idempotents, then this recursive operator is self-adjoint and idempotent.
Local fix (Figure-11 fusion coisometry): The source embedding \(\widetilde U_N\) is \(W_N^\dagger \), the adjoint of the retained-row coisometry used above. Exact reverse fusion uses the active-support reconstruction of Appendix C.4, lines 2020–2029. See docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
Scope restriction (fixed initial label): This theorem concerns one chosen initial label and list of appended labels. The following theorem performs the vertical-canonical assembly over all labels and multiplicity coordinates. The commutator remains outside the present result; see docs/paper-gaps/cpsv16_topological_projector_recursion.tex.
The base case has no appended labels and represents the one-site chain; its fusion history is unique. For the induction step, tensor the preceding coisometry with the identity on the new bond, apply the induction hypothesis, and then apply the pairwise fusion identity in each history block. Reindexing the nested direct sum adjoins the new final label and multiplicity coordinate to the preceding history. Summing (??) over equal physical indices gives (??). Under length independence every accumulated coefficient \(w(h)\) is one, so self-adjointness and idempotence hold block by block.
Let \(V\) be a one-site matrix, not necessarily square. Applying \(V^{\otimes N}\) to both physical legs of the length-\(N\) operator generated by \(M\) is the operator generated by the tensor whose one-site physical matrices are conjugated by \(V\).
Expand a matrix entry as a sum over the original ket and bra configurations. The coefficient factors site by site, and the remaining virtual product is the word evaluation of the original tensor.
Let \(M\) be a matrix product density operator, and choose a vertical canonical decomposition with basis tensors \(M_\gamma \) and positive diagonal multiplicity matrices \(\mu _\gamma \). Then every terminal transfer matrix \(T_\gamma =\operatorname{tr}(M_\gamma )\) is positive semidefinite. This supplies the positivity needed for the spectral decomposition invoked in [ CPGSV16 , lines 1010–1016 ] ; it asserts neither that \(T_\gamma \) is a projection nor the subsequent spectral, Hamiltonian, or Gibbs decomposition.
Local fix (terminal trace orientation): The trace closes the horizontal operator leg and leaves the two bond indices of \(M_\gamma \) open. See docs/paper-gaps/cpsv16_topological_projector_recursion.tex.
At chain length one, conjugating the positive density operator by the retained-row vertical coisometry preserves positivity. Fix a label \(\gamma \) and a diagonal coordinate \(q\) of \(\mu _\gamma \). The corresponding principal block is \((\mu _\gamma )_{q,q}\, T_\gamma \). Since \((\mu _\gamma )_{q,q}{\gt}0\), multiplication by its reciprocal proves \(T_\gamma \geq 0\).
For a chain of positive length \(N\), choose at every site a BNT label \(\alpha _j\) and a diagonal coordinate \(q_j\) of \(\mu _{\alpha _j}\). Equivalently, putting \(L=N-1\), the chain has length \(L+1=N\) and contains \(L\) appended labels after the initial site. Define
The first label is the initial label of the recursive fusion, while the remaining labels are appended in the physical order \(\alpha _1,\ldots ,\alpha _{N-1}\). The recursive specification records this sequence in the reverse order \((\alpha _{N-1},\ldots ,\alpha _1)\), so removing its first entry removes the final physical site. This reversal affects only the recursive notation: the tensor product and the sequential fusion coisometry remain in the physical order \(\alpha _0,\ldots ,\alpha _{N-1}\). Let \(W_{N,c}\) be the resulting retained-row sequential coisometry for the configuration \(c\), and set \(Q_{N,c}=Q_N((\operatorname{tr}(M_\gamma ))_\gamma )\). The corresponding density block is \(m(\boldsymbol \alpha ,\boldsymbol q) W_{N,c}^\dagger Q_{N,c}W_{N,c}\). Thus the embedding \(\widetilde U\) in the source formula is \(W_{N,c}^\dagger \). The all-label recursive density operator is the direct sum of these blocks over every sitewise label and diagonal coordinate; denote it by \(R_N\). On the same direct-sum coordinates, define
Its scalar coefficients are precisely the selected diagonal coefficients of the factor \(\mu ^{\otimes N}\) in [ CPGSV16 , line 999 ] ; the identities mentioned there act on the complementary tensor factors of each block.
