21 Mixed States: General Case and Algebraic Structure

In the general mixed-state renormalization analysis of [ 13 , Section 4 ] , an MPDO tensor in vertical canonical form generates closed-chain operators whose algebraic structure is governed by diagonal $\chi $-matrices and fusion isometries $U_{\alpha ,\beta }$.

21.1 Closed-chain operators

The vertical canonical form of an MPDO tensor is $\widetilde M = U^\dagger B U$ with $B=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha $ (Theorem 16.4.105), as in [ 13 , Section 4 ] .

The closed-chain operators $O_L(M_\alpha )$ contract cyclically the BNT bond indices of $L$ copies in the vertical reading. Decomposing each vertical physical index into a pair $(a,b)$ of horizontal bond indices and writing $\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}$, the matrix entries for index strings $\mathbf a=(a_1,\ldots ,a_L)$ and $\mathbf b=(b_1,\ldots ,b_L)$ are

\begin{align} [O_L(M_\alpha )]_{\mathbf a,\mathbf b} & = \operatorname{tr}\! \left(\widehat M_\alpha ^{a_1b_1}\cdots \widehat M_\alpha ^{a_Lb_L}\right) \notag \\ & = [\rho ^{(L)}(\widehat M_\alpha )]_{\mathbf a,\mathbf b}, \notag \end{align}

where the trace runs over the BNT bond space. These closed-chain operators $O_L(M_\alpha )=\rho ^{(L)}(\widehat M_\alpha )$ act on the horizontal bond strings $\mathbf a$ and $\mathbf b$ and enter the operator algebra of [ 13 , Theorem 4.14 ] .

21.2 Algebra structure

Definition 21.2.1 Support algebras and blocking maps

✓

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

\begin{align} m_n(x,y) & = xy, \notag \\ \iota _n(x) & = x. \notag \end{align}

Definition 21.2.2 Algebras of blocked adjoint fixed points

✓

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

\begin{align} \mathcal{A}_n & = \operatorname{Fix}(E_n^{\dagger }), \notag \end{align}

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

Theorem 21.2.3 Idempotent transfer maps and stationary support algebras

✓

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

\begin{align} \mathcal{A}_n & = \operatorname{Fix}(E_1^{\dagger }), \notag \end{align}

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

Proof ▶

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 family $\mathcal{A}_n = \operatorname{Fix}(E_1^{\dagger })$.

For the one-site and two-site vertical canonical forms, let $\{ M_\alpha \} _\alpha $ and $\{ A_\alpha \} _\alpha $ denote their respective BNTs. Their boundary contractions have the same local form:

$M_\alpha (X)=$ $\begin{tenkz}[rows={wire}, cols=2, boundary=periodic] \tn[skin=box]{X} & \tn[skin=mpo, label pos=45, ports={90:physical, 270:physical}]{M_\alpha} \end{tenkz}$ $A_\alpha (X)=$ $\begin{tenkz}[rows={wire}, cols=2, boundary=periodic] \tn[skin=box]{X} & \tn[skin=mpo, label pos=45, ports={90:physical, 270:physical}]{A_\alpha} \end{tenkz}$

In each cell, $X$ is inserted on the virtual bond and the virtual indices are closed by $\operatorname{tr}$; the physical legs remain open. In [ 13 , Appendix C.4, lines 1974–1980 ] , the weighted direct sums of these contractions are fixed by compositions of the reduced tpCPM. Products of these fixed operators are then used to prove that the compositions are identities on the corresponding direct sums of matrix algebras. This argument concerns the vertical canonical forms, not the fixed-point algebras $\operatorname{Fix}(E_n^\dagger )$ of Definition 21.2.2.

Theorem 21.2.4 Stabilized blocked adjoint fixed points yield support algebras

✓

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

\begin{align} \operatorname{Fix}(E_n^{\dagger }) & = \operatorname{Fix}(E_1^{\dagger }), \notag \end{align}

then there is a compatible family of support algebras.

Proof ▶

The trace-preserving map and its positive-definite fixed point give a $*$-subalgebra $\operatorname{Fix}(E_1^\dagger )$. Take $\mathcal{A}_n := \operatorname{Fix}(E_1^{\dagger })$ for every $n$, with ambient multiplication and inclusion. 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 $M$.

Stabilization of adjoint fixed points does not establish the converse direction of [ 13 , Theorem 4.14 ] . In particular, [ 13 , Theorem 4.14(ii) ] requires positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$ for the multiplication of the vertical operators $O_L(M_\alpha )$ and a vector $m_\alpha =\operatorname{tr}(\mu _\alpha )$ that is an idempotent for the multiplication induced by $c^{(1)}$. By contrast, choosing a basis in each $\mathcal{A}_n$ gives coordinates for multiplication and inclusion between different blocking sizes. These coordinates alone do not identify the vertical BNT or its structure coefficients.

Definition 21.2.5 Coordinates in the support algebras

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

\begin{align} x & = \sum _{i \in I_n} x_i\, b_i^{(n)}. \notag \end{align}

Theorem 21.2.6 Reconstruction from coordinates

✓

Reconstructing a coordinate family $(a_i)_{i\in I_n}$ gives $\sum _i a_i b_i^{(n)}$ in $\mathcal{A}_n$.

Proof ▶

This is the finite-basis reconstruction formula for the chosen basis of $\mathcal{A}_n$.

Definition 21.2.7 Multiplication coefficients between blocking sizes

✓

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

\begin{align} m_n(b_i^{(n)},b_j^{(n)}) & = \sum _{k \in I_{2n}} c^{(n)}_{i,j,k}\, b_k^{(2n)}. \notag \end{align}

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

Theorem 21.2.8 Reconstruction of multiplication from its coefficients

✓

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

Proof ▶

Apply the reconstruction formula of Theorem 21.2.6 to the element $m_n(b_i^{(n)},b_j^{(n)}) \in \mathcal{A}_{2n}$.

Definition 21.2.9 Inclusion coefficients between blocking sizes

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

\begin{align} \iota _n(b_i^{(n)}) & = \sum _{j \in I_{n+1}} d^{(n)}_{ij}\, b_j^{(n+1)}. \notag \end{align}

Theorem 21.2.10 Reconstruction of inclusion from its coefficients

✓

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

Proof ▶

Coefficient extraction and reconstruction are mutually inverse coordinate maps for the chosen basis of $\mathcal{A}_{n+1}$. Applying them successively to $\iota _n(b_i^{(n)})$ therefore returns the same element.

Theorem 21.2.11 Reconstruction of compatible adjoint fixed points

✓

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

Proof ▶

Compatibility identifies $\mathcal{A}_n$ with the adjoint fixed-point algebra of $E_n$, so $X$ belongs to $\mathcal{A}_n$. Theorem 21.2.6 then reconstructs it.

Theorem 21.2.12 Descent of blocked adjoint fixed points

✓

If an MPO tensor has a compatible family of support algebras, then for every positive blocking size $n$ and every $X$, $E_n^{\dagger }X=X$ implies $E_1^{\dagger }X=X$.

Proof ▶

Compatibility at size $n$ places $X$ in $\mathcal{A}_n$. The inclusion $\iota _n$ places the same ambient matrix $X$ in $\mathcal{A}_{n+1}$. Compatibility at size $n+1$ therefore gives $E_{n+1}^\dagger X=X$. Since $E_{n+1}=E_n\circ E_1$, it follows that $X=E_{n+1}^\dagger X=E_1^\dagger E_n^\dagger X=E_1^\dagger X$.

Lemma 21.2.13 Reverse inclusion of adjoint fixed points

✓

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

Proof ▶

Induct 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 argument does not require a compatible family of support algebras.

Lemma 21.2.14 Blocked powers of adjoint eigenvectors

✓

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

Proof ▶

The case $n=0$ is $E_0=\operatorname{id}$. For the induction step, $E_{n+1}=E_n\circ E_1$, hence

\begin{align} E_{n+1}^{\dagger }X & = E_1^{\dagger }E_n^{\dagger }X = E_1^{\dagger }(\lambda ^n X) = \lambda ^n E_1^{\dagger }X = \lambda ^{n+1}X. \notag \end{align}

Theorem 21.2.15 Finite-order adjoint eigenvalues under compatibility

✓

Assume there is a compatible family of support algebras. If $X\ne 0$, $E_1^\dagger X=\lambda X$, and $\lambda ^n=1$ for some $n{\gt}0$, then $\lambda =1$.

Proof ▶

By Lemma 21.2.14, $E_n^\dagger X=\lambda ^n X=X$. Theorem 21.2.12 gives $E_1^\dagger X=X$. Thus $\lambda X=X$, so $(\lambda -1)X=0$. Since $X\ne 0$, one obtains $\lambda =1$.

Theorem 21.2.16 Equality of compatible adjoint fixed-point algebras

✓

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

Proof ▶

Combine Theorem 21.2.12 with Lemma 21.2.13.

Theorem 21.2.17 Compatibility of the stationary family of support algebras

✓

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

Theorem 21.2.18 Compatible support algebras do not imply an idempotent transfer map

✓

There exists an MPO tensor of physical and bond dimension two whose doubled-tensor transfer map is trace-preserving and has a positive-definite fixed point, and which has a compatible family of support algebras, although its doubled-tensor transfer map is not idempotent.

Proof ▶

Take the phase-flip channel with Kraus operators $K_0=\frac35 \mathbb {1}$ and $K_1=\frac45 Z$, where $Z=\operatorname{diag}(1,-1)$. Its transfer map satisfies

\begin{align} E(X)_{ij} & = \begin{cases} X_{ij}, & i=j, \\ -\frac7{25}X_{ij}, & i\neq j. \end{cases} \notag \end{align}

The identity is a positive-definite fixed point, and $K_0^\dagger K_0+K_1^\dagger K_1=\mathbb {1}$ proves trace preservation. 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 a compatible family of support algebras exists. On the other hand, for $X_{01}\ne 0$, $E^2(X)_{01}=\frac{49}{625}X_{01}\neq -\frac7{25}X_{01}=E(X)_{01}$. Therefore $E^2\neq E$, so the doubled-tensor ZCL condition fails.

Remark 21.2.19 Stabilized fixed points and the vertical algebra condition

Write $E=E_1$ for the one-site transfer map. Compatibility gives $\operatorname{Fix}((E^n)^\dagger )=\operatorname{Fix}(E^\dagger )$ for every $n{\gt}0$. Hence a nonzero matrix $X$ 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 21.2.15. Compatibility does not imply $E^2=E$, as Theorem 21.2.18 shows. More generally, on $M_{D}(\mathbb {C})$ with $D\ge 2$, let $\Pi _{\mathrm{diag}}$ be the projection onto diagonal matrices. For $0{\lt}\varepsilon {\lt}1$, set $E(X)=\varepsilon X+(1-\varepsilon )\Pi _{\mathrm{diag}}(X)$, so that

\begin{align} E^n(X) & = \Pi _{\mathrm{diag}}(X) + \varepsilon ^n(X-\Pi _{\mathrm{diag}}(X)), \notag \\ E^2-E & = (\varepsilon ^2-\varepsilon ) (\operatorname{id}-\Pi _{\mathrm{diag}}). \notag \end{align}

The fixed-point algebra is diagonal for every $n{\gt}0$, but $E^2\ne E$.

The coefficient argument in [ 13 , Appendix C.4 ] instead uses the vertical canonical form of an MPDO tensor $M$ in CF. Write its positive diagonal multiplicity matrices as $\mu _\alpha $ and its BNT as $\{ M_\alpha \} _\alpha $. The operators $O_L(M_\alpha )$ are obtained by contracting $L$ tensors in the vertical direction, with the original physical and virtual roles exchanged. The required same-length product law is

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L}) O_L(M_\gamma ), \label{eq:mpdo_chi_product_law} \end{align}

where the $\chi _{\alpha ,\beta ,\gamma }$ are positive diagonal matrices. With $m_\alpha =\operatorname{tr}(\mu _\alpha )$, the vector $m$ must be an idempotent for the multiplication induced by $c^{(1)}$, namely

\begin{align} m_\gamma & = \sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \label{eq:mpdo_idempotent_trace_identity} \end{align}

The proof of [ 13 , Theorem 4.14 ] uses (3) and (4), positivity, and the simple-matrix lemma [ 13 , lines 1155–1163 ] to identify the diagonal entries of the two-site multiplicity matrix $\nu _\gamma $ as a multiset:

\begin{align} \{ \nu _{\gamma ,k}\} _k & = \{ \mu _{\alpha ,k_1}\mu _{\beta ,k_2} \chi _{\alpha ,\beta ,\gamma ,k_3}\} _{\alpha ,\beta ,k_1,k_2,k_3}, \notag \\ \operatorname{tr}(\nu _\gamma ) & =m_\gamma . \label{eq:mpdo_nu_chi_witness} \end{align}

The reconstruction maps in [ 13 , lines 2065–2085 ] then use

\begin{align} \widetilde R_2\left(\bigoplus _\gamma X_\gamma \right) & = \bigoplus _\gamma \frac{\nu _\gamma }{m_\gamma }\otimes X_\gamma , \notag \\ \widetilde R_1\left(\bigoplus _\gamma X_\gamma \right) & = \bigoplus _\gamma \frac{\mu _\gamma }{m_\gamma }\otimes X_\gamma . \label{eq:mpdo_reconstruction_maps} \end{align}

Thus $\operatorname{tr}(\nu _\gamma )=m_\gamma $ and $m_\gamma =\operatorname{tr}(\mu _\gamma )$ normalize the factors $\nu _\gamma /m_\gamma $ and $\mu _\gamma /m_\gamma $, respectively, in $T=\widetilde R_2\widetilde T R_1$ and $S=\widetilde R_1\widetilde S R_2$. Here $R_1$ and $R_2$ compress to the canonical sectors and trace out their multiplicity factors, while $\widetilde T$ and $\widetilde S$ conjugate by the block unitaries relating the BNTs. These compositions are tpCPM on the retained spaces. When discarded zero-sector complements are present, extend the compressions by trace-and-prepare branches on the discarded input supports to obtain tpCPM on the full physical spaces. These additional branches vanish on the tensor letters, so the extended maps still satisfy $T[M_1(X)]=M_2(X)$ and $S[M_2(X)]=M_1(X)$, as required for an RFP.

The scalar Newton–Girard identity relates traces of matrix powers to characteristic-polynomial coefficients; see the corresponding theorem in the companion quantum-channel volume [ 216 , “Newton–Girard trace recursion” ] . Such scalar identities do not supply the vertical canonical decomposition. The trace-power conclusion of [ 13 , Theorem 4.14(ii) ] concerns a common BNT and positive diagonal matrices satisfying (3) and (4). Relating these objects to the one-site and two-site vertical canonical forms is required for the maps in (6); it does not follow from a choice of bases in the support algebras. In particular, Theorem 21.2.18 rules out deriving idempotence of the doubled-tensor transfer map from compatibility alone.

21.2.1 Diagonal $\chi $-matrices and the trace-power formula

In [ 13 , Theorem 4.14(ii) ] , the structure coefficients of the length-$L$ operator algebra are traces of powers of positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$. In Definition 21.2.1.1, arbitrary complex diagonal entries are permitted to separate the scalar trace-power identity from positivity and operator multiplication.

Definition 21.2.1.1 Diagonal $\chi $ matrices

✓

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

\begin{align} \chi _{\alpha ,\beta ,\gamma } & = \operatorname{diag}\bigl( (\chi _{\alpha ,\beta ,\gamma ,k})_{0\le k{\lt}r_{\alpha ,\beta ,\gamma }} \bigr). \notag \end{align}

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

Theorem 21.2.1.2 Trace of a power of a diagonal matrix

✓

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

\begin{align} \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) & = \sum _{0\le k{\lt}r_{\alpha ,\beta ,\gamma }} \chi _{\alpha ,\beta ,\gamma ,k}^{L}, \notag \end{align}

where $r_{\alpha ,\beta ,\gamma }$ is the size of $\chi _{\alpha ,\beta ,\gamma }$ and $\chi _{\alpha ,\beta ,\gamma ,k}$ are its diagonal entries.

Proof ▶

Diagonal matrices raise to powers entrywise and the trace of a diagonal matrix is the sum of its entries.

Definition 21.2.1.3 Trace-power form of structure coefficients

✓

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

\begin{align} c^{(L)}_{\alpha ,\beta ,\gamma } & = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) = \sum _{0\le k{\lt}r_{\alpha ,\beta ,\gamma }} \chi _{\alpha ,\beta ,\gamma ,k}^{L}. \notag \end{align}

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

21.2.2 Structure coefficients and the idempotent condition

Definition 21.2.2.1 Structure coefficients indexed by a BNT

✓

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

Definition 21.2.2.2 Operators indexed by a BNT

✓

For each length $L$, let $O_L(M_\alpha )$ be operators indexed by the fixed BNT labels $\alpha $. Their ambient complex vector space, equipped with multiplication, may depend on $L$ and is not identified with a chosen blocked support algebra. Write $\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}$ for the vertical reading. The vertical indices of the $L$ copies are contracted cyclically, while the horizontal legs remain open, as in [ 13 , lines 962–967 ] . For $\mathbf a=(a_1,\ldots ,a_L)$ and $\mathbf b=(b_1,\ldots ,b_L)$,

\begin{align} [O_L(M_\alpha )]_{\mathbf a,\mathbf b} & = \operatorname{tr}\! \left(\widehat M_\alpha ^{a_1b_1}\cdots \widehat M_\alpha ^{a_Lb_L}\right). \notag \end{align}

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

$\begin{tenkz}[rows={op}, cols=4, boundary=periodic] \tn[skin=mpo, size=l, ports={90:physical:$a_1$, 270:physical:$b_1$}]{\widehat M_\alpha} & \tn[skin=mpo, size=l, ports={90:physical:$a_2$, 270:physical:$b_2$}]{\widehat M_\alpha} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, ports={90:physical:$a_L$, 270:physical:$b_L$}]{\widehat M_\alpha} \end{tenkz}$

Definition 21.2.2.3 Product of BNT operators at the same length

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma } O_L(M_\gamma ). \notag \end{align}

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

$\begin{tenkz}[rows={op,op}, cols=4, boundary=periodic, bonds=none] \tn[skin=mpo, size=l, name=a1, ports={90:physical:$a_1$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_\alpha} & \tn[skin=mpo, size=l, name=a2, ports={90:physical:$a_2$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_\alpha} & \tn[skin=dots, name=ad, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, name=aL, ports={90:physical:$a_L$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_\alpha} \\ \tn[skin=mpo, size=l, name=b1, ports={90:physical, 270:physical:$b_1$, 180:virtual, 0:virtual}]{\widehat M_\beta} & \tn[skin=mpo, size=l, name=b2, ports={90:physical, 270:physical:$b_2$, 180:virtual, 0:virtual}]{\widehat M_\beta} & \tn[skin=dots, name=bd, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, name=bL, ports={90:physical, 270:physical:$b_L$, 180:virtual, 0:virtual}]{\widehat M_\beta} \tnwire{a1.0}{a2.180} \tnwire{a2.0}{ad.180} \tnwire{ad.0}{aL.180} \tnwire{b1.0}{b2.180} \tnwire{b2.0}{bd.180} \tnwire{bd.0}{bL.180} \tnwire{a1.270}{b1.90} \tnwire{a2.270}{b2.90} \tnwire{aL.270}{bL.90} \end{tenkz}$ $=\displaystyle \sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }$ $\begin{tenkz}[rows={op}, cols=4, boundary=periodic] \tn[skin=mpo, size=l, ports={90:physical:$a_1$, 270:physical:$b_1$}]{\widehat M_\gamma} & \tn[skin=mpo, size=l, ports={90:physical:$a_2$, 270:physical:$b_2$}]{\widehat M_\gamma} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, ports={90:physical:$a_L$, 270:physical:$b_L$}]{\widehat M_\gamma} \end{tenkz}$

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

Definition 21.2.2.4 Scalars indexed by a BNT

✓

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

Definition 21.2.2.5 Idempotent for the multiplication induced by the coefficients

✓

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

\begin{align} m_\gamma & = \sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta . \notag \end{align}

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

Definition 21.2.2.6 Assignment of BNT labels to blocked bases

✓

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

Definition 21.2.2.7 Comparison of BNT structure coefficients with blocked coefficients

✓

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

\begin{align} c^{(n)}_{i,j,k} & = c^{(n)}_{\sigma _n(i),\sigma _n(j),\tau _n(k)} \notag \end{align}

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

Definition 21.2.2.8 Trace-power formula at positive lengths

✓

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

\begin{align} c^{(L)}_{\alpha ,\beta ,\gamma } & = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}). \notag \end{align}

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

Lemma 21.2.2.9 Structure coefficient as a trace power at positive length

✓

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

Proof ▶

Combine the defining positive-length trace-power identity with the trace formula for powers of a diagonal matrix.

Lemma 21.2.2.10 Coefficients defined by traces of diagonal matrix powers

✓

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

Proof ▶

This follows from the definition of the coefficients.

Definition 21.2.2.11 Positive diagonal matrices giving the structure coefficients

✓

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

Lemma 21.2.2.12 Trace powers of the positive diagonal matrices

✓

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

Proof ▶

Apply the trace reformulation of the trace-power identity at positive lengths.

Theorem 21.2.2.13 Product of BNT operators with trace-power coefficients

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L}) O_L(M_\gamma ). \notag \end{align}

Proof ▶

Substitute the positive-length trace-power formula into the same-length BNT product expansion.

Theorem 21.2.2.14 Product of BNT operators for coefficients defined by traces

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L}) O_L(M_\gamma ). \notag \end{align}

Proof ▶

Substitute the defining trace-power formula for the coefficients into the same-length BNT product expansion.

Theorem 21.2.2.15 Idempotent condition in terms of traces

✓

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

\begin{align} m_\gamma & = \sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \notag \end{align}

Proof ▶

Substitute the length-one trace-power formula into the idempotent coefficient identity.

Theorem 21.2.2.16 Idempotent condition for coefficients defined by traces

✓

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

\begin{align} m_\gamma & = \sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \notag \end{align}

Proof ▶

Substitute the defining length-one trace formula into the idempotent coefficient identity.

Lemma 21.2.2.17 Blocked coefficients inherit BNT trace powers

✓

\begin{align} c^{(n)}_{i,j,k} & = \operatorname{tr}\! \left(\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}^{\, n}\right). \notag \end{align}

Proof ▶

Use the comparison equality to replace the blocked-basis coefficient by the corresponding BNT structure coefficient, and then apply the trace-power formula. In the special case where the compared BNT coefficients are defined by traces of powers of $\chi $, the same conclusion follows directly from their definition.

Definition 21.2.2.18 Diagonal matrices for blocked multiplication coefficients

✓

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

\begin{align} \chi _{n,i,j,k} & = \operatorname{diag}(\chi _{n,i,j,k,s})_{s=0}^{r_{n,i,j,k}-1}. \notag \end{align}

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

Lemma 21.2.2.19 Trace of a power of a blocked diagonal matrix

✓

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

Proof ▶

Diagonal matrices raise to powers entrywise and the trace of a diagonal matrix is the sum of its entries.

Definition 21.2.2.20 Trace-power form for blocked multiplication coefficients

✓

The blocked multiplication coefficients are in trace-power form when, for every positive blocking length $n$ and every multiplication triple $(i,j,k)$, the coefficient $c^{(n)}_{i,j,k}$ equals $\sum _{s=0}^{r_{n,i,j,k}-1} \chi _{n,i,j,k,s}^{\, n}$. Equivalently, by Lemma 21.2.2.19, it is the trace of the $n$-th power of the corresponding diagonal matrix. The exponent $n$ is the blocking length of the two factors in $\mathcal A_n$; the product is expanded in the basis of $\mathcal A_{2n}$.

Lemma 21.2.2.21 Blocked coefficient as a trace power

✓

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

Proof ▶

Combine the defining trace-power identity with the trace formula for powers of a diagonal matrix.

Definition 21.2.2.22 Positive diagonal matrices giving the blocked coefficients

✓

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

Theorem 21.2.2.23 Positive BNT diagonal matrices give positive blocked diagonal matrices

✓

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

Proof ▶

Reindex the positive diagonal matrices 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 trace-power formula.

Lemma 21.2.2.24 Constant power sums have entries zero or one

✓

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

Proof ▶

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.

Definition 21.2.2.25 Length-independent coefficients

✓

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

Theorem 21.2.2.26 Length-independent coefficients are natural numbers

✓

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

Proof ▶

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 21.2.2.24 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.

Theorem 21.2.2.27 Unit entries give length independence

✓

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

Proof ▶

The power sums of a family of ones do not depend on the exponent.

Theorem 21.2.2.28 Three consecutive positive power sums

✓

Let $x_0,\ldots ,x_{r-1}{\gt}0$ and $p_n=\sum _k x_k^n$. For every integer $n\geq 0$,

\begin{align} p_{n+2}-2p_{n+1}+p_n =\sum _k x_k^n(x_k-1)^2. \label{eq:mpdo_positive_second_difference} \end{align}

The second difference vanishes if and only if every $x_k$ equals one. In particular, equality of three consecutive trace powers of a positive diagonal matrix forces that matrix to be the identity.

Proof ▶

Expanding the squares gives (7). Every summand is nonnegative, and every factor $x_k^n$ is strictly positive. Thus the sum vanishes exactly when all $x_k-1$ vanish.

Theorem 21.2.2.29 Finite criterion for length independence

✓

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

Proof ▶

Length independence implies both equalities. Conversely, apply Theorem 21.2.2.28 with $n=1$ to each diagonal matrix. All its entries equal one, so Theorem 21.2.2.27 gives length independence.

Remark 21.2.2.30 Topological order and length dependence

Starting from the same-length product law and the condition that the vector $m$ is an idempotent for the multiplication induced by $c^{(1)}$, 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 [ 8 ] . The RFP density operators correspond to the boundary theories of the RFP representatives of those phases, which yields a large family of non-trivial examples. The source asks whether there exist RFPs whose structure coefficients depend on the length [ 13 , Section 4.5 ] . The example below gives such coefficients for its displayed BNT presentation.

Theorem 21.2.2.31 An RFP with length-dependent BNT structure coefficients

✓

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

Proof ▶

The bond-dimension-two tensor with a one-element vertical basis of normal tensors generates positive operators on every nonempty chain, is in canonical form, and satisfies the displayed multiplication law and the corresponding tensor decomposition. The reverse implication of Theorem 21.2.6.3 supplies the two trace-preserving completely positive maps required for a renormalization fixed point. Finally, $\sqrt{1/2}\ne 1/2=(\sqrt{1/2})^2$, which proves the length dependence.

Theorem 21.2.2.32 Length dependence under every uniform positive rescaling

✓

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

\begin{align} A^0 & =\frac45E_{00}, & A^1 & =\frac45E_{11}, & A^2 & =\frac35E_{01}, & A^3 & =\frac35E_{10}. \notag \end{align}

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

\begin{align} R^{pq}_{ab} & =\frac{25}{32} \left(A^a\otimes \overline{A^b}\right)_{p,q}. \notag \end{align}

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

\begin{align} \mathcal T_R & =\frac{25}{32}|t\rangle \! \langle t|, & t & =\left(\frac45,\frac45,0,0\right), \notag \end{align}

where $t_a=\operatorname{tr}(A^a)$. It satisfies $\mathcal T_R^2=\mathcal T_R$, the zero-correlation-length condition of [ 13 , Definition 4.2, lines 736–741 ] . Define the positive diagonal matrix $\chi =\operatorname{diag}(1,\frac{7}{25})$ and its trace powers by

\begin{align} c^{(L)} & =\operatorname{tr}(\chi ^L)=1+\left(\frac{7}{25}\right)^L. \notag \end{align}

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

\begin{align} c_s^{(L)} & =\operatorname{tr}\! \left((s\chi )^L\right) =s^L\left(1+\left(\frac{7}{25}\right)^L\right), \notag \end{align}

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

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

Proof ▶

The $(a,b)$ entry of $\mathcal T_R$ is $\frac{25}{32}\operatorname{tr}(A^a)\operatorname{tr}(A^b)$, and $\sum _a\operatorname{tr}(A^a)^2=2\cdot (\frac45)^2=\frac{32}{25}$, so $\mathcal T_R^2=\mathcal T_R$. The formulas for $c^{(L)}$ and $c_s^{(L)}$ follow by summing the powers of the diagonal entries of $\chi $ and $s\chi $. Write $\lambda =\frac{7}{25}$. If $c_s$ were length independent, comparing lengths one and two and lengths two and three would give

\begin{align} s & =\frac{1+\lambda }{1+\lambda ^2}, \notag \\ s & =\frac{1+\lambda ^2}{1+\lambda ^3}. \notag \end{align}

Hence $(1+\lambda )(1+\lambda ^3)=(1+\lambda ^2)^2$, that is, $\lambda (1-\lambda )^2=0$. This fails at $\lambda =\frac{7}{25}$.

Definition 21.2.2.33 The $\mathbb {Z}_2$-twisted quantum dimer

✓

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

\begin{align} M^{(l,r,f),(l',r',f')} & =\sum _{k\in \{ 0,1\} }\frac12\, C_k[l,l’]\, (\tau _k)_{ff} \delta _{ff'}\, E_{(l,l',k),(r,r',k)}. \notag \end{align}

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

Theorem 21.2.2.34 Closures and physical-trace transfer of the twisted dimer

✓

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

\begin{align} [M_1(X)]_{(l,r,f),(l',r',f')} & =\sum _k\frac12\, C_k[l,l’]\, (\tau _k)_{ff} \delta _{ff'} X_{(r,r',k),(l,l',k)}. \notag \end{align}

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

Proof ▶

The trace of a matrix unit against $X$ picks one entry of $X$, and the product of two matrix units is a matrix unit or zero according to whether the inner indices agree. Since the block label is part of the bond index, only common labels survive. Summing the diagonal physical indices over the flag value gives $\sum _f(\tau _k)_{ff}$, which is $2$ for $k=0$ and $0$ for $k=1$.

Theorem 21.2.2.35 Positivity of the twisted quantum dimer

✓

The closed operator $\rho ^{(N)}(M)$ of the twisted quantum dimer is positive semidefinite for every positive chain length $N$, so $M$ generates MPDOs. Writing $l_n$, $r_n$, $f_n$ for the three bits at site $n$ of a physical string $\sigma $, and using primes for those of $\tau $, the matrix element is

\begin{align} \rho ^{(N)}(M)_{\sigma \tau } & =\frac1{2^N} \mathbb {1}[\text{cyclic bond match}](\sigma ,\tau ) \delta _{f f'}\left(\prod _n C_0[l_n,l’_n] +(-1)^{\lvert f\rvert }\prod _n C_1[l_n,l’_n]\right). \notag \end{align}

The indicator is one exactly when both $\sigma $ and $\tau $ match their bond bits around the ring, $\delta _{ff'}$ is one exactly when the two flag strings agree, and $\lvert f\rvert $ is the number of sites carrying flag one. Thus $\rho ^{(N)}(M)$ is $2^{-N}$ times the Hadamard product of the rank-one bond-matching indicator with the pullback, along the flag and left bits, of the sum of the $N$-fold Kronecker powers of the two local four-by-four factors $g_k[(f,l),(f',l')]=\delta _{ff'}\, C_k[l,l']\, (\tau _k)_{ff}$. The Schur product theorem then gives positivity.

