For every finite-dimensional unitary representation \(\rho \) of a finite group,

Hence \(P_\rho \) is an orthogonal projection.

Every matrix product unitary on nonempty chains is a matrix product unitary in the sense of Definition 18.1.2.

If \(\mathcal U\) is a matrix product unitary on nonempty chains and \(L{\gt}0\), then \(\mathcal U^{[L]}\) is also a matrix product unitary on nonempty chains.

Let \(\{ \mathcal U_g\} _{g\in G}\) be an injective family of MPUs with positive bond dimensions, satisfying (7) and (8) at every positive chain length. Simplicity is not required. Then \(d{\gt}0\) and

These are the index properties used in the finite-order argument of [ 197 , line 1547 ] .

Under the hypotheses of Theorem 19.2.6, suppose that \(G\) is finite. Then \(\operatorname {ind}(\mathcal U_g)=0\) for every \(g\in G\). This is the finite-order conclusion in [ 197 , line 1547 ] .

Let \(G\) be finite and let \(\{ \mathcal U_g\} _{g\in G}\) be an exact representation by simple injective MPU tensors of physical dimension \(d\). For every \(g\in G\), the two source-leg dimensions are

This is the conclusion of [ 197 , line 1547 ] .

Let \(G\) be a finite group and \(\{ \mathcal U_g\} _{g\in G}\) matrix product operator tensors whose periodic operators satisfy \(U_g^{(N)}U_h^{(N)}=U_{gh}^{(N)}\) for all \(g,h\in G\) and \(N\geq 1\). Then \(\{ \mathcal U_g\} \) is a fusion algebra with structure constants

and every label \(g\) is invertible with respect to the unit \(e\), with inverse \(g^{-1}\) [ 199 , Section 4 ] . In particular this holds for a representation as in Definition 19.2.4; only the multiplicative law (8) is used.

For every \(g\in G\), every \(n\in \mathbb {N}\), and every \(N\geq 1\),

Suppose that \(G\) is finite. For every \(g\in G\) and every \(N\geq 1\), the represented operator has finite order dividing the order of \(g\):

This is the finite-order assertion in [ 197 , line 1547 ] .

Let \(\mathcal U_g\mathbin {\cdot }\mathcal U_h\) denote the product MPO tensor, whose virtual bond dimension is \(D_gD_h\). For every \(N\geq 1\),

Thus the product tensor and the selected tensor \(\mathcal U_{gh}\) generate the same nonempty-chain periodic operators, although their virtual bond dimensions may differ.

For every \(g\in G\) and every \(N\geq 1\), the physical adjoint of \(\mathcal U_g\) generates the operator representing \(g^{-1}\):

This is the first assertion in [ 197 , line 1552 ] . It requires no finiteness assumption on \(G\).

View \(\mathcal U_g^\sharp \) and \(\mathcal U_{g^{-1}}\) as MPS tensors by pairing their two physical indices. They generate the same matrix product vectors at every positive length:

Let \(M\) and \(N\) be MPO tensors with bond dimensions \(D_M\) and \(D_N\). Write a left bond index of the product tensor as a pair \((\alpha ,\gamma )\), with \(\alpha \) in the bond space of \(M\) and \(\gamma \) in the bond space of \(N\), and a right bond index as \((\beta ,\delta )\) likewise. Then, for every pair of physical indices \(i,k\),

The two bond spaces need not have the same dimension, so the exchange of the two factors of a product bond space is recorded here for unequal dimensions. This is the inverse-product step used in the proof of the inversion identities for \(\zeta \) and \(\omega \) in  [ 197 , lines 1662–1677 ] , for the dagger operation described there, which conjugates every entry and exchanges the two legs lying on the same side of a tensor.

Let \(G\) be a group, and let \(\{ \mathcal U_g\} _{g\in G}\) be a representation by simple injective MPU tensors whose normalized flattenings are left-canonical. Then

After identifying these bond spaces, for every \(g\in G\) there is a unitary matrix \(T_g\in \operatorname {Mat}_{D_g}(\mathbb {C})\) such that

Here \(\mathcal U_g^\dagger \) is the physical-adjoint tensor. This is the dagger–inverse gauge relation in [ 197 , lines 1552–1557 ] .

The gauges in Theorem 19.2.17 may be chosen together with scalars \(\sigma _g\in \mathbb {C}\) satisfying

Every such scalar lies in \(U(1)\), and

If \(g^{-1}=g\), the chosen gauge at \(g^{-1}\), transported to the bond coordinates at \(g\), equals \(T_g\). Thus the same chosen scalar satisfies \(T_gT_g^*=\sigma _g\mathbf1\), and \(\sigma _g\in \{ 1,-1\} \). These are the dagger–inverse scalar relation and its immediate consequences in [ 197 , lines 1557–1567 ] .

Let \(\mathcal U\) be a tensor of physical dimension \(d\) and bond dimension \(D\). Suppose that \(T\) is unitary on \(\mathbb {C}^D\) and that, for every physical pair \(i,j\), \((\mathcal U^\sharp )^{ij}=T^\dagger \mathcal U^{ij}T\). Then, for every physical pair \(i,j\), \(\mathcal U^{ij}=T(\mathcal U^\sharp )^{ij}T^\dagger \). This reverses the gauge relation used in [ 197 , lines 5391–5410 ] .

Let \(\mathcal U\) be a tensor of physical dimension \(d\) and bond dimension \(D\). Suppose that \(T\) is unitary and that, for every physical pair \(i,j\), \((\mathcal U^\sharp )^{ij}=T^\dagger \mathcal U^{ij}T\). With \(A=I_d\otimes \overline T\) and \(B=T^{\mathsf T}\otimes I_d\), the source cuts satisfy \(M_1(\mathcal U)=AM_2(\mathcal U)^\dagger B\). This is the cut identity underlying [ 197 , lines 5391–5431 ] .

Let \(\mathcal U\) be a tensor and \(T\) a unitary virtual matrix such that, for every physical pair \(i,j\), \((\mathcal U^\sharp )^{ij}=T^\dagger \mathcal U^{ij}T\). The candidates \(X,Y\) of Definition 19.2.20 satisfy \(M_1(\mathcal U)=XY\). This is the candidate decomposition of [ 197 , line 5432 ] .