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form and suppose that it satisfies the renormalization fixed-point condition. There is one vertical canonical decomposition and one compatible family of fusion coisometries such that, for every \(N{\gt}0\), the rectangular vertical coisometry \(V\) satisfies the two exact equations
No additional compatibility hypothesis is imposed. The statement is restricted to positive lengths and does not assert the commutator at line 1001, length independence, or the subsequent spectral and Gibbs decomposition.
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
Local fix (rectangular vertical map): The second equality uses exact reconstruction from the retained nonzero sectors of Proposition 4.13 and Appendix C.4, lines 2020–2029, not a square-unitary identity. See docs/paper-gaps/cpgsv17_vertical_isometry_zero_sector.tex.
Under the normalized BNT-refined horizontal hypothesis, the vertical canonical-form theorem gives a rectangular coisometry onto the retained nonzero vertical sectors, together with exact reconstruction from those sectors. In the retained coordinates, one-site matrix entries are block diagonal in the label and multiplicity coordinate. Hence a closed chain vanishes between different sitewise configurations. Within one configuration, the scalar entries multiply to \(m(\boldsymbol \alpha ,\boldsymbol q)\), while the remaining horizontal matrices form the fixed-label product tensor. Apply (??) and (??), then sum over all configurations. The local reconstruction identity, applied at every site, gives (??).
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form and suppose that it satisfies the renormalization fixed-point condition. One choice of the vertical canonical decomposition and fusion coisometries satisfies, for every \(N{\gt}0\),
Equivalently, \([A_N,T_N]=0\), as asserted in [ CPGSV16 , lines 1000–1002 ] . No additional compatibility hypothesis is imposed. This statement makes no length-independence, terminal spectral, projection, Hamiltonian, or Gibbs-state assertion.
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
In every sitewise copy configuration, the factor \(A_N\) is the scalar \(m(c)\) times the identity on the corresponding simple-bond space. It therefore commutes with \(W_{N,c}^{\dagger }Q_{N,c}W_{N,c}\). Multiplication of the two direct sums proves (??) and (??). The renormalization fixed-point condition supplies the same fusion family used in the density decomposition.
For an MPDO in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition, fix the fusion clause selected by that construction. For every terminal label \(\gamma \), close the horizontal operator leg of \(M_\gamma \) to obtain a matrix \(K_\gamma \) on its bond space. Its spectral indices are pairs \(s=(\gamma ,k)\) with \(0\leq k{\lt}D_\gamma \). Write \(\lambda _s\) for the corresponding eigenvalue and \(E_s\) for the rank-one spectral projection, extended by zero on terminal labels different from \(\gamma \). Recursively transport this terminal family through every fusion history, then through the adjoint sequential fusion coisometry, and assemble over all sitewise label and multiplicity configurations. Denote the resulting all-label operator at positive chain length \(N\) by \(P_s^{(N)}\).
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition. Fix internally the fusion clause supplied by that construction, and suppose that its positive trace-power coefficients are independent of the positive chain length. Then \(K_\gamma \) is positive semidefinite for every terminal label, so \(\lambda _s\geq 0\). Write the total positive chain length as \(N=n+1\), where \(n\in \mathbb {N}\) is its predecessor. For every \(N{\gt}0\),
For distinct spectral indices \(s\) and \(t\),
The recursive factor and multiplicity weights satisfy
Consequently,
The length-independence assumption here is clause-relative: it is stated on the coefficients of the fixed clause selected from the renormalization fixed-point construction, not on an arbitrary fusion clause.
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
This theorem stops at [ CPGSV16 , lines 1003–1012 ] ; it does not assert the commuting nearest-neighbor Hamiltonian or Gibbs-state conclusion beginning at line 1013.
Positivity of the one-site density operator passes to every fixed BNT block; division by its strictly positive multiplicity weight shows that each \(K_\gamma \) is positive semidefinite. Apply the finite-dimensional spectral theorem to obtain the non-negative eigenvalues and rank-one orthogonal projections. Length independence makes every retained fusion coefficient equal to one, so the recursive direct sum preserves idempotence and pairwise orthogonality. Write \(Q_{N,c,s}\) for the recursive spectral block in copy configuration \(c\). The coisometry identity \(W_{N,c}W_{N,c}^\dagger =1\) gives
Thus transport through the adjoint sequential coisometry preserves the projector identities. Linearity gives (??). Finally, \(A_N\) is scalar on every copy-configuration block, hence it commutes with each \(P_s^{(N)}\) and gives (??).