Proof ▶

Each bond matrix is a sum of two matrix units, one per block label, so an ordered product keeps a common block label and vanishes unless consecutive physical indices match on their bond bits. The surviving product is a single matrix unit whose row is determined by the left bits at the first site and whose column is determined by the right bits at the last site. Taking the trace imposes the wraparound match and leaves the sum over the block label of the product of the one-site coefficients. Each coefficient is half of $g_k$ evaluated at the flag and left bits, so the product along the ring is $2^{-N}$ times an entry of the $N$-fold Kronecker power of $g_k$.

Neither the sum nor the difference of the two Kronecker powers is a Kronecker power, so both are proved positive semidefinite together by induction on the chain length. Separating the last tensor factor gives

\begin{align} g_0^{\otimes (N+1)}\pm g_1^{\otimes (N+1)} & =\frac12\Bigl[ (g_0^{\otimes N}+g_1^{\otimes N})\otimes (g_0\pm g_1) +(g_0^{\otimes N}-g_1^{\otimes N})\otimes (g_0\mp g_1) \Bigr]. \notag \end{align}

Both local combinations are positive semidefinite: each is block diagonal in the flag bit, with $C_0+C_1=\frac74P_+$ in one flag sector and $C_0-C_1=\frac14P_-$ in the other, the two sectors being exchanged between the sum and the difference. A Kronecker product of positive semidefinite matrices is positive semidefinite, and so are sums, nonnegative multiples, and pullbacks along an arbitrary reindexing.

Definition 21.2.2.36 Controlled Bell-sign flip

✓

On two bond qubits let

\begin{align} |\Phi _\pm \rangle & =\frac{|00\rangle \pm |11\rangle }{\sqrt2}, \notag \\ \Pi _\pm & =|\Phi _\pm \rangle \! \langle \Phi _\pm |. \notag \end{align}

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

\begin{align} V & =(\mathbb {1}-\Pi _-)\otimes \mathbb {1}+\Pi _-\otimes X. \notag \end{align}

Lemma 21.2.2.37 Bond-state matrix entries

✓

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

\begin{align} \langle r,l|\sigma _k|r’,l’\rangle & = \begin{cases} C_k[l,l’] & \text{if } r=l \text{ and } r’=l’, \\ 0 & \text{otherwise}. \end{cases} \notag \end{align}

Proof ▶

Expand the two Bell projections in the computational basis.

Lemma 21.2.2.38 Local unitary identities

✓

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

\begin{align} V(\sigma \otimes \mathbb {1})V^\dagger & =\sigma \otimes \mathbb {1}, \notag \\ V(\sigma \otimes Z)V^\dagger & =\sigma ’\otimes Z. \notag \end{align}

Proof ▶

Evaluate the $8\times 8$ matrices entry by entry in the computational basis. This gives $V^\dagger =V$, $V^2=\mathbb {1}$, $V(\sigma \otimes \mathbb {1})=(\sigma \otimes \mathbb {1})V$, and $V(\sigma \otimes Z)=(\sigma '\otimes Z)V$. Multiplying each intertwining identity on the right by $V^\dagger $ gives the stated conjugation identities.

Definition 21.2.2.39 Bond–flag regrouping

✓

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

\begin{align} \omega _N & =2^{-N}(\mathbb {1}^{\otimes N}+Z^{\otimes N}), \notag \\ D_N & =E_N^\dagger S_N^\dagger (\sigma ^{\otimes N}\otimes \omega _N)S_NE_N, \notag \\ U_N & =E_N^\dagger V^{\otimes N}E_N. \notag \end{align}

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

Theorem 21.2.2.40 Unitary bond–flag factorization

✓

# Discussion

For every positive length $N$, the operator $U_N$ of Definition 21.2.2.39 is self-adjoint and obeys $U_NU_N^\dagger =U_N^\dagger U_N=\mathbb {1}$. The closed operator of the twisted quantum dimer satisfies $\rho ^{(N)}(M)=U_ND_NU_N^\dagger $.

Proof ▶

Take tensor products of the identities in Lemma 21.2.2.38. After separating the bond and flag strings, we obtain

\begin{align} S_NE_NU_ND_NU_N^\dagger E_N^\dagger S_N^\dagger & =2^{-N}\left(\sigma ^{\otimes N}\otimes \mathbb {1}^{\otimes N} +(\sigma ’)^{\otimes N}\otimes Z^{\otimes N}\right). \notag \end{align}

By Lemma 21.2.2.37, both bond operators vanish outside the span of $|00\rangle $ and $|11\rangle $; on this span their matrices are $C_0$ and $C_1$, respectively. In physical coordinates the entries therefore equal the cyclic bond-matching formula of Theorem 21.2.2.35.

The gates in this factorization act on an incoming bond and its receiving flag across neighboring physical sites. After the unitary conjugation by $U_N$, the bond and flag factors decouple, yielding the state-level product form.

Theorem 21.2.2.41 Identification of the flag factor

✓

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

Proof ▶

Substitute the identity and Pauli $Z$ matrices in the definition of $\omega _N$. Its expression $2^{-N}(\mathbb {1}^{\otimes N}+Z^{\otimes N})$ is the closed operator of $\widehat M$ by Theorem 20.16.8.53 and the factor $2^{-N}$; the empty case is evaluated directly. Positivity, normalization, and the two channel equations follow from Theorem 20.16.8.61.

Theorem 21.2.2.42 Normalization of the bond–flag factorization

✓

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

Proof ▶

The Bell projectors are positive rank-one operators. By Lemma 21.2.2.37, $\operatorname{tr}\sigma =C_0[0,0]+C_0[1,1]=1$. Tensor products preserve positivity, and their traces multiply. Unitary changes of coordinates preserve both positivity and trace. Apply these facts to $D_N$ and then use Theorem 21.2.2.40.

The flag factor is identified using the density-normalized tensor $\widehat M$; the constant multiplication coefficients for the unnormalized matrices $\mathbb {1},Z$ differ by scaling from those for the spectrally normalized representatives $\mathbb {1}/\sqrt2,Z/\sqrt2$.

Definition 21.2.2.43 The mixed-Bell bond dimer

✓

Let

\begin{align} C & =\begin{pmatrix} 1/2 & 3/8 \\ 3/8 & 1/2 \end{pmatrix}. \notag \end{align}

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

\begin{align} Q^{(l,r),(l',r')} & =C_{ll'}E_{(l,l'),(r,r')}, \notag \end{align}

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

\begin{align} B_N|((l_m,r_m))_m\rangle & =|((r_{m-1},l_m))_m\rangle , \notag \end{align}

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

Theorem 21.2.2.44 Closed operators of the mixed-Bell dimer

✓

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

\begin{align} \rho ^{(N)}(Q) & =B_N^\dagger \sigma ^{\otimes N}B_N. \notag \end{align}

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

$\begin{tenkz}[rows={op}, cols=7, boundary=periodic, size=l] \tn[at=(1,1), skin=mpo, wide=2, ports={90@1:physical:$l_1$, 90@2:physical:$r_1$, 270@1:physical, 270@2:physical}]{Q} \tn[at=(1,3), skin=mpo, wide=2, ports={90@1:physical:$l_2$, 90@2:physical:$r_2$, 270@1:physical, 270@2:physical}]{Q} \tn[at=(1,5), skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,6), skin=mpo, wide=2, ports={90@1:physical:$l_N$, 90@2:physical:$r_N$, 270@1:physical, 270@2:physical}]{Q} \end{tenkz}$
$=$ $\begin{tenkz}[rows={op}, cols=7, bonds=none, size=l] \tn[at=(1,1), skin=box, wide=2, ports={90@1:physical:$r_N$, 90@2:physical:$l_1$, 270@1:physical, 270@2:physical}]{\sigma} \tn[at=(1,3), skin=box, wide=2, ports={90@1:physical:$r_1$, 90@2:physical:$l_2$, 270@1:physical, 270@2:physical}]{\sigma} \tn[at=(1,5), skin=dots]{} \tn[at=(1,6), skin=box, wide=2, ports={90@1:physical:$r_{N-1}$, 90@2:physical:$l_N$, 270@1:physical, 270@2:physical}]{\sigma} \end{tenkz}$

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

Proof ▶

Fix independent configurations $s_m=(l_m,r_m)$ and $t_m=(l'_m,r'_m)$, with $m=0,\ldots ,N-1$ and cyclic successors. The cyclic trace expansion of Lemma 21.2.2.56 has only one possibly nonzero virtual path, namely $(l_m,l'_m)_m$. Hence

\begin{align} \operatorname{tr}\! \left(\prod _{m=0}^{N-1}Q^{(l_m,r_m),(l'_m,r'_m)}\right) & =\begin{cases} \prod _{m=0}^{N-1}C_{l_m l'_m} & \text{if } r_m=l_{m+1} \text{ and } r’_m=l’_{m+1} \text{ for every }m, \\ 0 & \text{otherwise}. \end{cases} \label{eq:mpdo_dimer_closed_entries} \end{align}

By Lemma 21.2.2.37, the $(s,t)$ entry of $B_N^\dagger \sigma ^{\otimes N}B_N$ is $\prod _m\sigma _{(r_{m-1},l_m),(r'_{m-1},l'_m)}$. Its matching conditions and value agree with (12) after cyclic relabeling. Positivity and trace one follow from Theorem 21.2.2.42.

Definition 21.2.2.45 The normalized mixed-Bell representative

✓

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

Theorem 21.2.2.46 One-site injectivity of the mixed-Bell factor

✓

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

Proof ▶

Each matrix unit $E_{(l,l'),(r,r')}$ is a nonzero scalar multiple of $Q^{(l,r),(l',r')}$, since every entry of $C$ is nonzero. Summing the adjoint squares gives the stated scalar identity: the coefficient at each diagonal matrix unit is $2(1/2)^2+2(3/8)^2=25/32$.

Lemma 21.2.2.47 Normality and trace of the mixed-Bell representative

✓

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

Proof ▶

The adjoint-square sum of $A$ is $c^2(25/32)\mathbb {1}=\mathbb {1}$. Nonzero scalar multiplication preserves injectivity, so Theorem 7.6.3 gives normality. The one-site density operator of $Q$ has trace one; hence $\operatorname{tr}\mathcal B_A=c\operatorname{tr}\mathcal B_Q=c$.

Theorem 21.2.2.48 Simplicity of the exact mixed-Bell factor

✓

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

Proof ▶

After one-site blocking, relabel the ket-bra alphabet and use the single representative $A$, with one copy of weight $c^{-1}$. The scalar word law recovers the positive-length matrix product vectors of $Q$. The singleton normal family is eventually linearly independent by Theorem 8.2.13. A nilpotent complex matrix has zero trace, whereas $\operatorname{tr}\mathcal B_A=c\ne 0$. Thus the only representative has nonnilpotent physical-trace transfer, as required.

Definition 21.2.2.49 Bond insertion and removal

✓

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

\begin{align} J|(l,r),(u,v)\rangle & =|(l,u),(v,r)\rangle . \notag \end{align}

Define the refinement and coarse-graining maps by

\begin{align} \mathcal T_Q(Y) & =J(Y\otimes \sigma )J^\dagger , \notag \\ \mathcal S_Q(W) & =\operatorname{tr}_{\mathrm{bond}}(J^\dagger WJ). \notag \end{align}

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

Theorem 21.2.2.50 Channel equations for the mixed-Bell dimer

✓

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

\begin{align} \mathcal T_Q(Q_1(X)) & =Q_2(X), \notag \\ \mathcal S_Q(Q_2(X)) & =Q_1(X). \notag \end{align}

Thus $Q$ is a renormalization fixed point in the sense of Definition 20.2.1.

Proof ▶

Preparation of the positive trace-one state $\sigma $, permutation of basis coordinates, and partial trace are trace-preserving completely positive maps. Tracing out the prepared state proves $\mathcal S_Q(\mathcal T_Q(Y))=Y$. For the refinement equation, fix independent ket and bra indices $l,r,l',r',u,v,u',v'\in \{ 0,1\} $. Multiplication of matrix units gives

\begin{align} Q^{(l,r),(l',r')}Q^{(u,v),(u',v')} & =\begin{cases} C_{ll'}C_{uu'}E_{(l,l'),(v,v')} & \text{if } r=u \text{ and } r’=u’, \\ 0 & \text{otherwise}. \end{cases} \notag \end{align}

For every boundary matrix $X\in M_{4}(\mathbb {C})$, the one-site closure has entries $[Q_1(X)]_{(l,r),(l',r')}=C_{ll'}X_{(r,r'),(l,l')}$. Put $a=((l,r),(u,v))$ and $b=((l',r'),(u',v'))$. The definition of the insertion map and Lemma 21.2.2.37 give

\begin{align} [\mathcal T_Q(Q_1(X))]_{a,b} & =[Q_1(X)]_{(l,v),(l',v')}\sigma _{(r,u),(r',u')} \notag \\ & =\begin{cases} C_{ll'}C_{uu'}X_{(v,v'),(l,l')} & \text{if } r=u \text{ and } r’=u’, \\ 0 & \text{otherwise} \end{cases} \notag \\ & =[Q_2(X)]_{a,b}. \notag \end{align}

Thus the refinement equation holds for arbitrary boundary matrices. Applying $\mathcal S_Q\mathcal T_Q=\operatorname{id}$ gives the coarse-graining equation.

These channel equations establish that the mixed-Bell bond dimer is a renormalization fixed point, with nonzero bond-state eigenvalues $7/8$ and $1/8$.

Definition 21.2.2.51 On-site separation of bond qubits and flags

✓

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

\begin{align} H_N|((l_m,r_m,f_m))_m\rangle & =|((l_m,r_m))_m\rangle \otimes |(f_m)_m\rangle . \notag \end{align}

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

Theorem 21.2.2.52 Factorization into two RFP families

✓

Let $Q$ be the mixed-Bell bond tensor and let $\widehat M$ be the density-normalized tensor identified in Theorem 21.2.2.41. For every $N{\gt}0$,

\begin{align} D_N & =H_N^\dagger (\rho ^{(N)}(Q)\otimes \rho ^{(N)}(\widehat M))H_N, \notag \\ \rho ^{(N)}(M) & =U_NH_N^\dagger (\rho ^{(N)}(Q)\otimes \rho ^{(N)}(\widehat M)) H_NU_N^\dagger . \notag \end{align}

Both factor tensors generate MPDOs and are RFPs, by Theorems 21.2.2.44, 21.2.2.50, and 21.2.2.41.

Proof ▶

The coordinate permutations satisfy $(B_N\otimes \mathbb {1})H_N=S_NE_N$: both read the incoming bonds $(r_{m-1},l_m)$ and the unchanged flags $f_m$. Substitute the bond-product formula and the flag identification into the definition of $D_N$. The formula for $\rho ^{(N)}(M)$ then follows from Theorem 21.2.2.40. The RFP channel equations for the two factors were proved independently in the cited theorems.

The separation $H_N$ is on-site, while the conjugating operator $U_N$ couples neighboring sites; at positive chain length $N \ge 1$, this decouples the bond qubits from the site flags.

Theorem 21.2.2.53 The twisted quantum dimer is a renormalization fixed point

✓

The twisted quantum dimer of Definition 21.2.2.33 is a renormalization fixed point in the sense of Definition 20.2.1. The refinement map $\mathcal T$ is completely positive and trace preserving, with eight Kraus operators, labelled by the two prepared site flags $f_1,f_2$ and a bond label $\varepsilon $. It prepares those two flags on the output sites together with the bond state $(|00\rangle +(-1)^\varepsilon |11\rangle )/\sqrt2$ on their shared bond, with weight $x$ for $\varepsilon =0$ and $y$ for $\varepsilon =1$, and is supported on the incoming flag $f=f_1+f_2+\varepsilon $. Flag addition is modulo two. The coarse-graining map $\mathcal S$ is completely positive and trace preserving, with sixteen Kraus operators, labelled by a member of the orthonormal bond basis $(|00\rangle \pm |11\rangle )/\sqrt2$, $|01\rangle $, $|10\rangle $ together with the two site flags. It measures the shared bond pair in that basis, reads the two flags, and returns the outer site qubits with flag $f_1+f_2+s$, where $s$ is one for the antisymmetric bond outcome and zero otherwise. These maps satisfy $\mathcal T[M_1(X)]=M_2(X)$ and $\mathcal S[M_2(X)]=M_1(X)$ for every bond matrix $X$. This example is motivated by the question of [ 13 , lines 995–1010 ] .

Proof ▶

Fix a one-site basis index $(l,r,f)$. The two-site indices in the support of a refinement Kraus operator carry the outer bits $l$ and $r$, the two prepared flags, and a free shared bond bit. Each of the four labels with $f_1+f_2+\varepsilon =f$ therefore contributes twice the squared amplitude $w_\varepsilon /4$, where $w_0=x$ and $w_1=y$. Two of the four labels carry each value of $\varepsilon $, and the total is $x+y=1$. Supports belonging to distinct one-site basis indices are disjoint, so the off-diagonal entries vanish and the Kraus operators resolve the identity. For the intertwining relation, the flag signs multiply as $(\tau _k)_{f_1f_1}(\tau _k)_{f_2f_2}=(\tau _k)_{gg}$ with $g=f_1+f_2$. For shared bond bits $b,b'$, the two prepared bond states give

\begin{align} \frac{x}{4}+(-1)^{b+b'}\frac{y}{4} & =\frac12\, C_0[b,b’], \notag \\ \frac{x}{4}-(-1)^{b+b'}\frac{y}{4} & =\frac12\, C_1[b,b’]. \notag \end{align}

These are exactly the coefficients in the two-site closure of Theorem 21.2.2.34.

For the coarse-graining map, write $B_v(b_1,b_2)$ for the amplitude of the bond-basis vector $v$ at $(b_1,b_2)$. Its orthonormality gives

\begin{align} \sum _v B_v(b_1,b_2)B_v(c_1,c_2) & =\delta _{b_1c_1}\delta _{b_2c_2}. \notag \end{align}

The sum over the two site flags is

\begin{align} \sum _{f_1,f_2} \delta _{f_1F_1}\delta _{f_2F_2} \delta _{f_1F'_1}\delta _{f_2F'_2} & =\delta _{F_1F'_1}\delta _{F_2F'_2}. \notag \end{align}

Orthonormality of the bond basis and the flag Kronecker deltas identify all six bits of the two-site indices and give the resolution of the identity. On the support of the two-site closure, the two bond qubits of each argument agree, so the outcomes $|01\rangle $ and $|10\rangle $ contribute nothing. The symmetric and antisymmetric Bell outcomes contribute $\frac12(\pm 1)^{b+b'}$. Summing over the two flag pairs compatible with the output flag leaves, for each block label $k$, the identity

\begin{align} \frac12\Bigl(\sum _{b,b'}C_k[b,b’] +(\tau _k)_{11}\sum _{b,b'}(-1)^{b+b'}C_k[b,b’]\Bigr) & =1. \notag \end{align}

The left side is $\frac12(\frac74+\frac14)=1$ in both cases: for $k=0$ the alternating sum is $\frac14$ and the flag sign is $1$, and for $k=1$ it is $-\frac14$ and the flag sign is $-1$. The remaining factor is the one-site coefficient $\frac12\, C_k[l,l']\, (\tau _k)_{ff} \delta _{ff'}$ of Theorem 21.2.2.34.

Theorem 21.2.2.54 Two-label structure coefficients whose length dependence survives positive rescaling

✓

Let the labels be $\{ 0,1\} $, with addition modulo two, and define positive diagonal matrices of size one, identified with their entries, by

\begin{align} \chi _{ff',\, f+f'} & =\alpha =\frac{7}{10}, \notag \\ \chi _{ff',\, f+f'+1} & =\beta =\frac1{10}. \notag \end{align}

Then $c^{(L)}_{ff'g}=\operatorname{tr}(\chi _{ff'g}^L)$ equals $\alpha ^L$ when $g=f+f'$ and $\beta ^L$ otherwise. These coefficients are not length independent. For every pair of positive rescalings $(s_0,s_1)$, replacing $\chi _{ff'g}$ by $(s_fs_{f'}/s_g)\chi _{ff'g}$ still gives coefficients that are not length independent. Unlike the one-label families of Theorems 21.2.2.31 and 21.2.2.32, the obstruction here is not a diagonal matrix with two distinct entries but the inconsistency of the equations $s_fs_{f'}\chi _{ff'g}=s_g$ over the eight triples of labels. This statement concerns the coefficients alone. Theorem 21.2.2.58 identifies them as the structure coefficients for the closed operators of the flag sectors of the twisted quantum dimer.

Proof ▶

If the powers of a positive real number $t$ agree at lengths one and two, then $t=t^2$, hence $t=1$. The triples $(0,0,0)$ and $(0,1,0)$ give $s_0\alpha =1$ and $s_1\beta =1$; the triple $(0,0,1)$ then gives $s_0^2\beta /s_1=\beta ^2/\alpha ^2=1$, contradicting $\alpha \ne \beta $.

Definition 21.2.2.55 Flag sectors of the twisted quantum dimer

✓

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

\begin{align} \widehat M_f^{(p,p',k),(q,q',k')} & =[k=k’]\, \frac1{2\mu }\, C_k[p,p’]\, (\tau _k)_{ff} E_{pp'}\otimes E_{qq'}, \notag \end{align}

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

Lemma 21.2.2.56 Cyclic path expansion of a trace

✓

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

\begin{align} \operatorname{tr}(M_0\cdots M_{L-1}) & =\sum _{t_0,\ldots ,t_{L-1}}\prod _{n=0}^{L-1}(M_n)_{t_n t_{n+1}}, \notag \end{align}

with the cyclic convention $t_L=t_0$.

Proof ▶

The open expansion is an induction on $L$: splitting a path into its first step and the remaining path gives $(M_0(M_1\cdots M_{L-1})v)_a =\sum _c(M_0)_{ac}((M_1\cdots M_{L-1})v)_c$. For the trace, separate the last factor $M_{L-1}$ and apply the open expansion to the vector $c\mapsto (M_{L-1})_{ca}$, using the empty product when $L=1$. Splitting a cyclic assignment into its first entry $a$ and the remaining entries identifies the two sides term by term, with the last factor closing the cycle back to $a$.

Lemma 21.2.2.57 Closed operators of the flag sectors

✓

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

\begin{align} O_L(\widehat M_f)_{\sigma \tau } & =\mathbb {1}[\text{cyclic bond match}](\sigma ,\tau ) \prod _n[k_n=k’_n] \prod _n\frac{C_{k_n}[p_n,p'_n]\, (\tau _{k_n})_{ff}}{2\mu }. \notag \end{align}

Proof ▶

A matrix unit multiplied on the left by a tensor matrix is a matrix unit or zero according to whether the right bits of that matrix match the row of the unit. The open-chain formula follows by induction on the length of the words, as for the twisted dimer itself in Theorem 21.2.2.35. Taking the trace imposes the cyclic bond match. For a flag sector, the coefficient at a bond pair is the one-site weight of the first index when the block labels agree and zero otherwise. Thus the product of coefficients along the ring is the product of the weights of the first word if the two label strings agree, and zero otherwise.

Theorem 21.2.2.58 Product law for the closed operators of the flag sectors

✓

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

\begin{align} O_L(\widehat M_f)\, O_L(\widehat M_{f'}) & =\alpha ^L\, O_L(\widehat M_{f+f'}) +\beta ^L\, O_L(\widehat M_{f+f'+1}), \notag \end{align}

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

$\begin{tenkz}[rows={op,op}, cols=4, boundary=periodic, bonds=none] \tn[skin=mpo, size=l, name=a1, ports={90:physical:$a_1$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_f} & \tn[skin=mpo, size=l, name=a2, ports={90:physical:$a_2$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_f} & \tn[skin=dots, name=ad, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, name=aL, ports={90:physical:$a_L$, 270:physical, 180:virtual, 0:virtual}]{\widehat M_f} \\ \tn[skin=mpo, size=l, name=b1, ports={90:physical, 270:physical:$b_1$, 180:virtual, 0:virtual}]{\widehat M_{f'}} & \tn[skin=mpo, size=l, name=b2, ports={90:physical, 270:physical:$b_2$, 180:virtual, 0:virtual}]{\widehat M_{f'}} & \tn[skin=dots, name=bd, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, name=bL, ports={90:physical, 270:physical:$b_L$, 180:virtual, 0:virtual}]{\widehat M_{f'}} \tnwire{a1.0}{a2.180} \tnwire{a2.0}{ad.180} \tnwire{ad.0}{aL.180} \tnwire{b1.0}{b2.180} \tnwire{b2.0}{bd.180} \tnwire{bd.0}{bL.180} \tnwire{a1.270}{b1.90} \tnwire{a2.270}{b2.90} \tnwire{aL.270}{bL.90} \end{tenkz}$
$=\alpha ^L$ $\begin{tenkz}[rows={op}, cols=4, boundary=periodic] \tn[skin=mpo, size=l, ports={90:physical:$a_1$, 270:physical:$b_1$}]{\widehat M_g} & \tn[skin=mpo, size=l, ports={90:physical:$a_2$, 270:physical:$b_2$}]{\widehat M_g} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, ports={90:physical:$a_L$, 270:physical:$b_L$}]{\widehat M_g} \end{tenkz}$ $+\, \beta ^L$ $\begin{tenkz}[rows={op}, cols=4, boundary=periodic] \tn[skin=mpo, size=l, ports={90:physical:$a_1$, 270:physical:$b_1$}]{\widehat M_{\bar g}} & \tn[skin=mpo, size=l, ports={90:physical:$a_2$, 270:physical:$b_2$}]{\widehat M_{\bar g}} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, size=l, ports={90:physical:$a_L$, 270:physical:$b_L$}]{\widehat M_{\bar g}} \end{tenkz}$

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

Proof ▶

Fix cyclically bond-matched words $\sigma $, $\tau $ with the common block labels $k_n$ and left bits $p_n$ of $\sigma $; all other entries of both sides vanish by Lemma 21.2.2.57. The entry of the product is a sum over a middle word $\rho $, which must be cyclically bond matched with the same labels $k_n$, hence of the form $\rho _n=(t_n,t_{n+1},k_n)$ for a bit string $t$. Its contribution factors as the weight of $\sigma $ in the sector $f$ times $(2\mu )^{-L}\prod _n(\tau _{k_n})_{f'f'}\prod _n C_{k_n}[t_n,t_{n+1}]$. By Lemma 21.2.2.56, the sum over $t$ of the last product is $\operatorname{tr}(C_{k_0}\cdots C_{k_{L-1}})$. Since $C_k=xP_++(-1)^kyP_-$ with orthogonal rank-one idempotents $P_\pm $, the ordered product is $x^LP_++(-1)^{\sum _n k_n}y^LP_-$, with trace $x^L+(-1)^{\sum _n k_n}y^L$; this is verified entrywise through the sign expansion $C_k[p,q]=\frac12(x+(-1)^{k+p+q}y)$. The flag signs multiply as $(\tau _k)_{ff}(\tau _k)_{f'f'}=(\tau _k)_{gg}$ with $g=f+f'$, and the sign $(-1)^{\sum _n k_n}=\prod _n(\tau _{k_n})_{11}$ of the second summand shifts the label to $g+1$. With $x/(2\mu )=\frac{7}{10}$ and $y/(2\mu )=\frac1{10}$, the two summands are $\alpha ^L$ and $\beta ^L$ times the entries of $O_L(\widehat M_{f+f'})$ and $O_L(\widehat M_{f+f'+1})$. These are precisely the coefficients for $g=f+f'$ and $g=f+f'+1$ in Theorem 21.2.2.54.

Theorem 21.2.2.59 Horizontal canonical form of the twisted quantum dimer

✓

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

\begin{align} M^{ij} & =U^\dagger \Bigl(\tfrac 58\, \widehat B_0^{ij}\oplus \tfrac 58\, \widehat B_1^{ij}\Bigr)U. \notag \end{align}

The physical-trace transfer of $\widehat B_1$ vanishes.

Proof ▶

The matrix of $B_k$ at the physical pair $((p,q,0),(p',q',0))$ is the matrix unit at the bond pair $((p,p'),(q,q'))$ with the nonzero coefficient $\frac12 C_k[p,p']$, so these matrices span the bond algebra. The product $(\widehat B_k^{ij})^\dagger \widehat B_k^{ij}$ is the diagonal matrix unit at the right bits $(r,r')$ of $i$ and $j$, with coefficient $\frac{64}{25}$ times the squared coefficient of the tensor matrix. Grouping by the right bits, the coefficients at fixed $(r,r')$ sum to $\frac{64}{25}\cdot 2\cdot \frac14(2C_k[0,0]^2+2C_k[0,1]^2)=1$, so the sum of the products is the identity. Theorem 7.6.3 gives normality with spectral radius one. The reconstruction is the block decomposition of Definition 21.2.2.33 entry by entry, with $\frac58\cdot \frac85=1$ restoring the original blocks. The matrix $U$ is a permutation matrix, hence a coisometry. The physical-trace transfer of $\widehat B_1$ is $\frac85$ times that of $B_1$, which vanishes by Theorem 21.2.2.34.

Lemma 21.2.2.60 Blocking raises the physical-trace transfer to a power

✓

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

Proof ▶

The physical-trace transfer of the blocking is the sum over all length-$L$ configurations of the diagonal word evaluations, which is the $L$-th power of the one-site loop by Lemma 16.4.56.

Theorem 21.2.2.61 The twisted quantum dimer is not simple

✓

The twisted quantum dimer of Definition 21.2.2.33 is not simple in the sense of Definition 20.12.6: for every positive blocking length $L$ and every basis-of-normal-tensors sector presentation of the doubled-index tensor of the $L$-site blocking, some representative has nilpotent physical-trace transfer.

Proof ▶

Fix $L{\gt}0$ and a presentation $P$. Blocking the canonical form of Theorem 21.2.2.59 by $L$ sites (Theorem 7.6.15) and relabeling the doubled alphabet along the pairing of the blocked ket and bra words (Theorems 7.6.8 and 20.15.2) gives a canonical form of the blocked doubled-index tensor whose retained blocks are the $L$-site blockings of $\widehat B_0$ and $\widehat B_1$. Its basis-of-normal-tensors refinement (Theorem 8.6.1.9) is a basis-of-normal-tensors sector presentation $Q$ (Theorem 8.6.1.12) in which every retained block is related to the representative of its phase class by a similarity transformation and a phase. The blocked $\widehat B_1$ has vanishing physical-trace transfer by Lemma 21.2.2.60 together with Lemma 20.12.2, so by Theorem 20.12.4 the representative of its phase class has nilpotent physical-trace transfer. Theorem 20.12.5 transfers this to $P$.

