Let \(\sigma \) be the positive definite fixed point of \(\mathcal{E}_A^*\) and pass to the similar tensor \(B=\sigma ^{1/2}A\sigma ^{-1/2}\) of [ 191 , eq. unital, line 316 ] , whose map is trace preserving and still irreducible of spectral radius one. Its adjoint map is unital, and the positive definite Perron fixed point of the map itself is the positive definite fixed point required of the adjoint of that unital map. Write \(B^\dagger \) for the family with matrices \((B^i)^\dagger \), so that \(\mathcal{E}_{B^\dagger }=\mathcal{E}_B^*\). By [ 244 , Theorem 6.6 ] , there are a positive integer \(m\) and a primitive \(m\)-th root of unity \(\gamma \) such that

Passing back from the adjoint conjugates the peripheral spectrum. Since complex conjugation preserves the set of \(m\)-th roots of unity,

Finally, similarity gives \(\operatorname {Spec}_{\mathrm{per}}(\mathcal{E}_A)=\operatorname {Spec}_{\mathrm{per}}(\mathcal{E}_B)\).

In bond dimension zero the matrix space and its endomorphism algebra are zero-dimensional, so the transfer map has spectral radius zero. This contradicts the spectral-radius-one hypothesis.

Conjugation by \(X\mapsto CXC^\dagger \) gives a linear equivalence of the matrix space. Transporting eigenvectors through this equivalence preserves every eigenvalue and hence the unit-circle eigenvalue set.

The transfer map \(\mathcal{E}_A\) is positive and trace-preserving on a nonzero matrix space, so \(\rho _{\operatorname{spec}}(\mathcal{E}_A)=1\). By Theorem 22.1.2, \(\operatorname {Spec}_{\mathrm{per}}(\mathcal{E}_A)=\{ \mu \in \mathbb {C}:\mu ^m=1\} \) for some \(m\ge 1\), which together with irreducibility and left-canonicity is the definition of an \(m\)-periodic tensor.

Apply Theorem 8.2.2. The transfer map of \(B\) is similar to that of \(A\), so Lemma 22.1.4 preserves the prescribed set of \(m\)-th roots of unity.

Repeat \(X_j\) on every copy indexed by a diagonal entry of \(R_j\), and take their block-diagonal direct sum. On the \((j,q)\) block, conjugation by this matrix sends \(\mu _{j,q}A_j^i\) to \(\mu _{j,q}B_j^i\).

Apply the positive Perron gauge to each representative and combine them via Theorem 22.1.9. Since every spectral radius is already one, no representative is rescaled.

The prescribed peripheral spectrum contains the eigenvalue \(1\). In bond dimension zero the matrix space is zero-dimensional and admits no eigenvector, which is a contradiction.

The bond dimension of \(A\) is positive. A left-canonical tensor on a bond space of positive dimension has positive physical dimension.

This is the unitary freedom of Kraus representations from [ 244 , Theorem 2.1(4) ] .

Expand \(\sum _i(B^i)^\dagger B^i\), where \(B^i=\sum _j u_{ij}A^j\), swap sums, apply orthogonality \(u^\dagger u=\mathbb {1}\), and collapse to \(\sum _j(A^j)^\dagger A^j=\mathbb {1}\).

Contrapositive: any nontrivial invariant projection \(P\) for the rotated tensor yields \((\mathbb {1}-P)A^kP=0\) for all \(k\) via the orthogonality of \(u\), contradicting irreducibility of \(A\).

This combines transfer-map invariance (Theorem 22.1.16), left-canonical preservation (Theorem 22.1.17), and irreducibility (Theorem 22.1.18).

Induction on the word length with a strengthened hypothesis carrying an arbitrary prefix matrix, reducing each step to the hypothesis via word concatenation.

Pointwise matrix-entry computation using linearity of the block-diagonal embedding and reindexing.

Each block remains periodic by Theorem 22.1.19, equality of MPV families transfers via Theorem 22.1.20, and block-diagonal compatibility follows from Theorem 22.1.21.

Apply the assumed periodic equal-case theorem to \(A\) and \(B\). Its \(Z\)-gauge conclusion supplies \(m\), \(Y\), and \(Z\) with the three displayed identities by Definition 22.1.1.7.

Write the \(\mathbb {Z}_m\)-gauge relation as \(ZA^i=YB^iY^{-1}\) with \(Z^m=\mathbb {1}\) and \([A^i,Z]=0\). For a blocked letter, commute one factor of \(Z\) past each physical letter in the associated length-\(m\) word. The total prefactor becomes \(Z^m=\mathbb {1}\), so the entire blocked word is related by the same ordinary gauge \(Y\).