Let \(\mathcal U\) be any tensor and \(T\) any unitary virtual matrix. For the candidate \(X=AY_2^\dagger \) on the second-cut space \(\mathbb {C}^\ell \), the matrix \(Z_2^\dagger A^\dagger \) is a left inverse: \((Z_2^\dagger A^\dagger )X=I_\ell \). No adjoint gauge relation is required. This verifies the left-invertibility used in [ 197 , line 5432 ] .

Let \(\mathcal U\) be any tensor and \(T\) any unitary virtual matrix. For the candidate \(Y=X_2^\dagger B\) on the second-cut space \(\mathbb {C}^\ell \), the matrix \(B^\dagger X_2\) is a right inverse: \(Y(B^\dagger X_2)=I_\ell \). No adjoint gauge relation is required. This verifies the right-invertibility used in [ 197 , line 5432 ] .

Let \(G\) be a group and \(\{ \mathcal U_a\} _{a\in G}\) a representation by simple injective MPUs. Assume that the normalized flattening of \(\mathcal U_a\) is left-canonical for every \(a\in G\). If \(g^{-1}=g\), the chosen unitary gauge \(T_g\) of Theorem 19.2.17 satisfies, for every physical pair \(i,j\), \((\mathcal U_g^\sharp )^{ij}=T_g^\dagger \mathcal U_g^{ij}T_g\). This specializes the gauge relation of [ 197 , lines 1552–1557 ] to the involutive case discussed in [ 197 , lines 1933–1955 ] .

Let \(G\) be a group and \(\{ \mathcal U_a\} _{a\in G}\) a representation by simple injective MPUs whose normalized flattenings are left-canonical for every \(a\in G\). Let \(g^{-1}=g\), and take the chosen gauge \(T_g\) of Theorem 19.2.17. For \(\mathcal U_g\), put \(\ell =\operatorname {rank}M_2(\mathcal U_g)\), \(A=I_d\otimes \overline{T_g}\), \(B=T_g^{\mathsf T}\otimes I_d\), \(X=AY_2^\dagger \), and \(Y=X_2^\dagger B\), using its second-cut factors \(X_2,Y_2,Z_2\). Then

This is the first algebraic construction of [ 197 , lines 5390–5432 ] , with the gauge chosen from the representation rather than supplied separately.

Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\) and bond dimension \(D\). Suppose that \(T\) is unitary on \(\mathbb {C}^D\) and \((\mathcal U^\sharp )^{ij}=T^\dagger \mathcal U^{ij}T\) for every \(0\leq i,j{\lt}d\). Then its source-cut ranks satisfy \(r=\operatorname {rank}M_1(\mathcal U) =\operatorname {rank}M_2(\mathcal U)=\ell \). No positivity, simplicity, or positive-dimension assumption is required.

Let \(\mathcal U\) be a simple MPU of physical dimension \(d\) and bond dimension \(D\) in canonical form II. Suppose that \(T\) is unitary on \(\mathbb {C}^D\) and that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every \(0\leq i,j{\lt}d\). Then \(\ell =r=d\), and the resulting bijection of \(\{ 0,\ldots ,\ell -1\} \) with \(\{ 0,\ldots ,d-1\} \) carries the raw source-cut index to the physical index. Neither the phase relation \(T\overline T=\sigma I_D\), nor simplicity of \(\mathcal U^\dagger \), nor a further virtual gauge enters. This is the self-inverse case of [ 197 , lines 1552–1563 ] , in which the general finite-group index argument of line 1547 is not needed.

Let \(\mathcal U\) be an MPO tensor of dimensions \(d,D\), let \(T\) be unitary on \(\mathbb {C}^D\), and suppose that \((\mathcal U^\sharp )^{ij}=T^\dagger \mathcal U^{ij}T\) for every \(0\leq i,j{\lt}d\). Let \(\rho \) be positive definite on \(\mathbb {C}^D\). Write \(\widetilde X,\widetilde Y,\widetilde Z\) for the first source factors with weight \(\rho \), and \(X_2,Y_2,Z_2\) for the second-cut factors. Set

With \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\), the matrices \(K\colon \mathbb {C}^\ell \to \mathbb {C}^r\) and \(J\colon \mathbb {C}^r\to \mathbb {C}^\ell \) satisfy

The two source-coordinate spaces are retained separately, without identifying their bases along the equality \(r=\ell \). These are inverse identities, not adjoint or unitarity identities.

Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\) and bond dimension \(D\), let \(\rho \) be positive definite on \(\mathbb {C}^D\), and let \(T\) be unitary on \(\mathbb {C}^D\). Suppose that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\). Write \(\widetilde X_1,\widetilde Y_1\) for the first source factors with weight \(\rho \), and \(X_2,Y_2\) for the second source factors. Put \(A=I_d\otimes \overline T\) and take the weighted comparison \(K=\widetilde X_1^\dagger (I_d\otimes \rho )AY_2^\dagger \) of Definition 19.2.1.3. Then

This is the adjointed comparison used in [ 197 , lines 5440–5443 ] , in the maintained physical–virtual column order.

Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\) and bond dimension \(D\), let \(\rho \) be positive definite on \(\mathbb {C}^D\), and let \(T\) be unitary on \(\mathbb {C}^D\). Suppose that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\), and that \(T\overline T=\sigma I_D\) for a scalar \(\sigma \in \mathbb {C}\). Write \(\widetilde X_1,\widetilde Y_1\) for the first source factors constructed with this \(\rho \), and \(X_2,Y_2\) for the second source factors. Set \(r=\operatorname {rank}M_1(\mathcal U)\), \(\ell =\operatorname {rank}M_2(\mathcal U)\), and \(K=\widetilde X_1^\dagger (I_d\otimes \rho ) (I_d\otimes \overline T)Y_2^\dagger \colon \mathbb {C}^\ell \to \mathbb {C}^r\). For the source gates \(u,v\) of Definition 18.4.59 formed from \(\widetilde X_1,\widetilde Y_1,X_2,Y_2\),

The row orders of \(u\) and \(v^\dagger \) are respectively \((\ell ,r)\) and \((r,\ell )\); the Kronecker product maps between these orders without an additional swap or identification of the rank coordinates. This is the contraction in [ 197 , lines 5444–5487 ] , retaining the diagram’s \(\widetilde Y_1\) [ 137 ] . Neither canonical form, simplicity, gate unitarity, nor \(|\sigma |=1\) is a hypothesis. The conclusion is the algebraic identity, not unitarity of \(K\) or the gates.