Theorem 21.2.2.62 Vertical canonical form of the twisted quantum dimer

✓

The twisted quantum dimer $M$ of Definition 21.2.2.33 has a vertical decomposition in the sense of Definition 16.4.6, with the two rescaled flag sectors $\widehat M_0$, $\widehat M_1$ of Definition 21.2.2.55, multiplicity one each, and weight $\mu =\frac58$. Let $U$ be the permutation matrix reordering the original physical index $(l,r,f)\mapsto (f,(l,r))$, so that $U$ maps $\mathbb {C}^8$ onto the direct sum of the two flag subspaces, each isomorphic to $\mathbb {C}^4$. Writing $\widetilde M_{ab}$ for the matrices in the vertical direction, we have $UU^\dagger =U^\dagger U=\mathbb {1}$ and, for every horizontal bond pair $(a,b)$,

\begin{align} U\, \widetilde M_{ab}\, U^\dagger & =\mu \, \widehat M_0^{ab}\oplus \mu \, \widehat M_1^{ab}, \notag \\ \widetilde M_{ab} & =U^\dagger (\mu \, \widehat M_0^{ab}\oplus \mu \, \widehat M_1^{ab})U. \notag \end{align}

The first identity is the vertical decomposition drawn in [ 13 , Proposition 4.13, lines 945–959 ] :

$\begin{tenkz}[rows={op,op,op}, cols=1, bonds=none] \tn[skin=box, size=l, name=U, ports={90:physical, 270:physical}]{U} \\ \tn[skin=mpo, size=l, name=M, ports={90:physical, 270:physical, 180:virtual:$a$, 0:virtual:$b$}]{\widetilde M} \\ \tn[skin=box, size=l, name=Ud, ports={90:physical, 270:physical}]{U^\dagger} \tnwire{U.270}{M.90} \tnwire{M.270}{Ud.90} \end{tenkz}$ $ = \mu $ $\begin{tenkz}[cols=1] \tn[skin=mpo, size=l, ports={90:physical, 270:physical, 180:virtual:$a$, 0:virtual:$b$}]{\widehat M_0} \end{tenkz}$ $ \oplus \, \mu $ $\begin{tenkz}[cols=1] \tn[skin=mpo, size=l, ports={90:physical, 270:physical, 180:virtual:$a$, 0:virtual:$b$}]{\widehat M_1} \end{tenkz}$

Each flag sector is injective at one site and therefore proportional to a normal tensor with spectral radius one. At every positive length $N$, the matrix product vector of the vertically viewed tensor is $\mu ^N$ times the sum of those of the two sectors, and the matrix product vectors of the two sectors are linearly independent. The fact that these sectors already have the normalization required of normal tensors in the CPSV sense is verified below.

Proof ▶

The matrix of the twisted dimer between two physical indices with different flag values vanishes, because its coefficient contains $\delta _{ff'}$. Between two indices with common flag value $f$, it is $\mu $ times the matrix of $\widehat M_f$ by Definition 21.2.2.55. Conjugating by the permutation matrix therefore places the two sectors on the diagonal blocks, and the reconstruction follows from $U^\dagger U=\mathbb {1}$. Traces of products are unchanged by conjugation, so $|V^{(N)}(\widetilde M)\rangle =\mu ^N\bigl(|V^{(N)}(\widehat M_0)\rangle +|V^{(N)}(\widehat M_1)\rangle \bigr)$.

For every matrix unit $E_{(p,q),(p',q')}$ of the two-qubit space, the matrix of $\widehat M_f$ at the bond pair $((p,p',0),(q,q',0))$ is $C_0[p,p']/(2\mu )$ times that matrix unit, with $C_0[p,p']\in \{ \frac12,\frac38\} $ nonzero. Hence the matrices of each sector span the full matrix algebra, giving one-site injectivity. For linear independence, evaluate the closed sector operators of Lemma 21.2.2.57 on two cyclically bond-matched words: on the constant word of bond index $(0,0,0)$ both sectors give $(2/5)^N$, since the one-site weight is $C_0[0,0]/(2\mu )=\frac25$; on the word with bond index $(0,0,1)$ at the first site and $(0,0,0)$ elsewhere, the sector $f$ gives $(\tau _1)_{ff}\frac{3}{10}(2/5)^{N-1}$, since $C_1[0,0]/(2\mu )=\frac3{10}$, and the two flag signs are opposite. A vanishing linear combination $g_0|V^{(N)}(\widehat M_0)\rangle +g_1|V^{(N)}(\widehat M_1)\rangle =0$ thus forces $g_0+g_1=0$ and $g_0-g_1=0$.

Theorem 21.2.2.63 Vertical BNT algebra of the twisted quantum dimer

✓

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

\begin{align} \sum _{a,b}(\widehat M_f^{ab})^\dagger \widehat M_f^{ab} & =\mathbb {1}, \notag \end{align}

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

\begin{align} m_g & =\sum _{f,f'}c^{(1)}_{ff'g}\, m_fm_{f'} =2(\alpha +\beta )\mu ^2=\frac85\cdot \frac{25}{64}=\frac58. \notag \end{align}

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

Proof ▶

The adjoint of a sector matrix times that matrix is the diagonal matrix unit at the right bits $(p',q')$ of the bond pair with the squared modulus of the coefficient, and the flag sign squares to one. Summing over the bond pairs $((p,p',k),(q,q',k))$ with the right bits fixed gives, at each diagonal entry,

\begin{align} \frac{1}{(2\mu )^2}\sum _{k}\sum _{p,q}C_k[p,p’]^2 & =\frac{2\cdot 2\cdot (\tfrac 14+\tfrac 9{64})}{4\cdot \tfrac {25}{64}} =1, \notag \end{align}

and the off-diagonal entries vanish. Together with one-site injectivity from Theorem 21.2.2.62, this makes each sector a normal tensor with spectral radius one by Theorem 7.6.3. The spanning and linear independence are those of Theorem 21.2.2.62. The product law is Theorem 21.2.2.58. For each output label $g$, two pairs $(f,f')$ contribute $\alpha $ and two contribute $\beta $, so $\sum _{f,f'}c^{(1)}_{ff'g}m_fm_{f'}=2(\alpha +\beta )\mu ^2=\mu =m_g$.

Lemma 21.2.2.64 Doubled-index matrices and their rank-one factorization

✓

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

\begin{align} R^{pq}_{ab} & =\frac{25}{32}A^a_{p_1q_1}A^b_{p_2q_2}, \notag \end{align}

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

Proof ▶

The doubled-index reading gives $R^{pq}$ by definition. Expanding the Kronecker-product definition of $R$ and using that the entries of the four matrices $A^a$ are real gives $R^{pq}=\frac{25}{32}u_{pq}v_{pq}^{\mathsf T}$.

Lemma 21.2.2.65 Rank-one form of the dimer transfer map

✓

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

\begin{align} \lambda (X) & =\left(\frac{25}{32}\right)^2 \sum _{b=0}^3s_b^2X_{bb}. \notag \end{align}

Then the transfer map of $R$ has rank-one form

\begin{align} \mathcal{E}_R(X) & =\lambda (X)F, \notag \\ \lambda (F) & =\frac{337}{512}. \notag \end{align}

Proof ▶

By Lemma 21.2.2.64, the doubled-index matrices are $R^{pq}=\frac{25}{32}u_{pq}v_{pq}^{\mathsf T}$. Substituting these factors into the transfer-map sum and summing over the physical indices gives $\mathcal{E}_R(X)=\lambda (X)F$. Substituting $X=F$ and evaluating the four diagonal entries gives $\lambda (F)=337/512$.

Lemma 21.2.2.66 One-site transfer quasi-idempotence of the dimer tensor

✓

The transfer map of the dimer tensor $R$ satisfies $\mathcal{E}_R^2=\frac{337}{512}\mathcal{E}_R$.

Proof ▶

By Lemma 21.2.2.65,

\begin{align} \mathcal{E}_R(\mathcal{E}_R(X)) & =\lambda (X)\lambda (F)F =\frac{337}{512}\mathcal{E}_R(X). \notag \end{align}

Theorem 21.2.2.67 Positivity of the rescaling-stable MPDO

✓

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

\begin{align} \rho ^{(N)}(R)_{\sigma \tau } & =\left(\frac{25}{32}\right)^N \mathbb {1}[\text{cyclic bond match}](\sigma ,\tau ) \prod _n W(\sigma _n,\tau _n), \notag \end{align}

where the indicator is one exactly when both $\sigma $ and $\tau $ satisfy the cyclic bond-matching condition: consecutive letters share their bond bit around the ring. In the product, $W$ is evaluated on the corresponding bond bits. It is the local factor from Theorem 21.2.2.32, with eigenvalues $\{ 1,7/25\} $. Thus $\rho ^{(N)}(R)$ is $(25/32)^N$ times the Hadamard product of the rank-one bond-matching indicator with the pullback of the $N$-fold Kronecker power of $W$.

Proof ▶

The entrywise factorization follows by induction on the chain length, using the rank-one contraction identity

\begin{align} \operatorname{tr}((u_1v_1^{\mathsf T})\cdots (u_Nv_N^{\mathsf T})) & =(v_N^{\mathsf T}u_1)\prod _{n=1}^{N-1}(v_n^{\mathsf T}u_{n+1}), \notag \end{align}

where $u_n,v_n$ are the column vectors in the rank-one factorization of the $n$-th bond matrix. Closing the trace produces the cyclic bond-matching condition. The bond-matching indicator is positive semidefinite because it is an outer product. The pullback of the Kronecker power of $W$ is positive semidefinite by congruence. The Schur product theorem and the positive scalar $(25/32)^N$ therefore give positivity.

Theorem 21.2.2.68 Unitality of the transfer map

✓

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

Proof ▶

Evaluate the four matrix entries of $\mathcal{E}_A(\mathbb {1})$ from the matrices $A^a$. The diagonal entries are one and the off-diagonal entries vanish.

Theorem 21.2.2.69 Traceless diagonal eigenvector of the transfer map

✓

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

Proof ▶

Evaluate $\sum _a A^aZ(A^a)^\dagger $ entrywise. Its off-diagonal entries vanish, while its diagonal entries are $7/25$ and $-7/25$.

Theorem 21.2.2.70 Annihilation of the first off-diagonal matrix unit

✓

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

Proof ▶

Substitution of the four matrices $A^a$ shows that every entry of $\sum _a A^aE_{01}(A^a)^\dagger $ is zero.

Theorem 21.2.2.71 Annihilation of the second off-diagonal matrix unit

✓

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

Proof ▶

Substitution of the four matrices $A^a$ shows that every entry of $\sum _a A^aE_{10}(A^a)^\dagger $ is zero.

Theorem 21.2.2.72 Spectra of the local factor and the transfer map

✓

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

Proof ▶

The uniform and alternating vectors are eigenvectors of the explicit $2\times 2$ matrix $W$:

\begin{align} W\begin{pmatrix} 1 \\ 1 \end{pmatrix}& =\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \notag \\ W\begin{pmatrix} 1 \\ -1 \end{pmatrix}& =\frac{7}{25}\begin{pmatrix} 1 \\ -1 \end{pmatrix}. \notag \end{align}

Direct expansion of the letter matrices gives the action on a basis of $M_{2}(\mathbb {C})$:

\begin{align} \mathcal{E}_A(\mathbb {1}) & =\mathbb {1}, \notag \\ \mathcal{E}_A(\operatorname{diag}(1,-1)) & =\frac{7}{25}\operatorname{diag}(1,-1), \notag \\ \mathcal{E}_A(E_{01}) & =0, \notag \\ \mathcal{E}_A(E_{10}) & =0. \notag \end{align}

Lemma 21.2.2.73 Injectivity of $R^{\bullet \bullet }$

✓

The $16$ letters $R^{pq}$ of the doubled-index tensor $R^{\bullet \bullet }$ span the full bond matrix algebra $M_{4}(\mathbb {C})$.

Proof ▶

Each letter $R^{pq}$ is, up to a nonzero scalar, one of the $16$ matrix units of $M_{4}(\mathbb {C})$. Indeed, the outer-product structure of $R^{pq}$ gives a single matrix unit because each factor $A^a$, $A^b$ of the vertical letters $B^{ab}=A^a\otimes \overline{A^b}$ is a scaled matrix unit of $M_{2}(\mathbb {C})$. Hence the standard matrix-unit basis lies in the span of the letters.

Theorem 21.2.2.74 Canonical form of $R$

✓

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

Proof ▶

By Lemma 21.2.2.73, the $16$ letters span the full bond matrix algebra. Thus $R^{\bullet \bullet }$ is injective, and no nontrivial invariant orthogonal projection exists.

The transfer map $\mathcal{E}_{R^{\bullet \bullet }}(X)=\sum _{p,q}R^{pq}X(R^{pq})^\dagger $ has rank one: every image is a scalar multiple of a fixed diagonal matrix, and the scalar evaluated at that matrix is $337/512$. Rescaling the tensor by $\mu ^{-1}=\sqrt{512/337}$ makes its transfer map idempotent. Together with injectivity, this gives a primitive transfer map with spectral radius one; all its other eigenvalues are zero. Thus $\mu ^{-1}R^{\bullet \bullet }$ is a normal tensor with the required normalization. The equality $R^{\bullet \bullet }=\mu (\mu ^{-1}R^{\bullet \bullet })$ gives exact reconstruction with the identity coisometry.

Theorem 21.2.2.75 Non-nilpotency of the retained horizontal block

✓

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

\begin{align} \mathcal T_{\widehat R} & =\mu ^{-1}\mathcal T_R, \notag \\ \mathcal T_R^2 & =\mathcal T_R, \notag \\ (\mathcal T_R)_{00} & =\frac12. \notag \end{align}

Consequently neither $\mathcal T_R$ nor $\mathcal T_{\widehat R}$ is nilpotent. Thus the retained basis element satisfies the non-nilpotency condition in the definition of simplicity.

Proof ▶

A nilpotent idempotent is zero, but the $(0,0)$ entry of $\mathcal T_R$ is

\begin{align} \frac{25}{32}\operatorname{tr}(A^0)^2 & =\frac{25}{32}\left(\frac45\right)^2 =\frac12, \notag \end{align}

so $\mathcal T_R$ is not nilpotent. The diagonal physical contraction is linear in the tensor, which gives $\mathcal T_{\widehat R}=\mu ^{-1}\mathcal T_R$. Since $\mu {\gt}0$, a nilpotent $\mathcal T_{\widehat R}$ would remain nilpotent after multiplication by $\mu $, forcing $\mathcal T_R$ to be nilpotent.

Theorem 21.2.2.76 Simplicity of the rescaling-stable dimer

✓

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

Proof ▶

Theorem 21.2.2.67 gives the MPDO condition. Choose the one-site blocking. Its doubled-index tensor is the canonical physical relabeling of $R^{\bullet \bullet }$. Refine the canonical form from Theorem 21.2.2.74 to its basis-of-normal-tensors sector presentation. The only representative is $\widehat R$, whose physical-trace transfer is non-nilpotent by Theorem 21.2.2.75.

Theorem 21.2.2.77 The rescaling-stable tensor is a renormalization fixed point

✓

The tensor $R$ of Theorem 21.2.2.32 is a renormalization fixed point in the sense of Definition 20.2.1. The refinement map $T$ from one site to two sites is completely positive and trace preserving, with two Kraus operators built from the eigenvectors of $W$, with amplitude $5/8$ on the uniform eigenvector and $\sqrt7/8$ on the alternating eigenvector. The coarse-graining map $S$ from two sites to one site is completely positive and trace preserving, with four Kraus operators. Two use the amplitude $1/\sqrt2$ and the eigenvectors of $W$ on physical-index pairs satisfying the bond-matching condition. The other two assign the pairs that fail this condition deterministically, so that the four Kraus operators resolve the identity. These maps satisfy $T[M_1(X)]=M_2(X)$ and $S[M_2(X)]=M_1(X)$ for every boundary matrix $X$, with $M=R$.

Proof ▶

Identify each physical index $p\in \{ 0,1,2,3\} $ with a pair of bits $(p',p'')\in \{ 0,1\} ^2$, and write $i_1\ast i_2:=(i_1',i_2'')$ for the physical index obtained from the first bit of $i_1$ and the second bit of $i_2$. For $M=R$, the two-site physical operator satisfies

\begin{align} M_2(X)_{(i_1,i_2)(j_1,j_2)} & =\frac{25}{32} \mathbb {1}[i_1”=i_2’]\, \mathbb {1}[j_1”=j_2’] W(i_1”,j_1”)\, M_1(X)_{i_1\ast i_2,\, j_1\ast j_2}, \notag \end{align}

and therefore vanishes whenever either bond match fails. Write $e_0=(1,1)$ and $e_1=(1,-1)$ for the eigenvectors of $W$, with eigenvalues $\lambda _0=1$ and $\lambda _1=7/25$ (Theorem 21.2.2.72). The two Kraus operators of $T$ have amplitudes $\sqrt{25\lambda _s/64}$ on $e_s$. For $b,b'\in \{ 0,1\} $, the eigendecomposition of $W$ gives

\begin{align} \sum _{s\in \{ 0,1\} }\frac{25}{64}\lambda _s e_s(b)e_s(b’) & =\frac{25}{32}W(b,b’). \notag \end{align}

This reproduces the coefficient in $M_2(X)$, so $T[M_1(X)]=M_2(X)$.

The two Kraus operators of $S$ supported on bond-matching pairs have amplitude $1/\sqrt2$ on $e_0,e_1$. For $b,b'\in \{ 0,1\} $,

\begin{align} \frac12\sum _{s\in \{ 0,1\} }e_s(b)e_s(b’) & =\begin{cases} 1,& b=b’,\\ 0,& b\ne b’.\end{cases} \notag \end{align}

The diagonal case gives the resolution of the identity on the bond-matching subspace, and the off-diagonal case gives cancellation between these two Kraus operators. The other two Kraus operators assign the bond-mismatched pairs deterministically. They do not contribute to $S[M_2(X)]$, because $M_2(X)$ vanishes on these pairs. Together the four Kraus operators resolve the identity on the full two-site domain and give $S[M_2(X)]=M_1(X)$.

Theorem 21.2.2.78 Vertical BNT and algebra structure for $R$

✓

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

\begin{align} M_0^{(a,b)} & =A^a\otimes \overline{A^b}. \notag \end{align}

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

$\begin{tenkz}[cols=1, west=open, east=open] \tn[skin=mpo, size=l, ports={90:physical:$a$, 270:physical:$b$}]{M_0} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={ket,bra}, cols=1, west=open, east=open, bonds=none, size=l] \tn[skin=box, ports={90:physical:$a$, 180:virtual, 0:virtual}]{A} \\ \tn[skin=box, ports={270:physical:$b$, 180:virtual, 0:virtual}]{\overline A} \end{tenkz}$

The positive diagonal multiplicity and coefficient matrices are

\begin{align} \mu _0 & =\begin{pmatrix} \frac{25}{32} \end{pmatrix}, \notag \\ \chi _{0,0,0} & =\operatorname{diag}\! \left(1,\frac{7}{25}\right). \notag \end{align}

The vertical tensor is reconstructed exactly as

\begin{align} \operatorname {vertical}(R) & =\frac{25}{32}M_0. \notag \end{align}

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

\begin{align} O_L(M_0)^2 & =\left(1+\left(\frac{7}{25}\right)^L\right)O_L(M_0) =\operatorname{tr}\! \left(\chi _{0,0,0}^L\right)O_L(M_0). \notag \end{align}

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

\begin{align} \frac{25}{32} & =\frac{32}{25}\left(\frac{25}{32}\right)^2. \notag \end{align}

Proof ▶

Each letter of $M_0$ is a nonzero multiple of a matrix unit, so the letters span $M_{4}(\mathbb {C})$. The tensor $A$ is left-canonical; hence $M_0$ is left-canonical and algebraically normal, and therefore is a normal tensor. The identity coisometry gives both the vertical decomposition and its exact reconstruction.

The vertical operator of $M_0=A\otimes \overline A$ is $|V^{(L)}(A)\rangle \! \langle V^{(L)}(A)|$. Its square is its self-overlap times itself. A direct transfer-map computation gives $\langle V^{(L)}(A) | V^{(L)}(A) \rangle =\operatorname{tr}(W^L)$, while Walsh–Hadamard diagonalization gives $\operatorname{tr}(W^L)=\operatorname{tr}(\chi _{0,0,0}^L)=1+(7/25)^L$. Finally, $c^{(1)}_{0,0,0}m_0^2=(32/25)(25/32)^2=m_0$.

The square computed in this proof is the following contraction. The two middle rings form the self-overlap of the periodic vector of $A$, a closed network equal to $\operatorname{tr}(W^L)=\operatorname{tr}(\chi _{0,0,0}^L)$:

$\begin{tenkz}[rows={ket,bra,ket,bra}, cols=4, boundary=periodic, bonds=none, size=l] \tn[skin=box, name=k1, ports={90:physical:$a_1$, 180:virtual, 0:virtual}]{A} & \tn[skin=box, name=k2, ports={90:physical:$a_2$, 180:virtual, 0:virtual}]{A} & \tn[skin=dots, name=kd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=kL, ports={90:physical:$a_L$, 180:virtual, 0:virtual}]{A} \\ \tn[skin=box, name=p1, ports={270:physical, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=box, name=p2, ports={270:physical, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=dots, name=pd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=pL, ports={270:physical, 180:virtual, 0:virtual}]{\overline A} \\ \tn[skin=box, name=q1, ports={90:physical, 180:virtual, 0:virtual}]{A} & \tn[skin=box, name=q2, ports={90:physical, 180:virtual, 0:virtual}]{A} & \tn[skin=dots, name=qd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=qL, ports={90:physical, 180:virtual, 0:virtual}]{A} \\ \tn[skin=box, name=b1, ports={270:physical:$b_1$, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=box, name=b2, ports={270:physical:$b_2$, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=dots, name=bd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=bL, ports={270:physical:$b_L$, 180:virtual, 0:virtual}]{\overline A} \tnwire{k1.0}{k2.180} \tnwire{k2.0}{kd.180} \tnwire{kd.0}{kL.180} \tnwire{p1.0}{p2.180} \tnwire{p2.0}{pd.180} \tnwire{pd.0}{pL.180} \tnwire{q1.0}{q2.180} \tnwire{q2.0}{qd.180} \tnwire{qd.0}{qL.180} \tnwire{b1.0}{b2.180} \tnwire{b2.0}{bd.180} \tnwire{bd.0}{bL.180} \tnwire{p1.270}{q1.90} \tnwire{p2.270}{q2.90} \tnwire{pL.270}{qL.90} \end{tenkz}$ $=\operatorname{tr}\bigl(\chi _{0,0,0}^{L}\bigr)$ $\begin{tenkz}[rows={ket,bra}, cols=4, boundary=periodic, bonds=none, size=l] \tn[skin=box, name=k1, ports={90:physical:$a_1$, 180:virtual, 0:virtual}]{A} & \tn[skin=box, name=k2, ports={90:physical:$a_2$, 180:virtual, 0:virtual}]{A} & \tn[skin=dots, name=kd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=kL, ports={90:physical:$a_L$, 180:virtual, 0:virtual}]{A} \\ \tn[skin=box, name=b1, ports={270:physical:$b_1$, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=box, name=b2, ports={270:physical:$b_2$, 180:virtual, 0:virtual}]{\overline A} & \tn[skin=dots, name=bd, ports={180:virtual, 0:virtual}]{} & \tn[skin=box, name=bL, ports={270:physical:$b_L$, 180:virtual, 0:virtual}]{\overline A} \tnwire{k1.0}{k2.180} \tnwire{k2.0}{kd.180} \tnwire{kd.0}{kL.180} \tnwire{b1.0}{b2.180} \tnwire{b2.0}{bd.180} \tnwire{bd.0}{bL.180} \end{tenkz}$

Remark 21.2.2.79 Vertical and horizontal contractions

Let $\operatorname {rot}(\operatorname {vertical}(R))$ denote the MPO obtained by separating the $16$-valued physical index of the vertical tensor into two four-valued indices. Then

\begin{align} \rho ^{(L)} \left(\operatorname {rot}(\operatorname {vertical}(R))\right) & =\left(\frac{25}{32}\right)^L O_L(M_0). \notag \end{align}

This is a rotated vertical contraction, not the original horizontal tensor $R$, even though both physical and bond dimensions equal $4$. At length one, $\rho ^{(1)}(R)$ has rank two, whereas $O_1(M_0)$ has rank one, so these operators are not proportional. In particular, no identity $\rho ^{(L)}(R)=(25/32)^L O_L(M_0)$ is asserted.

Theorem 21.2.2.80 Closed-operator factorization and local diagonalization for $R$

✓

For every positive chain length $N$, the horizontal operator factors as

\begin{align} \rho ^{(N)}(R) & =\left(\frac{25}{32}\right)^N B\, W_N\, B^{\mathsf T}, \notag \end{align}

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

Proof ▶

Let $\phi $ read the first bits of a cyclic-bond-matching string. Given a bit string $a$, each adjacent pair of entries of $a$ determines one physical index around the ring. This gives the unique matching string whose image under $\phi $ is $a$, and defines $B$. Injectivity gives $B^{\mathsf T}B=\mathbb {1}$. Rewriting the entrywise formula of Theorem 21.2.2.67 through this embedding gives $\rho ^{(N)}(R)=(25/32)^N B W_N B^{\mathsf T}$. The identity $H\chi H=2W$ follows by direct $2\times 2$ computation.

Theorem 21.2.2.81 Trace-power formula for the structure coefficients

✓

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

\begin{align} c^{(L)}_{0,0,0} & =\operatorname{tr}(\chi ^L)=\operatorname{tr}(W^L)=1+(7/25)^L. \notag \end{align}

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

Proof ▶

The coefficients are defined by the diagonal entries of $\chi $. Since these entries are positive, Lemma 21.2.2.12 gives $c^{(L)}_{0,0,0}=\operatorname{tr}(\chi ^L)$ for every $L{\gt}0$. The Walsh–Hadamard identity $H\chi H=2W$ and $HH=2\mathbb {1}$ give $H\chi =WH$. Induction then gives $H\chi ^L=W^L H$. Conjugating by $H$, whose inverse is $H/2$, and using cyclicity of the trace yields $\operatorname{tr}(W^L)=\operatorname{tr}(\chi ^L)$ for every $L$.

Definition 21.2.2.82 Linearly independent operator family

✓

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

Theorem 21.2.2.83 Uniqueness of the structure coefficients

✓

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

Proof ▶

Applying the product law to both systems of structure coefficients gives

\begin{align} \sum _\delta c^{(L)}_{\alpha ,\beta ,\delta }\, O_L(M_\delta ) & = O_L(M_\alpha )O_L(M_\beta ) = \sum _\delta c’^{(L)}_{\alpha ,\beta ,\delta }\, O_L(M_\delta ). \notag \end{align}

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 vanish.

Definition 21.2.2.84 Change of vertical representatives

✓

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

\begin{align} O_L(M’_\alpha ) & =s_\alpha ^L\Phi _L(O_L(M_{e(\alpha )})). \notag \end{align}

Definition 21.2.2.85 Structure coefficients under a change of normalization

✓

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

\begin{align} \widetilde c^{(L)}_{\alpha ,\beta ,\gamma } & =\left(\frac{s_\alpha s_\beta }{s_\gamma }\right)^L c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}. \notag \end{align}

For this definition, use the convention $z/0=0$.

Theorem 21.2.2.86 Product law under a change of vertical representatives

✓

A change of vertical representatives carries a product law with structure coefficients $c^{(L)}_{\alpha ,\beta ,\gamma }$ to a product law with structure coefficients $\widetilde c^{(L)}_{\alpha ,\beta ,\gamma }$.

Proof ▶

For every positive length $L$, multiplication of the two new representatives and the original product law give

\begin{align} O_L(M’_\alpha )O_L(M’_\beta ) & =(s_\alpha s_\beta )^L \Phi _L(O_L(M_{e(\alpha )})O_L(M_{e(\beta )})) \notag \\ & =(s_\alpha s_\beta )^L \sum _\delta c^{(L)}_{e(\alpha ),e(\beta ),\delta } \Phi _L(O_L(M_\delta )). \notag \end{align}

Reindexing by $\delta =e(\gamma )$ and using $s_\gamma \ne 0$ yields

\begin{align} O_L(M’_\alpha )O_L(M’_\beta ) & =\sum _\gamma \left(\frac{s_\alpha s_\beta }{s_\gamma }\right)^L c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )} \, s_\gamma ^L\Phi _L(O_L(M_{e(\gamma )})) \notag \\ & =\sum _\gamma \widetilde c^{(L)}_{\alpha ,\beta ,\gamma }O_L(M’_\gamma ). \notag \end{align}

Theorem 21.2.2.87 Covariance of structure coefficients for operator families

✓

Suppose that two operator families satisfy product laws and are related by a change of vertical representatives. At a positive length where the operators of the second family are linearly independent, their structure coefficients equal the transformed coefficients $\widetilde c^{(L)}_{\alpha ,\beta ,\gamma }$ of the first family.

Proof ▶

The product law for the second family and Theorem 21.2.2.86 give

\begin{align} \sum _\gamma c’^{(L)}_{\alpha ,\beta ,\gamma }O_L(M’_\gamma ) & =O_L(M’_\alpha )O_L(M’_\beta ) =\sum _\gamma \widetilde c^{(L)}_{\alpha ,\beta ,\gamma }O_L(M’_\gamma ). \notag \end{align}

Linear independence of these operators identifies the two systems of structure coefficients term by term.

Theorem 21.2.2.88 Covariance of structure coefficients for vertical BNT decompositions

✓

Suppose that two vertical basis-of-normal-tensors decompositions of tensors satisfy the algebra condition and are related by a change of vertical representatives. At every positive length $L$ at which the operators of the second decomposition are linearly independent, their structure coefficients satisfy

\begin{align} {c’}^{(L)}_{\alpha ,\beta ,\gamma } & =\left(\frac{s_\alpha s_\beta }{s_\gamma }\right)^L c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}. \notag \end{align}

Theorem 21.2.2.92 shows why equality of the horizontal periodic operator families alone cannot replace the specified relation between vertical representatives.

Proof ▶

The algebra conditions for the two vertical basis-of-normal-tensors decompositions give the product laws required by the covariance theorem for operator families. Hence

\begin{align} {c’}^{(L)}_{\alpha ,\beta ,\gamma } & =\widetilde c^{(L)}_{\alpha ,\beta ,\gamma } =\left(\frac{s_\alpha s_\beta }{s_\gamma }\right)^L c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}. \notag \end{align}

Theorem 21.2.2.89 Equality of structure coefficients under compatible normalization

✓

Under the hypotheses of Theorem 21.2.2.87, if $s_\alpha s_\beta =s_\gamma $, then ${c’}^{(L)}_{\alpha ,\beta ,\gamma } =c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}$.

Proof ▶

The compatible normalization and $s_\gamma \ne 0$ give $\left(s_\alpha s_\beta /s_\gamma \right)^L =\left(s_\gamma /s_\gamma \right)^L=1$. Substitution in the covariance formula gives the result.

Theorem 21.2.2.90 Equality of vertical BNT structure coefficients under compatible normalization

✓

Under the hypotheses of Theorem 21.2.2.88, if $s_\alpha s_\beta =s_\gamma $, then ${c’}^{(L)}_{\alpha ,\beta ,\gamma } =c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}$.

Proof ▶

For the vertical basis-of-normal-tensors decompositions, the covariance formula gives

\begin{align} {c’}^{(L)}_{\alpha ,\beta ,\gamma } & =\left(\frac{s_\alpha s_\beta }{s_\gamma }\right)^L c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )} =c^{(L)}_{e(\alpha ),e(\beta ),e(\gamma )}, \notag \end{align}

where the last equality uses $s_\alpha s_\beta =s_\gamma $ and $s_\gamma \ne 0$.