Let \(\mu \) be the strictly positive diagonal multiplicity matrix in the retained one-site coordinates. Define \(h=\operatorname{diag}(-\log \mu )\otimes I\) on two adjacent sites. For a chain of length \(L=N+2\), let \(H_L=\sum _{i=1}^{L}h_{i,i+1}\), with periodic indices.
The terminal spectral family has cardinality at most the physical one-site dimension \(d\). Fix an injection of its indices into \(\{ 1,\ldots ,d\} \) and extend the remaining eigenvalues and projectors by zero.
Scope restriction (retained vertical coordinates): the Hamiltonian and projectors act on the retained nonzero vertical sectors. Applying the adjoint retained-row map reconstructs the physical density operator, but does not provide a Hamiltonian or projectors on the original physical space. This restriction is recorded in docs/paper-gaps/cpsv16_topological_gibbs_physical_complement.tex.
Scope restriction (positive chains of length at least two): the local interaction is defined on two spins, so these data are indexed by \(L=N+2\). The length-one boundary is recorded in docs/paper-gaps/cpsv16_topological_gibbs_length_one.tex.
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form satisfying the renormalization fixed-point condition. Select the fusion clause supplied by that condition and suppose that its positive trace-power coefficients are independent of the positive chain length. There are non-negative numbers \(\lambda _1,\ldots ,\lambda _d\), independent of the chain length. For every \(L\geq 2\), the same two-site matrix \(h\) is Hermitian, its periodic translates commute pairwise, and
There are orthogonal projections \(P_1^{(L)},\ldots ,P_d^{(L)}\) such that
Applying the adjoint retained-row map at every site reconstructs \(\rho ^{(L)}(M)\) from the right-hand side of (??).
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
Scope restriction (retained vertical coordinates): the formula is an equality for \(R_L\) on the retained vertical space, not the literal physical-space equality asserted in [ CPGSV16 , lines 1013–1016 ] . Exact reconstruction by the adjoint retained-row map is weaker than the existence of physical projectors and a finite physical Hamiltonian. See docs/paper-gaps/cpsv16_topological_gibbs_physical_complement.tex.
Scope restriction (positive chains of length at least two): the conclusion is proved for \(L=N+2\). Definition 4.8 of [ CPGSV16 ] defines \(h\) on two spins but does not state a length-one convention; see docs/paper-gaps/cpsv16_topological_gibbs_length_one.tex.
Each diagonal entry of \(\mu \) is strictly positive, so its negative logarithm is real. The translated terms are diagonal, and each site contributes once to their sum. Exponentiating gives (??).
Positivity of the multiplicities embeds one terminal mode for every terminal spectral index into the retained coordinates. The rank of the retained-row coisometry is at most \(d\), so the terminal family has at most \(d\) members. Extend the eigenvalues and spectral projectors by zero along an injection into \(\{ 1,\ldots ,d\} \). The added zero projectors preserve self-adjointness, idempotence, commutation, and the spectral sum. Combine that sum with (??) and the exact adjoint reconstruction of the retained density operator.
Let \(M\) be in normalized BNT-refined horizontal form and satisfy the hypotheses of Theorem 27.3.54. There are non-negative numbers \(\lambda _1,\ldots ,\lambda _d\) and one fixed Hermitian two-site matrix \(\widehat h\), all independent of the chain length. For every \(L\geq 2\), let \(\widehat H_L=\sum _{j=1}^{L}\widehat h_{j,j+1}\). The translates of \(\widehat h\) commute pairwise, and there are orthogonal projections \(\widehat P_1^{(L)},\ldots ,\widehat P_d^{(L)}\) on the physical chain such that
Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.
Local fix (physical complement): the retained one-site energy is compressed to physical coordinates and extended by zero energy on the orthogonal complement. The retained projectors are transported through the adjoint sitewise coisometry. See docs/paper-gaps/cpsv16_topological_gibbs_physical_complement.tex.