Apply Theorem 22.1.1.9 and then the gauge invariance of MPV families.

Because each \(p_i\) divides \(L = \operatorname{lcm}_j(p_j)\), Theorem D.4.5 implies that the transfer map \(\mathcal{E}_{B_i^{[L]}}\) is primitive for every \(i \in I\).

Theorem 22.1.2 yields a positive period \(m_j\) for each block \(A_j\) and \(n_k\) for each block \(B_k\). Applying Theorem 22.2.2.3 with these derived periods gives every conclusion.

Normalize each representative by its positive-definite adjoint Perron fixed point, which is the source construction at [ 191 , lines 313–332 ] . The normalized representatives are periodic with trace-preserving block maps and are blockwise similar to the originals, so the multiplicities and their power sums are unchanged, the non-repetition of the two bases is preserved, and the two direct-sum tensors generate the same matrix-product vectors as before. Theorem 22.2.2.14 applies to the normalized pair.

Transport the conclusion back. The bijection, the scalars, the periods and the blockwise multiplicity gauges concern only the multiplicity data, which the normalization leaves untouched, and the repeated-block relations compose with the two blockwise similarities. The multiplicity gauge is unchanged: writing \(X=\bigoplus _jX_j\) for the normalizing similarity, Lemma 22.2.2.11 gives

Only the similarity between the two tensors is conjugated, by the two normalizing similarities.

Take \(Z=\operatorname{diag}(\mu _i/\nu _i)_{i\in I}\) and apply the two diagonal identities from Lemmas 22.1.1.5 and 22.1.1.6.

Apply Theorem 22.2.2.4 to obtain \(Z\), and apply the Newton–Girard power-sum identity to recover the multiplicity multiset.

The entries \(\mu _{j,q}^{m}\) are again nonzero, so they form a sector weight family over the same basis, and the displayed identity is the exponent rule.

Raise all multiplicity entries of both decompositions to the power \(m\). By Lemma 22.2.2.6 the hypothesis becomes a comparison of the coefficients of the two resulting decompositions at every exponent beyond the threshold, with scalar \(\zeta ^{m}\). Applying Lemma 8.7.1.2 to those decompositions gives the multiplicity equality and the bijection, and its conclusion is the displayed identity.

Expand both vectors along the basis \(\{ A_j\} _{j\in J}\) by Lemma 8.6.1.4, rewriting each \(|V^{(N)}(B_{\pi (j)})\rangle \) as \(\zeta _j^{-N}|V^{(N)}(A_j)\rangle \). Equality of the two vectors makes the combination of the \(|V^{(N)}(A_j)\rangle \) with coefficients \(\operatorname{tr}(R_j^N)-\zeta _j^{-N}\operatorname{tr}(S_{\pi (j)}^N)\) vanish. By Theorem 7.4 the term for an index with \(m_j\nmid N\) contributes the zero vector, so the combination runs over the indices with \(m_j\mid N\), and by Theorem 22.2.3.45 those vectors are independent for every large \(N\). Each coefficient therefore vanishes.

Fix \(j\). Theorem 22.2.2.9 compares the power sums of \(R_j\) and \(S_{\pi (j)}\) at every multiple of \(m_j\) beyond \(N_0\), with scalar \(\zeta _j^{-1}\). Feeding that progression to Lemma 22.2.2.7 gives the equality of sizes, the reordering \(\tau \), and the entry identity \(\nu _{\pi (j),\tau (q)}^{m_j}=(\zeta _j\mu _{j,q})^{m_j}\). Since the entries \(\zeta _j\mu _{j,q}\) are nonzero, Theorem 22.2.2.4 produces the diagonal \(Z_j\) with the two displayed properties.

On the \(k\)-th block the two factors are \(z_k\mathbb {1}_{D_k}\) and \(X_k\), and a scalar matrix is central.

The multiplicity gauge acts by the scalar \(z_{j,l}\) on the bond space of the copy \((j,l)\), so it commutes with every block-diagonal matrix and its \(L\)-th power is the identity. Multiplying \(A^i\) by it rescales the multiplicity entries by the \(z_{j,l}\), and the hypothesis on the \(z_{j,l}\) makes the rescaled entries equal to the matched entries of \(B\). The matched blocks are therefore related copy by copy, so the two direct-sum tensors are related by the permutation of the matched copies followed by the direct sum of the blockwise similarities. If every \(Y_j\) is unitary, then, since each \(T_j\) is a permutation matrix,

and likewise \(Y^\dagger Y=\mathbb {1}\). For the matrix-product vectors, a copy whose period divides the length contributes the factor \(z_{j,l}^N=1\), and a copy whose period does not divide the length contributes the zero vector by Theorem 7.4.