Definition 21.2.2.91 Horizontally gauge-equivalent tensors with distinct vertical normalizations

✓

Let

\begin{align} P & =\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \notag \\ X & =\begin{pmatrix} 1 & -3/4 \\ 0 & 1 \end{pmatrix}, \notag \\ Q & =XPX^{-1} =\begin{pmatrix} 1 & 3/4 \\ 0 & 0 \end{pmatrix}. \notag \end{align}

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

$\begin{tenkz}[boundary=open, cols=1] \tn[skin=mpo, size=l, ports={90:physical, 270:physical}]{M_Q} \end{tenkz}$ $ = $ $\begin{tenkz}[boundary=open] \tn[skin=ring]{X} & \tn[skin=mpo, size=l, ports={90:physical, 270:physical}]{M_P} & \tn[skin=ring]{X^{-1}} \end{tenkz}$

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

\begin{align} A_P & =P, \notag \\ m_P & =1, \notag \\ A_Q & =\frac45Q, \notag \\ m_Q & =\frac54. \notag \end{align}

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

Theorem 21.2.2.92 Horizontal periodic equality alone does not determine vertical BNT coefficients

✓

The tensors $M_P$ and $M_Q$ are related by the nonunitary gauge $X$. For every positive length $N$, their unnormalized periodic operators satisfy $\rho ^{(N)}(M_P)=\rho ^{(N)}(M_Q)$. In particular, both tensors generate matrix product density operators. Their chosen vertical basis-of-normal-tensors decompositions nevertheless have structure coefficients

\begin{align} c_P^{(L)} & =1, \notag \\ c_Q^{(L)} & =\left(\frac45\right)^L. \notag \end{align}

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

\begin{align} c_Q^{(L)}(\alpha ,\beta ,\gamma ) & =c_P^{(L)}(e(\alpha ),e(\beta ),e(\gamma )). \notag \end{align}

Indeed, the two sides at length one are $4/5$ and $1$.

While Theorem 21.2.2.83 guarantees uniqueness of structure coefficients for a fixed operator family, this counterexample shows that periodic operator equality alone does not constrain the structure coefficients of independent vertical decompositions across nonunitary gauge changes.

Proof ▶

The identities $P^2=P$, $Q^2=Q$, and $\operatorname{tr}(P)=\operatorname{tr}(Q)=1$ prove the periodic equality. The Frobenius norms of $P$ and $Q$ are $1$ and $5/4$, respectively, which gives the normalized representatives and multiplicity weights. Their products satisfy

\begin{align} A_P^2 & =A_P, \notag \\ A_Q^2 & =\frac45A_Q. \notag \end{align}

Taking the $L$-fold vertical operators gives $c_P^{(L)}=1$ and $c_Q^{(L)}=(4/5)^L$. Their values at $L=1$ exclude the only possible relabelling.

Theorem 21.2.2.93 Length independence is not preserved by horizontal periodic equality alone

✓

The structure coefficients in the vertical decomposition of $M_P$ are length independent, whereas those in the chosen vertical decomposition of $M_Q$ are not. More precisely,

\begin{align} c_P^{(L)} & =1, \notag \\ c_Q^{(1)} & =\frac45, \notag \\ c_Q^{(2)} & =\frac{16}{25}. \notag \end{align}

Proof ▶

The first family is constant, whereas $c_Q^{(1)}\ne c_Q^{(2)}$.

Theorem 21.2.2.94 Associativity constraint on the structure coefficients

✓

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

\begin{align} \sum _\delta c^{(L)}_{\alpha ,\beta ,\delta } c^{(L)}_{\delta ,\gamma ,\epsilon } & = \sum _\delta c^{(L)}_{\beta ,\gamma ,\delta } c^{(L)}_{\alpha ,\delta ,\epsilon }. \notag \end{align}

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

Proof ▶

Applying the product law twice in each order gives

\begin{align} (O_L(M_\alpha )O_L(M_\beta ))O_L(M_\gamma ) & = \sum _{\varepsilon '} \left(\sum _\delta c^{(L)}_{\alpha ,\beta ,\delta } c^{(L)}_{\delta ,\gamma ,\varepsilon '}\right) O_L(M_{\varepsilon '}), \notag \\ O_L(M_\alpha )(O_L(M_\beta )O_L(M_\gamma )) & = \sum _{\varepsilon '} \left(\sum _\delta c^{(L)}_{\beta ,\gamma ,\delta } c^{(L)}_{\alpha ,\delta ,\varepsilon '}\right) O_L(M_{\varepsilon '}). \notag \end{align}

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 $\varepsilon '$.

Theorem 21.2.2.95 Associativity constraint in trace-power form

✓

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

\begin{align} \sum _\delta \operatorname{tr}(\chi _{\alpha ,\beta ,\delta }^{\, L}) \operatorname{tr}(\chi _{\delta ,\gamma ,\epsilon }^{\, L}) & = \sum _\delta \operatorname{tr}(\chi _{\beta ,\gamma ,\delta }^{\, L}) \operatorname{tr}(\chi _{\alpha ,\delta ,\epsilon }^{\, L}). \notag \end{align}

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

Proof ▶

Substitute the trace-power form of the coefficients into the associativity constraint.

Theorem 21.2.2.96 The dimer RFP has length-dependent coefficients in its chosen vertical BNT decomposition

✓

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

\begin{align} c^{(L)}_{0,0,0} & =1+\left(\frac{7}{25}\right)^L. \notag \end{align}

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

Proof ▶

Theorem 21.2.2.67 and Theorem 21.2.2.74 show that $R$ generates MPDOs and has the direct-sum form with normalized normal tensor blocks. Theorem 21.2.2.77 constructs the completely positive trace-preserving maps $S,T$ satisfying $S[R_2(X)]=R_1(X)$ and $T[R_1(X)]=R_2(X)$ for all $X$, while Theorem 21.2.2.78 provides its single-element vertical basis-of-normal-tensors decomposition satisfying the algebra condition. Its structure coefficients are $c^{(L)}_{0,0,0}=1+(7/25)^L$. The values at lengths one and two differ, so these coefficients are not length independent.

Definition 21.2.2.97 Algebra conditions for a basis of normal tensors

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & =\sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }O_L(M_\gamma ), \notag \end{align}

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

$\begin{tenkz}[rows={op,op}, cols=4, west=trace, east=trace, bonds=none] \tn[skin=mpo, size=l, ports={90:physical:$i_1$, 270:physical}]{M_\alpha} & \tn[skin=mpo, size=l, ports={90:physical, 270:physical}]{M_\alpha} & \tn[skin=dots]{} & \tn[skin=mpo, size=l, ports={90:physical:$i_L$, 270:physical}]{M_\alpha} \\ \tn[skin=mpo, ports={90:physical, 270:physical:$j_1$}]{M_\beta} & \tn[skin=mpo, ports={90:physical, 270:physical}]{M_\beta} & \tn[skin=dots]{} & \tn[skin=mpo, ports={90:physical, 270:physical:$j_L$}]{M_\beta} \tnwire{(1,1)}{(1,2)} \tnwire{(1,2)}{(1,3)} \tnwire{(1,3)}{(1,4)} \tnwire{(2,1)}{(2,2)} \tnwire{(2,2)}{(2,3)} \tnwire{(2,3)}{(2,4)} \tnwire{(1,1)}{(2,1)} \tnwire{(1,2)}{(2,2)} \tnwire{(1,4)}{(2,4)} \tnmark[form=bracket]{(2,1) .. (2,4)}{$L\text{ sites}$} \end{tenkz}$ $=\sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }$ $\begin{tenkz}[cols=4, west=trace, east=trace] \tn[skin=mpo, ports={90:physical:$i_1$, 270:physical:$j_1$}]{M_\gamma} & \tn[skin=mpo, ports={90:physical, 270:physical}]{M_\gamma} & \tn[skin=dots]{} & \tn[skin=mpo, ports={90:physical:$i_L$, 270:physical:$j_L$}]{M_\gamma} \tnmark[form=bracket]{(1,1) .. (1,4)}{$L\text{ sites}$} \end{tenkz}$

together with

\begin{align} m_\gamma & =\sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta . \notag \end{align}

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

Definition 21.2.2.98 Algebra conditions for a vertical canonical decomposition

✓

Choose a vertical canonical decomposition

\begin{align} U\widetilde M U^\dagger & =\bigoplus _\alpha \mu _\alpha \otimes M_\alpha , \notag \\ \mu _\alpha & =\operatorname{diag}(\omega _{\alpha ,0},\ldots ,\omega _{\alpha ,r_\alpha -1}). \notag \end{align}

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

\begin{align} \widetilde M & =U^\dagger \left(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \right)U. \notag \end{align}

These tensor identities hold for every vertical physical index. In particular, any complementary sector is zero. Separating the two factors of the doubled physical index of $M_\alpha $ gives an MPO tensor $\widehat M_\alpha $. Define

\begin{align} O_L(M_\alpha ) & =\operatorname {mpo}(\widehat M_\alpha ,L), \notag \\ m_\alpha & =\sum _q\omega _{\alpha ,q}=\operatorname{tr}(\mu _\alpha ). \notag \end{align}

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

Theorem 21.2.2.99 Positivity of the physical trace in a retained vertical sector

✓

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

\begin{align} T_\gamma & =\sum _{a=0}^{D-1}M_\gamma ^{(a,a)}. \label{eq:rfp_algebra_terminal_transfer} \end{align}

$\begin{tenkz}[rows={wire}, cols=1, west=open, east=open] \tn[skin=box]{T_\gamma} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={wire}, cols=1, west=open, east=open, trace=physical, physical=updown] \tn[skin=mpo]{M_\gamma} \end{tenkz}$

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

Proof ▶

Apply the retained physical coisometry to the positive one-site operator. For a positive diagonal coordinate $q$ of the multiplicity matrix $\mu _\gamma $, the corresponding principal submatrix is $(\mu _\gamma )_{q,q}T_\gamma $. Principal submatrices remain positive semidefinite, and $(\mu _\gamma )_{q,q}{\gt}0$; division by this scalar proves $T_\gamma \succeq 0$.

Lemma 21.2.2.100 Physical traces of a nonunitary similarity pair

✓

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

\begin{align} T_A & =\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \notag \\ T_B & =\begin{pmatrix} 1 & 0 \\ \tfrac 12 & 1 \end{pmatrix}. \notag \end{align}

Proof ▶

Sum the two diagonal physical letters $A^{(0,0)}+A^{(1,1)}$ and then use $B^v=XA^vX^{-1}$ with $X=\operatorname{diag}(2,1)$.

Lemma 21.2.2.101 The physical traces of the nonunitary similarity pair are not positive semidefinite

✓

Neither $T_A$ nor $T_B$ in Lemma 21.2.2.100 is positive semidefinite.

Proof ▶

A positive semidefinite matrix is Hermitian. In both matrices the $(0,1)$ entry is zero, whereas the $(1,0)$ entry is respectively $1$ and $1/2$. Hence neither matrix is Hermitian.

Theorem 21.2.2.102 The nonunitary similarity pair is excluded from retained positive sectors

✓

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

Proof ▶

For any such retained sector, closing the physical legs gives a positive semidefinite matrix by Theorem 21.2.2.99. Lemma 21.2.2.101 gives the contradiction for both tensors.

Lemma 21.2.2.103 The nonunitary similarity pair has no positive isometric realization

✓

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

Proof ▶

The identity letter forces $c=1$. The matrix $S=V^\dagger X$ then commutes with all four letters of $A$, which span $M_{2}(\mathbb {C})$, and hence $S$ is scalar. Together with $V^\dagger V=\mathbb {1}$, this would make $X^\dagger X=\operatorname{diag}(4,1)$ scalar, a contradiction.

Definition 21.2.2.104 Linear extension on each virtual block

✓

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

Theorem 21.2.2.105 Multiplicativity of the linear extension on each virtual block

✓

The linear extension on block $k$ satisfies $T_k(MN)=T_k(M)T_k(N)$ for all $M,N\in M_{D_k}(\mathbb {C})$.

Proof ▶

The letters of $A_k$ span $M_{D_k}(\mathbb {C})$, and equality of the closed matrix product vectors gives the identity on products of spanning letters.

Theorem 21.2.2.106 Bijectivity of the linear extension on each virtual block

✓

The linear extension $T_k$ on block $k$ is bijective.

Proof ▶

Multiplicativity makes the kernel an ideal of the simple matrix algebra. The map is nonzero: otherwise every letter of $B_k$ would vanish, contradicting the nonvanishing trace pairing for the injective family. Thus the kernel is zero, and finite dimensionality gives surjectivity.

Theorem 21.2.2.107 Positive linear extensions on virtual blocks are unitarily implemented

✓

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

Proof ▶

The map on each block is a bijective algebra endomorphism. Positivity makes it preserve adjoints, so unitary Skolem–Noether implements it by conjugation with a unitary matrix.

Theorem 21.2.2.108 Closed matrix product vectors do not force a positive virtual extension

✓

# Discussion

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

Proof ▶

If the linear extension were positive, its unitary-implementation theorem would give a unitary $U$ satisfying $B^i=UA^iU^\dagger $ for every letter $i$. Hence $U^\dagger B^iU=A^i$, giving the forbidden realization with $c=1$ in Lemma 21.2.2.103.

Definition 21.2.2.109 Bond-two vertical tensor and Bell matrix

✓

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

\begin{align} A^{((a,b),(c,d))} & =\begin{cases} E_{a,b}, & (a,b)=(c,d), \\ 0, & (a,b)\ne (c,d). \end{cases} \notag \end{align}

Its physical trace is $J=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$. Define $|\Phi \rangle =|0,0\rangle +|1,1\rangle $ and $P=|\Phi \rangle \! \langle \Phi |$. Thus $P$ is an unnormalized rank-one positive operator, not an orthogonal projection. The tensor $A$ is the vertical tensor of the physical model in Definition 21.2.3.1. Theorem 21.2.3.4 identifies $P$ with its two-site operator after reindexing. For $X=\frac12\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$, set $B^v=XA^vX^{-1}$ and define

\begin{align} J_X & =XJX^{-1}, \notag \\ P_X & =(X\otimes X)P(X^{-1}\otimes X^{-1}). \notag \end{align}

Lemma 21.2.2.110 Positive physical trace and non-Hermitian Bell similarity

✓

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

\begin{align} \sum _{u\in I\times I} A^{(u,u)} & =\sum _{u\in I\times I} B^{(u,u)}=J\succeq 0, \notag \\ J_X & =J. \notag \end{align}

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

Proof ▶

The four diagonal physical letters of $A$ are the matrix units, so $A$ is injective. Similarity preserves every cyclic contraction. Direct multiplication gives $XJ=JX$, hence $J_X=J\succeq 0$, whereas $X^\dagger X\ne \mathbb {1}$. In the Bell calculation, the $(00,01)$ and $(01,00)$ entries of the real matrix $P_X$ are unequal, so $P_X$ is not Hermitian.

Theorem 21.2.2.111 Physical-trace and Bell Hermiticity force a scalar Gram matrix

✓

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

Proof ▶

Put $G=Y^\dagger Y$, which is positive definite. Hermiticity of $YJY^{-1}$ gives $GJ=JG$. Hermiticity of the Bell similarity gives $G_{00}G_{01}+G_{10}G_{11}=0$. The commutator identity implies $G_{01}=G_{10}$ and $G_{00}=G_{11}$. Since $G_{00}{\gt}0$, the remaining equation forces $G_{01}=G_{10}=0$, and therefore $G=G_{00}\mathbb {1}$.

21.2.3 A physical MPDO with a single vertical BNT element

Definition 21.2.3.1 Physical tensor with a bond-two vertical tensor

✓

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

\begin{align} M^{i,j} & =E_{(i,j),(i,j)}. \notag \end{align}

This model is motivated by [ 13 , Appendix C.4, lines 2048–2057 ] ; it is not a tensor stated there.

Theorem 21.2.3.2 Vertical tensor identity

✓

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

Proof ▶

A doubled bond coordinate $(a,b)$ selects $E_{a,b}$ when the two physical pair labels agree, and gives zero otherwise.

Theorem 21.2.3.3 Positive-length GHZ formula in the physical direction

✓

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

\begin{align} \rho ^{(N)}(M) & =|\mathrm{GHZ}_N\rangle \! \langle \mathrm{GHZ}_N|. \notag \end{align}

Hence $M$ generates MPDOs. These are unnormalized rank-one positive operators, not orthogonal projections. At $N=0$, the closed MPO contraction instead gives $4\mathbb {1}$, whereas the empty-word outer product is defined to be $\mathbb {1}$. Thus the positive-length restriction is necessary.

Proof ▶

The cyclic equality weights force the ket and bra words to be independently constant. The result is an outer product and is therefore positive semidefinite.

Theorem 21.2.3.4 Two-site Bell identity under reindexing

✓

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

\begin{align} \operatorname{reindex}_{\varepsilon ,\varepsilon } (\rho ^{(2)}(M)) & =|\Phi \rangle \! \langle \Phi |. \notag \end{align}

Thus $P$ is exactly the two-site physical operator in ordered-pair coordinates.

Proof ▶

Specialize Theorem 21.2.3.3 to $N=2$ and evaluate the coordinate identification.

Definition 21.2.3.5 Physical similarity of the MPO tensor

✓

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

\begin{align} M_X^{i,j} & =\sum _{a,b\in I}X_{i,a}(X^{-1})_{b,j}M^{a,b}. \notag \end{align}

$\begin{tenkz}[rows={wire}, cols=1, west=open, east=open, physical=updown] \tn[skin=mpo, size=l, ports={90:physical:$i$, 270:physical:$j$}]{M_X} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={op,op,op}, cols=1, west=none, east=none, physical=updown] \tn[skin=ring, ports={90:physical:$i$, 270:physical}]{X} \\ \tn[skin=mpo, ports={180:virtual, 0:virtual}]{M} \\ \tn[skin=ring, ports={90:physical, 270:physical:$j$}]{X^{-1}} \end{tenkz}$

This family is motivated by the sector comparison in [ 13 , Appendix C.4, lines 2048–2057 ] and the physical-sector argument of [ 13 , Proposition 4.13, lines 1898–1921 ] . Neither passage states this deformation or the results below.

Theorem 21.2.3.6 Vertical tensor under physical similarity

✓

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

$\begin{tenkz}[rows={wire}, cols=1, west=open, east=open] \tn[label pos=s, ports={90:physical:$v$}]{\widetilde M_X} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{X} & \tn[label pos=s, ports={90:physical:$v$}]{A} & \tn[skin=ring]{X^{-1}} \end{tenkz}$

Proof ▶

Left and right multiplication on the physical indices become matrix multiplication on the vertically viewed tensor. Now use Theorem 21.2.3.2.

Theorem 21.2.3.7 One-site physical-similarity identity

✓

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

\begin{align} \operatorname{reindex}(\rho ^{(1)}(M_X)) & =XJX^{-1}. \notag \end{align}

Proof ▶

Expand the one-site contraction. The four diagonal physical letters sum to $J$, and the two physical multiplications give conjugation by $X$.

Theorem 21.2.3.8 Two-site physical-similarity identity

✓

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

\begin{align} \operatorname{reindex}_{\varepsilon ,\varepsilon } (\rho ^{(2)}(M_X)) & =(X\otimes X)P(X^{-1}\otimes X^{-1}). \notag \end{align}

Proof ▶

Evaluate the two-site contraction in ordered-pair coordinates. Each ket index contributes $X$, each bra index contributes $X^{-1}$, and the undeformed contraction is $P$.

Theorem 21.2.3.9 MPDO positivity forces a scalar physical Gram matrix

✓

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

Proof ▶

Positivity of the one- and two-site operators implies Hermiticity. Apply the two exact reindexing identities and then Theorem 21.2.2.111.

Lemma 21.2.3.10 The concrete physical Gram matrix is nonscalar

✓

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

Proof ▶

The $(0,1)$ entry of $X^\dagger X$ equals $3/2$, whereas every scalar matrix has vanishing $(0,1)$ entry.

Theorem 21.2.3.11 The concrete physical similarity does not generate MPDOs

✓

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

Proof ▶

Otherwise Theorem 21.2.3.9 would make $X^\dagger X$ scalar, contrary to Lemma 21.2.3.10.

Theorem 21.2.3.12 Canonical form in the original physical direction

✓

As an MPS with four doubled physical symbols, $M$ is the direct sum of four bond-one normal tensors, one supported on each symbol, all with weight one. Hence $M$ itself is in canonical form, without a change of basis or blocking.

Proof ▶

Each block has identity transfer map, and their direct sum is the diagonal matrix-unit tensor $M$.

Theorem 21.2.3.13 Cyclic-edge projection in the vertical direction

✓

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

\begin{align} O_L(A) & =\operatorname{diag}(q_L), \notag \\ O_L(A)^\dagger & =O_L(A)=O_L(A)^2. \notag \end{align}

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

Proof ▶

The cyclic indicator takes only the values zero and one, so its diagonal matrix is Hermitian and idempotent.

Theorem 21.2.3.14 Normalized one-element vertical BNT and its algebra

✓

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

\begin{align} O_L(M_0) & =s^L O_L(A), \notag \\ O_L(M_0)^2 & =s^L O_L(M_0). \notag \end{align}

$\begin{tenkz}[rows={op,op}, cols=4, west=trace, east=trace, bonds=none] \tn[skin=mpo, ports={90:physical:$i_1$, 270:physical}]{M_0} & \tn[skin=mpo, ports={90:physical, 270:physical}]{M_0} & \tn[skin=dots]{} & \tn[skin=mpo, ports={90:physical:$i_L$, 270:physical}]{M_0} \\ \tn[skin=mpo, ports={90:physical, 270:physical:$j_1$}]{M_0} & \tn[skin=mpo, ports={90:physical, 270:physical}]{M_0} & \tn[skin=dots]{} & \tn[skin=mpo, ports={90:physical, 270:physical:$j_L$}]{M_0} \tnwire{(1,1)}{(1,2)} \tnwire{(1,2)}{(1,3)} \tnwire{(1,3)}{(1,4)} \tnwire{(2,1)}{(2,2)} \tnwire{(2,2)}{(2,3)} \tnwire{(2,3)}{(2,4)} \tnwire{(1,1)}{(2,1)} \tnwire{(1,2)}{(2,2)} \tnwire{(1,4)}{(2,4)} \tnmark[form=bracket]{(2,1) .. (2,4)}{$L\text{ sites}$} \end{tenkz}$ $=s^L$ $\begin{tenkz}[cols=4, west=trace, east=trace] \tn[skin=mpo, ports={90:physical:$i_1$, 270:physical:$j_1$}]{M_0} & \tn[skin=mpo, ports={90:physical, 270:physical}]{M_0} & \tn[skin=dots]{} & \tn[skin=mpo, ports={90:physical:$i_L$, 270:physical:$j_L$}]{M_0} \tnmark[form=bracket]{(1,1) .. (1,4)}{$L\text{ sites}$} \end{tenkz}$

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

Proof ▶

The identity $s^2=1/2$ gives left-canonical normalization. Scaling every local tensor contributes $s^L$ to a length-$L$ contraction, and $s^{-1}$ cancels the one-site scaling in the weighted reconstruction.

Theorem 21.2.3.15 Coisometry identifying the vertical sector coordinates

✓

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

\begin{align} WZW^\dagger & =\operatorname{reindex}(Z), \notag \\ W^\dagger \operatorname{reindex}(Z)W & =Z. \notag \end{align}

Proof ▶

Each row and column of $W$ contains one nonzero entry, equal to one.

Theorem 21.2.3.16 One-element vertical BNT algebra for the physical tensor

✓

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

Definition 21.2.3.17 BNT multiplicity-spectrum comparison

✓

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

Definition 21.2.3.18 Two-site multiplicity spectrum

✓

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

\begin{align} \widetilde{M^{(2)}} & =\bigoplus _\delta \nu _\delta \otimes A_\delta , \notag \end{align}

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

\begin{align} \bigl\{ \nu _{\sigma (\gamma ),r}:r\bigr\} & = \bigl\{ \mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k}:\alpha ,\beta ,i,j,k\bigr\} . \notag \end{align}

This is the decomposition and comparison used in [ 13 , Appendix C.4, lines 2046–2058 ] .

Definition 21.2.3.19 Exact invertible gauges for the two-site sectors

✓

Exact invertible gauges for a two-site multiplicity spectrum require, for every one-site sector $\gamma $, equality of the paired bond dimensions $d_\gamma =d_{\sigma (\gamma )}$ and an invertible matrix $Z_\gamma \in \mathrm{GL}(d_\gamma ,\mathbb {C})$ such that, for every physical index $i$ of the vertical tensor,

\begin{align} A_{\sigma (\gamma )}^i & =Z_\gamma M_\gamma ^i Z_\gamma ^{-1}. \notag \end{align}

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

$\begin{tenkz}[rows={wire}, cols=1, west=open, east=open] \tn[label pos=s, ports={90:physical:$i$}]{A_{\sigma(\gamma)}} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{Z_\gamma} & \tn[label pos=s, ports={90:physical:$i$}]{M_\gamma} & \tn[skin=ring]{Z_\gamma^{-1}} \end{tenkz}$

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

Theorem 21.2.3.20 Two-site multiplicity spectrum from the BNT algebra condition

✓

Let $M$ generate MPDOs, and suppose that its MPS tensor obtained by pairing the physical indices is in canonical form. Suppose also that $M$ has a vertical decomposition $\widetilde M=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha $ satisfying the BNT algebra condition. There is a vertical canonical decomposition of the two-site blocking,

\begin{align} \widetilde{M^{(2)}} & =\bigoplus _\delta \nu _\delta \otimes A_\delta . \notag \end{align}

Its normal-tensor sectors are in bijection, via $\sigma $, with the sectors of the one-site product expansion, in which every sector occurs. The tensors $M_\gamma $ and $A_{\sigma (\gamma )}$ generate the same matrix product vectors at every positive length and, for every $\gamma $,

\begin{align} \bigl\{ \nu _{\sigma (\gamma ),r}:r\bigr\} & = \bigl\{ \mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k}:\alpha ,\beta ,i,j,k\bigr\} \label{eq:rfp_multiplicity_spectrum} \end{align}

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

\begin{align} A_{\sigma (\gamma )}^i & =Z_\gamma M_\gamma ^i Z_\gamma ^{-1}. \label{eq:rfp_sector_conjugacy} \end{align}

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

Proof ▶

Multiplying the two one-site decompositions and using the BNT algebra law expresses the two-site closed chain in terms of $O_L(M_\gamma )$, with coefficient

\begin{align} \sum _{\alpha ,\beta ,i,j,k} \bigl(\mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k}\bigr)^L. \notag \end{align}

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 (23). 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 (22).

Theorem 21.2.3.21 Physical compression onto a conjugate two-site sector

✓

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

\begin{align} V_\gamma ^\dagger \widetilde{M^{(2)}}^vV_\gamma & =c_\gamma Z_\gamma M_\gamma ^vZ_\gamma ^{-1}. \label{eq:mpdo_bnt_exact_sector_gauge_blocked_corner} \end{align}

$\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{V_\gamma^\dagger} & \tn[label pos=s, ports={90:physical:$v$}]{\widetilde{M^{(2)}}} & \tn[skin=ring]{V_\gamma} \end{tenkz}$ $ = c_\gamma $ $\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{Z_\gamma} & \tn[label pos=s, ports={90:physical:$v$}]{M_\gamma} & \tn[skin=ring]{Z_\gamma^{-1}} \end{tenkz}$

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

\begin{align} (Z_\gamma ^\dagger Z_\gamma )M_\gamma ^v (Z_\gamma ^\dagger Z_\gamma )^{-1} \notag \end{align}

equals the corresponding reflected-adjoint marked-chain coefficient in the adjoint two-site blocking. This is one of the two marked comparisons in Figures 7–8 of [ 13 , Proposition 4.13, lines 1909–1919 ] , applied to the sector matching of Appendix C.4, lines 2048–2057.

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

\begin{align} (Z_\gamma ^\dagger Z_\gamma )M_\gamma ^v (Z_\gamma ^\dagger Z_\gamma )^{-1} & =M_\gamma ^v. \notag \end{align}

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

Proof ▶

Choose the first positive-weight copy of the matched two-site sector. Its inclusion is isometric and its compression selects the corresponding weighted normal tensor. Substituting the exact sector conjugacy gives (24), with the selected weight as $c_\gamma $. The reflected marked-chain identity for this compression, conjugated on its open indices by $Z_\gamma ^\dagger $ and $(Z_\gamma ^{-1})^\dagger $, gives the stated equality of marked-chain coefficients; the gauge factors cancel on the reflected side.

Theorem 21.2.3.22 Physical compression onto a matched one-site sector

✓

In the setting above, each sector $\gamma $ admits an isometry $W_\gamma \colon \mathbb {C}^{d_\gamma }\to \mathbb {C}^d$ and a scalar $a_\gamma {\gt}0$ such that, for every vertical letter $v$,

\begin{align} W_\gamma ^\dagger \widetilde M^vW_\gamma & =a_\gamma M_\gamma ^v. \label{eq:mpdo_bnt_exact_sector_one_site_raw_corner} \end{align}

Here the bond space of $M_\gamma $ is identified with that of its matched two-site sector. This compression selects a positive-weight copy in the one-site vertical decomposition of [ 13 , Proposition 4.13, statement lines 943–951, and Appendix C.4, lines 2046–2057 ] .

Proof ▶

Select the first copy of $M_\gamma $ in the one-site vertical direct sum. Its inclusion is isometric. Compressing the reconstruction identity by this inclusion gives (25), with $a_\gamma $ equal to the positive multiplicity weight of the selected copy. Identify the paired bond spaces using equality of their dimensions.

Theorem 21.2.3.23 Reflected marked chains with a two-site initial block

✓

Let $M$ generate matrix product density operators, let $A$ be a vertical tensor, and suppose that an isometry $V$ selects from the two-site blocking a positive multiple of an invertible conjugate of $A$. For some scalar $b{\gt}0$, this means that, for every vertical letter $v$,

\begin{align} V^\dagger \widetilde{M^{(2)}}^vV & =b\, X A^vX^{-1}. \label{eq:mpdo_two_site_prefix_gauge_corner} \end{align}

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

\begin{align} (X^\dagger X)A^v(X^\dagger X)^{-1} \notag \end{align}

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

Keeping the tail unblocked supplies the common comparison needed when [ 13 , Proposition 4.13, lines 1909–1919 ] is invoked in [ 13 , Appendix C.4, lines 2048–2057 ] . This correction to the direct two-site comparison is recorded in [ 28 ] .

Proof ▶

Compress the first two physical sites of the density operator of length $N+2$ by $V$. Hermiticity exchanges the open bond indices and the two physical words. Taking adjoints reverses only the remaining $N$ one-site factors. Finally conjugate the open indices by $X^\dagger $ and $(X^{-1})^\dagger $. The left side becomes conjugation by $X^\dagger X$, while the gauge factors cancel on the reflected side.

Theorem 21.2.3.24 Two-sided compression onto a matched two-site sector

✓

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

\begin{align} L_\gamma & =V_\gamma (Z_\gamma ^{-1})^\dagger , \label{eq:mpdo_bnt_identity_oblique_maps} \\ R_\gamma & =V_\gamma Z_\gamma . \notag \end{align}

Then, for every vertical letter $v$,

\begin{align} L_\gamma ^\dagger \widetilde{M^{(2)}}^vR_\gamma & =c_\gamma M_\gamma ^v, \label{eq:mpdo_bnt_identity_oblique_compression} \\ R_\gamma ^\dagger \widetilde{M^{(2)}}^vL_\gamma & =c_\gamma (Z_\gamma ^\dagger Z_\gamma )M_\gamma ^v (Z_\gamma ^\dagger Z_\gamma )^{-1}. \label{eq:mpdo_bnt_identity_oblique_opposite} \end{align}

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

$\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{L_\gamma^\dagger} & \tn[label pos=s, ports={90:physical:$v$}]{\widetilde{M^{(2)}}} & \tn[skin=ring]{R_\gamma} \end{tenkz}$ $ = c_\gamma $ $\begin{tenkz}[rows={wire}, cols=1, west=open, east=open] \tn[label pos=s, ports={90:physical:$v$}]{M_\gamma} \end{tenkz}$

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

\begin{align} f^\gamma _{(r,s),(i,j)} & =c_\gamma ^{-1} \overline{(L_\gamma )_{ir}}(R_\gamma )_{js}. \label{eq:mpdo_bnt_identity_physical_coefficients} \end{align}

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

\begin{align} \sum _{i,j}f^\gamma _{(r,s),(i,j)}(M^{[2]})^{ij} & =\mathcal{R}(M_\gamma )^{rs}. \label{eq:mpdo_bnt_identity_physical_span} \end{align}

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

In general $L_\gamma \ne R_\gamma $, so (28) is not an isometric compression onto a positive multiple of $M_\gamma $. At the marked-chain level, adjoint reflection exchanges the two maps together with the bond-pair and tail reversal. The reflected orientation is therefore governed by (29), rather than by a second copy of $M_\gamma $. The linear-span identity alone gives neither a common reflected tensor nor the Gram identity

\begin{align} (Z_\gamma ^\dagger Z_\gamma )M_\gamma ^v (Z_\gamma ^\dagger Z_\gamma )^{-1} & =M_\gamma ^v. \notag \end{align}

This prevents the direct two-site comparison from proving the Gram identity. The theorem below instead compares the one-site compression onto $M_\gamma $ with the two-site compression onto $Z_\gamma M_\gamma Z_\gamma ^{-1}$, keeping the same unblocked tail.