Put \(V\) for the retained-row coisometry and \(k\) for the retained logarithmic multiplicity energy. The physical energy \(K=V^\dagger kV\) is Hermitian and
Taking the fixed two-site term \(K\otimes I\), its periodic translates act on separate one-site factors, hence commute, and their exponential is the tensor power of (??).
For the sitewise coisometry \(W=V^{\otimes L}\), transport each retained projector by \(\widehat P_i^{(L)}=W^\dagger P_i^{(L)}W\). The identity \(WW^\dagger =I\) preserves self-adjointness, idempotence, and pairwise orthogonality. The one-site intertwining identities transport the retained commutator to the physical Gibbs factor. Applying the same identities term by term to (??), and then using exact adjoint reconstruction, gives (??).
Definition 4.8 of [ CPGSV16 ] lets a local operator act on the first two spins and translates it around the periodic chain. Thus its construction has domain \(L\geq 2\). The periodic identification \(L+1=1\) identifies site labels, but supplies neither a map from two-site operators to one-site operators nor a multiplicity for the coincident edge.
Theorem 4.15 is printed without an explicit lower bound on \(L\), but its \(L=1\) instance is therefore undefined without an author or erratum convention. The source construction is defined for \(L\geq 2\), and no length-one theorem is asserted. The physical-coordinate theorem above proves the result on this domain; no \(L=1\) statement follows without an additional convention.
For a full-support fusion family, let \(P_\gamma \) act on the bond space of \(M_\gamma \). Define
The terminal contraction is the same as in the source operator. The additional condition is that the active support is the whole product bond space.
Scope restriction (full-support fusion family): Full support is not an additional hypothesis of the arbitrary-chain source statement. This distinction is documented in docs/paper-gaps/cpsv16_topological_projector_recursion.tex.
Suppose that the labelled tensors form a basis of normal tensors and the positive trace-power coefficients are independent of the positive chain length. If each \(P_\gamma \) is a self-adjoint idempotent on the bond space, then \(Q^{\mathrm{fs}}_{\alpha ,\beta }\) is a self-adjoint idempotent and \(\widehat Q^{\mathrm{fs}}_{\alpha ,\beta }\) is an orthogonal projection.
Length independence gives \(\chi _{\alpha ,\beta ,\gamma }=1\), hence \((Q^{\mathrm{fs}}_{\alpha ,\beta })^2 =Q^{\mathrm{fs}}_{\alpha ,\beta }\) and \((Q^{\mathrm{fs}}_{\alpha ,\beta })^\dagger =Q^{\mathrm{fs}}_{\alpha ,\beta }\). The basis-of-normal-tensors completeness theorem gives \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1\), so
Under the hypotheses of Theorem 27.3.40, \(U^{\mathrm L}_{\alpha ,\beta ,\gamma } (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^{\dagger }=1\).
Pair fusion completeness gives \(\sum _\delta r_{a,b}^{\delta }D_\delta =D_aD_b\). Applying this equality first to \((\delta ,\gamma )\) and then to \((\alpha ,\beta )\) shows that the source and target dimensions of \(U^{\mathrm L}_{\alpha ,\beta ,\gamma }\) coincide. A square isometry is surjective.
Under the same hypotheses, \(U^{\mathrm R}_{\alpha ,\beta ,\gamma } (U^{\mathrm R}_{\alpha ,\beta ,\gamma })^{\dagger }=1\).
The right-associated dimension calculation is
The right iterated fusion map is therefore a square isometry.
Under the same hypotheses, the comparison of the full direct sums satisfies \(C_{\alpha ,\beta ,\gamma }C_{\alpha ,\beta ,\gamma }^{\dagger }=1\).
The left range identity reduces the product to \(U^{\mathrm L}(U^{\mathrm L})^{\dagger }\), which is the identity by Theorem 27.3.59.
Under the same hypotheses, \(C_{\alpha ,\beta ,\gamma }^{\dagger }C_{\alpha ,\beta ,\gamma }=1\). This proves invertibility on the full direct sums. It does not yet prove invertibility of a fixed-final multiplicity matrix.
Apply the right range identity and Theorem 27.3.60.