Let \(\mathcal U\) be a simple MPU of physical dimension \(d\) and bond dimension \(D\), with a canonical-form-II presentation and its recorded positive-definite matrix \(\rho \). Let \(T\) be unitary on \(\mathbb {C}^D\) and let \(\sigma \in \mathbb {C}\). Suppose that, for every physical pair \(i,j\), \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\), and that \(T\overline T=\sigma I_D\). Construct the first source factor \(\widetilde X\) with this same \(\rho \) and the second source factor \(Y_2\). Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). The comparison

is unitary between these spaces: \(K^\dagger K=I_\ell \) and \(KK^\dagger =I_r\). This is the comparison-unitarity step of [ 197 , lines 5444–5487 ] . The comparison carries the weight of the chosen normalization [ 16 , lines 487–494 ] , which the printed definition of \(K\) at [ 197 , lines 5436–5439 ] omits; the omission and its repair are recorded in [ 137 ] .

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\), with a canonical-form-II presentation and recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary on \(\mathbb {C}^D\), that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\), and that \(T\overline T=\sigma I_D\) for a scalar \(\sigma \in \mathbb {C}\). Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). Form the first source factors with this same \(\rho \), the candidate \(X=(I_d\otimes \overline T)Y_2^\dagger \), and the comparisons \(K\colon \mathbb {C}^\ell \to \mathbb {C}^r\) and \(J\colon \mathbb {C}^r\to \mathbb {C}^\ell \) of Definitions 19.2.1.3 and 19.2.1.4. Then

These give the adjoint identification and first weighted normalization in [ 197 , lines 5432–5487 ] , retaining the source normalization in [ 16 , lines 487–494 ] .

Let \(G\) be a group and let \(\{ \mathcal U_a\} _{a\in G}\) be a representation by simple injective MPUs of physical dimension \(d\). Suppose that the normalized flattening of \(\mathcal U_a\) is left-canonical for every \(a\in G\). Fix \(g\in G\) with \(g^{-1}=g\), and give \(\mathcal U_g\) a canonical-form-II presentation with recorded positive-definite matrix \(\rho _g\). Take the chosen gauge \(T_g\) of Theorem 19.2.17. For \(\mathcal U_g\), set \(r=\operatorname {rank}M_1(\mathcal U_g)\) and \(\ell =\operatorname {rank}M_2(\mathcal U_g)\). Construct \(K_g,J_g\) with this \(T_g\) and this same \(\rho _g\), and put \(X_g=(I_d\otimes \overline{T_g})Y_2^\dagger \), using its second-cut factor \(Y_2\). Then

Canonical form II is required only at \(g\); left-canonicity of the normalized flattenings is required throughout the family.

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary on \(\mathbb {C}^D\), that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\), and that \(T\overline T=\sigma I_D\) for a scalar \(\sigma \in \mathbb {C}\). Let \(\widetilde X_1,\widetilde Y_1\) be the chosen first source factors at this same weight, and let \(\widehat X_1,\widehat Y_1\) be the inverse-compatible first factors of Definition 19.2.3.2. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). Take the weighted comparison \(K\colon \mathbb {C}^\ell \to \mathbb {C}^r\) of Definition 19.2.1.3 and the permutation matrix \(P_e\colon \mathbb {C}^\ell \to \mathbb {C}^r\) of Definition 19.2.3.1. The square matrix \(C=KP_e^\dagger \) is unitary on \(\mathbb {C}^r\), and

Consequently, the Gram matrices on the external first-cut spaces are unchanged:

This is the transport of the comparison in [ 197 , lines 5432–5443 ] , after its unitarity is established at lines 5444–5487. It asserts no transport formula for the chosen first right inverse and no further pleasant properties.

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). Let \(\widehat Y_1,\widehat Y_2\) be the inverse-compatible source factors at this same weight. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). For physical indices \(p,q\), one has

These are the physical closures of [ 197 , lines 869–917 ] , transported as in lines 5486–5487. Simplicity of \(\mathcal U^\dagger \) is not required.

Let \(\mathcal U\) be an MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Assume that both \(\mathcal U\) and \(\mathcal U^\dagger \) are simple. Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). Let \(\widehat Y_1,\widehat Y_2\) be its inverse-compatible source factors at \(\rho \), with cut ranks \(r,\ell \). Then

These are the rank normalizations of [ 197 , lines 1064–1119 ] , transported at lines 5486–5487. The adjoint-simplicity hypothesis is retained.

Let \(\mathcal U\) be an MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Assume that both \(\mathcal U\) and \(\mathcal U^\dagger \) are simple. Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). Let \(\widehat X_1,\widehat X_2\) be its inverse-compatible source factors at \(\rho \), with cut ranks \(r,\ell \). For physical indices \(i,j\),

These are the physical left-factor closures of [ 197 , lines 1120–1162 ] , transported at lines 5486–5487. The canonical weight is not replaced by the identity.

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) with a canonical-form-II presentation and recorded positive-definite matrix \(\rho \). Let \(T\) be unitary on \(\mathbb {C}^D\) and let \(\sigma \in \mathbb {C}\) satisfy \(T\overline T=\sigma I_D\). Suppose that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\). Let \(S_0\) be the chosen source factors at this \(\rho \), and let \(S\) be the inverse-compatible source factors of Definition 19.2.3.2 for the same data. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). Let \(e\colon \{ 0,\ldots ,\ell -1\} \to \{ 0,\ldots ,r-1\} \) be the rank identification of Definition 19.2.3.1. Reindex the columns of the comparison \(K\) by \(e\) to obtain \(C\in \operatorname {Mat}_r(\mathbb {C})\), with \(C_{a,b}=K_{a,e^{-1}(b)}\). Then the crossed movement gates satisfy

The right gate has row order \((\text{physical},\text{first rank})\) and column order \((\text{first rank},\text{physical})\); no leg permutation occurs. These are the crossed contractions of [ 197 , lines 811–869 ] , transported by the unitary comparison as in [ 197 , lines 5486–5487 ] .

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) with a canonical-form-II presentation and recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary on \(\mathbb {C}^D\), that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\), and that \(T\overline T=\sigma I_D\) for a scalar \(\sigma \in \mathbb {C}\). Let \(S\) be its inverse-compatible source factors constructed with this same \(\rho \). With source-cut ranks \(r,\ell \), both movement gates are unitary maps between their indicated spaces:

No simplicity hypothesis on the physical adjoint is required.