Proof ▶

Taking adjoints in the left factor of (27) and substituting (24) gives

\begin{align} L_\gamma ^\dagger \widetilde{M^{(2)}}^vR_\gamma & =Z_\gamma ^{-1} (V_\gamma ^\dagger \widetilde{M^{(2)}}^vV_\gamma ) Z_\gamma =c_\gamma M_\gamma ^v. \notag \end{align}

Reversing the two maps instead gives (29). Finally, taking the $(r,s)$ matrix entry of a general two-sided compression yields

\begin{align} [L^\dagger \widetilde{M^{(2)}}^vR]_{rs} & =\sum _{i,j}\overline{L_{ir}}R_{js}(M^{[2]})^{ij}_v. \notag \end{align}

Divide by $c_\gamma $ and use (28) to obtain (31).

Theorem 21.2.3.25 Normalization of conjugation by a scalar Gram matrix

✓

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

Proof ▶

The Gram identity identifies $Z^{-1}$ with $\omega ^{-1}Z^\dagger $; hence, for every matrix $B$ of the same size, $(\omega ^{-1/2}Z)B(\omega ^{-1/2}Z)^\dagger =ZBZ^{-1}$.

Theorem 21.2.3.26 Unitary sector conjugacy from a positive scalar Gram matrix

✓

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

\begin{align} A_{\sigma (\gamma )}^v & =U_\gamma M_\gamma ^v U_\gamma ^\dagger . \label{eq:mpdo_bnt_conditional_unitary_conjugacy} \end{align}

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

Proof ▶

The positive scalar Gram identity gives $U_\gamma ^\dagger U_\gamma =\mathbb {1}$ and $Z_\gamma ^{-1}=\omega _\gamma ^{-1}Z_\gamma ^\dagger $. Consequently $U_\gamma M_\gamma ^vU_\gamma ^\dagger =Z_\gamma M_\gamma ^vZ_\gamma ^{-1}$. The exact sector conjugacy then gives (32).

Theorem 21.2.3.27 Gram identity and unitary sector conjugacy

✓

Let $M$ generate MPDOs, and suppose that its MPS tensor obtained by pairing the physical indices is in canonical form. Let the one-site and two-site sectors be paired by the exact gauges $Z_\gamma $ above. Then, for every sector $\gamma $ and every vertical letter $v$,

\begin{align} (Z_\gamma ^\dagger Z_\gamma )M_\gamma ^v (Z_\gamma ^\dagger Z_\gamma )^{-1} & =M_\gamma ^v. \label{eq:mpdo_bnt_mixed_prefix_gram_identity} \end{align}

$\begin{tenkz}[rows={wire}, cols=3, west=open, east=open] \tn[skin=ring]{Z_\gamma^\dagger Z_\gamma} & \tn[label pos=s, ports={90:physical:$v$}]{M_\gamma} & \tn[skin=ring]{(Z_\gamma^\dagger Z_\gamma)^{-1}} \end{tenkz}$ $ = $ $\begin{tenkz}[rows={wire}, cols=1, west=open, east=open] \tn[label pos=s, ports={90:physical:$v$}]{M_\gamma} \end{tenkz}$

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

\begin{align} Z_\gamma ^\dagger Z_\gamma & =\omega _\gamma \mathbb {1}, \label{eq:mpdo_bnt_mixed_prefix_scalar_gram} \end{align}

and there is a unitary $U_\gamma $ satisfying

\begin{align} A_{\sigma (\gamma )}^v & =U_\gamma M_\gamma ^vU_\gamma ^\dagger . \label{eq:mpdo_bnt_mixed_prefix_unitary_conjugacy} \end{align}

The proof compares a one-site compression onto $M_\gamma $ with a two-site compression onto $Z_\gamma M_\gamma Z_\gamma ^{-1}$, keeping the same unblocked tail. This supplies the common comparison missing from the direct application of [ 13 , Proposition 4.13, lines 1909–1919 ] in [ 13 , Appendix C.4, lines 2048–2057 ] . The correction to the direct two-site argument is recorded in [ 28 ] .

Proof ▶

Put $G_\gamma =Z_\gamma ^\dagger Z_\gamma $. The two-site initial-block identity applied to the compression onto $Z_\gamma M_\gamma Z_\gamma ^{-1}$ identifies the marked chains of $G_\gamma M_\gamma G_\gamma ^{-1}$ with the reflected-adjoint chains of $M_\gamma $, followed by an arbitrary unblocked tail of $M^\dagger $. The ordinary one-site reflected identity applied to the compression onto $M_\gamma $ gives the same reflected-adjoint chains.

Both marked tensors are linear combinations of the one-site physical letters. Indeed, the mark $M_\gamma $ is obtained from the pair $(W_\gamma ,W_\gamma )$, whereas the mark $G_\gamma M_\gamma G_\gamma ^{-1}$ is obtained from the pair $(W_\gamma G_\gamma ^\dagger ,W_\gamma G_\gamma ^{-1})$. Their marked traces agree against every positive-length tail. The original-coordinate form of Lemma L therefore identifies the two marked tensors and proves (33). Normality of $M_\gamma $ gives (34), and normalizing $Z_\gamma $ by $\omega _\gamma ^{-1/2}$ gives the unitary in (35).

Lemma 21.2.3.28 Idempotence in terms of the diagonal matrices

✓

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

\begin{align} m_\gamma & =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \notag \end{align}

Proof ▶

The trace-power 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.

Definition 21.2.3.29 Multiplicity spectra from eventual equality of power sums

✓

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

\begin{align} \sum _r \nu _{\gamma ,r}^{L} & = \sum _{\alpha ,\beta ,i,j,k} \bigl(\mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k}\bigr)^L, \notag \end{align}

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

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

\begin{align} \sum _r \nu _{\gamma ,r}^{L} & = \sum _{\alpha ,\beta ,i,j,k} \bigl(\mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k}\bigr)^L. \notag \end{align}

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

Theorem 21.2.3.30 Multiplicity trace normalization

✓

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

\begin{align} \operatorname{tr}(\nu _\gamma ) & =\sum _{\alpha ,\beta ,i,j,k} \mu _{\alpha ,i}\mu _{\beta ,j} \chi _{\alpha ,\beta ,\gamma ,k} =\sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta =m_\gamma . \label{eq:rfp_trace_normalization} \end{align}

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

Proof ▶

Sum the multiset equality of the two spectra. For each $\alpha ,\beta ,\gamma $, the resulting product sum satisfies

\begin{align} \sum _{i,j,k} \mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k} & = \left(\sum _i\mu _{\alpha ,i}\right) \left(\sum _j\mu _{\beta ,j}\right) \left(\sum _k\chi _{\alpha ,\beta ,\gamma ,k}\right). \notag \end{align}

The identities

\begin{align} \sum _i\mu _{\alpha ,i} & =m_\alpha , \notag \\ \sum _j\mu _{\beta ,j} & =m_\beta , \notag \\ \sum _k\chi _{\alpha ,\beta ,\gamma ,k} & =c^{(1)}_{\alpha ,\beta ,\gamma } \notag \end{align}

therefore give

\begin{align} \operatorname{tr}(\nu _\gamma ) & =\sum _{\alpha ,\beta } m_\alpha m_\beta c^{(1)}_{\alpha ,\beta ,\gamma } =m_\gamma , \notag \end{align}

where the final equality is the length-one idempotent law.

Definition 21.2.3.31 BNT algebra with a blocked-basis comparison

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & =\sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }O_L(M_\gamma ). \notag \end{align}

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

\begin{align} m_\gamma & =\sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta . \notag \end{align}

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

\begin{align} c^{(n)}_{i,j,k} & =c^{(n)}_{\sigma _n(i),\sigma _n(j),\tau _n(k)}. \notag \end{align}

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

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

Theorem 21.2.3.32 BNT structure coefficients and product formulas

✓

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

\begin{align} c^{(L)}_{\alpha ,\beta ,\gamma } & =c^{\chi ,(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}\! \left(\chi _{\alpha ,\beta ,\gamma }^{\, L}\right), \notag \end{align}

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

$c^{(L)}_{\alpha ,\beta ,\gamma }={}$ $\begin{tenkz}[cols=4, west=trace, east=trace] \tn[skin=box, label pos=90]{\chi_{\alpha,\beta,\gamma}} & \tn[skin=box, label pos=90]{\chi_{\alpha,\beta,\gamma}} & \tn[skin=dots]{} & \tn[skin=box, label pos=90]{\chi_{\alpha,\beta,\gamma}} \tnmark[form=bracket]{(1,1) .. (1,4)}{$L\text{ factors}$} \end{tenkz}$

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & =\sum _\gamma \operatorname{tr}\! \left(\chi _{\alpha ,\beta ,\gamma }^{\, L}\right) O_L(M_\gamma ), \notag \end{align}

and the idempotent identity reads

\begin{align} m_\gamma & =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \notag \end{align}

Proof ▶

The assumed positive diagonal matrices satisfy $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L})$ for $L{\gt}0$. Substitution into the product law, and into the idempotent law at length one, gives the stated formulas.

Theorem 21.2.3.33 Positive diagonal matrices for the blocked coefficients

✓

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

Proof ▶

Apply the construction of positive diagonal matrices for blocked coefficients to the comparison maps and the positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$ representing the BNT structure coefficients.

Corollary 21.2.3.34 Blocked coefficient trace-power formulas

✓

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

\begin{align} c^{(n)}_{i,j,k} & = \operatorname{tr}\! \left(\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}^{\, n}\right) =\operatorname{tr}(\chi _{n,i,j,k}^{\, n}). \notag \end{align}

Proof ▶

The blocked-basis comparison and the trace-power representation identify the coefficient with the trace of the $n$th power of the matrix selected by the source and target maps. By definition, the pulled-back matrix $\chi _{n,i,j,k}$ is this same matrix.

Definition 21.2.3.35 BNT operators and structure coefficients

✓

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

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

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

Theorem 21.2.3.36 Product and idempotent equations for BNT operators

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma } O_L(M_\gamma ), \label{eq:mpdo_bnt_same_len_product} \\ m_\gamma & = \sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta . \label{eq:mpdo_bnt_idempotent_law} \end{align}

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & = \sum _\gamma \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{\, L}) O_L(M_\gamma ), \label{eq:mpdo_bnt_same_len_product_chi} \\ m_\gamma & = \sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }) m_\alpha m_\beta . \label{eq:mpdo_bnt_idempotent_law_chi} \end{align}

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

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

\begin{align} c^{(n)}_{i,j,k} & = c^{(n)}_{\sigma _n(i),\sigma _n(j),\tau _n(k)} = \operatorname{tr}\! \left(\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}^{\, n}\right). \notag \end{align}

Proof ▶

Choose the assumed operators, scalars, coefficients, and diagonal matrices. Their product and idempotent identities give (37) and (38); substitute the trace-power identity to obtain (39) and (40). The blocked-basis comparison gives the final coefficient formula.

Theorem 21.2.3.37 Positive diagonal matrices for blocked coefficients

✓

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

\begin{align} \chi _{n,i,j,k} & = \chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}, \notag \\ c^{(n)}_{i,j,k} & = \operatorname{tr}(\chi _{n,i,j,k}^{\, n}). \notag \end{align}

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

Proof ▶

Apply the blocked-basis trace-power consequence to the chosen operators, scalars, coefficients, and diagonal matrices.

Corollary 21.2.3.38 Blocked coefficients as traces of powers

✓

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

\begin{align} c^{(n)}_{i,j,k} & = \operatorname{tr}\! \left(\chi _{\sigma _n(i),\sigma _n(j),\tau _n(k)}^{\, n}\right) = \operatorname{tr}(\chi _{n,i,j,k}^{\, n}). \notag \end{align}

Proof ▶

Apply the trace-power identity for the positive diagonal matrices $\chi _{n,i,j,k}$ obtained from the assumed BNT matrices and blocked-basis comparison.

21.2.4 Direct-sum sector conjugations

Definition 21.2.4.1 Direct-sum unitary conjugations

✓

Work in the setting of Definition 21.2.3.19. Let

\begin{align} \mathcal A_1 & =\bigoplus _\gamma M_{d_\gamma }(\mathbb {C}), & \mathcal A_2^\sigma & =\bigoplus _\gamma M_{\widetilde d_{\sigma (\gamma )}}(\mathbb {C}) \end{align}

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

\begin{align} A_{\sigma (\gamma )}^i & =U_\gamma M_\gamma ^i U_\gamma ^* \label{eq:mpdo_bnt_sector_unitary_conjugacy} \end{align}

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

\begin{align} \widetilde T\colon \mathcal A_1 & \longrightarrow \mathcal A_2^\sigma , & [\widetilde T(X)]_\gamma & =U_\gamma \iota _\gamma (X_\gamma )U_\gamma ^*, \label{eq:mpdo_bnt_direct_sum_unitary_T} \\ \widetilde S\colon \mathcal A_2^\sigma & \longrightarrow \mathcal A_1, & [\widetilde S(Y)]_\gamma & =\iota _\gamma ^{-1} (U_\gamma ^*Y_\gamma U_\gamma ). \label{eq:mpdo_bnt_direct_sum_unitary_S} \end{align}

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

Theorem 21.2.4.2 Properties of the direct-sum unitary conjugations

✓

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

\begin{align} \widetilde S\widetilde T & =\operatorname{id}_{\mathcal A_1}, & \widetilde T\widetilde S & =\operatorname{id}_{\mathcal A_2^\sigma }. \label{eq:mpdo_bnt_direct_sum_unitary_inverses} \end{align}

They also carry the matched tensor letters in both directions:

\begin{align} \widetilde T\bigl((M_\gamma ^i)_\gamma \bigr) & =\bigl(A_{\sigma (\gamma )}^i\bigr)_\gamma , \label{eq:mpdo_bnt_direct_sum_unitary_tensor_letters_fwd} \\ \widetilde S\bigl((A_{\sigma (\gamma )}^i)_\gamma \bigr) & =\bigl(M_\gamma ^i\bigr)_\gamma . \label{eq:mpdo_bnt_direct_sum_unitary_tensor_letters_rev} \end{align}

Proof ▶

On each summand, conjugation by $U_\gamma $ and the coordinate identification $\iota _\gamma $ are completely positive and trace-preserving. Their direct sums therefore have the same properties. Unitarity gives

\begin{align} U_\gamma ^*U_\gamma & =I, & U_\gamma U_\gamma ^* & =I, \end{align}

which proves (45) sector by sector. Identity (46) is precisely (42); applying $\widetilde S$ gives (47).

21.2.5 Ambient sector retractions and normalized embeddings

Definition 21.2.5.1 Ambient sector retractions and normalized embeddings

✓

Let

\begin{align} \mathcal A_1 & =\bigoplus _\gamma M_{d_\gamma }(\mathbb {C}), & \mathcal A_2^\sigma & =\bigoplus _\gamma M_{\widetilde d_{\sigma (\gamma )}}(\mathbb {C}) \notag \end{align}

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

\begin{align} \mathcal K_1 & =\bigoplus _\gamma \left(\mathbb {C}^{r_\gamma }\otimes \mathbb {C}^{d_\gamma }\right), & \mathcal K_2^\sigma & =\bigoplus _\gamma \left(\mathbb {C}^{\widetilde r_{\sigma (\gamma )}} \otimes \mathbb {C}^{\widetilde d_{\sigma (\gamma )}}\right). \notag \end{align}

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

\begin{align} \kappa _\ast \colon M_{d^2}(\mathbb {C}) & \longrightarrow M_{d}(\mathbb {C})\otimes M_{d}(\mathbb {C}) \notag \end{align}

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

\begin{align} m_\gamma & =\operatorname{tr}(\mu _\gamma ) =\operatorname{tr}(\nu _{\sigma (\gamma )}). \label{eq:mpdo_bnt_ambient_sector_trace_weights} \end{align}

Define

\begin{align} R_1\colon M_{d}(\mathbb {C}) & \longrightarrow \mathcal A_1, & [R_1(X)]_\gamma & =\operatorname{tr}_{\mathrm{mult}} \left(\Pi _{1,\gamma }W_1XW_1^\dagger \Pi _{1,\gamma }\right), \label{eq:mpdo_bnt_ambient_sector_R1} \\ R_2\colon M_{d}(\mathbb {C})\otimes M_{d}(\mathbb {C}) & \longrightarrow \mathcal A_2^\sigma , & [R_2(Y)]_\gamma & =\operatorname{tr}_{\mathrm{mult}} \left(\Pi _{2,\gamma }^\sigma W_2\kappa _\ast ^{-1}(Y)W_2^\dagger \Pi _{2,\gamma }^\sigma \right), \label{eq:mpdo_bnt_ambient_sector_R2} \\ \widetilde R_1\colon \mathcal A_1 & \longrightarrow M_{d}(\mathbb {C}), & \widetilde R_1((X_\gamma )_\gamma ) & =W_1^\dagger \left(\bigoplus _\gamma \frac{\mu _\gamma }{m_\gamma }\otimes X_\gamma \right)W_1, \label{eq:mpdo_bnt_ambient_sector_R1_tilde} \\ \widetilde R_2\colon \mathcal A_2^\sigma & \longrightarrow M_{d}(\mathbb {C})\otimes M_{d}(\mathbb {C}), & \widetilde R_2((Y_\gamma )_\gamma ) & =\kappa _\ast \! \left[ W_2^\dagger \left(\bigoplus _\gamma \frac{\nu _{\sigma (\gamma )}}{m_\gamma }\otimes Y_\gamma \right)W_2 \right]. \label{eq:mpdo_bnt_ambient_sector_R2_tilde} \end{align}

In Appendix C.4 of [ 13 ] , lines 2067–2068 define $R_1$ by (50), and lines 2078–2079 define $R_2$ by (51). Lines 2081–2083 give the normalized map $\widetilde R_1$ in (52). Lines 2058–2071 give $\widetilde R_2$, including its common denominator, in (53). These four definitions use only the two vertical decompositions, their sector relabelling, and (49).

Theorem 21.2.5.2 Complete positivity of the ambient sector maps

✓

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

Proof ▶

Each raw retraction is the composition of compression by $W_i$, a unitary relabelling of coordinates, sectorwise partial trace, and block-diagonal inclusion. Each normalized embedding is the composition of diagonal-block extraction, preparation of the positive matrix $\mu _\gamma /m_\gamma $ or $\nu _{\sigma (\gamma )}/m_\gamma $, coordinate relabelling, and expansion by $W_i^\dagger $. Every constituent has a Kraus representation, and compositions of such maps are completely positive.

Theorem 21.2.5.3 Trace of the ambient sector maps

✓

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

\begin{align} \operatorname {Tr}_{\mathrm{sec}}(R_1(X)) & =\operatorname{tr}(W_1XW_1^\dagger ), \label{eq:mpdo_bnt_ambient_sector_R1_trace} \\ \operatorname {Tr}_{\mathrm{sec}}(R_2(Y)) & =\operatorname{tr}\! \left(W_2\kappa _\ast ^{-1}(Y)W_2^\dagger \right). \label{eq:mpdo_bnt_ambient_sector_R2_trace} \end{align}

If $X$ and $Y$ are positive semidefinite, then

\begin{align} \operatorname {Tr}_{\mathrm{sec}}(R_1(X)) & \leq \operatorname{tr}(X), & \operatorname {Tr}_{\mathrm{sec}}(R_2(Y)) & \leq \operatorname{tr}(Y). \label{eq:mpdo_bnt_ambient_sector_trace_loss} \end{align}

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

\begin{align} \operatorname{tr}(\widetilde R_1(Z)) & =\operatorname {Tr}_{\mathrm{sec}}(Z), & \operatorname{tr}(\widetilde R_2(Z’)) & =\operatorname {Tr}_{\mathrm{sec}}(Z’). \label{eq:mpdo_bnt_ambient_sector_embedding_trace} \end{align}

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

The inequalities in (56) may be strict: $W_i^\dagger W_i$ projects onto the retained nonzero sectors, so positive elements supported on the discarded orthogonal complement lose trace under the compression.

Proof ▶

Sectorwise partial trace and block-diagonal inclusion preserve the total trace, which gives (54) and (55). For $P_i=W_i^\dagger W_i$, cyclicity of trace gives

\begin{align} \operatorname{tr}(W_i ZW_i^\dagger ) & =\operatorname{tr}(P_i Z)\leq \operatorname{tr}(Z) \notag \end{align}

when $Z\succeq 0$. The relabelling $\kappa _\ast $ preserves positivity and trace, proving (56). Finally, each normalized multiplicity matrix has trace one, and $W_i W_i^\dagger =I$ gives $\operatorname{tr}(W_i^\dagger BW_i)=\operatorname{tr}(B)$. This proves (57); complete positivity is supplied by Theorem 21.2.5.2.

Theorem 21.2.5.4 Retraction identities for the ambient sector maps

✓

The raw retractions are left inverses of their normalized embeddings:

\begin{align} R_1\widetilde R_1 & =\operatorname{id}_{\mathcal A_1}, & R_2\widetilde R_2 & =\operatorname{id}_{\mathcal A_2^\sigma }. \label{eq:mpdo_bnt_ambient_sector_retractions} \end{align}

Proof ▶

Compressing an expansion by $W_i^\dagger $ gives the original retained block-diagonal matrix because $W_i W_i^\dagger =I$. On every sector,

\begin{align} \operatorname{tr}_{\mathrm{mult}} \left(\frac{\mu _\gamma }{m_\gamma }\otimes X_\gamma \right) & =X_\gamma , \notag \\ \operatorname{tr}_{\mathrm{mult}} \left(\frac{\nu _{\sigma (\gamma )}}{m_\gamma }\otimes Y_\gamma \right) & =Y_\gamma , \label{eq:mpdo_bnt_ambient_retract_nu} \end{align}

where (59) follows from $\operatorname{tr}(\nu _{\sigma (\gamma )})=m_\gamma $. The two coordinate relabellings in the two-site composite cancel. These identities prove (58).

Theorem 21.2.5.5 Action on the weighted tensor letters

✓

For every vertical letter $v$, the one-site maps satisfy

\begin{align} R_1(\widetilde M_v) & =\bigl(m_\gamma M_\gamma ^v\bigr)_\gamma , \label{eq:mpdo_bnt_ambient_sector_one_site_fwd} \\ \widetilde R_1\bigl((m_\gamma M_\gamma ^v)_\gamma \bigr) & =\widetilde M_v. \label{eq:mpdo_bnt_ambient_sector_one_site_rev} \end{align}

The relabelled two-site maps satisfy

\begin{align} R_2\! \left(\kappa _\ast (\widetilde{M^{[2]}}_v)\right) & =\bigl(m_\gamma A_{\sigma (\gamma )}^v\bigr)_\gamma , \label{eq:mpdo_bnt_ambient_sector_two_site_fwd} \\ \widetilde R_2\bigl((m_\gamma A_{\sigma (\gamma )}^v)_\gamma \bigr) & =\kappa _\ast (\widetilde{M^{[2]}}_v). \label{eq:mpdo_bnt_ambient_sector_two_site_rev} \end{align}

Identities (60) and (61) combine the one-site weighted sector form of [ 13 , Appendix C.4, lines 2048–2051 ] with, respectively, [ 13 , Appendix C.4, lines 2067–2068 ] and [ 13 , Appendix C.4, lines 2081–2083 ] . Identity (62) combines the matched weighted sector form of [ 13 , Appendix C.4, lines 2048–2064 ] with the map $R_2$ of [ 13 , Appendix C.4, lines 2078–2079 ] . Identity (63) combines the same sector form with the normalized map $\widetilde R_2$ of [ 13 , Appendix C.4, lines 2070–2071 ] .

Proof ▶

The vertical reconstruction identities give

\begin{align} \widetilde M_v & =W_1^\dagger \left(\bigoplus _\gamma \mu _\gamma \otimes M_\gamma ^v\right)W_1, \notag \\ \widetilde{M^{[2]}}_v & =W_2^\dagger \left(\bigoplus _\gamma \nu _{\sigma (\gamma )}\otimes A_{\sigma (\gamma )}^v\right)W_2. \notag \end{align}

Compression removes the two outer coisometries, and partial trace over the multiplicity coordinate multiplies the simple-sector letter by $\operatorname{tr}(\mu _\gamma )=m_\gamma $ or $\operatorname{tr}(\nu _{\sigma (\gamma )})=m_\gamma $. This proves (60) and (62). Preparing the normalized multiplicity matrix cancels the factor $m_\gamma $, yielding (61) and (63).

21.2.6 Trace-preserving completions and physical channels

Theorem 21.2.6.1 Trace-preserving completion outside a projection

✓

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

\begin{align} \mathcal D_\rho (X) & =\operatorname{tr}(X)\rho , & \mathcal C_{P,\rho }(X) & =\mathcal D_\rho \bigl((I-P)X(I-P)^\dagger \bigr). \label{eq:matrix_support_complement_map} \end{align}

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

\begin{align} \operatorname{tr}(\mathcal E(X))=\operatorname{tr}(PX) \label{eq:matrix_support_map_trace} \end{align}

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

\begin{align} \widehat{\mathcal E}(X) & =\mathcal E(X)+\mathcal C_{P,\rho }(X) \label{eq:matrix_support_completion} \end{align}

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

\begin{align} PXP=X \quad \Longrightarrow \quad \widehat{\mathcal E}(X)=\mathcal E(X). \label{eq:matrix_support_completion_agreement} \end{align}

Proof ▶

The map $\mathcal D_\rho $ factors as state preparation, interchange of the two tensor factors, and partial trace. The map $X\mapsto (I-P)X(I-P)^\dagger $ has a single Kraus operator. Thus the two maps in (64) are completely positive. Since $P=P^\dagger =P^2$,

\begin{align} \operatorname{tr}\bigl(\mathcal C_{P,\rho }(X)\bigr) & =\operatorname{tr}((I-P)X). \notag \end{align}

Adding this identity to (65) gives $\operatorname{tr}(\widehat{\mathcal E}(X))=\operatorname{tr}(X)$. Finally, $PXP=X$ implies $(I-P)X(I-P)=0$, which proves (67).

Theorem 21.2.6.2 Physical channels from unitary sector conjugacies

✓

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

\begin{align} A_{\sigma (\gamma )}^i & =U_\gamma M_\gamma ^i U_\gamma ^\dagger \label{eq:mpdo_bnt_physical_channel_unitary_conjugacy} \end{align}

for every sector $\gamma $ and every vertical letter $i$. With the ambient sector maps and unitary conjugations defined above, set

\begin{align} \mathcal T_0 & =\widetilde R_2\circ \widetilde T\circ R_1, & \mathcal S_0 & =\widetilde R_1\circ \widetilde S\circ R_2. \label{eq:mpdo_bnt_active_physical_maps} \end{align}

Let

\begin{align} P_1 & =W_1^\dagger W_1, & P_2 & =\kappa _\ast (W_2^\dagger W_2), & Q_i & =I-P_i. \label{eq:mpdo_bnt_physical_active_projections} \end{align}

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

\begin{align} \mathcal T(X) & =\mathcal T_0(X)+\operatorname{tr}(Q_1XQ_1)\tau _2, \notag \\ \mathcal S(Y) & =\mathcal S_0(Y)+\operatorname{tr}(Q_2YQ_2)\tau _1. \label{eq:mpdo_bnt_completed_physical_maps} \end{align}

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

\begin{align} \mathcal T[M_1(Z)] & =M_2(Z), & \mathcal S[M_2(Z)] & =M_1(Z). \label{eq:mpdo_bnt_completed_physical_map_actions} \end{align}

Hence $M$ is a renormalization fixed point in the sense of Definition 20.2.1.

The compositions in (69) are the formulas of [ 13 , Appendix C.4, lines 2065–2085 ] . Those formulas determine the action after compression to the retained physical sectors. With rectangular coisometries, the literal composites annihilate the discarded zero-sector complements and fail to preserve trace there; Formula (71) provides a trace-preserving completion on those complements when the physical space is nonzero, yielding the same action on the physical operators generated by $M$ for any choice of reference states $\tau _1,\tau _2$.

Proof ▶

Complete positivity of the four outer maps and of the two direct-sum conjugations gives complete positivity of $\mathcal T_0$ and $\mathcal S_0$. Their traces are

\begin{align} \operatorname{tr}(\mathcal T_0(X)) & =\operatorname{tr}(P_1X), & \operatorname{tr}(\mathcal S_0(Y)) & =\operatorname{tr}(P_2Y). \label{eq:mpdo_bnt_active_physical_map_traces} \end{align}

Suppose first that $d{\gt}0$. Since $Q_i$ is a projection and $\tau _i$ has trace one, the corresponding complementary term is completely positive and has trace $\operatorname{tr}(Q_iXQ_i)=\operatorname{tr}(Q_iX)$. Adding (73) and using $P_i+Q_i=I$ proves trace preservation of the completed maps. If $d=0$, every ambient matrix is zero, so the zero maps are completely positive and trace preserving.

The reconstruction identities for the two vertical decompositions imply

\begin{align} P_1M_1(Z)P_1 & =M_1(Z), & P_2M_2(Z)P_2 & =M_2(Z). \label{eq:mpdo_bnt_physical_closure_support} \end{align}

Thus both complementary terms vanish on the corresponding physical operators. The retraction formulas extract the families $(m_\gamma M_\gamma ^i)_\gamma $ and $(m_\gamma A_{\sigma (\gamma )}^i)_\gamma $; unitary conjugation carries these families into one another, and the normalized restorations recover the one-site and two-site letters. Linearity in the virtual matrix gives (72).

Theorem 21.2.6.3 BNT algebra clause implies a renormalization fixed point

✓

Let $M$ be an MPDO whose doubled-index tensor is in literal CPSV canonical form. If $M$ has a tensor-attached basis-of-normal-tensors algebra clause, then there are trace-preserving completely positive maps

\begin{align} \mathcal T\colon M_{d}(\mathbb {C}) & \longrightarrow M_{d^2}(\mathbb {C}), & \mathcal S\colon M_{d^2}(\mathbb {C}) & \longrightarrow M_{d}(\mathbb {C}) \notag \end{align}

such that, for every virtual matrix $X$,

\begin{align} \mathcal T[M_1(X)] & =M_2(X), & \mathcal S[M_2(X)] & =M_1(X). \label{eq:mpdo_bnt_algebra_tensor_rfp_maps} \end{align}

Thus $M$ is a renormalization fixed point in the sense of Definition 20.2.1. This is implication (ii)$\Rightarrow $(i) of [ 13 , Theorem 4.14, lines 972–993 ] .