Suppose the positive trace-power coefficients are independent of the positive chain length and common word selectors exist. If \(\varepsilon \ne \varepsilon '\), then the corner of the full comparison from the right final sector \(\varepsilon '\) to the left final sector \(\varepsilon \) vanishes: \(P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}=0\). The assertion is conditional on the displayed word-selector identities. It neither states that a diagonal corner is invertible nor identifies one with an \(F\)-matrix, and it does not assert the pentagon identity.
Fix one left multiplicity index and one right multiplicity index. Taking the corresponding bond-space corner of the full letterwise identity gives \(M_\varepsilon ^{ij}X=X M_{\varepsilon '}^{ij}\). Induction on the word \(w\) gives \(M_\varepsilon ^w X = X M_{\varepsilon '}^w\). Multiply these identities by \(c_\varepsilon (w)\) and sum over all words of length \(S\). The two selector identities give \(X=0\). Since the multiplicity indices were arbitrary, the whole off-diagonal final-sector corner vanishes.
Fix a final label \(\varepsilon \). The multiplicity spaces of the two bracketings are
These spaces have the bracketing pattern of the \((e,\mu ,\nu )\) and \((f,\lambda ,\sigma )\) index spaces in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] . Identifying the dimensions of the positive diagonal matrices with the fusion multiplicities of that equation requires the length-independent integer specialization and is not asserted here. The corresponding fixed-final row spaces are \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\).
Suppose that the positive trace-power coefficients are independent of the positive chain length and that the labelled operators are eventually linearly independent. Then, for each final label \(\varepsilon \),
Consequently a matrix between the two fixed-final multiplicity spaces is square. This is the numerical associativity statement of [ BMW\(^{+}\)17 , lines 237–241 ] ; it does not assert completeness of the fusion maps or invertibility of the fixed-final comparison.
Choose a positive length beyond the linear-independence threshold. Associativity of the same-length product law gives equality of the two sums of products of structure coefficients. Length independence and positivity identify every coefficient with the size of its corresponding diagonal \(\chi \)-matrix. The two sums are precisely the cardinalities of the displayed dependent sums.
The maps \(U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }\) and \(U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon }\) are obtained by restricting the two iterated fusion isometries to the summands with final label \(\varepsilon \). Their adjoints, after the canonical reassociation of the triple bond space, have the orientation and bracketing pattern of the weighted analogs of the two sides in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] .
For every final label \(\varepsilon \), the left-associated restriction satisfies the positive-diagonal weighted identity
Restrict Theorem 27.3.31 to the rows and columns whose final label is \(\varepsilon \). Reindexing the conjugating matrix commutes with conjugation, and the outer block-diagonal matrix retains precisely the displayed \(\varepsilon \)-blocks.
For every final label \(\varepsilon \), the right-associated restriction satisfies the positive-diagonal weighted identity
Restrict Theorem 27.3.34 to the rows and columns whose final label is \(\varepsilon \). Reindexing the conjugating matrix commutes with conjugation, and the outer block-diagonal matrix retains precisely the displayed \(\varepsilon \)-blocks.
Suppose that \(M_\varepsilon \) is injective. Let \(C\) be a matrix from the right fixed-final-sector space to the left fixed-final-sector space. Under the canonical identifications with \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\), assume that, for every pair of physical indices \(i,j\),
Then there is a rectangular matrix \(F:\mathcal H^{\mathrm R}_{\varepsilon }\to \mathcal H^{\mathrm L}_{\varepsilon }\) such that \(C=F\otimes 1_{D_\varepsilon }\). This conclusion is conditional on the supplied matrix \(C\). It does not assert the existence or invertibility of \(C\) or \(F\), and it does not identify the dimensions of the positive diagonal matrices \(\chi \) with the fusion multiplicities in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] .
Apply Lemma 2.1.6 to the family \(\{ M_\varepsilon ^{ij}\} _{i,j}\). This family spans the full matrix algebra by injectivity, and the assumed relation is precisely the amplified intertwining identity of that lemma.