Let \(\mathcal U\) be an MPO tensor, \(\rho \) a positive-definite virtual matrix, and \(T\) a unitary virtual matrix satisfying \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair. Let \(u\) be its source gate at weight \(\rho \) and \(J\) its reverse comparison at that weight. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). Then \(u_{\mathrm{new}}=(I_\ell \otimes J)u\). This is the comparison transport in [ 197 , lines 5432–5443 and 5486–5487 ] ; it does not require \(J=K^\dagger \) or gate unitarity.

Let \(\mathcal U\) be an MPO tensor, \(\rho \) a positive-definite virtual matrix, and \(T\) a unitary virtual matrix satisfying \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair. Let \(v\) be its source gate at weight \(\rho \) and \(K\) its weighted comparison. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\). Then \(v_{\mathrm{new}}=v(K\otimes I_\ell )\). This is the comparison transport used in [ 197 , lines 5432–5443 and 5486–5487 ] .

Let \(\mathcal U\) be a simple MPU in canonical form II, with recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary, \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). The literal gates of Definition 19.2.4.3 are unitary between \(\mathbb {C}^d\otimes \mathbb {C}^d\) and \(\mathbb {C}^\ell \otimes \mathbb {C}^\ell \), and satisfy \(u_{\mathrm{new}}=\sigma v_{\mathrm{new}}^\dagger \). This is the new-gate conclusion of [ 197 , lines 5440–5487 ] . The equality uses the common second-cut coordinate space, not the differently ordered rank spaces of the gates formed after transporting the first factors.

Let \(G\) be a group and \(\{ \mathcal U_a\} _{a\in G}\) a representation by simple injective MPUs. Assume that the normalized flattening of \(\mathcal U_a\) is left-canonical for every \(a\in G\). Fix \(g\) with \(g^{-1}=g\) and give \(\mathcal U_g\) a canonical-form-II presentation. Use its chosen gauge \(T_g\) and scalar \(\sigma _g\) from Theorems 19.2.17 and 19.2.18. The literal new gates of \(\mathcal U_g,T_g\) are unitary and satisfy \(u_{\mathrm{new},g}=\sigma _g v_{\mathrm{new},g}^\dagger \) on the common second-cut rank-pair space. Canonical form II is needed only at \(g\); left-canonicity of normalized flattenings is required for the whole family. This specializes the construction of [ 197 , lines 5440–5487 ] using its dagger–inverse gauges at lines 1552–1562.

Let \(G\) be a group and \(\{ \mathcal U_a\} _{a\in G}\) a representation by simple injective MPUs whose normalized flattenings are all left-canonical. Fix \(g\) with \(g^{-1}=g\), give \(\mathcal U_g\) a canonical-form-II presentation with recorded positive-definite matrix \(\rho \), and build the inverse-compatible source factors \(\widehat X_1,\widehat Y_1,\widehat X_2,\widehat Y_2\) from the chosen gauge \(T_g\) and scalar \(\sigma _g\) of Theorems 19.2.17 and 19.2.18. Write \(r,\ell \) for the two source-cut ranks of \(\mathcal U_g\) and \(D_g\) for its bond dimension. Then all three pairs of source properties hold: for physical indices \(p,q\),

the two rank normalizations

and, for physical indices \(i,j\),

Simplicity of \(\mathcal U_g^\dagger \), which [ 197 , line 5344 ] assumes, is here a consequence of the canonical-form-II presentation at \(g\), establishing the properties labelled eq:MPUnice2–eq:MPUnice4 in [ 197 , lines 1064–1162 ] , transported as at lines 5486–5487.

Let \(\mathcal U\) be an MPO tensor of dimensions \(d,D\) and \(T\) a unitary matrix on \(\mathbb {C}^D\). Suppose that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair \(i,j\). For equal-length physical words \(p,q\), including the empty words, their ordered matrix products satisfy

This iterates the local gauge relation of [ 197 , lines 1552–1557 ] .

Let \(\mathcal U\) be any MPO tensor and \(T\) any unitary virtual matrix. For \(A=I_d\otimes \overline T\) and the candidate \(X=AY_2^\dagger \), one has \(Y_2=X^\dagger A\). This is the endpoint recovery from the candidate in [ 197 , line 5432 ] .

Let \(\mathcal U\) be an MPO tensor and \(T\) a unitary virtual matrix. Suppose that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair and that \(T\overline T=\sigma I_D\) for a scalar \(\sigma \in \mathbb {C}\). For every \(N\geq 0\), the endpoint matrices of Definition 19.2.4.11 satisfy \(A_N=\sigma B_N^\dagger \). The scalar occurs once, independently of \(N\). Adjointing exchanges the physical configurations without reversing either word or inserting an additional swap. No canonical form, simplicity, comparison matrix, or rank transport is required. This is an all-word local extension of [ 197 , lines 5444–5487 ] , not the final truncated-chain sewing identity.

Let \(L{\gt}0\). The commonly blocked family \(\{ \mathcal U_g^{[L]}\} _{g\in G}\) is again a group representation by simple injective MPUs. Under the canonical identification of blocked configurations with words of length \(NL\),

for every \(N\geq 1\). Physical blocking also commutes with tensor products:

Let \(\{ \mathcal U_g\} _{g\in G}\) be a raw MPU representation. Then \(L_*{\gt}0\), and every \(\mathcal U_g^{[L_*]}\) is simple. For every \(K{\gt}0\), each further block \((\mathcal U_g^{[L_*]})^{[K]}\) is simple. Consequently the blocked family at length \(L_*\) satisfies Definition 19.2.4, and every further common positive blocking does as well.

Every MPU representation by simple injective tensors admits a family of selected reductions.

Let \(\{ \mathcal U_g\} _{g\in G}\) be an MPU representation by simple injective tensors with a family of selected reductions, and let \(L{\gt}0\) satisfy \(N_0(g,h)\leq L+1\) for every pair. Then the blocked family of selected reductions of \(\{ \mathcal U_g^{[L]}\} _{g\in G}\) has exterior buffer length \(1\): for all \(g,h\in G\) and all nonempty blocked words \(\mathbf p,\mathbf c,\mathbf q\),

The common buffer length is such an \(L\). Consequently, every MPU representation of a finite group by simple injective tensors admits a positive \(L\) such that \(\{ \mathcal U_g^{[L]}\} _{g\in G}\) is again an MPU representation by simple injective tensors and has a family of selected reductions with exterior buffer length \(1\).