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

Proof ▶

The tensor-attached algebra clause determines the two-site vertical decomposition, the sector relabelling, and exact invertible gauges. Theorem 21.2.3.27 normalizes these gauges simultaneously to unitaries. Apply Theorem 21.2.6.2 to obtain the two trace-preserving completely positive maps and (75).

21.3 Transfer retracts and fusion coisometries

A linear factorization of the doubled-index blocked transfer map $E_n$ through a subspace with the retract identities of (76) exists precisely when $E_n$ is idempotent. Requiring such a factorization at every positive blocked size is equivalent to idempotence of $E_1$. This transfer-retract condition is distinct from the physical renormalization fixed-point condition in Definition 20.2.1 and [ 13 , Definition 4.1, lines 638–660 ] . For an MPDO tensor $M$ in CF, the latter requires tpCPMs $S,T$ on the physical indices satisfying $S[M_2(X)]=M_1(X)$ and $T[M_1(X)]=M_2(X)$ for every boundary matrix $X$, acting on physical indices as in [ 13 , lines 1974–2085 ] . By contrast, the fusion maps in [ 13 , Theorem 4.14(ii)–(iii), lines 972–991 ] act on the vertically read tensors, distinguishing support restrictions and exact reconstruction from full-support isometries.

Definition 21.3.1 Transfer-retract datum

✓

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

\begin{align} T_n & :M_{D}(\mathbb {C})\longrightarrow \mathcal{A}_n, & S_n & :\mathcal{A}_n\longrightarrow M_{D}(\mathbb {C}), \notag \\ T_n\circ S_n & =\operatorname{id}_{\mathcal{A}_n}, & S_n\circ T_n & =E_n, \label{eq:rfp_transfer_retract} \end{align}

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

Theorem 21.3.2 Transfer-retract data imply transfer-map idempotence

✓

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

Proof ▶

By (76),

\begin{align} E_1^2 & =S_1T_1S_1T_1 =S_1(T_1S_1)T_1 =S_1T_1 =E_1. \notag \end{align}

Theorem 21.3.3 One-site transfer-retract criterion

✓

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

Proof ▶

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.

Theorem 21.3.4 Transfer retracts at positive blocked sizes

✓

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

Proof ▶

Every positive blocked transfer map equals a positive power of the idempotent one-site map and is therefore idempotent. Factoring it through its range gives the transfer-retract datum.

Definition 21.3.5 Operator product of MPO tensors

✓

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:

\begin{align} (M\mathbin {\cdot }N)^{ik} & =\sum _j M^{ij}\otimes N^{jk}. \label{eq:rfp_product_tensor} \end{align}

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

Definition 21.3.6 Canonical reassociation of three bond spaces

✓

For three bond spaces, let

\begin{align} A_{D_1,D_2,D_3}: \mathbb {C}^{D_1}\otimes (\mathbb {C}^{D_2}\otimes \mathbb {C}^{D_3}) & \longrightarrow (\mathbb {C}^{D_1}\otimes \mathbb {C}^{D_2})\otimes \mathbb {C}^{D_3} \notag \end{align}

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

Theorem 21.3.7 Associativity of the product tensor

✓

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

\begin{align} ((M\mathbin {\cdot }N)\mathbin {\cdot }P)^{ik} A_{D_1,D_2,D_3} & =A_{D_1,D_2,D_3} (M\mathbin {\cdot }(N\mathbin {\cdot }P))^{ik}. \notag \end{align}

Proof ▶

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.

Lemma 21.3.8 Word evaluation of the product tensor

✓

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:

\begin{align} (M\mathbin {\cdot }N)^{\sigma \tau } & =\sum _\rho M^{\sigma \rho }\otimes N^{\rho \tau }. \label{eq:rfp_product_word} \end{align}

Proof ▶

Induction on the length, using the mixed-product property of the tensor product to merge one letter at a time.

Theorem 21.3.9 The operator family is multiplicative

✓

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

Proof ▶

Take traces in (78): 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.

Lemma 21.3.10 Word products of isometrically conjugated letters

✓

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

\begin{align} (UF_1U^\dagger )(UF_2U^\dagger )\cdots (UF_LU^\dagger ) & =U F_1F_2\cdots F_L U^\dagger . \notag \end{align}

Proof ▶

The inner factors $U^\dagger U$ telescope to the identity.

Lemma 21.3.11 Word products of block-diagonal tensor-product letters

✓

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

\begin{align} \prod _{l=1}^{L} \bigoplus _{\gamma }X_\gamma \otimes G_{\gamma ,l} & = \bigoplus _{\gamma }X_\gamma ^{L}\otimes \prod _{l=1}^{L}G_{\gamma ,l}. \notag \end{align}

Proof ▶

Blocks multiply blockwise and tensor products multiply factorwise, so the fixed factors accumulate into the matrix power $X_\gamma ^L$.

Lemma 21.3.12 Blockwise conjugation and restriction

✓

Let $F=\bigoplus _a F_a$ and $G=\bigoplus _a G_a$. Then

\begin{align} FGF^\dagger & =\bigoplus _a F_aG_aF_a^\dagger . \label{eq:matrix_block_conjugation} \end{align}

If $r$ is a bijective reindexing of the row space of a matrix $A$, then

\begin{align} A_rXA_r^\dagger & =(AXA^\dagger )_{r,r}, \label{eq:matrix_row_reindexing_conjugation} \end{align}

where $A_r$ denotes the corresponding row reindexing.

Proof ▶

Identity (79) follows because block-diagonal matrices multiply blockwise. Identity (80) follows by commuting bijective row reindexing with multiplication and adjunction.

Lemma 21.3.13 Cancellation of a nonempty identity factor

✓

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 $F$.

Proof ▶

Evaluate the amplified identity on one basis vector of the nonzero factor. Multiplicativity of the Kronecker product gives the product and adjoint specializations.

Definition 21.3.14 Fusion isometries on the full product bond space

✓

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

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} U_{\alpha ,\beta }^\dagger & = \bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}. \label{eq:rfp_fusion_identity} \end{align}

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

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

Definition 21.3.15 Fusion coisometries onto the retained direct sum

✓

For the same labelled tensors and positive diagonal matrices, put $G_{\alpha ,\beta }^{ij} =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}$. Fusion coisometries onto the retained direct sum are maps $U_{\alpha ,\beta }$ satisfying $U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1$, the forward fusion identity

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} U_{\alpha ,\beta }^\dagger & =G_{\alpha ,\beta }^{ij}, \notag \end{align}

and exact reconstruction

\begin{align} (M_\alpha M_\beta )^{ij} & =U_{\alpha ,\beta }^\dagger G_{\alpha ,\beta }^{ij}U_{\alpha ,\beta }. \label{eq:rfp_coisometry_reconstruction} \end{align}

Zero-dimensional retained sums are permitted. If in addition $U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1$ for every pair, these maps are fusion isometries on the full product bond space.

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

With $G_{\alpha ,\beta }^{ij}$ denoting the retained direct-sum letter, [ 13 , Theorem 4.14(iii), equation (89), lines 986–991 ] gives the following fusion identity:

$\begin{tenkz}[rows={op,op}, cols=3] \tnfuse[at=(1,1), name=U, skin=pill]{U_{\alpha,\beta}} \tn[at=(1,2), skin=mpo, ports={90:physical:$i$, 270:physical}]{M_\alpha} \tn[at=(2,2), skin=mpo, ports={90:physical, 270:physical:$j$}]{M_\beta} \tnfuse[at=(1,3), name=Ud, skin=pill]{U_{\alpha,\beta}^{\dagger}} \tnwire{open w}{U.180} \tnwire{Ud.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[cols=1, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i$, 270:physical:$j$}]{G_{\alpha,\beta}} \end{tenkz}$

Lemma 21.3.16 Weighted zipper identities on the active support

✓

A fusion coisometry onto the retained direct sum satisfies

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} & =G_{\alpha ,\beta }^{ij}U_{\alpha ,\beta }, \notag \\ (M_\alpha M_\beta )^{ij}U_{\alpha ,\beta }^\dagger & =U_{\alpha ,\beta }^\dagger G_{\alpha ,\beta }^{ij}. \notag \end{align}

Proof ▶

Substitute exact reconstruction (82) into the left-hand sides and use $U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1$.

Theorem 21.3.17 Weighted zipper identities

✓

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

The corresponding identities in [ 8 , Equation (21) ] have identity matrices on the multiplicity spaces in place of the positive diagonal matrices $\chi $. Their unweighted form requires the length-independent integer specialization and is not a consequence of the present hypotheses. Even after that specialization, comparison of the two triple-fusion orders and separation of the final labels are still needed to construct an invertible $F$-move.

Proof ▶

Multiply the fusion identity (81) on the right by $U_{\alpha ,\beta }$ and on the left by $U_{\alpha ,\beta }^\dagger $, respectively, and use $U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1$.

Theorem 21.3.18 Length-independent zipper identities and reconstruction

✓

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

\begin{align} H_{\alpha ,\beta }^{ij} & = \bigoplus _\gamma 1_{r_{\alpha ,\beta ,\gamma }}\otimes M_\gamma ^{ij}, \notag \end{align}

the fusion isometries satisfy

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} & =H_{\alpha ,\beta }^{ij}U_{\alpha ,\beta }, \notag \\ (M_\alpha M_\beta )^{ij}U_{\alpha ,\beta }^\dagger & =U_{\alpha ,\beta }^\dagger H_{\alpha ,\beta }^{ij}, \notag \\ (M_\alpha M_\beta )^{ij} & =U_{\alpha ,\beta }^\dagger H_{\alpha ,\beta }^{ij}U_{\alpha ,\beta }. \label{eq:rfp_unweighted_reconstruction} \end{align}

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

Proof ▶

Length independence makes every positive diagonal entry of $\chi _{\alpha ,\beta ,\gamma }$ equal to one. Substitute the resulting identity matrices into Theorem 21.3.17. For reconstruction, 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 (83).

Theorem 21.3.19 Common positive blocking of BNT fusion isometries

✓

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

For this same $L$, write

\begin{align} P_{\alpha ,\beta }^{IJ} & =(M_\alpha ^{[L]}M_\beta ^{[L]})^{IJ}, \notag \\ H_{\alpha ,\beta }^{IJ} & =\bigoplus _\gamma 1_{r_{\alpha ,\beta ,\gamma }} \otimes (M_\gamma ^{[L]})^{IJ}, \notag \end{align}

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

\begin{align} U_{\alpha ,\beta }P_{\alpha ,\beta }^{IJ} U_{\alpha ,\beta }^\dagger & =H_{\alpha ,\beta }^{IJ}, \label{eq:rfp_blocked_fusion} \end{align}

the two blocked zipper identities

\begin{align} U_{\alpha ,\beta }P_{\alpha ,\beta }^{IJ} & =H_{\alpha ,\beta }^{IJ}U_{\alpha ,\beta }, \notag \\ P_{\alpha ,\beta }^{IJ}U_{\alpha ,\beta }^\dagger & =U_{\alpha ,\beta }^\dagger H_{\alpha ,\beta }^{IJ}, \notag \end{align}

and exact reconstruction

\begin{align} P_{\alpha ,\beta }^{IJ} & =U_{\alpha ,\beta }^\dagger H_{\alpha ,\beta }^{IJ}U_{\alpha ,\beta }. \notag \end{align}

Proof ▶

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 $\chi _{\alpha ,\beta ,\gamma }$ by the identity on its multiplicity space. The two zipper identities and reconstruction then follow by multiplying the blocked fusion identity (84) by $U_{\alpha ,\beta }$ or $U_{\alpha ,\beta }^\dagger $ on the appropriate side.

The three displayed diagrams represent, 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 relies directly on the isometry condition $U_{\alpha ,\beta }^{\dagger }U_{\alpha ,\beta }=1$. The zipper orientation is that of [ 8 , Section 3, Equations (19)–(21) ] ; the MPDO identity is given in [ 13 , Theorem 4.14(iii), lines 986–991 ] , with identity weights in the length-independent case discussed in [ 13 , line 1010 ] .

$\begin{tenkz}[rows={op,op}, cols=2, east=open] \tnfuse[at=(1,1), name=U, skin=pill]{U_{\alpha,\beta}} \tn[at=(1,2), skin=mpo, ports={90:physical:$i$, 270:physical}]{M_\alpha} \tn[at=(2,2), skin=mpo, ports={90:physical, 270:physical:$j$}]{M_\beta} \tnwire{open w}{U.180} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op,op}, cols=2, bonds=none] \tn[at=(1,1), skin=mpo, wires=2, name=H, ports={90:physical:$i$, 270:physical:$j$, 180:virtual, 0:virtual}]{H} \tnfuse[at=(1,2), name=U, skin=pill]{U_{\alpha,\beta}} \tnwire{open w}{H.180} \tnwire{H.0}{U.180} \tnwire{U.0@1}{open e} \tnwire{U.0@2}{open e} \end{tenkz}$

$\begin{tenkz}[rows={op,op}, cols=2, west=open] \tn[at=(1,1), skin=mpo, ports={90:physical:$i$, 270:physical}]{M_\alpha} \tn[at=(2,1), skin=mpo, ports={90:physical, 270:physical:$j$}]{M_\beta} \tnfuse[at=(1,2), name=Ud, skin=pill]{U_{\alpha,\beta}^{\dagger}} \tnwire{Ud.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op,op}, cols=2, bonds=none] \tnfuse[at=(1,1), name=Ud, skin=pill]{U_{\alpha,\beta}^{\dagger}} \tn[at=(1,2), skin=mpo, wires=2, name=H, ports={90:physical:$i$, 270:physical:$j$, 180:virtual, 0:virtual}]{H} \tnwire{open w}{Ud.180@1} \tnwire{open w}{Ud.180@2} \tnwire{Ud.0}{H.180} \tnwire{H.0}{open e} \end{tenkz}$

$\begin{tenkz}[rows={op,op}, cols=1, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i$, 270:physical}]{M_\alpha} \\ \tn[skin=mpo, ports={90:physical, 270:physical:$j$}]{M_\beta} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op,op}, cols=3, bonds=none] \tnfuse[at=(1,1), name=Ud, skin=pill]{U_{\alpha,\beta}^{\dagger}} \tn[at=(1,2), skin=mpo, wires=2, name=H, ports={90:physical:$i$, 270:physical:$j$, 180:virtual, 0:virtual}]{H} \tnfuse[at=(1,3), name=U, skin=pill]{U_{\alpha,\beta}} \tnwire{open w}{Ud.180@1} \tnwire{open w}{Ud.180@2} \tnwire{Ud.0}{H.180} \tnwire{H.0}{U.180} \tnwire{U.0@1}{open e} \tnwire{U.0@2}{open e} \end{tenkz}$

Definition 21.3.20 Operators generated by tensors with fusion isometries

✓

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

\begin{tenkz}[rows={op,op}, cols=4, west=trace, east=trace, bonds=none]
\tn[skin=mpo, ports={90:physical:$i_1$, 270:physical}]{M_\alpha} &
\tn[skin=mpo, ports={90:physical, 270:physical}]{M_\alpha} &
\tn[skin=dots]{} &
\tn[skin=mpo, ports={90:physical:$i_L$, 270:physical}]{M_\alpha} \\
\tn[skin=mpo, ports={90:physical, 270:physical:$j_1$}]{M_\beta} &
\tn[skin=mpo, ports={90:physical, 270:physical}]{M_\beta} &
\tn[skin=dots]{} &
\tn[skin=mpo, ports={90:physical, 270:physical:$j_L$}]{M_\beta}
\tnwire{(1,1)}{(1,2)} \tnwire{(1,2)}{(1,3)} \tnwire{(1,3)}{(1,4)}
\tnwire{(2,1)}{(2,2)} \tnwire{(2,2)}{(2,3)} \tnwire{(2,3)}{(2,4)}
\tnwire{(1,1)}{(2,1)} \tnwire{(1,2)}{(2,2)} \tnwire{(1,4)}{(2,4)}
\tnmark[form=bracket]{(2,1) .. (2,4)}{$L\text{ sites}$}
\end{tenkz} — Figure 21.1 Closing an $L$-site chain of the local fusion identity. On the left, the bra legs of the $M_\alpha $ operator are contracted with the ket legs of the $M_\beta $ operator. On the right, the multiplicity loop and the $M_\gamma $ loop are disjoint; the former contributes $\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$. Thus the diagram is the trace-power product law of [ 13 , Theorem 4.14(ii)–(iii), equations (87) and (89), lines 962–991 ] .

\begin{tenkz}[cols=4, west=trace, east=trace]
\tn[skin=box]{\chi} & \tn[skin=box]{\chi} & \tn[skin=dots]{} &
\tn[skin=box]{\chi}
\end{tenkz} — Figure 21.1 Closing an $L$-site chain of the local fusion identity. On the left, the bra legs of the $M_\alpha $ operator are contracted with the ket legs of the $M_\beta $ operator. On the right, the multiplicity loop and the $M_\gamma $ loop are disjoint; the former contributes $\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$. Thus the diagram is the trace-power product law of [ 13 , Theorem 4.14(ii)–(iii), equations (87) and (89), lines 962–991 ] .

Theorem 21.3.21 Fusion identities give the operator product law

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & =\sum _{\gamma } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})O_L(M_\gamma ). \label{eq:rfp_same_length_product} \end{align}

This derives the product law of [ 13 , Theorem 4.14(ii) ] from the isometry statement of [ 13 , Theorem 4.14(iii) ] . The active-support formulation is the one used in [ 13 , Appendix C.4, lines 2020–2029 ] .

Proof ▶

The product of the operators is generated by the product tensor. Write $P^w$ for a nonempty word in that tensor and $D^w$ for the corresponding word in the weighted direct sum. For a coisometry onto the active sectors, exact reconstruction and $U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1$ give $P^w=U_{\alpha ,\beta }^\dagger D^w U_{\alpha ,\beta }$. Cyclicity of trace then gives $\operatorname{tr}(P^w)=\operatorname{tr}(D^w)$. For an isometry on the full product bond space, $D^w=U_{\alpha ,\beta }P^wU_{\alpha ,\beta }^\dagger $ and $\operatorname{tr}(D^w)=\operatorname{tr}(P^wU_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }) =\operatorname{tr}(P^w)$. By the fusion identity, the letters of $D$ are block-diagonal tensor products. 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 (85).

Definition 21.3.22 Algebra relations from fusion isometries

✓

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

\begin{align} O_L(M_\alpha )O_L(M_\beta ) & =\sum _\gamma c^{(L)}_{\alpha ,\beta ,\gamma }O_L(M_\gamma ), \notag \\ m_\gamma & =\sum _{\alpha ,\beta } c^{(1)}_{\alpha ,\beta ,\gamma }m_\alpha m_\beta . \notag \end{align}

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

Definition 21.3.23 Active-sector fusion for a vertical canonical decomposition

✓

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

\begin{align} U\widetilde M U^\dagger & =\bigoplus _\alpha \mu _\alpha \otimes M_\alpha . \notag \end{align}

Here $U$ is a coisometry onto the retained sectors, and its adjoint reconstructs $\widetilde M$, including any common zero-sector complement. Active-sector fusion for this decomposition consists of positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$ and coisometries $U_{\alpha ,\beta }$ onto the active product sectors, satisfying

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} U_{\alpha ,\beta }^\dagger & =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}. \label{eq:rfp_active_fusion} \end{align}

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

\begin{align} (M_\alpha M_\beta )^{ij} & =U_{\alpha ,\beta }^\dagger \left(\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}\right) U_{\alpha ,\beta }. \notag \end{align}

The vector with entries $m_\alpha =\operatorname{tr}(\mu _\alpha )$ is an idempotent for the multiplication induced by $c^{(1)}_{\alpha ,\beta ,\gamma }=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })$. This is the active-support formulation of statement (iii) of [ 13 , Theorem 4.14, lines 986–993 ] , for the chosen decomposition of [ 13 , Proposition 4.13, lines 943–951 ] .

Theorem 21.3.24 Idempotent relation from paired decompositions and given fusion coisometries

✓

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

\begin{align} U_{\alpha ,\beta }(M_\alpha M_\beta )^{ij} U_{\alpha ,\beta }^{\dagger } & =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}, \notag \\ (M_\alpha M_\beta )^{ij} & =U_{\alpha ,\beta }^{\dagger } \left(\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes M_\gamma ^{ij}\right) U_{\alpha ,\beta }. \notag \end{align}

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

Proof ▶

The sector correspondence expresses the blocked closed-chain operator in the one-site normal-tensor basis. Multiplication of the one-site vertical decomposition gives a second expression in the same basis. At every sufficiently large length, linear independence equates their coefficients. Equality of the resulting positive power sums identifies the multiplicity multisets, and summing their entries gives $m_\gamma =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta $.

Theorem 21.3.25 Grouped product inputs give active-sector fusion

✓

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

Proof ▶

Obtain the one-site and two-site vertical decompositions from the grouped hypotheses. Transport the renormalization maps to compare their normal sectors. The positive product decomposition supplies the fusion matrices and coisometries; the paired-decomposition argument gives the length-one relation.

Theorem 21.3.26 Renormalization fixed points in CF have active-sector fusion

✓

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

\begin{align} m_\gamma & =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta . \notag \end{align}

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

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

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

Each fusion map is a coisometry onto the active direct sum, and its adjoint reconstructs the whole product tensor, including any discarded common zero corner. This corrects the orientation of the printed map. See [ 125 ] .

Proof ▶

Literal canonical form supplies the grouped vertical product hypotheses; apply the grouped active-sector fusion theorem.

Theorem 21.3.27 Horizontal-form specialization of active-sector fusion

✓

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

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

Proof ▶

Normalized BNT-refined horizontal form supplies the grouped vertical product hypotheses; apply the grouped active-sector fusion theorem.

Theorem 21.3.28 Active-sector fusion implies the algebra relations

✓

Suppose a chosen vertical canonical decomposition satisfies the active-sector fusion and idempotent relations. Then its operators $O_L(M_\alpha )$ satisfy the algebra relations. For every positive chain length $L$, the structure coefficients are $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, determined by the same diagonal matrices. Thus statement (iii) implies statement (ii) of [ 13 , Theorem 4.14, lines 972–993 ] , with the active-support interpretation of (iii).

Proof ▶

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 idempotent identity is $m_\gamma =\sum _{\alpha ,\beta } \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma })m_\alpha m_\beta $. These are the product and idempotent relations for the chosen vertical decomposition, with the same diagonal matrices.

Theorem 21.3.29 Printed algebra and fusion characterization

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

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

Theorem 21.3.30 BNT characterizations of MPDO renormalization fixed points

✓

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

$M$ is an RFP: there are tpCPM $S,T$ with $S[M_2(X)]=M_1(X)$ and $T[M_1(X)]=M_2(X)$ for every $X$.
A vertical canonical decomposition of $M$ has positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$ such that the $O_L(M_\alpha )$ satisfy the product law with $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$ for every $L{\gt}0$, and the vector with entries $m_\alpha =\operatorname{tr}(\mu _\alpha )$ is an idempotent for the multiplication induced by $c^{(1)}$.
A vertical canonical decomposition of $M$ has active-sector fusion coisometries with exact reconstruction and positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$, and the vector with entries $m_\alpha =\operatorname{tr}(\mu _\alpha )$ is an idempotent for the multiplication induced by $c^{(1)}$.

Then

\begin{align} \textnormal{(i)} & \iff \textnormal{(ii)}, \notag \\ \textnormal{(i)} & \iff \textnormal{(iii}_{\mathrm{act}}\textnormal{)}. \label{eq:mpdo_cpsv_bnt_rfp_equivalence} \end{align}

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

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

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

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

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

Proof ▶

An RFP has active-sector fusion and the idempotent relation by Theorem 21.3.26. These imply the algebra and idempotent relations by Theorem 21.3.28. Finally, the algebra and idempotent relations imply that $M$ is an RFP by Theorem 21.2.6.3. These three implications prove both equivalences in (87).

$Q={}$ $\begin{tenkz}[rows={wire,wire,wire,wire,wire,wire}, cols=19, bonds=none, metrics=compact] % Nineteen columns at the base metric outrun the compiled page and % lose the terminal trace; the compact profile keeps the whole % recursion on one line. \tn[at=(5,1), skin=box, name=X1, ports={45:physical, 0:physical, -20:physical, -45:physical}]{X} \tn[at=(4,5), skin=box, name=X2, ports={45:physical, 0:physical, -20:physical, -45:physical}]{X} \tn[at=(3,9), skin=box, name=X3, ports={45:physical, 0:physical, -20:physical, -45:physical}]{X} \tn[at=(2,14), skin=box, name=X4, ports={45:physical, 0:physical, -20:physical, -45:physical}]{X} \tn[at=(4,2), name=k1, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(5,3), name=a1, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(6,4), name=b1, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(3,6), name=k2, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(4,7), name=a2, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(5,8), name=b2, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(6,7), name=g1, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(2,10), name=k3, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(3,11), name=a3, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(4,12), name=b3, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(5,11), name=g2, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(4,13), name=g3, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(1,15), name=k4, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(2,16), name=a4, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(3,17), name=b4, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(3,18), name=g4, ports={180:physical, 90:physical, 270:physical}]{} \tn[at=(4,18), name=sel, ports={90:physical, 270:physical, 0:physical}]{} \tn[at={1 e of sel}, skin=box, name=M, label pos=0, ports={180:physical, 90:virtual, 270:virtual, 225:physical, 315:physical}]{M_{\gamma_r}} \tn[at=(5,19), skin=none, name=tr, ports={135:physical, 45:physical}]{} \tn[at={midway (2,12) and (3,13)}, skin=dots, name=dd]{} \tnwire{X1.45}{k1.180} \tnwire{X1.0}{a1.180} \tnwire{X1.-45}{b1.180} \tnwire{X1.-20}{g1.180} \tnwire{X2.45}{k2.180} \tnwire{X2.0}{a2.180} \tnwire{X2.-45}{b2.180} \tnwire{X2.-20}{g2.180} \tnwire{X3.45}{k3.180} \tnwire{X3.0}{a3.180} \tnwire{X3.-45}{b3.180} \tnwire{X3.-20}{g3.180} \tnwire{X4.45}{k4.180} \tnwire{X4.0}{a4.180} \tnwire{X4.-45}{b4.180} \tnwire{X4.-20}{g4.180} \tnwire{k1.90}{open n} \tnwire{k1.270}{open s} \tnwire{a1.90}{open n} \tnwire{a1.270}{open s} \tnwire{b1.90}{open n} \tnwire{b1.270}{open s} \tnwire{k2.90}{open n} \tnwire{k2.270}{open s} \tnwire{a2.90}{open n} \tnwire{a2.270}{g1.90} \tnwire{g1.270}{open s} \tnwire{b2.90}{open n} \tnwire{b2.270}{open s} \tnwire{k3.90}{open n} \tnwire{k3.270}{open s} \tnwire{a3.90}{open n} \tnwire{a3.270}{g2.90} \tnwire{g2.270}{open s} \tnwire{b3.90}{open n} \tnwire{b3.270}{open s} \tnwire{g3.90}{open n} \tnwire{g3.270}{open s} \tnwire{k4.90}{open n} \tnwire{k4.270}{open s} \tnwire{a4.90}{open n} \tnwire{a4.270}{open s} \tnwire{b4.90}{open n} \tnwire{b4.270}{open s} \tnwire{g4.90}{open n} \tnwire{g4.270}{sel.90} \tnwire{sel.270}{open s} \tnwire{M.180}{sel.0} \tnwire{M.90}{open n} \tnwire{M.270}{open s} \tnwire{M.225}{tr.135} \tnwire{M.315}{tr.45} \end{tenkz}$

Figure 21.2 The recursive contraction defining the structure operator $Q$, with terminal factor $\operatorname{tr}(M_{\gamma _r})$ [ 13 , lines 999–1008 ] .

Associativity of the algebra product in [ 13 , Equation (87) ] constrains the diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$ and induces compatibility between the fusion isometries of the two bracketings $(M_\alpha M_\beta ) M_\gamma $ and $M_\alpha (M_\beta M_\gamma )$ [ 13 , lines 995–999 ] . Solutions constructed from the $F$-symbols of unitary fusion categories connect to the boundary theories of [ 8 ] . On the full product bond spaces, the tensor identities for the two bracketings determine a comparison between the full direct sums, which restricts to a fixed final label in Theorem 21.3.80.

\begin{align} \tntree {((\alpha \, \beta )_\delta \, \gamma )_\varepsilon } & \xrightarrow {F_{\varepsilon }^{\alpha \beta \gamma }} \tntree {(\alpha \, (\beta \, \gamma )_\eta )_\varepsilon }. \notag \end{align}

Figure 21.3 The two ways of fusing three labels to a fixed final label $\varepsilon $, using the printed-$F$ row/column convention of [ 8 , Section 3.4 ] . The solid fusion trees represent the left- and right-bracketed fusion maps $U^{\mathrm L}$ and $U^{\mathrm R}$ of Definitions 21.3.31 and 21.3.34. The arrow is the change of basis $F_{\varepsilon }^{\alpha \beta \gamma }$ between left- and right-bracketed multiplicity coordinates, whose two-sided unitarity is established in Theorem 21.3.82.

Definition 21.3.31 Iterated fusion isometry of a triple product, left bracketing

✓

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

Theorem 21.3.32 The left iterated fusion isometry is an isometry

✓

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

Proof ▶

The composite of two isometries is an isometry.

Theorem 21.3.33 The left iterated fusion isometry conjugates a triple product onto a doubly weighted direct sum

✓

Site by site,

\begin{align} & U^{\mathrm L}_{\alpha ,\beta ,\gamma } ((M_\alpha M_\beta )M_\gamma )^{ij} (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^\dagger \notag \\ & \qquad =\bigoplus _{\delta }\bigoplus _{\varepsilon } \chi _{\alpha ,\beta ,\delta } \otimes \bigl(\chi _{\delta ,\gamma ,\varepsilon } \otimes M_\varepsilon ^{ij}\bigr). \notag \end{align}

Proof ▶

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.

Definition 21.3.34 Iterated fusion isometry of a triple product, right bracketing

✓

For labels $\alpha , \beta , \gamma $, the right iterated fusion isometry $U^{\mathrm R}_{\alpha ,\beta ,\gamma }$ is obtained by applying $U_{\beta ,\gamma }$, tensored with the identity on the bond space of $M_\alpha $, and then applying $U_{\alpha ,\delta }$ for every intermediate label $\delta $. Its codomain is indexed by the intermediate and final labels $(\delta ,\varepsilon )$, the multiplicity indices of $\chi _{\beta ,\gamma ,\delta }$ and $\chi _{\alpha ,\delta ,\varepsilon }$, and the bond index of $M_\varepsilon $. This is the right-bracketed composite in the associativity equation of [ 8 , Section “Associativity and the pentagon equation” ] , before the change of basis by an $F$-matrix.

Theorem 21.3.35 The right iterated fusion isometry is an isometry

✓

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

Proof ▶

The two successive fusion maps are isometries on each intermediate sector, so their block-diagonal composite is an isometry.