Suppose the positive trace-power coefficients are independent of the positive chain length and common word selectors separate the final labels. Fix labels \(\alpha ,\beta ,\gamma ,\varepsilon \), and suppose that \(M_\varepsilon \) is injective at the present blocking. Write \(C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma } =P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}\) for the corner from the right final sector \(\varepsilon '\) to the left final sector \(\varepsilon \). There is a rectangular matrix \(F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }\) such that, after the canonical index identifications, \(C^{\varepsilon ,\varepsilon }_{\alpha ,\beta ,\gamma } =F_\varepsilon \otimes 1_{D_\varepsilon }\). For every \(\varepsilon '\ne \varepsilon \), \(C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma }=0\). The selector identities and injectivity at the present blocking are additional hypotheses here. The theorem neither asserts that \(F_\varepsilon \) is square or invertible nor identifies it with the printed \(F\)-matrix of [ BMW\(^{+}\)17 , Section 3.4 ] ; no pentagon identity is asserted.
Taking the \(\varepsilon \)–\(\varepsilon \) corner of the full comparison identity gives, for every pair \(i,j\),
Injectivity and the amplified commutant result give \(C^{\varepsilon ,\varepsilon }=F_\varepsilon \otimes 1_{D_\varepsilon }\). The common word selectors give \(C^{\varepsilon ,\varepsilon '}=0\) whenever \(\varepsilon '\ne \varepsilon \) by Theorem 27.3.63.
Suppose the positive trace-power coefficients are independent of the positive chain length and common selectors exist at a positive length. For every final label \(\varepsilon \), write \(C_\varepsilon =P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon }\). Then
The conclusion is conditional on the simultaneous selector identities. It is the fixed-output-channel invertibility consequence of the simultaneous inverse used in [ BMW\(^{+}\)17 , lines 247–277 ] .
Order each full direct sum first by its final label. In the product \(CC^\dagger \), the \(\varepsilon \)-diagonal corner is \(\sum _{\varepsilon '} C^{\varepsilon ,\varepsilon '} (C^{\varepsilon ,\varepsilon '})^\dagger \). Every term with \(\varepsilon '\ne \varepsilon \) vanishes, while \(CC^\dagger =1\). Hence \(C_\varepsilon C_\varepsilon ^\dagger =1\). Taking instead the \(\varepsilon \)-diagonal corner of \(C^\dagger C=1\) gives the second identity.
Assume the hypotheses of Theorem 27.3.71. Suppose in addition that \(M_\varepsilon \) is injective at the present blocking and that \(D_\varepsilon {\gt}0\). There is a matrix \(F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }\) such that
Thus \(F_\varepsilon \) is invertible. Its orientation is opposite to the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] ; no pentagon identity is asserted.
Injectivity gives \(C_\varepsilon =F_\varepsilon \otimes 1_{D_\varepsilon }\). Substituting this expression into (??) and (??) gives
Since \(D_\varepsilon {\gt}0\), Lemma 27.3.16 removes the identity factor and gives the two asserted identities.
Let \(M_c^{il}\in M_{D_c}(\mathbb {C})\) be a finite labelled MPO tensor family. A simultaneous block left inverse consists of coefficients \(C_{(c,x,y),(i,l)}\) such that
This is the common block inverse used in [ CPGSV16 , Appendix C.2, lines 1666–1676 ] and [ BMW\(^{+}\)17 , lines 269–277 ] .
Let \(A_c^q\in M_{D_c}(\mathbb {C})\) be a finite labelled tensor family. If the one-letter tuples \(q\longmapsto (A_c^q)_c\) span the full product algebra \(\prod _c M_{D_c}(\mathbb {C})\), then there are coefficients \(C_{(c,x,y),q}\) such that
For an MPO tensor the letter is the ket–bra pair \(q=(i,k)\), so the same coefficients form the common block-letter left inverse used in [ BMW\(^{+}\)17 , lines 269–277 ] . The full product-algebra span is obtained after the common blocking described in [ BMW\(^{+}\)17 , lines 427–431 ] .
Fix \((c,x,y)\) and consider the product-algebra element which is the matrix unit \(E_{xy}\) in block \(c\) and zero in every other block. By the spanning hypothesis it is a linear combination of the one-letter tuples. Reading its \((x',y')\) entry in block \(d\) gives the displayed identity. Reindexing the doubled physical letter by the ket–bra pair gives the MPO form.