This is the assertion of [ 197 ] , main.tex line 1498, and of the footnote to Theorem 1 of [ 200 ] , that the nilpotency length can be reduced to \(1\) by blocking, read as the one-site exterior identity (162). Blocking need not make a blocked residual letter vanish, and the blocked nilpotency length \(\ell (B_{g,h}^{[L]},A_{g,h}^{[L]};V_{g,h},W_{g,h})\) of Definition 8 in [ 221 ] need not be \(1\) [ 74 ] .

Let \((V,W)\) be a reduction from \(B\) to \(A\) and let \(\beta \in \mathbb {C}^\times \). Then \((\beta ^{-1}V,\beta W)\) is again a reduction from \(B\) to \(A\); its residual letters \(N^i=B^i-WA^iV\) are unchanged, hence so are its residual-word nilpotency bounds and its residual nilpotency length \(\ell (B,A;V,W)\); and a natural number is an exterior buffer length for \((\beta ^{-1}V,\beta W)\) if and only if it is one for \((V,W)\).

Consequently, let \(\beta \colon G^2\to \mathbb {C}^\times \) be a scalar \(2\)-cochain and let \(\{ \mathcal U_g\} _{g\in G}\) have a family of selected reductions. The rescaled matrices of (262), namely \(W_{g,h}\mapsto \beta _{g,h}W_{g,h}\) and \(V_{g,h}\mapsto \beta _{g,h}^{-1}V_{g,h}\), again form a family of selected reductions of \(\{ \mathcal U_g\} _{g\in G}\). It has exterior buffer length \(m\) exactly when the original family does, and rescaling commutes with blocking by any length \(L\).

This is the scalar gauge freedom of the fusion tensors in [ 197 , lines 1500–1504 ] under the identification (163); uniqueness up to boundary dressing is given by Theorem 29.11.28.

Let \(\mathcal U\) be a specified MPO tensor of positive physical dimension \(d\), with positive right and left source-cut ranks \(r\) and \(\ell \). If

and \(\operatorname {ind}_{\mathrm{src}}(\mathcal U)=0\), then

This is the arithmetic conclusion in [ 197 , line 1547 ] , under the displayed hypotheses.

Let \(I\) be a nonempty finite set, let \(T,S\in \operatorname {Mat}_I(\mathbb {C})\) be unitary, and let \(\sigma \in \mathbb {C}\). If

then

This is the scalar phase conclusion following [ 197 , lines 1557–1562 ] once the matrix relation has been supplied.

Let \(I\) be a nonempty finite set, let \(T,S\in \operatorname {Mat}_I(\mathbb {C})\) be unitary, and let \(\sigma ,\tau \in \mathbb {C}\). If

then

This is the conditional matrix-algebra content of \(\sigma _g\sigma _{g^{-1}}=1\) in [ 197 , lines 1559–1563 ] .

Let \(I\) be a nonempty finite set, let \(T\in \operatorname {Mat}_I(\mathbb {C})\) be unitary, and let \(\sigma \in \mathbb {C}\). If

then

This is the sign step \(\mathbb {1}=xx^\dagger =e^{2i\phi }\mathbb {1}\) in [ 16 , lines 1036–1038 ] , and the involution conclusion in [ 197 , lines 1563–1567 ] , conditional on the self relation.

Let \(n\geq 0\), and let \(D_0,\ldots ,D_n\) be horizontal bond dimensions. For each \(0\leq i{\lt}n\), let \(A_i\) be an injective MPS tensor with square bond dimension \(D_i\), and let \(B_i,B'_i\) be tensors whose left and right horizontal bond dimensions are \(D_i\) and \(D_{i+1}\). Suppose that every \(B_i\) is nonzero and that the connected open networks agree:

Here the equality retains the two horizontal end legs and all physical legs. Then there are scalars \(\beta _i\in \mathbb {C}\), for \(0\leq i{\lt}n\), such that

This is the nondegenerate corrected form of Lemma lemma:1 in [ 197 , lines 2296–2395 ] . The printed wording omits nonvanishing. The paper’s truncated symmetry \(U^L_g\) is unitary at line 2099; lines 2444–2466 identify the side used at line 2567 with \((U^L_{g^{-1}})^\dagger \), while lines 3033–3053 identify the side used at line 3157 with \(\sigma _g(U^L_{g^{-1}})^\dagger \) for \(\sigma _g\in \{ 1,-1\} \). Both displayed networks are therefore nonzero. A vanishing intervening factor would make its entire network zero, so taking the factors on these sides as the \(B_i\) supplies \(B_i\ne 0\) at both applications. This nondegeneracy condition is recorded in [ 75 ] .

Let \(M\) and \(N\) be composable complex matrices between pairs of index sets. If their common index sets and the two outer index sets are finite, then \((MN)_{23}=M_{23}N_{23}\). The identity matrix satisfies \(I_{23}=I\).

Let \(\mathcal U\) have source factors as in Definition 19.4.1, and form \(u,v,w_L,w_R\) from those same factors. Then

No simplicity or unitarity assumption on \(\mathcal U\) is required. This is the first diagram in the proof of the staircase-gate corollary of [ 197 , lines 918–968 ] .

Let \(\mathcal U\) have source factors as in Definition 19.4.1, and form \(u,v,w_L,w_R\) from those same factors. If \(u\) is a coisometry, so that \(uu^\dagger =I_{d_ld_r}\), then

The second expression uses the regrouping of Definition 19.4.2. This is the displayed network identity in [ 197 , lines 969–974 ] .

Let \(A,B,C,E\) be finite index sets, let \(B',C'\) be index sets, and let \(M\) be a complex matrix with rows indexed by \(B\times C\) and columns by \(B'\times C'\). If \(M^\dagger M=I\), then \((M_{23})^\dagger M_{23}=I\), with the middle-leg placement of Definition 19.4.2.