Theorem 21.3.36 The right iterated fusion isometry conjugates a triple product onto a doubly weighted direct sum

✓

Site by site,

\begin{align} & U^{\mathrm R}_{\alpha ,\beta ,\gamma } (M_\alpha (M_\beta M_\gamma ))^{ij} (U^{\mathrm R}_{\alpha ,\beta ,\gamma })^\dagger \notag \\ & \qquad =\bigoplus _{\delta }\bigoplus _{\varepsilon } \chi _{\beta ,\gamma ,\delta } \otimes \bigl(\chi _{\alpha ,\delta ,\varepsilon } \otimes M_\varepsilon ^{ij}\bigr). \notag \end{align}

Proof ▶

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 $.

Definition 21.3.37 Full comparison of the two triple-fusion orders

✓

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

\begin{align} C_{\alpha ,\beta ,\gamma } & =U^{\mathrm L}_{\alpha ,\beta ,\gamma } A_{D_\alpha ,D_\beta ,D_\gamma } (U^{\mathrm R}_{\alpha ,\beta ,\gamma })^\dagger . \label{eq:rfp_triple_comparison} \end{align}

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

Lemma 21.3.38 Left range projection of the full triple-fusion comparison

✓

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

\begin{align} C_{\alpha ,\beta ,\gamma }C_{\alpha ,\beta ,\gamma }^{\dagger } & =U^{\mathrm L}_{\alpha ,\beta ,\gamma } (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^{\dagger }. \notag \end{align}

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

Proof ▶

Substitute (88). 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 $.

Lemma 21.3.39 Right range projection of the full triple-fusion comparison

✓

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

\begin{align} C_{\alpha ,\beta ,\gamma }^{\dagger }C_{\alpha ,\beta ,\gamma } & =U^{\mathrm R}_{\alpha ,\beta ,\gamma } (U^{\mathrm R}_{\alpha ,\beta ,\gamma })^{\dagger }. \notag \end{align}

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

Proof ▶

Substitute (88). 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 $.

Theorem 21.3.40 The full triple-fusion comparison intertwines the unweighted direct sums

✓

Suppose the structure coefficients $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, with positive diagonal $\chi _{\alpha ,\beta ,\gamma }$, are independent of the positive chain length. Write $r_{\alpha ,\beta }^{\gamma }$ for the dimension of $\chi _{\alpha ,\beta ,\gamma }$. For every pair of physical indices $i,k$, let

\begin{align} D^{\mathrm L}_{ik} & =\bigoplus _{\delta ,\varepsilon } 1_{r_{\alpha ,\beta }^{\delta }}\otimes \bigl(1_{r_{\delta ,\gamma }^{\varepsilon }} \otimes M_\varepsilon ^{ik}\bigr), \notag \end{align}

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

\begin{align} D^{\mathrm L}_{ik}C_{\alpha ,\beta ,\gamma } & =C_{\alpha ,\beta ,\gamma }D^{\mathrm R}_{ik}. \notag \end{align}

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

Proof ▶

Length independence makes every positive diagonal factor equal to the identity. Substitute the left and right triple-fusion identities into (88), cancel the two isometries, and apply Theorem 21.3.7.

Definition 21.3.41 Common word selectors for the final labels

✓

Fix a common length $S$. The tensors admit common word selectors for the final labels at length $S$ if, for each final label $\varepsilon $, there are coefficients $c_\varepsilon (w)$, indexed by the words $w$ of length $S$ in the doubled physical alphabet, such that

\begin{align} \sum _{|w|=S}c_\varepsilon (w)M_\varepsilon ^w & =1, \notag \\ \sum _{|w|=S}c_\varepsilon (w)M_{\varepsilon '}^w & =0 \notag \end{align}

for every $\varepsilon '\ne \varepsilon $. This is the finite-word form of the simultaneous inverse relation $B_d^+B_{d'}=\delta _{d,d'}1$ used in the fixed-channel argument of [ 8 ] .

Theorem 21.3.42 Pair fusion completeness from positive-length selectors

✓

Suppose the structure coefficients $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, with positive diagonal $\chi _{\alpha ,\beta ,\gamma }$, are independent of the positive chain length, and suppose that common selectors exist at a positive length. Then every pair fusion isometry is surjective: $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1$. The common selectors are the finite-word form of the simultaneous inverse used in [ 8 , line 269 ] .

Proof ▶

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 $P=1$.

Theorem 21.3.43 Pair fusion completeness from a BNT

✓

Suppose that the tensors $\{ M_\alpha \} _\alpha $ with the given fusion isometries form a BNT and that the structure coefficients $c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, with positive diagonal $\chi _{\alpha ,\beta ,\gamma }$, are independent of the positive chain length. Then every pair fusion isometry is surjective: $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1$. In particular, common final-label selectors and the coisometry identity follow from these hypotheses. This is a consequence of the common BNT word span in [ 13 , lines 317–345 ] , the fusion identity at lines 986–993, and length independence at lines 995–1010.

Proof ▶

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 21.3.42.

Definition 21.3.44 One fusion layer of the topological projector

✓

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

\begin{align} Q_{\alpha ,\beta } & =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes P_\gamma . \label{eq:rfp_one_fusion} \end{align}

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

\begin{align} U_{\alpha ,\beta }\operatorname{tr}(M_\alpha M_\beta ) U_{\alpha ,\beta }^{\dagger } & =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes \operatorname{tr}(M_\gamma ). \label{eq:rfp_terminal_trace} \end{align}

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

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

Theorem 21.3.45 Projection property of one fusion layer

✓

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

\begin{align} Q_{\alpha ,\beta } & =\bigoplus _\gamma \mathbb {1}_{r_{\alpha ,\beta }^{\gamma }}\otimes P_\gamma \notag \end{align}

is self-adjoint and idempotent, where $r_{\alpha ,\beta }^{\gamma }$ is the dimension of $\chi _{\alpha ,\beta ,\gamma }$. This is the single recursive fusion step in [ 13 , lines 999–1010 ] . Since the fusion map onto the nonzero sectors satisfies $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=\mathbb {1}$, the operator $U_{\alpha ,\beta }^{\dagger }Q_{\alpha ,\beta }U_{\alpha ,\beta }$ is also an orthogonal projection on the product bond space.

Proof ▶

The trace-power hypothesis and length independence force every diagonal entry of $\chi _{\alpha ,\beta ,\gamma }$ to equal one. Thus $\chi _{\alpha ,\beta ,\gamma }=\mathbb {1}$, while $c_{\alpha ,\beta ,\gamma }^{(L)}=r_{\alpha ,\beta }^{\gamma }$ need not equal one. Hence

\begin{align} Q_{\alpha ,\beta }^{\dagger } & =\bigoplus _\gamma (\mathbb {1}\otimes P_\gamma ^{\dagger }) =Q_{\alpha ,\beta }, \notag \\ Q_{\alpha ,\beta }^{2} & =\bigoplus _\gamma (\mathbb {1}\otimes P_\gamma ^{2}) =Q_{\alpha ,\beta }. \notag \end{align}

Finally, the coisometry identity $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=\mathbb {1}$ and Theorem [ 216 , “Coisometric transport of a star projection” ] show that $U_{\alpha ,\beta }^{\dagger }Q_{\alpha ,\beta }U_{\alpha ,\beta }$ is an orthogonal projection.

Definition 21.3.46 Recursive fusion histories and their sequential circuit

✓

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

\begin{align} w(h) & =\prod _r (\chi _{\gamma _{r-1},\beta _r,\gamma _r})_{k_r,k_r}. \label{eq:rfp_history_weight} \end{align}

For terminal bond operators $P_\gamma $, define

\begin{align} Q_N(P) & =\bigoplus _h w(h)P_{\gamma (h)}. \label{eq:rfp_recursive_q} \end{align}

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

Theorem 21.3.47 Fixed-label recursive fusion identity

✓

For every initial label and every finite list of appended labels, let $N\geq 1$ be one plus the length of that list, and let $M_{\alpha ,\boldsymbol \beta }$ denote the left-associated product tensor. The sequential fusion map $W_N$ is a coisometry onto the retained direct sum, so $W_NW_N^\dagger =\mathbb {1}$. For every pair of horizontal operator indices $i,j$, it carries the product tensor to the recursive direct sum:

\begin{align} W_N M_{\alpha ,\boldsymbol \beta }^{i,j}W_N^\dagger & =Q_N((M_\gamma ^{i,j})_\gamma ). \label{eq:rfp_recursive_fusion} \end{align}

Consequently, after closing the horizontal operator index,

\begin{align} W_N\operatorname{tr}(M_{\alpha ,\boldsymbol \beta })W_N^\dagger & =Q_N((\operatorname{tr}(M_\gamma ))_\gamma ). \label{eq:rfp_recursive_trace} \end{align}

Exact reconstruction gives the reverse identities

\begin{align} M_{\alpha ,\boldsymbol \beta }^{i,j} & =W_N^\dagger Q_N((M_\gamma ^{i,j})_\gamma ) W_N, \label{eq:rfp_reverse_fusion} \\ \operatorname{tr}(M_{\alpha ,\boldsymbol \beta }) & =W_N^\dagger Q_N((\operatorname{tr}(M_\gamma ))_\gamma ) W_N. \label{eq:rfp_reverse_trace} \end{align}

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

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

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

Proof ▶

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 (93) over equal horizontal operator indices gives (94). Under length independence every accumulated coefficient $w(h)$ is one, so self-adjointness and idempotence hold block by block.

For $N=3$, with appended labels $\beta _1,\beta _2$, the forward identity (93) is displayed below on the history block whose intermediate label is $\gamma _1$. The inner fusion map fuses the first two bonds, and the outer one fuses the result with the third bond; the right-hand side is the corresponding block of $Q_3$.

$\begin{tenkz}[rows={op,op,op}, cols=5, bonds=none] \tnfuse[at=(1,1), name=V, skin=pill, wires=3]{U_{\gamma_1,\beta_2}} \tnfuse[at=(1,2), name=U, skin=pill]{U_{\alpha,\beta_1}} \tn[at=(1,3), skin=mpo, size=l, name=A, ports={180:virtual, 0:virtual, 90:physical:$i$, 270:physical}]{M_\alpha} \tn[at=(2,3), skin=mpo, size=l, name=B1, ports={180:virtual, 0:virtual, 90:physical, 270:physical}]{M_{\beta_1}} \tn[at=(3,3), skin=mpo, size=l, name=B2, ports={180:virtual, 0:virtual, 90:physical, 270:physical:$j$}]{M_{\beta_2}} \tn[at=(3,2), skin=none]{} \tn[at=(3,4), skin=none]{} \tnfuse[at=(1,4), name=Ud, skin=pill]{U_{\alpha,\beta_1}^{\dagger}} \tnfuse[at=(1,5), name=Vd, skin=pill, wires=3]{U_{\gamma_1,\beta_2}^{\dagger}} \tnwire{A.270}{B1.90} \tnwire{B1.270}{B2.90} \tnwire{open w}{V.180} \tnwire{V.0@1}{U.180} \tnwire{V.0@3}{B2.180} \tnwire{U.0@1}{A.180} \tnwire{U.0@2}{B1.180} \tnwire{A.0}{Ud.180@1} \tnwire{B1.0}{Ud.180@2} \tnwire{Ud.0}{Vd.180@1} \tnwire{B2.0}{Vd.180@3} \tnwire{Vd.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[cols=1, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i$, 270:physical:$j$}]{Q_3} \end{tenkz}$

Lemma 21.3.48 Sitewise change of physical coordinates

✓

Let $V$ be a one-site matrix, not necessarily square. The operator $V^{\otimes N}\rho ^{(N)}(M)(V^{\otimes N})^\dagger $ is generated by the tensor obtained by applying $X\mapsto VXV^\dagger $ to the one-site physical matrices of $M$.

Proof ▶

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.

Theorem 21.3.49 Positivity of the terminal transfer matrices

✓

Let $M$ generate MPDOs, and choose a vertical canonical decomposition with BNT $\{ M_\gamma \} _\gamma $ and positive diagonal multiplicity matrices $\mu _\gamma $. Then every terminal transfer matrix $T_\gamma =\operatorname{tr}(M_\gamma )$ is positive semidefinite. This supplies the positivity needed for the spectral decomposition invoked in [ 13 , lines 1010–1016 ] .

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

Proof ▶

The fusion identities imply the algebra identities for the same vertical decomposition. Apply Theorem 21.2.2.99.

Definition 21.3.50 Recursive density operator summed over all labels

✓

For a chain of positive length $N$, choose at every site a BNT label $\alpha _j$ and a diagonal coordinate $q_j$ of $\mu _{\alpha _j}$. There are $N-1$ appended labels after the initial site. Define

\begin{align} m(\boldsymbol \alpha ,\boldsymbol q) & =\prod _{j=0}^{N-1}(\mu _{\alpha _j})_{q_j,q_j}. \label{eq:rfp_multiplicity_weight} \end{align}

The first label is the initial label of the recursive fusion, while the remaining labels are appended in the physical order $\alpha _1,\ldots ,\alpha _{N-1}$. The recursive specification records this sequence in the reverse order $(\alpha _{N-1},\ldots ,\alpha _1)$, so removing its first entry removes the final physical site. This reversal affects only the recursive notation: the tensor product and the sequential fusion coisometry remain in the physical order $\alpha _0,\ldots ,\alpha _{N-1}$. Let $c=(\boldsymbol \alpha ,\boldsymbol q)$ denote the configuration, let $W_{N,c}$ be its sequential coisometry onto the retained direct sum, and set $Q_{N,c}=Q_N((\operatorname{tr}(M_\gamma ))_\gamma )$. The corresponding density block is $m(\boldsymbol \alpha ,\boldsymbol q) W_{N,c}^\dagger Q_{N,c}W_{N,c}$. The embedding $\widetilde U$ in [ 13 , line 999 ] corresponds to $W_{N,c}^\dagger $ on this block. The recursive density operator $R_N$ is the direct sum of these blocks over all sitewise labels and diagonal coordinates. On the same direct-sum coordinates, define

\begin{align} A_N & =\bigoplus _c m(c)\mathbb {1}_c, \notag \\ T_N & =\bigoplus _c W_{N,c}^{\dagger }Q_{N,c}W_{N,c}, \notag \\ R_N & =A_NT_N. \label{eq:rfp_density_factors} \end{align}

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

Theorem 21.3.51 Density decomposition from fusion on the nonzero sectors

✓

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

\begin{align} V^{\otimes N}\rho ^{(N)}(M)(V^{\otimes N})^\dagger & =R_N, \notag \\ ((V^\dagger )^{\otimes N})R_NV^{\otimes N} & =\rho ^{(N)}(M). \notag \end{align}

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

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

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

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

Proof ▶

In retained coordinates, the one-site entries are block diagonal in the label and multiplicity coordinates. Within each configuration, multiply the scalar weights and apply the forward and adjoint recursive fusion identities. Summing over all configurations gives the forward equality; exact sitewise reconstruction from the retained sectors gives the reverse equality.

Theorem 21.3.52 Density decomposition of an RFP MPDO

✓

Let $M$ generate MPDOs and be in normalized BNT-refined horizontal form. Suppose that $M$ is an RFP. There is one vertical canonical decomposition and one compatible family of fusion coisometries such that, for every $N{\gt}0$, the rectangular vertical coisometry $V$ satisfies

\begin{align} V^{\otimes N}\rho ^{(N)}(M)(V^{\otimes N})^\dagger & =R_N, \label{eq:rfp_density_forward} \\ ((V^\dagger )^{\otimes N})R_NV^{\otimes N} & =\rho ^{(N)}(M). \label{eq:rfp_density_reverse} \end{align}

The statement applies to positive lengths under the normalized BNT-refined horizontal hypothesis.

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

Proof ▶

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 coordinates. 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 (93) and (95), then sum over all configurations. Local reconstruction applied at every site gives (100).

Corollary 21.3.53 Commuting density factors from fusion on the nonzero sectors

✓

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

\begin{align} R_N & =A_NT_N, \notag \\ A_NT_N & =T_NA_N. \notag \end{align}

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

Proof ▶

In every sitewise label and multiplicity configuration, $A_N$ is a scalar multiple of the identity on the corresponding product bond space, so it commutes with the reconstructed recursive factor. Taking the direct sum gives both identities.

Corollary 21.3.54 Commuting density factors of an RFP MPDO

✓

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

\begin{align} R_N & =A_NT_N, \label{eq:rfp_factorization} \\ A_NT_N & =T_NA_N. \label{eq:rfp_factor_commutation} \end{align}

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

Proof ▶

In every sitewise label and multiplicity configuration, the factor $A_N$ is the scalar $m(c)$ times the identity on the corresponding product bond space. It therefore commutes with $W_{N,c}^{\dagger }Q_{N,c}W_{N,c}$. Multiplication of the two direct sums proves (101) and (102). The renormalization fixed-point condition supplies the same fusion family used in the density decomposition.

Definition 21.3.55 Terminal spectral refinement of a fusion decomposition

✓

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

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

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

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

Closing the horizontal operator leg of $M_\gamma $ produces the terminal bond matrix, and the chosen orthonormal eigenbasis expands it as a sum over the spectral indices $s=(\gamma ,k)$ with first entry $\gamma $.

$\begin{tenkz}[rows={op}, cols=1, boundary=periodic] \tn[skin=mpo, ports={180:virtual, 0:virtual, 90:physical, 270:physical}]{M_\gamma} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op}, cols=1] \tn[skin=mpo, ports={90:physical, 270:physical}]{K_\gamma} \end{tenkz}$ $=\displaystyle \sum _{s=(\gamma ,k)}\lambda _{s}$ $\begin{tenkz}[rows={op}, cols=1] \tn[skin=mpo, ports={90:physical, 270:physical}]{E_s} \end{tenkz}$

Definition 21.3.56 Terminal spectral refinement for an RFP MPDO

✓

For an MPDO in normalized BNT-refined horizontal form satisfying the RFP condition, fix the fusion decomposition supplied by that construction. For every terminal label $\gamma $, close the horizontal operator leg of $M_\gamma $ to obtain a matrix $K_\gamma $ on its bond space. Choose an orthonormal eigenbasis of each $K_\gamma $, with spectral indices $s=(\gamma ,k)$ satisfying $0\leq k{\lt}D_\gamma $. Write $\lambda _s$ for the corresponding eigenvalue and $E_s$ for the rank-one orthogonal projection onto the chosen eigenvector, extended by zero on terminal labels different from $\gamma $. Recursively transport this terminal family through every fusion history, then through the adjoint sequential fusion coisometry, and take the direct sum over all sitewise label and multiplicity configurations. Denote the resulting operator at positive chain length $N$ by $P_s^{(N)}$.

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

Theorem 21.3.57 Terminal spectral projectors from a fusion decomposition

✓

Let $M$ generate MPDOs, fix a fusion decomposition on the nonzero product sectors, and suppose that its structure coefficients $c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$ are independent of $L$ for all positive lengths, with $\chi _{\alpha ,\beta ,\gamma }$ positive diagonal. Then every terminal eigenvalue $\lambda _s$ is non-negative. For every $N{\gt}0$, the operators $P_s^{(N)}$ are pairwise orthogonal projections and

\begin{align} T_N & =\sum _s\lambda _sP_s^{(N)}, \notag \\ A_NP_s^{(N)} & =P_s^{(N)}A_N, \notag \\ R_N & =\sum _s\lambda _sA_NP_s^{(N)}. \notag \end{align}

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

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

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

Proof ▶

Positivity of the physical density operator makes each terminal bond matrix positive semidefinite. Its spectral theorem supplies the non-negative eigenvalues and orthogonal rank-one projections. Length independence makes every retained diagonal entry of $\chi _{\alpha ,\beta ,\gamma }$ equal to one, so recursive coisometric transport preserves orthogonality and the spectral sum. Corollary 21.3.53 gives the final two identities.

Theorem 21.3.58 Terminal spectral projectors for an RFP MPDO with length-independent structure coefficients

✓

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

\begin{align} (P_s^{(N)})^\dagger & =P_s^{(N)}, \notag \\ (P_s^{(N)})^2 & =P_s^{(N)}. \notag \end{align}

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

\begin{align} T_N & =\sum _s\lambda _s P_s^{(N)}, \label{eq:rfp_terminal_spectral_sum} \\ A_NP_s^{(N)} & =P_s^{(N)}A_N. \label{eq:rfp_spectral_commutation} \end{align}

Consequently,

\begin{align} R_N & =\sum _s\lambda _s A_NP_s^{(N)}. \label{eq:rfp_density_spectral_sum} \end{align}

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

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

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

Proof ▶

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 diagonal entry of $\chi _{\alpha ,\beta ,\gamma }$ 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 $W_{N,c}W_{N,c}^\dagger =\mathbb {1}$ gives

\begin{align} (W_{N,c}^\dagger Q_{N,c,s}W_{N,c})^\dagger & =W_{N,c}^\dagger Q_{N,c,s}W_{N,c}, \notag \\ (W_{N,c}^\dagger Q_{N,c,s}W_{N,c})^2 & =W_{N,c}^\dagger Q_{N,c,s}W_{N,c}. \notag \end{align}

Thus transport through the adjoint sequential coisometry preserves the projector identities. Linearity gives (103). Finally, $A_N$ is scalar on every copy-configuration block, hence it commutes with each $P_s^{(N)}$ and gives (105).

Definition 21.3.59 Commuting Gibbs form on the retained vertical space for chains of at least two sites

✓

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

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

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

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

Theorem 21.3.60 Fusion decomposition gives a commuting Gibbs decomposition on the retained vertical space for chains of at least two sites

✓

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

\begin{align} H_L & =\sum _{j=0}^{L-1}h_{j,j+1}, \label{eq:rfp_gibbs_hamiltonian} \\ e^{-H_L} & =\mu ^{\otimes L}. \label{eq:rfp_gibbs_factor} \end{align}

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

\begin{align} [P_i^{(L)},e^{-H_L}] & =0, \label{eq:rfp_gibbs_projector_commutation} \\ R_L & =\sum _{i=0}^{d-1}\lambda _iP_i^{(L)}e^{-H_L}. \label{eq:rfp_retained_gibbs_sum} \end{align}

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

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

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

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

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

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

Proof ▶

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 (107).

Positivity of the multiplicities embeds one terminal mode for every terminal spectral index into the retained coordinates. The rank of the coisometry restricted to the retained rows 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 $\{ 0,\ldots ,d-1\} $. The added zero projectors preserve self-adjointness, idempotence, commutation, and the spectral sum. Combine that sum with (107) and the exact adjoint reconstruction of the retained density operator.

Theorem 21.3.61 Commuting Gibbs decomposition on the retained vertical space for chains of at least two sites

✓

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

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

Proof ▶

Select the fusion decomposition on the nonzero product sectors supplied by the RFP condition and apply Theorem 21.3.60 with the assumed length independence of its structure coefficients.

Theorem 21.3.62 Fusion decomposition gives a physical commuting Gibbs decomposition for chains of at least two sites

✓

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

\begin{align} [\widehat h_{j-1,j},\widehat h_{j,j+1}] & =0, \notag \\ {}[\widehat P_i^{(L)},e^{-\widehat H_L}]& =0, \notag \\ \rho ^{(L)}(M) & =\sum _{i=0}^{d-1}\lambda _i\widehat P_i^{(L)} e^{-\widehat H_L}. \label{eq:rfp_physical_gibbs_sum} \end{align}

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

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

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

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

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

Proof ▶

Let $V$ be the coisometry restricted to the retained rows and let $k$ be the retained logarithmic multiplicity energy. The physical energy $K=V^\dagger kV$ is Hermitian and

\begin{align} e^{-K} & =V^\dagger e^{-k}V+(\mathbb {1}-V^\dagger V). \label{eq:rfp_physical_weight} \end{align}

Taking the fixed two-site term $K\otimes \mathbb {1}$, its periodic translates act on separate one-site factors, hence commute, and their exponential is the tensor power of (111).

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 $WW^\dagger =\mathbb {1}$ 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 (109), and then using exact adjoint reconstruction, gives (110).

Theorem 21.3.63 Physical commuting Gibbs decomposition for chains of at least two sites

✓

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

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

Proof ▶

Select the fusion decomposition on the nonzero product sectors supplied by the RFP condition and apply Theorem 21.3.62 with the assumed length independence of its structure coefficients.

Definition 21.3.64 Fusion decomposition for an RFP tensor in CPSV canonical form

✓

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

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

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

Theorem 21.3.65 Physical commuting Gibbs decomposition in CPSV canonical form for chains of at least two sites

✓

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

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

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

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

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

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

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

Proof ▶

Choose the fusion decomposition on the nonzero product sectors supplied by Theorem 21.3.26. Its structure coefficients are length independent by hypothesis. Apply Theorem 21.3.62.

Remark 21.3.66 Length-one boundary of the two-site construction

The local-interaction definition in [ 13 ] lets a local operator act on the first two spins and translates it around the periodic chain. Thus its construction has domain $L\geq 2$. Periodic identification of site labels supplies neither a map from two-site operators to one-site operators nor a multiplicity for the coincident edge.

The source states its Gibbs-decomposition theorem without an explicit lower bound on $L$, whereas the two-site interaction convention requires $L\geq 2$ (see [ 122 ] ).

Definition 21.3.67 One fusion layer with full support

✓

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

\begin{align} Q^{\mathrm{fs}}_{\alpha ,\beta } & =\bigoplus _\gamma \chi _{\alpha ,\beta ,\gamma }\otimes P_\gamma , \notag \\ \widehat Q^{\mathrm{fs}}_{\alpha ,\beta } & =U_{\alpha ,\beta }^{\dagger } Q^{\mathrm{fs}}_{\alpha ,\beta }U_{\alpha ,\beta }. \notag \end{align}

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

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

Theorem 21.3.68 Projection property with full support

✓

Suppose that $\{ M_\alpha \} _\alpha $ is a BNT and that the structure coefficients $c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, with positive diagonal $\chi _{\alpha ,\beta ,\gamma }$, are independent of $L{\gt}0$. If each $P_\gamma $ is a self-adjoint idempotent on the bond space, then $Q^{\mathrm{fs}}_{\alpha ,\beta }$ is a self-adjoint idempotent and $\widehat Q^{\mathrm{fs}}_{\alpha ,\beta }$ is an orthogonal projection.

Proof ▶

Length independence gives $\chi _{\alpha ,\beta ,\gamma }=1$, hence $(Q^{\mathrm{fs}}_{\alpha ,\beta })^2 =Q^{\mathrm{fs}}_{\alpha ,\beta }$ and $(Q^{\mathrm{fs}}_{\alpha ,\beta })^\dagger =Q^{\mathrm{fs}}_{\alpha ,\beta }$. Completeness for a BNT gives $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=1$, so

\begin{align} (\widehat Q^{\mathrm{fs}}_{\alpha ,\beta })^2 & =U_{\alpha ,\beta }^{\dagger }Q^{\mathrm{fs}}_{\alpha ,\beta } (U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }) Q^{\mathrm{fs}}_{\alpha ,\beta }U_{\alpha ,\beta } =\widehat Q^{\mathrm{fs}}_{\alpha ,\beta }. \notag \end{align}

Conjugation also preserves self-adjointness.

Theorem 21.3.69 Left iterated fusion completeness

✓

Under the hypotheses of Theorem 21.3.42, $U^{\mathrm L}_{\alpha ,\beta ,\gamma } (U^{\mathrm L}_{\alpha ,\beta ,\gamma })^{\dagger }=1$.

Proof ▶

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.

Theorem 21.3.70 Right iterated fusion completeness

✓

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

Proof ▶

The right-associated dimension calculation is

\begin{align} \sum _{\delta ,\varepsilon } r_{\beta ,\gamma }^{\delta }r_{\alpha ,\delta }^{\varepsilon } D_\varepsilon & =D_\alpha (D_\beta D_\gamma ). \notag \end{align}

The right iterated fusion map is therefore a square isometry.

Theorem 21.3.71 Right adjoint inverse for the full triple-fusion comparison

✓

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

Proof ▶

The left range identity reduces the product to $U^{\mathrm L}(U^{\mathrm L})^{\dagger }$, which is the identity by Theorem 21.3.69.

Theorem 21.3.72 Left adjoint inverse for the full triple-fusion comparison

✓

Under the same hypotheses, $C_{\alpha ,\beta ,\gamma }^{\dagger }C_{\alpha ,\beta ,\gamma }=1$. This proves invertibility on the full direct sums, but not yet invertibility of the multiplicity matrix for a fixed final label.

Proof ▶

Apply the right range identity and Theorem 21.3.70.

Theorem 21.3.73 Vanishing between distinct final sectors

✓

Proof ▶

Fix one left multiplicity index and one right multiplicity index. Taking the corresponding bond-space block 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 block vanishes.

Definition 21.3.74 Multiplicity spaces for a fixed final label

✓

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

\begin{align} \mathcal H^{\mathrm L}_{\varepsilon } & =\bigoplus _{\delta } \mathbb {C}^{\dim \chi _{\alpha ,\beta ,\delta }} \otimes \mathbb {C}^{\dim \chi _{\delta ,\gamma ,\varepsilon }}, \notag \\ \mathcal H^{\mathrm R}_{\varepsilon } & =\bigoplus _{\delta } \mathbb {C}^{\dim \chi _{\beta ,\gamma ,\delta }} \otimes \mathbb {C}^{\dim \chi _{\alpha ,\delta ,\varepsilon }}. \notag \end{align}

These spaces have the bracketing pattern of the $(e,\mu ,\nu )$ and $(f,\lambda ,\sigma )$ index spaces in the row and column convention for $F$ in [ 8 , Section 3.4 ] ; no equality with the fusion multiplicities of that source is asserted. The corresponding spaces for a fixed final label are $\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }$ and $\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }$.

Theorem 21.3.75 Equality of multiplicity dimensions for a fixed final label

✓

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

\begin{align} \dim \mathcal H^{\mathrm L}_{\varepsilon } & =\sum _{\delta } \dim \chi _{\alpha ,\beta ,\delta } \dim \chi _{\delta ,\gamma ,\varepsilon } =\sum _{\delta } \dim \chi _{\beta ,\gamma ,\delta } \dim \chi _{\alpha ,\delta ,\varepsilon } =\dim \mathcal H^{\mathrm R}_{\varepsilon }. \notag \end{align}

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

Proof ▶

Choose a positive length beyond the linear-independence threshold. Associativity of the product of the operators $O_L(M_\alpha )$ 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 dimensions of the two multiplicity spaces.

Definition 21.3.76 Triple-fusion maps with fixed final label

✓

The maps $U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }$ and $U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon }$ are obtained by restricting the rows of the two iterated fusion isometries to the summands with final label $\varepsilon $. After the canonical reassociation of the triple bond space, their adjoints have the orientation and bracketing pattern of weighted versions of the two fusion maps in [ 8 , Section 3.4 ] .

Theorem 21.3.77 Left triple fusion at a fixed final label

✓

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

\begin{align} U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon } ((M_\alpha M_\beta )M_\gamma )^{ij} (U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon })^\dagger & =\bigoplus _{\delta } \chi _{\alpha ,\beta ,\delta }\otimes \bigl(\chi _{\delta ,\gamma ,\varepsilon }\otimes M_\varepsilon ^{ij}\bigr). \notag \end{align}

Proof ▶

Restrict Theorem 21.3.33 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 summands with final label $\varepsilon $.

Theorem 21.3.78 Right triple fusion at a fixed final label

✓

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

\begin{align} U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon } (M_\alpha (M_\beta M_\gamma ))^{ij} (U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon })^\dagger & =\bigoplus _{\delta } \chi _{\beta ,\gamma ,\delta }\otimes \bigl(\chi _{\alpha ,\delta ,\varepsilon }\otimes M_\varepsilon ^{ij}\bigr). \notag \end{align}

Proof ▶

Restrict Theorem 21.3.36 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 summands with final label $\varepsilon $.