Let \(\Lambda \) be a finite set. A complete zipper fusion family consists of positive bond dimensions \(D_a\), MPO blocks \(B_a^{ij}\in M_{D_a}(\mathbb {C})\), fusion multiplicities \(N_{ab}^{c}\), and matrices
The matrices \(X^{c+}_{ab,\mu }\) are independently chosen left inverses, not adjoints. For all labels and multiplicity indices,
The opposite product \(P_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }\) is the projector onto the product-MPO support; it need not be the identity on the ambient bond space. Every product-MPO letter is reconstructed on this support:
Writing \(B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}\), the two zipper identities are
Each block is injective, and the labelled block-letter matrix has a common left inverse: its contraction with \(B_d^{ij}\) is \(\delta _{cd}E_{xy}\) on the block selected by \((c,x,y)\). These are the fusion-tensor, reconstruction, zipper, and simultaneous inverse relations of [ BMW\(^{+}\)17 , lines 161, 181–200, and 269–277 ] .
Suppose that the labelled tensors of a length-independent fusion family form a source basis of normal tensors. There is one common positive blocking at which they determine a complete zipper fusion family. Its bond dimensions are the original labelled bond dimensions, its fusion multiplicities are \(N_{\alpha \beta }^{\gamma }=\dim \chi _{\alpha ,\beta ,\gamma }\), its synthesis map is the adjoint of \(U_{\alpha ,\beta }\), and its analysis map is \(U_{\alpha ,\beta }\), after the canonical identification \(\mathbb {C}^{D_\alpha D_\beta } \cong \mathbb {C}^{D_\alpha }\otimes \mathbb {C}^{D_\beta }\). In particular, the two zipper identities, literal reconstruction, injectivity, and the simultaneous block-letter inverse are conclusions; none is an additional hypothesis.
Length independence makes every chi matrix the identity. Use the common positive BNT blocking for the labelled tensors and transport the fusion equation through it. Pair completeness gives \(U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=\mathbb {1}\), so analysis followed by synthesis is the identity. The other three matrix identities are the transported zipper identities (??) and (??) and the reconstruction identity (??). The common one-letter product-algebra span supplies both block injectivity and the simultaneous block-letter inverse. These data are exactly those of Definition 27.3.75.
Fix a final label \(d\). Summing the diagonal matrix-unit coefficients of the simultaneous block inverse defines
Here \(C_{(d,x,y),(i,j)}\) denotes the coefficient of the simultaneous block inverse corresponding to the matrix unit \(E_{xy}\) in block \(d\).
The one-letter selector satisfies
Thus one physical letter selects the identity on the chosen final block and annihilates every other final block.
Apply the simultaneous block inverse entrywise and sum over the diagonal matrix units of block \(d\). If \(e=d\), their sum is \(\mathbb {1}_{D_d}\); if \(e\ne d\), every term vanishes.
Fix \(a,b,c,d\in \Lambda \). There is an invertible matrix
whose rows are indexed by \((f,\lambda ,\sigma )\) and whose columns are indexed by \((e,\mu ,\nu )\). Its entries satisfy
The matrix in the opposite orientation is its two-sided inverse. This is the printed \(F\)-move of [ BMW\(^{+}\)17 , Section 3.4 ] .
Form the full left- and right-associated synthesis matrices \(L\) and \(R\) and their analysis matrices \(L^+\) and \(R^+\). Applying (??) twice to the two associative decompositions gives
Contracting (??) with the one-letter final-block selector from Definition 27.3.77, the two direct sums \(D_{\mathrm L}\) and \(D_{\mathrm R}\) both contract to the identity. Consequently,
Hence \(R^+L\) changes left-tree coordinates into right-tree coordinates and
Its opposite-oriented comparison is \(L^+R\). Using (??),
The zipper identities (??) and (??) show that \(R^+L\) intertwines the direct sums of the final-block letters. The simultaneous block inverse annihilates its corners with different final labels. On the \(d\)-diagonal corner, injectivity gives \((R^+L)_d=F^{abc}_d\otimes \mathbb {1}_{D_d}\). Restricting \(R(R^+L)=L\) to this corner gives (??). The corresponding fixed-final corner of \(L^+R\), together with \((R^+L)(L^+R)=\mathbb {1}=(L^+R)(R^+L)\), gives amplified inverse identities. Lemma 27.3.16 removes the nonempty bond-space identity factor and yields the two inverse identities for \(F^{abc}_d\).