Let \(\mathcal U\) have source factors as in Definition 19.4.1, and form \(u,v,w_L\) from those same factors. Write \(d_l,d_r\) for the source-cut ranks. Under the natural associativity identifications of the three tensor factors,

Both sides map \(\mathbb {C}^d\otimes \mathbb {C}^d\otimes \mathbb {C}^{d_l}\) to \(\mathbb {C}^{d_l}\otimes \mathbb {C}^d\otimes \mathbb {C}^d\). Explicitly, for \(0\leq l,k{\lt}d_l\) and \(0\leq i,j,a,b{\lt}d\),

This is the second network in [ 197 , lines 977–1011 ] . It uses two copies of \(w_L\) and requires no simplicity or unitarity hypothesis.

Let \(\mathcal U\) have source factors as in Definition 19.4.1. If the associated operators \(u\) and \(v\) are unitary between their respective spaces, then

Let \(\mathcal U\) have source factors as in Definition 19.4.1, with left source-cut rank \(d_l\). If \(u^\dagger u=I\) and \(v^\dagger v=I\), then \(K=(w_L\otimes I_d)(I_d\otimes w_L)\) satisfies \(K^\dagger K=I\), with the three-leg associativity identifications understood.

Let \(\mathcal U\) be a simple MPU in canonical form II, with recorded positive diagonal fixed matrix \(\rho \). Construct the source factors from \(\mathcal U\) and this same \(\rho \), and let \(d_l,d_r\) be their source-cut ranks. Then the staircase gates are unitary maps

This is part (a) of the staircase-gate corollary in [ 197 , lines 811–1012 ] .

Let \(\mathcal U\) and the arbitrary endpoint matrices \(Y_2,Y_1\) have the dimensions in Definition 19.4.2.1. For endpoint indices \(l,r,a,b\) and the empty words \(\varnothing \),

This is total length two and requires no cut factorization. For normalized source factors, the right-hand side is \(u_{(l,r),(a,b)}\), with \(u\) as in Definition 18.4.59.

Let \(\mathcal U\) be an MPO tensor with physical dimension \(d\) and bond dimension \(D\), and put \(d_l=\operatorname {rank}M_2\) and \(d_r=\operatorname {rank}M_1\). Let \(Y_1\colon \mathbb {C}^D\otimes \mathbb {C}^d\to \mathbb {C}^{d_r}\), \(Y_2\colon \mathbb {C}^d\otimes \mathbb {C}^D\to \mathbb {C}^{d_l}\), and \(X_2\colon \mathbb {C}^{d_l}\to \mathbb {C}^D\otimes \mathbb {C}^d\) be complex matrices satisfying only \(M_2=X_2Y_2\), that is,

Define the crossed contraction

For \(N\geq 0\), physical words \(\sigma ,\tau \) of length \(N\), source indices \(l,r\), and physical indices \(i,a,b,c\),

The right endpoint \(Y_1\) is arbitrary and unchanged; no first-cut factorization is assumed. For normalized source factors, \(w_L\) is the movement operator of Definition 19.4.1. The identity is the left-end factorization step of [ 197 , lines 2101–2174 ] , using the second decomposition in [ 16 , lines 479–530 ] .

Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\) and bond dimension \(D\), with \(d_l=\operatorname {rank}M_2\) and \(d_r=\operatorname {rank}M_1\). Let \(X_1\colon \mathbb {C}^{d_r}\to \mathbb {C}^d\otimes \mathbb {C}^D\), \(Y_1\colon \mathbb {C}^D\otimes \mathbb {C}^d\to \mathbb {C}^{d_r}\), \(X_2\colon \mathbb {C}^{d_l}\to \mathbb {C}^D\otimes \mathbb {C}^d\), and \(Y_2\colon \mathbb {C}^d\otimes \mathbb {C}^D\to \mathbb {C}^{d_l}\) be complex matrices satisfying only \(M_1=X_1Y_1\) and \(M_2=X_2Y_2\). In coordinates,

Form \(U^{N+2}\) using the endpoints \(Y_2,Y_1\), and define the crossed contractions

For \(N\geq 0\), physical words \(\sigma ,\tau \) of length \(N\), source indices \(l,r\), and physical indices \(i,j,a,b,c,e\), one has

Concatenation \(i\sigma j\) inserts \(i\) before the word and \(j\) after it. This is the source-factorization identity in the growth diagram of [ 197 , lines 2101–2174 ] . It adds one bulk site at each end, using only the two decompositions in [ 16 , lines 479–530 ] . For normalized source factors, the crossed contractions yield the movement operators of Definition 19.4.1, extending the identity to the source-factor endpoint contraction. No simplicity or unitarity assumption on \(\mathcal U\) is needed. These algebraic identities do not assert unitarity for arbitrary factors, the finite-group physical specialization, or the complementary finite-chain movement relation.

Let \(A,B,C,E\) be finite sets, possibly empty, and let \(M\colon \mathbb {C}^B\to \mathbb {C}^A\) and \(P\colon \mathbb {C}^E\to \mathbb {C}^C\) be unitary maps: \(M^\dagger M=I_B\), \(MM^\dagger =I_A\), \(P^\dagger P=I_E\), and \(PP^\dagger =I_C\). Then \(M\otimes P\) is a unitary map from \(\mathbb {C}^{B\times E}\) to \(\mathbb {C}^{A\times C}\).

Let \(\mathcal U\) be a simple MPU with a canonical-form-II presentation and recorded positive diagonal fixed matrix \(\rho \). Construct the source factors from this tensor and this same \(\rho \). For every \(N\geq 0\), their endpoint contraction \(U^{N+2}\) is a unitary map

The subscripts of \(\mathcal H_{\mathrm{in},N}\) and \(\mathcal H_{\mathrm{out},N}\) record the bulk length \(N\). Equivalently,

This is the endpoint-coordinate unitarity part of the assertion following the truncated-symmetry diagram in [ 197 , lines 2062–2099 ] , between the displayed input and output spaces. The scope restriction is recorded in [ 48 ] . Neither simplicity of the physical adjoint nor unitarity of supplied gates is assumed. The assertion does not identify each endpoint source space with a physical site, specialize to a finite-group family, prove agreement with a global symmetry on interior observables, or establish the complementary finite-chain relation following the movement display at [ 197 , lines 2101–2174 ] .