Theorem 21.3.79 Intertwiners at an injective final sector

✓

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

\begin{align} (1_{\mathcal H^{\mathrm L}_{\varepsilon }}\otimes M_\varepsilon ^{ij})C & =C(1_{\mathcal H^{\mathrm R}_{\varepsilon }}\otimes M_\varepsilon ^{ij}). \notag \end{align}

Then there is a rectangular matrix $F:\mathcal H^{\mathrm R}_{\varepsilon }\to \mathcal H^{\mathrm L}_{\varepsilon }$ such that $C=F\otimes 1_{D_\varepsilon }$. Here the Kronecker decomposition holds for any given intertwiner $C$ between the fixed-final-label representations.

Proof ▶

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.

In the diagrams below the upper strand carries the multiplicity space and the lower strand the bond space $\mathbb {C}^{D_\varepsilon }$; the hypothesis on the left forces the factorization on the right.

$\begin{tenkz}[rows={wire,op}, cols=2, bonds=none] \tn[at=(2,1), skin=mpo, name=M, ports={180:virtual, 0:virtual, 90:physical:$i$, 270:physical:$j$}]{M_\varepsilon} \tn[at=(1,2), skin=box, wires=2, name=C, ports={180:virtual, 0:virtual}]{C} \tnwire{open w}{C.180@1} \tnwire{open w}{M.180} \tnwire{M.0}{C.180@2} \tnwire{C.0@1}{open e} \tnwire{C.0@2}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={wire,op}, cols=2, bonds=none] \tn[at=(1,1), skin=box, wires=2, name=C, ports={180:virtual, 0:virtual}]{C} \tn[at=(2,2), skin=mpo, name=M, ports={180:virtual, 0:virtual, 90:physical:$i$, 270:physical:$j$}]{M_\varepsilon} \tnwire{open w}{C.180@1} \tnwire{open w}{C.180@2} \tnwire{C.0@1}{open e} \tnwire{C.0@2}{M.180} \tnwire{M.0}{open e} \end{tenkz}$ $\Longrightarrow $ $\begin{tenkz}[rows={wire,wire}, cols=1, bonds=none] \tn[at=(1,1), skin=box, wires=2, name=C, ports={180:virtual, 0:virtual}]{C} \tnwire{open w}{C.180@1} \tnwire{open w}{C.180@2} \tnwire{C.0@1}{open e} \tnwire{C.0@2}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={wire,wire}, cols=1, bonds=none] \tn[at=(1,1), skin=box, name=F, ports={180:virtual, 0:virtual}]{F} \tn[at=(2,1), skin=none, name=n, ports={180:virtual, 0:virtual}]{} \tnwire{open w}{F.180} \tnwire{F.0}{open e} \tnwire{open w}{n.180} \tnwire{n.0}{open e} \end{tenkz}$

Theorem 21.3.80 Kronecker form and separation at a fixed final label

✓

Suppose that the structure coefficients $c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$, with positive diagonal $\chi _{\alpha ,\beta ,\gamma }$, are independent of $L{\gt}0$ and that common word selectors separate the final labels. Fix labels $\alpha ,\beta ,\gamma ,\varepsilon $, and suppose that $M_\varepsilon $ is injective at the present blocking. Write $C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma } =P^{\mathrm L}_{\varepsilon } C_{\alpha ,\beta ,\gamma } P^{\mathrm R}_{\varepsilon '}$ for the block from the right final sector $\varepsilon '$ to the left final sector $\varepsilon $. There is a rectangular matrix $F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }$ such that, after the canonical index identifications, $C^{\varepsilon ,\varepsilon }_{\alpha ,\beta ,\gamma } =F_\varepsilon \otimes 1_{D_\varepsilon }$. For every $\varepsilon '\ne \varepsilon $, $C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma }=0$.

Proof ▶

Taking the $\varepsilon $–$\varepsilon $ block of the full comparison identity gives, for every pair $i,j$,

\begin{align} (1_{\mathcal H^{\mathrm L}_\varepsilon }\otimes M_\varepsilon ^{ij})C^{\varepsilon ,\varepsilon } & =C^{\varepsilon ,\varepsilon } (1_{\mathcal H^{\mathrm R}_\varepsilon }\otimes M_\varepsilon ^{ij}). \notag \end{align}

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 21.3.73.

Theorem 21.3.81 Unitarity of the comparison at each final sector

✓

\begin{align} C_\varepsilon C_\varepsilon ^\dagger & =1, \label{eq:mpdo_final_sector_right_unitary} \\ C_\varepsilon ^\dagger C_\varepsilon & =1. \label{eq:mpdo_final_sector_left_unitary} \end{align}

The simultaneous selector identities are additional hypotheses. The conclusion gives invertibility at a fixed final label, as obtained from the simultaneous inverse in [ 8 , lines 247–277 ] .

Proof ▶

Order each full direct sum first by its final label. In the product $CC^\dagger $, the $\varepsilon $-diagonal block 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 block of $C^\dagger C=1$ gives (113).

Theorem 21.3.82 Unitarity of the multiplicity matrix at a fixed final label

✓

Assume the hypotheses of Theorem 21.3.81. Suppose in addition that $M_\varepsilon $ is injective at the present blocking and that $D_\varepsilon {\gt}0$. There is a matrix $F_\varepsilon :\mathcal H^{\mathrm R}_{\varepsilon } \to \mathcal H^{\mathrm L}_{\varepsilon }$ such that

\begin{align} C_\varepsilon & =F_\varepsilon \otimes 1_{D_\varepsilon }, \notag \\ F_\varepsilon F_\varepsilon ^\dagger & =1, \notag \\ F_\varepsilon ^\dagger F_\varepsilon & =1. \notag \end{align}

Thus $F_\varepsilon $ is invertible (with opposite orientation to the row and column convention for $F$ in [ 8 , Section 3.4 ] ).

Proof ▶

Injectivity gives $C_\varepsilon =F_\varepsilon \otimes 1_{D_\varepsilon }$. Substituting this expression into (112) and (113) gives

\begin{align} (F_\varepsilon F_\varepsilon ^\dagger ) \otimes 1_{D_\varepsilon } & =1, \notag \\ (F_\varepsilon ^\dagger F_\varepsilon ) \otimes 1_{D_\varepsilon } & =1. \notag \end{align}

Since $D_\varepsilon {\gt}0$, Lemma 21.3.13 removes the identity factor and gives the two asserted identities.

Definition 21.3.83 Simultaneous MPO block left inverse

✓

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

\begin{align} \sum _{i,l}C_{(c,x,y),(i,l)}(M_d^{il})_{x'y'} & =\delta _{cd}\delta _{xx'}\delta _{yy'}. \notag \end{align}

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

Theorem 21.3.84 Simultaneous inverse from a one-letter product-algebra span

✓

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

\begin{align} \sum _q C_{(c,x,y),q}(A_d^q)_{x'y'} & =\delta _{cd}\delta _{xx'}\delta _{yy'}. \notag \end{align}

For an MPO tensor the letter is the ket–bra pair $q=(i,k)$, so the same coefficients form the simultaneous block left inverse used in [ 8 , lines 269–277 ] . For a BNT, the full product-algebra span is obtained after the common blocking described in [ 8 , lines 427–431 ] .

Proof ▶

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 $\sum _q C_{(c,x,y),q}(A_d^q)_{x'y'} =\delta _{cd}\delta _{xx'}\delta _{yy'}$. Reindexing the doubled physical letter by the ket–bra pair gives the MPO form.

\begin{align} \textup{selector},\ S{\gt}0:\quad & \sum _{|w|=S}c_{\varepsilon }(w)M_{\varepsilon '}^{w} =\delta _{\varepsilon ,\varepsilon '}1_{D_\varepsilon } \notag \\[1.5em] \textup{sector cut},\ \varepsilon \ne \varepsilon ’:\quad & \underbrace{ \tntree {((\alpha \, \beta ) \gamma )_\varepsilon } }_{U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }} \xleftarrow { C^{\varepsilon ,\varepsilon '}_{\alpha ,\beta ,\gamma }} \underbrace{ \tntree {(\alpha \, (\beta \, \gamma ))_{\varepsilon '}} }_{U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon '}} =0 \notag \\[1.5em] \textup{diagonal corner}:\quad & \mathcal H^{\mathrm L}_{\varepsilon } \otimes \mathbb {C}^{D_\varepsilon } \xleftarrow {F_\varepsilon \otimes 1_{D_\varepsilon }} \mathcal H^{\mathrm R}_{\varepsilon } \otimes \mathbb {C}^{D_\varepsilon } \notag \\ & \Longrightarrow \quad F_\varepsilon F_\varepsilon ^\dagger =1, \qquad F_\varepsilon ^\dagger F_\varepsilon =1. \notag \end{align}

Figure 21.4 The fixed-final comparison in the graphical order of [ 8 , equation (pentagon3) and lines 269–277 ] . At a positive common length, the simultaneous word selector is the finite-word counterpart of the inverse $B_d^+$ in the source: it retains the chosen final label and annihilates every other one. Thus the comparison has no off-diagonal final-label corners. On the surviving corner, injectivity gives $C_\varepsilon =F_\varepsilon \otimes 1_{D_\varepsilon }$; two-sided unitarity of $C_\varepsilon $ and $D_\varepsilon {\gt}0$ give two-sided unitarity of $F_\varepsilon $. This is the associativity mechanism described in [ 13 , lines 995–1010 ] .

Definition 21.3.85 Complete zipper fusion family

✓

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

\begin{align} X^c_{ab,\mu } & :\mathbb {C}^{D_c}\longrightarrow \mathbb {C}^{D_a}\otimes \mathbb {C}^{D_b}, \notag \\ X^{c+}_{ab,\mu } & :\mathbb {C}^{D_a}\otimes \mathbb {C}^{D_b} \longrightarrow \mathbb {C}^{D_c}. \notag \end{align}

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

\begin{align} X^{d+}_{ab,\nu }X^c_{ab,\mu } & =\delta _{cd}\delta _{\mu \nu }\mathbb {1}_{D_c}. \label{eq:rfp_fusion_biorthogonality} \end{align}

The opposite product $P_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }$ is the projector onto the product-MPO support; it need not be the identity on the ambient bond space or an orthogonal projector. Write $B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}$. Every product-MPO matrix is reconstructed on this support:

\begin{align} B_{ab}^{ik} & =\sum _{c,\mu }X^c_{ab,\mu }B_c^{ik}X^{c+}_{ab,\mu }. \label{eq:rfp_fusion_reconstruction} \end{align}

The two zipper identities are

\begin{align} B_{ab}^{ik}X^c_{ab,\mu } & =X^c_{ab,\mu }B_c^{ik}, \label{eq:rfp_zipper_synthesis} \\ X^{c+}_{ab,\mu }B_{ab}^{ik} & =B_c^{ik}X^{c+}_{ab,\mu }. \label{eq:rfp_zipper_analysis} \end{align}

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

The three displayed diagrams are, in order, the zipper identities (116) and (117) and the reconstruction (115), with the index $j$ of $B_{ab}^{ik}$ on the vertical bond between $B_a$ and $B_b$.

$\begin{tenkz}[rows={op,op}, cols=2, west=open] \tn[at=(1,1), skin=mpo, ports={90:physical:$i$, 270:physical}]{B_a} \tn[at=(2,1), skin=mpo, ports={90:physical, 270:physical:$k$}]{B_b} \tnfuse[at=(1,2), name=X, skin=pill]{X^c_{ab,\mu}} \tnwire{X.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op,op}, cols=2, bonds=none] \tnfuse[at=(1,1), name=X, skin=pill]{X^c_{ab,\mu}} \tn[at=(1,2), skin=mpo, name=B, ports={90:physical:$i$, 270:physical:$k$, 180:virtual, 0:virtual}]{B_c} \tn[at=(2,2), skin=none]{} \tnwire{open w}{X.180@1} \tnwire{open w}{X.180@2} \tnwire{X.0}{B.180} \tnwire{B.0}{open e} \end{tenkz}$
$\begin{tenkz}[rows={op,op}, cols=2, east=open] \tnfuse[at=(1,1), name=Xp, skin=pill]{X^{c+}_{ab,\mu}} \tn[at=(1,2), skin=mpo, ports={90:physical:$i$, 270:physical}]{B_a} \tn[at=(2,2), skin=mpo, ports={90:physical, 270:physical:$k$}]{B_b} \tnwire{open w}{Xp.180} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op,op}, cols=2, bonds=none] \tn[at=(1,1), skin=mpo, name=B, ports={90:physical:$i$, 270:physical:$k$, 180:virtual, 0:virtual}]{B_c} \tn[at=(2,1), skin=none]{} \tnfuse[at=(1,2), name=Xp, skin=pill]{X^{c+}_{ab,\mu}} \tnwire{open w}{B.180} \tnwire{B.0}{Xp.180} \tnwire{Xp.0@1}{open e} \tnwire{Xp.0@2}{open e} \end{tenkz}$
$\begin{tenkz}[rows={op,op}, cols=1, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i$, 270:physical}]{B_a} \\ \tn[skin=mpo, ports={90:physical, 270:physical:$k$}]{B_b} \end{tenkz}$ $=\displaystyle \sum _{c,\mu }$ $\begin{tenkz}[rows={op,op}, cols=3, bonds=none] \tnfuse[at=(1,1), name=X, skin=pill]{X^c_{ab,\mu}} \tn[at=(1,2), skin=mpo, name=B, ports={90:physical:$i$, 270:physical:$k$, 180:virtual, 0:virtual}]{B_c} \tn[at=(2,2), skin=none]{} \tnfuse[at=(1,3), name=Xp, skin=pill]{X^{c+}_{ab,\mu}} \tnwire{open w}{X.180@1} \tnwire{open w}{X.180@2} \tnwire{X.0}{B.180} \tnwire{B.0}{Xp.180} \tnwire{Xp.0@1}{open e} \tnwire{Xp.0@2}{open e} \end{tenkz}$

Theorem 21.3.86 Complete zipper family from a BNT

✓

Suppose that the labelled tensors of a fusion-isometry family with length-independent structure coefficients $c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})$ form a BNT. There is a common positive blocking length at which they determine a complete zipper fusion family. Its bond dimensions are the original labelled bond dimensions, its fusion multiplicities are $N_{\alpha \beta }^{\gamma }=\dim \chi _{\alpha ,\beta ,\gamma }$, its synthesis map is the adjoint of $U_{\alpha ,\beta }$, and its analysis map is $U_{\alpha ,\beta }$, after the canonical identification $\mathbb {C}^{D_\alpha D_\beta } \cong \mathbb {C}^{D_\alpha }\otimes \mathbb {C}^{D_\beta }$. The two zipper identities, exact reconstruction, injectivity, and the simultaneous left inverse of the block coefficients follow as conclusions rather than additional hypotheses.

Proof ▶

Length independence makes each positive diagonal matrix $\chi _{\alpha ,\beta ,\gamma }$ the identity. Use a common positive blocking length for the BNT and carry the fusion equation through the blocking. Completeness of the pair decomposition gives $U_{\alpha ,\beta }U_{\alpha ,\beta }^{\dagger }=\mathbb {1}$, so the synthesis map followed by the analysis map is the identity. The other three matrix identities are the blocked zipper identities (116) and (117) and the reconstruction identity (115). At this blocking length, the labelled matrices span the direct sum of the block matrix algebras. This supplies both block injectivity and the simultaneous left inverse of the block coefficients. These maps satisfy Definition 21.3.85.

Theorem 21.3.87 Complete zipper family from split compressions

✓

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

Proof ▶

Biorthogonality (114) is $W_sV_t=\delta _{st}\mathbb {1}$. Since the remainder $B_{ab}^{ik}-\sum _sV_sB_{c(s)}^{ik}W_s$ vanishes, this sum is the reconstruction (115). Multiplying the reconstruction by $X^c_{ab,\mu }$ on the right, or by $X^{c+}_{ab,\mu }$ on the left, and using biorthogonality gives the two zipper identities. In the star-closed case the word-algebra module of each product is semisimple, so its multi-block compression can be chosen with vanishing remainder.

Theorem 21.3.88 Simultaneous left inverse after blocking

✓

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

Proof ▶

After a gauge and a nonzero rescaling each block has transfer map of spectral radius one. Rescaling a block multiplies its words of length $L$ by a nonzero constant, so it changes neither the joint span nor non-equivalence up to a scalar. For normal, pairwise inequivalent tensors the block-injectivity argument of [ 13 , lines 317–345 ] gives one length $L{\gt}0$ at which the word tuples span $\bigoplus _cM_{D_c}$. Grouping $L$ sites into one physical index turns the words of length $L$ into single letters. Projecting the span onto the summand $c$ shows that the letters of $B_c^{[L]}$ span $M_{D_c}$, so each blocked tensor is injective. The coefficients $C_{(c,x,y),(I,J)}$ expressing the matrix unit $E_{xy}$ of the summand $c$, with zero in every other summand, as a combination of the blocked letters are the left inverse of Definition 21.3.83, by Theorem 21.3.84. Each blocked letter of a product is a word of the unblocked letters. In the coordinates $g$ of a split compression every unblocked letter is block diagonal, $g\, B_{ab}^{ik}g^{-1}=\bigoplus _s B_{c(s)}^{ik}$, with one summand for each slot $s=(c,\mu )$. Hence for the word $w=(i_1k_1)\cdots (i_Lk_L)$ of the blocked letter $(I,J)$,

\begin{align} g\, (B_{ab}^{[L]})^{IJ}g^{-1} & =\prod _{t=1}^{L}\Bigl(\bigoplus _sB_{c(s)}^{i_tk_t}\Bigr) =\bigoplus _s\prod _{t=1}^{L}B_{c(s)}^{i_tk_t} =\bigoplus _s(B_{c(s)}^{[L]})^{IJ}. \notag \end{align}

The compression therefore remains split after blocking, with the same compression maps, and Theorem 21.3.87 applies to the blocked tensors.

Definition 21.3.89 Final-block selection from one physical index pair

✓

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

\begin{align} s_d(i,j) & =\sum _{x=0}^{D_d-1}C_{(d,x,x),(i,j)}. \notag \end{align}

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

Lemma 21.3.90 Final-block selection identity

✓

The coefficients $s_d$ satisfy

\begin{align} \sum _{i,j}s_d(i,j)B_e^{ij} & = \begin{cases} \mathbb {1}_{D_d}, & e=d, \\ 0, & e\ne d. \end{cases} \notag \end{align}

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

Proof ▶

Apply the simultaneous left 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.

Theorem 21.3.91 F-move and inverse

✓

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

\begin{align} F^{abc}_d: \bigoplus _e\mathbb {C}^{N_{ab}^{e}}\otimes \mathbb {C}^{N_{ec}^{d}} & \longrightarrow \bigoplus _f\mathbb {C}^{N_{bc}^{f}}\otimes \mathbb {C}^{N_{af}^{d}} \notag \end{align}

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

\begin{align} (X^e_{ab,\mu }\otimes \mathbb {1})X^d_{ec,\nu } & = \sum _{f,\lambda ,\sigma } (F^{abc}_d)^{f\lambda \sigma }_{e\mu \nu } (\mathbb {1}\otimes X^f_{bc,\lambda })X^d_{af,\sigma }. \label{eq:rfp_printed_fmove} \end{align}

The matrix in the opposite orientation is its two-sided inverse. This is the $F$-move of [ 8 , Section 3.4 ] .

Proof ▶

Form the full left- and right-associated synthesis matrices $L$ and $R$ and their analysis matrices $L^+$ and $R^+$. Applying (115) twice to the two associative decompositions gives

\begin{align} B_{abc}^{il} & =L D_{\mathrm L}^{il}L^+ =R D_{\mathrm R}^{il}R^+. \label{eq:rfp_triple_reconstruction} \end{align}

Contract (121) with the coefficients $s_d$ from Definition 21.3.89 and sum over the final labels $d$. The direct sums $D_{\mathrm L}$ and $D_{\mathrm R}$ both contract to the identity. Consequently,

\begin{align} P & :=LL^+=RR^+, \label{eq:rfp_triple_support} \\ L^+L & =R^+R=\mathbb {1}. \notag \end{align}

Hence $R^+L$ changes left-tree coordinates into right-tree coordinates and

\begin{align} R(R^+L) & =(RR^+)L=PL=(LL^+)L=L. \notag \end{align}

Its opposite-oriented comparison is $L^+R$. Using (122),

\begin{align} (R^+L)(L^+R) & =R^+PR=\mathbb {1}, \notag \\ (L^+R)(R^+L) & =L^+PL=\mathbb {1}. \notag \end{align}

The zipper identities (116) and (117) show that $R^+L$ intertwines the direct sums of the final-block matrices. The simultaneous left inverse annihilates its blocks with different final labels. On the $d$-diagonal block, injectivity gives $(R^+L)_d=F^{abc}_d\otimes \mathbb {1}_{D_d}$. Restricting $R(R^+L)=L$ to this block gives (120). The corresponding fixed-final block of $L^+R$, together with $(R^+L)(L^+R)=\mathbb {1}=(L^+R)(R^+L)$, gives the inverse identities tensored with $\mathbb {1}_{D_d}$. Lemma 21.3.13 removes the nonempty bond-space identity factor and yields the two inverse identities for $F^{abc}_d$.

\begin{align} \tntree {((a\, b)_{e,\mu }\, c)_{d,\nu }} & = \sum _{f,\lambda ,\sigma } (F^{abc}_{d})^{f\lambda \sigma }_{e\mu \nu } \tntree {(a\, (b\, c)_{f,\lambda })_{d,\sigma }}. \notag \end{align}

Figure 21.5 The source-oriented $F$-move. The left-associated fusion tensor is expanded in the right-associated fusion basis. The upper indices $(f,\lambda ,\sigma )$ label the rows of $F^{abc}_d$, and the lower indices $(e,\mu ,\nu )$ label its columns. This is the printed-$F$ row/column convention used here for [ 8 , Section 3.4 ] .

Theorem 21.3.92 Gauge covariance of the F-matrices

✓

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

\begin{align} X’^c_{ab,\mu } & =\sum _\nu (Y^c_{ab})_{\mu \nu }X^c_{ab,\nu }. \notag \end{align}

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

\begin{align} F’^{abc}_d & =G_{\mathrm R}^{-1}F^{abc}_d G_{\mathrm L}, & G_{\mathrm L} & =\bigoplus _e (Y^e_{ab})^{\mathsf T}\otimes (Y^d_{ec})^{\mathsf T}, & G_{\mathrm R} & =\bigoplus _f (Y^f_{bc})^{\mathsf T}\otimes (Y^d_{af})^{\mathsf T}. \label{eq:rfp_fmatrix_gauge} \end{align}

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

Proof ▶

Write $\mathcal B^{ik}_{ab}=\bigoplus _c\mathbb {1}_{N_{ab}^c}\otimes B_c^{ik}$, so that (115) reads $B_{ab}^{ik}=X_{ab}\mathcal B^{ik}_{ab}X^+_{ab}$. Since $\Pi _{ab}$ acts only on the multiplicity factors, it commutes with $\mathcal B^{ik}_{ab}$. The biorthogonality (114) and the reconstruction (115) of the new family are

\begin{align} (\Pi _{ab}^{-1}X^+_{ab})(X_{ab}\Pi _{ab}) & =\Pi _{ab}^{-1}(X^+_{ab}X_{ab})\Pi _{ab}=\mathbb {1}, \notag \\ (X_{ab}\Pi _{ab})\mathcal B^{ik}_{ab}(\Pi _{ab}^{-1}X^+_{ab}) & =X_{ab}\mathcal B^{ik}_{ab}\Pi _{ab}\Pi _{ab}^{-1}X^+_{ab} =X_{ab}\mathcal B^{ik}_{ab}X^+_{ab}=B_{ab}^{ik}. \notag \end{align}

The zipper identities (116) and (117) of the new family are

\begin{align} B_{ab}^{ik}X_{ab}\Pi _{ab} & =X_{ab}\mathcal B^{ik}_{ab}\Pi _{ab} =(X_{ab}\Pi _{ab})\mathcal B^{ik}_{ab}, \notag \\ \Pi _{ab}^{-1}X^+_{ab}B_{ab}^{ik} & =\Pi _{ab}^{-1}\mathcal B^{ik}_{ab}X^+_{ab} =\mathcal B^{ik}_{ab}(\Pi _{ab}^{-1}X^+_{ab}). \notag \end{align}

If $R(G\otimes \mathbb {1})=L$ for the fixed-final synthesis matrices $L$ and $R$, multiplying on the left by $R^+$ and using $R^+R=\mathbb {1}$ gives $G\otimes \mathbb {1}=F^{abc}_d\otimes \mathbb {1}$, hence $G=F^{abc}_d$. Expanding the composites shows that the new synthesis matrices are $L'=L(G_{\mathrm L}\otimes \mathbb {1})$ and $R'=R(G_{\mathrm R}\otimes \mathbb {1})$. For $G=G_{\mathrm R}^{-1}F^{abc}_dG_{\mathrm L}$, the $F$-move (120) of the original family gives

\begin{align} R’(G\otimes \mathbb {1}) & =R\bigl(G_{\mathrm R}G_{\mathrm R}^{-1}F^{abc}_dG_{\mathrm L} \otimes \mathbb {1}\bigr) =R(F^{abc}_d\otimes \mathbb {1})(G_{\mathrm L}\otimes \mathbb {1}) =L(G_{\mathrm L}\otimes \mathbb {1})=L’. \notag \end{align}

By the uniqueness just shown, applied to the new family, $F'^{abc}_d=G_{\mathrm R}^{-1}F^{abc}_dG_{\mathrm L}$, which is (123); multiplying on the left by $G_{\mathrm R}$ gives the intertwining form $G_{\mathrm R}F'^{abc}_d=F^{abc}_dG_{\mathrm L}$. For the inverse, let $H=G_{\mathrm L}^{-1}(F^{abc}_d)^{-1}G_{\mathrm R}$. By (123),

\begin{align} F’^{abc}_dH & =G_{\mathrm R}^{-1}F^{abc}_dG_{\mathrm L}G_{\mathrm L}^{-1} (F^{abc}_d)^{-1}G_{\mathrm R}=\mathbb {1}, \notag \\ (F’^{abc}_d)^{-1} & =(F’^{abc}_d)^{-1}\bigl(F’^{abc}_dH\bigr) =\bigl((F’^{abc}_d)^{-1}F’^{abc}_d\bigr)H=H. \notag \end{align}

Definition 21.3.93 Fourfold fusion maps and changes of basis

✓

Fix labels $a,b,c,d,e$. The five fusion trees in equation (33) of [ 8 ] determine five multiplicity spaces and five synthesis maps into the fourfold bond space. These spaces use the integer multiplicities $N_{ab}^{c}$ of the complete zipper family, rather than multiplicity coordinates weighted by the positive diagonal matrices $\chi _{\alpha ,\beta ,\gamma }$.

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

$\begin{tikzcd}[column sep=tiny] & \tntree{(a\,((b\,c)_h\,d)_i)_e} \arrow[dr, "F^{bcd}_{i}"] & \\ \tntree{((a\,(b\,c)_h)_g\,d)_e} \arrow[ur, "F^{ahd}_{e}"] & & \tntree{(a\,(b\,(c\,d)_j)_i)_e} \\ \tntree{(((a\,b)_f\,c)_g\,d)_e} \arrow[u, "F^{abc}_{g}"] \arrow[rr, "F^{fcd}_{e}"'] & & \tntree{((a\,b)_f\,(c\,d)_j)_e} \arrow[u, "F^{abj}_{e}"'] \end{tikzcd}$

Figure 21.6 The five complete zipper fusion trees of [ 8 , Section 3.4, equation (33) ] . Each arrow is a printed $F$-move, with right-tree coordinates indexing rows and left-tree coordinates indexing columns.

Theorem 21.3.94 Complete zipper fusion pentagon

✓

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

\begin{align} & \sum _{h,\sigma ,\lambda ,\omega } (F^{abc}_{g})^{f\mu \nu }_{h\sigma \lambda } (F^{ahd}_{e})^{g\lambda \rho }_{i\omega \kappa } (F^{bcd}_{i})^{h\sigma \omega }_{j\gamma \delta } \notag \\ & \hspace{3em}= \sum _{\tau } (F^{fcd}_{e})^{g\nu \rho }_{j\gamma \tau } (F^{abj}_{e})^{f\mu \tau }_{i\delta \kappa }. \label{eq:rfp_fusion_pentagon} \end{align}

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

Proof ▶

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 (114). This gives equality after tensoring with the identity on the nonzero final bond space, and Lemma 21.3.13 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 (124).

Theorem 21.3.95 Coherence of four bond-coordinate reassociations

✓

Fix four labels $\alpha ,\beta ,\gamma ,\delta $ in a BNT fusion-isometry family, and write $A=\{ 0,\ldots ,D_\alpha -1\} $, $B=\{ 0,\ldots ,D_\beta -1\} $, $C=\{ 0,\ldots ,D_\gamma -1\} $, and $D=\{ 0,\ldots ,D_\delta -1\} $ for their bond-coordinate sets. Let

\begin{align} r_{X,Y,Z}:((X\times Y)\times Z) & \longrightarrow X\times (Y\times Z) \notag \end{align}

denote the canonical reassociation. Define

\begin{align} r_1 & =r_{A,B,C}\times \mathbb {1}_D, \notag \\ r_2 & =r_{A,B\times C,D}, \notag \\ r_3 & =\mathbb {1}_A\times r_{B,C,D}, \notag \\ s_1 & =r_{A\times B,C,D}, \notag \\ s_2 & =r_{A,B,C\times D}. \notag \end{align}

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

\begin{align} r_3\circ r_2\circ r_1 & =s_2\circ s_1. \label{eq:rfp_bond_reassociation} \end{align}

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

\begin{align} P(r_1)P(r_2)P(r_3) & =P(s_1)P(s_2). \label{eq:rfp_bond_permutation} \end{align}

These are equalities of Cartesian bond-coordinate identifications only; they are distinct from the fusion-multiplicity identity in Theorem 21.3.94.

Proof ▶

Every element $(((a,b),c),d)$ is sent by either composite to $(a,(b,(c,d)))$. This proves (125). The identity $P(v\circ u)=P(u)P(v)$ then turns (125) into (126).

$\begin{tikzcd}[column sep=small] & (A\times((B\times C)\times D)) \arrow[dr, "r_3"] & \\ ((A\times(B\times C))\times D) \arrow[ur, "r_2"] & & (A\times(B\times(C\times D))) \\ (((A\times B)\times C)\times D) \arrow[u, "r_1"] \arrow[rr, "s_1"'] & & ((A\times B)\times(C\times D)) \arrow[u, "s_2"'] \end{tikzcd}$

Figure 21.7 The five parenthesizations of four bond-coordinate factors. The upper path is the three-edge composite $r_3\circ r_2\circ r_1$, and the lower path is the two-edge composite $s_2\circ s_1$, depicting canonical Cartesian reassociation of the bond coordinates.

Tensor Network Theory: A formalization blueprint

21 Mixed States: General Case and Algebraic Structure

21.1 Closed-chain operators

21.2 Algebra structure

21.2.1 Diagonal \(\chi \)-matrices and the trace-power formula

21.2.2 Structure coefficients and the idempotent condition

21.2.3 A physical MPDO with a single vertical BNT element

21.2.4 Direct-sum sector conjugations

21.2.5 Ambient sector retractions and normalized embeddings

21.2.6 Trace-preserving completions and physical channels

21.3 Transfer retracts and fusion coisometries