Fix labels \(a,b,c,d,e\). The five fusion trees in equation (33) of [ BMW\(^{+}\)17 ] determine five multiplicity spaces and five synthesis maps into the fourfold bond space. These spaces use the integer multiplicities \(N_{ab}^{c}\) of the complete zipper family, rather than the weighted positive-diagonal coordinates introduced earlier.
Each edge of the pentagon carries the printed \(F\)-matrix for the affected triple tensor factor and the identity on the untouched multiplicity factor. The three upper edges and the two lower edges define composites in the forward orientation. Replacing each edge by its inverse defines the opposite-oriented composites whose entries occur literally in equation (33).
![\begin{tenkzcd}[polygon=5, radius=34mm]
\tntree{(a\,((b\,c)_h\,d)_i)_e} &
\tntree{(a\,(b\,(c\,d)_j)_i)_e} &
\tntree{((a\,b)_f\,(c\,d)_j)_e} &
\tntree{(((a\,b)_f\,c)_g\,d)_e} &
\tntree{((a\,(b\,c)_h)_g\,d)_e}
\tnarrow[from=4, to=5]{F^{abc}_{g}}
\tnarrow[from=5, to=1]{F^{ahd}_{e}}
\tnarrow[from=1, to=2]{F^{bcd}_{i}}
\tnarrow*[from=4, to=3]{F^{fcd}_{e}}
\tnarrow*[from=3, to=2]{F^{abj}_{e}}
\end{tenkzcd}](tenkz_svg/tenkz-f0f560e56067c1c0.svg)
The three-edge and two-edge composites of the forward printed \(F\)-matrices are equal. The corresponding composites of the inverse \(F\)-matrices are also equal. In the upper/lower convention of [ BMW\(^{+}\)17 , equation (33) ] , the latter equality is
Every displayed entry denotes the inverse of the forward matrix in the \(F\)-move convention.
Apply the elementary three-block \(F\)-move along each of the five edges. Both paths give the same expansion in the right-associated fourfold fusion map. Its analysis map is a left inverse, by three successive applications of fusion-tensor biorthogonality (??). This gives equality after tensoring with the identity on the nonzero final bond space, and Lemma 27.3.16 gives equality of the forward composites. The reverse composites are two-sided inverses of this common matrix, hence they coincide. Taking a matrix entry gives (??).
Fix four labels \(\alpha ,\beta ,\gamma ,\delta \) in a BNT fusion-isometry family, and write \(A=\{ 1,\ldots ,D_\alpha \} \), \(B=\{ 1,\ldots ,D_\beta \} \), \(C=\{ 1,\ldots ,D_\gamma \} \), and \(D=\{ 1,\ldots ,D_\delta \} \) for their bond-coordinate sets. Let
denote the canonical reassociation. Define
The three-edge and two-edge reassociations from \((((A\times B)\times C)\times D)\) to \(A\times (B\times (C\times D))\) coincide:
If \(P(r)_{x,y}=1\) when \(r(x)=y\) and is zero otherwise, then the corresponding coordinate-permutation matrices satisfy
These are equalities of Cartesian bond-coordinate identifications only; they are distinct from the fusion-multiplicity identity in Theorem 27.3.81.
Every element \((((a,b),c),d)\) is sent by either composite to \((a,(b,(c,d)))\). This proves (??). The identity \(P(v\circ u)=P(u)P(v)\) then turns (??) into (??).
![\begin{tenkzcd}[polygon=5, radius=38mm]
\ensuremath{(A\times((B\times C)\times D))} &
\ensuremath{(A\times(B\times(C\times D)))} &
\ensuremath{((A\times B)\times(C\times D))} &
\ensuremath{(((A\times B)\times C)\times D)} &
\ensuremath{((A\times(B\times C))\times D)}
\tnarrow[from=4, to=5]{r_1}
\tnarrow[from=5, to=1]{r_2}
\tnarrow[from=1, to=2]{r_3}
\tnarrow*[from=4, to=3]{s_1}
\tnarrow*[from=3, to=2]{s_2}
\end{tenkzcd}](tenkz_svg/tenkz-aa4cc882a07b8b1b.svg)