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). Let \(S_0\) denote its chosen source factors at \(\rho \) and \(S\) its inverse-compatible source factors. Put \(r=\operatorname {rank}M_1(\mathcal U)\) and \(\ell =\operatorname {rank}M_2(\mathcal U)\), and take the square unitary comparison \(C=KP_e^\dagger \) of Theorem 19.2.3.3. For \(N\geq 0\), let \(\Omega _N\) be the set of physical configurations on \(N\) sites. Write \(U_0^{N+2}\) and \(\widehat U^{N+2}\) for the endpoint contractions of Definition 19.4.2.1 formed from \(S_0\) and \(S\), respectively. Their common domain and codomain, with the indicated association, are

Then

Explicitly, for source indices \(l,s\), configurations \(p,q\in \Omega _N\), and physical indices \(a,b\),

Only the right source coordinate changes. Here \(N\) counts bulk sites and the total source length is \(N+2\), including both endpoints. This applies the comparison transport of [ 197 , lines 5432–5487 ] to the endpoint contraction at lines 2062–2099. No simplicity assumption on the physical adjoint is needed.

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II, with recorded positive-definite matrix \(\rho \). Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). For every bulk length \(N\geq 0\), the endpoint contraction \(\widehat U^{N+2}\) formed from the inverse-compatible source factors is unitary from \(\mathcal H_N=\mathbb {C}^d\otimes (\mathbb {C}^{\Omega _N}\otimes \mathbb {C}^d)\) to \(\mathcal R_N=\mathbb {C}^\ell \otimes (\mathbb {C}^{\Omega _N}\otimes \mathbb {C}^r)\), where \(\Omega _N\) is the set of length-\(N\) physical configurations and \(r,\ell \) are the first and second cut ranks. The case \(N=0\) has total length two. No simplicity assumption on the physical adjoint is needed. This transports the unitarity assertion following the endpoint diagram of [ 197 , lines 2062–2099 ] , using its comparison argument at lines 5486–5487.

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II. Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). Construct the inverse-compatible source factors at the recorded canonical weight. Put \(r=\operatorname {rank}M_1(\mathcal U)\), \(\ell =\operatorname {rank}M_2(\mathcal U)\), and take the rank bijection \(e\colon \{ 0,\ldots ,\ell -1\} \to \{ 0,\ldots ,r-1\} \) of Definition 19.2.3.1. For every \(N\geq 0\), let \(\Omega _N\) be the length-\(N\) physical configurations. The raw and packaged source spaces and the physical space are, with right association,

Let \(E_N(l,(p,s))=(l,(p,e(s)))\), and let \(P_{E_N}\colon \mathcal R_N^{\mathrm{raw}}\to \mathcal R_N^{\mathrm{pkg}}\) be its permutation matrix. If \(\widehat U^{N+2}\) is the endpoint contraction of the inverse-compatible source factors, then the raw matrix \(A_N\) of Definition 19.2.4.11 satisfies

Equivalently, for raw source indices \(l,s\), configurations \(p,q\), and physical indices \(a,b\),

Thus \(A_N\colon \mathcal H_N\to \mathcal R_N^{\mathrm{raw}}\) is unitary. This transports only the right endpoint back along \(e^{-1}\); no spatial permutation is introduced. The total source length is \(N+2\), so \(N=0\) has total length two. This is coordinate transport of the endpoint construction in [ 197 , lines 2062–2099 and 5390–5487 ] .

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II. Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). For every bulk length \(N\geq 0\), take the raw endpoint matrices \(A_N\colon \mathcal H_N\to \mathcal R_N^{\mathrm{raw}}\) and \(B_N\colon \mathcal R_N^{\mathrm{raw}}\to \mathcal H_N\) of Definition 19.2.4.11, where \(\mathcal H_N=\mathbb {C}^d\otimes (\mathbb {C}^{\Omega _N}\otimes \mathbb {C}^d)\) and \(\mathcal R_N^{\mathrm{raw}}=\mathbb {C}^\ell \otimes (\mathbb {C}^{\Omega _N}\otimes \mathbb {C}^\ell )\). Here \(\Omega _N\) is the set of length-\(N\) physical configurations and \(\ell \) is the second-cut rank. Then

There is a single phase, independent of \(N\); the total source length is \(N+2\). These are local endpoint cancellations extending [ 197 , lines 5444–5487 ] , with the phase relation at lines 1559–1562.

Let \(\mathcal U\) be an MPO tensor with \(M_1(\mathcal U)=X_1Y_1\) and \(M_2(\mathcal U)=X_2Y_2\). Fix bulk lengths \(n,m\), marked physical indices \(a,a',b,b'\), and configurations \(p,p'\in \Omega _n\), \(q,q'\in \Omega _m\). Then

The same sum is the entry of the periodic operator \(\mathcal U^{(n+m+2)}\) at the concatenated configurations \((a,p,b,q)\) and \((a',p',b',q')\). The ring has \(n+m+2\) sites, including two distinct marked sites even when \(n=m=0\), and the two endpoint-inclusive arcs overlap at both marked sites. This identity requires only the two cut factorizations, without assuming canonicality, simplicity, normalization, or unitarity.

Let \(\mathcal U\) be an MPO tensor of dimensions \(d,D\) and \(T\) a unitary virtual matrix with \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair. For all bulk lengths \(n,m\),

Neither simplicity, canonical form, nor a phase relation is used; the complementary arc is the independently defined \(C_{n,m}\), not a rearrangement of \(M_{n,m}\).

Let \(\mathcal U\) be a simple MPU of dimensions \(d,D\) in canonical form II. Suppose that \(T\) is unitary, that \((\mathcal U^\dagger )^{ij}=T^\dagger \mathcal U^{ij}T\) for every physical pair, and that \(T\overline T=\sigma I_D\) for \(\sigma \in \mathbb {C}\). For all bulk lengths \(n,m\),

Exactly one phase occurs, independently of \(n\) and \(m\). The two endpoint-inclusive arcs share both marked sites, so this is a shared-boundary composition and not the disjoint-complement identity of [ 197 , lines 5486–5487 ] .

Let \(\mathcal U\) be a simple MPU in canonical form II, with physical dimension \(d\), bond dimension \(D\), and recorded fixed matrix \(\rho \). For \(0\leq p,q{\lt}d\),

This is the one-letter diagram in the proof of part (b) of the staircase-gate corollary in [ 197 , lines 1013–1030 ] .

Let \(\mathcal U\) be a simple MPU in canonical form II, with physical dimension \(d\), bond dimension \(D\), and recorded positive diagonal fixed matrix \(\rho \). Let \(Y_1,Y_2\) be the source factors constructed from this tensor and this same \(\rho \), with source-cut ranks \(d_r,d_l\). For \(0\leq p,q{\lt}d\),

The open indices are physical; these are identities on \(\mathbb {C}^d\), not on the source-cut spaces. They are the two diagrams of part (b) of the staircase-gate corollary in [ 197 , lines 869–917 ] .

Let \(M\) and \(N\) be nonempty sets with decidable equality, and let \(A\in \operatorname {Mat}_M(\mathbb {C})\) and \(B\in \operatorname {Mat}_N(\mathbb {C})\). If \(A\otimes B=\mathbf1\), then there is a nonzero \(c\in \mathbb {C}\) such that

Let \(M,N\) be nonempty finite sets, and let \(K\colon \mathbb {C}^N\to \mathbb {C}^M\) be a complex matrix. If \(K^\dagger \otimes K\colon \mathbb {C}^{M\times N}\to \mathbb {C}^{N\times M}\) is unitary, then \(K\) is unitary between its coordinate spaces:

No equality of \(|M|\) and \(|N|\) is assumed. This is the matrix implication used in [ 197 , lines 5444–5487 ] ; the tensor-network identity to which it is applied is a separate result.

Let \(M\) and \(N\) be nonempty finite sets. If \(K_1\in \operatorname {Mat}_M(\mathbb {C})\) and \(K_2\in \operatorname {Mat}_N(\mathbb {C})\) satisfy that \(K_1\otimes K_2\) is unitary, then there are a positive real number \(\delta \) and unitary matrices \(W_1,W_2\) such that

Equivalently, \(\delta ^{-1}K_1\) and \(\delta K_2\) are unitary. This is only the tensor-factor step in the decomposition result of  [ 197 ] ; it does not construct or compare the three-legged decompositions occurring there.

Let \(A_i\) and \(B_i\) be two families of square matrices of the same size, indexed by the same finite alphabet, and suppose that, for every \(i,j\),

If \(A\) is injective, or more generally normal, then there is a scalar \(c\in \mathbb {C}\) with \(B_i=cA_i\) for every \(i\). This is the commuting-tensor step in the proof of the source-factor contraction identities in  [ 197 , lines 1255–1325 ] , where the tensor commuting with a normal matrix product unitary tensor is shown to be a multiple of it.

Let \(\mathcal U\) be a simple MPU with a canonical-form-II presentation and recorded fixed matrix \(\rho \). Let \(W\) be its raw double layer, put \((\Phi |_{(a,b)}=\delta _{a,b}\) and \(|\rho )_{(a,b)}=\rho _{ba}\). For all physical indices \(i,j,k,l\),

These are the boundary removals used in the six-tensor argument of [ 197 , lines 1164–1254 ] .

Let \(\mathcal U\) be an MPU with a canonical-form-II presentation and recorded fixed matrix \(\rho \). Assume that \(\mathcal U^\dagger \) is simple. Let \(W_\dagger \) be the raw double layer of \(\mathcal U^\dagger \), put \((\Phi |_{(a,b)}=\delta _{a,b}\) and \(|\rho )_{(a,b)}=\rho _{ba}\). For all physical indices \(i,j,k,l\),

The simplicity assumption is precisely the additional hypothesis used for the reflected reduction in [ 197 , lines 1221–1254 ] .

Let \(\mathcal U\) have a canonical-form-II presentation with recorded fixed matrix \(\rho \), and assume that both \(\mathcal U\) and \(\mathcal U^\dagger \) are simple. Construct \(s\) from this same \(\rho \) by (277). For all physical indices \(i,j,k,l\),

This is the equality obtained from the two reductions in [ 197 , lines 1164–1254 ] .

Let \(\mathcal U\) have a canonical-form-II presentation with recorded fixed matrix \(\rho \), and suppose that both \(\mathcal U\) and \(\mathcal U^\dagger \) are simple. Then the tensor \(s\) constructed from this \(\rho \) satisfies

for a single scalar \(\delta \in \mathbb {C}\) and every physical pair \(i,j\). This is the proportionality conclusion in [ 197 , lines 1255–1325 ] .

The scalar \(\omega (g,h,k)\) exists, is nonzero, and is the only scalar with the dressed identity of Definition 19.5.1.3.

For every \(g,h,k,l\in G\),

Let \(\omega \) and \(\omega '\) come from two choices of fusion tensors, and let \(\beta _{g,h}\in \mathbb {C}^\times \) be the scalar with \(F^{\lt}_{g,h}B^{\mathbf w}=\beta _{g,h}F'^{\lt}_{g,h}B^{\mathbf w}\) for all long words of \(B=O_gO_h\). Then

so the cohomology class of \(\omega \) does not depend on the fusion tensors, as in (262).

For every word \(\sigma \) of length \(N\),

the sum running over words \(\tau \) of length \(N\). Consequently \(|V^{(N)}(T\cdot B)\rangle =\rho ^{(N)}(T)|V^{(N)}(B)\rangle \) for every \(N\).

If \(A\) in Definition 19.5.3.1 is normal of positive bond dimension, then action tensors for \(A\) exist [ 199 , lines 1064–1090 ] , and for every choice of them \(\alpha _{g,h}B^{\mathbf w}=L_{g,h}\, \varphi _{g,h}B^{\mathbf w}\) for all long words \(\mathbf w\) of \(B=(O_gO_h)\cdot A\).

In the setting of Theorem 19.5.3.2, for all \(g,h,k\in G\),

This is (417) for a one-point set of blocks.

Let \(\{ O_g\} \) be matrix product operator tensors with normal doubled-index tensors and \(O_gO_h=O_{gh}\) on every nonempty chain. If a normal matrix product state tensor of positive bond dimension satisfies \(O_g|V^{(N)}(A)\rangle =|V^{(N)}(A)\rangle \) for every \(g\) and every \(N\geq 1\), then the anomaly three-cocycle has trivial gauge class. Equivalently, a group with nontrivial anomaly has no invariant normal matrix product state. This is the statement that an anomalous representation cannot have an invariant injective matrix product state [ 197 , line 1545 ] , [ 199 , line 738 ] , with normal in place of injective and without unitarity or simplicity of the operators.