Repeated blocks have the same peripheral transfer spectrum,

and for a periodic tensor that set is the set of \(m\)-th roots of unity, so the two periods agree.

Equality of the generated matrix-product vectors is in particular proportionality, so Theorem 22.2.3.58 supplies non-decaying partners and Theorem 22.2.3.61 matches the two bases, each matched pair carrying a similarity with a unit-modulus scalar \(\xi _j\). Since the blocks are left canonical and irreducible, Lemma 9.4.1.2 replaces each similarity by a unitary \(U_j\) with the same scalar:

The matched periods agree by Theorem 22.2.2.13. That same \(\xi _j\) relates the matrix-product vectors of the matched pair at every length, so Theorem 22.2.2.9 compares the multiplicity power sums at every large multiple of \(m_j\), and Lemma 22.2.2.7 turns that comparison into the reordering \(T_j\) and the entry identities \(\nu _{\pi (j),\tau _j(l)}^{m_j}=(\xi _j\mu _{j,l})^{m_j}\). The ratios \(z_{j,l}=\nu _{\pi (j),\tau _j(l)}/(\xi _j\mu _{j,l})\) are therefore \(m_j\)-th roots of unity; they are collected into the diagonal \(Z_j\), and Theorem 22.2.2.12 constructs the global \(Z\) and the unitary similarity \(Y\).

The matched multiplicities satisfy \(z_{j,l}\xi _j\mu _{j,l}=\nu _{\pi (j),\tau _j(l)}\). Since \(z_{j,l}^{m_j}=1\) with \(m_j{\gt}0\) and \(|\xi _j|=1\), taking absolute values gives \(|\mu _{j,l}|=|\nu _{\pi (j),\tau _j(l)}|\), and the hypothesis \(|\nu _{\pi (j),\tau _j(l)}|=1\) gives \(|\mu _{j,l}|=1\).

For each sector choose the coordinate functional at the nonzero reference entry and normalize it so that it takes the value \(1\) on \(B_v^{i_v}\). Applying the tensor product of these functionals to the reference tensor identity gives \(\prod _v\kappa _v=z\). To prove the proportionality in sector \(v\), replace only the \(v\)-th functional by the coordinate functional at an arbitrary entry \((r,c)\) and apply the tensor identity with \(\sigma _v=i\) and all other letters fixed at their references. The product over the other sectors is nonzero because its product with \(\kappa _v\) is \(z\), so cancellation gives \((A_v^i)_{r,c}=(\kappa _v B_v^i)_{r,c}\) for every entry.

Apply Theorem 22.2.3.1 to obtain scalars \(\mu _v\) with \(\prod _v\mu _v=z\) and \(A_v^i=\mu _vB_v^i\). Since \(m{\gt}0\), the identities \(\xi ^m=z\) and \(z\ne 0\) imply \(\xi \ne 0\). Put \(\kappa _v=\mu _v/\xi \). Then \(\prod _v\kappa _v=(\prod _v\mu _v)/\xi ^m=z/z=1\), and \(A_v^i=(\kappa _v\xi )B_v^i\).

Left-canonicity of \(cA\) gives

where the last equality uses left-canonicity of \(A\). Since the bond dimension is positive, comparison of any diagonal entry gives \(|c|^2=1\), hence \(|c|=1\).

By Theorem 22.2.3.3, applied to \(B\) and the scalar \(\kappa \xi \), one has \(|\kappa \xi |=1\). Since \(|\xi |=1\), the multiplicativity of the complex norm gives \(|\kappa |=1\).

Put \(K_i=(A^i)^\dagger \) and let \(\Phi _K\) be the Kraus map of the family \(K\). Since \(A\) is left-canonical, \(\Phi _K\) is unital. The projection identities and the shift relation give

because both sides are equal to \(P_k\). The equality case in the Kadison–Schwarz inequality therefore gives \(P_{k+1}K_i^\dagger =K_i^\dagger \Phi _K(P_{k+1})\). Substituting \(K_i^\dagger =A^i\) and \(\Phi _K(P_{k+1})=P_k\) gives \(P_{k+1}A^i=A^iP_k\).

Insert \(\sum _u P_u=\mathbb {1}\) on the right of \(A^i\), then use Theorem 22.2.3.6 and \(P_u^2=P_u\) term by term.

Induct on the length of \(w\), using Theorem 22.2.3.6 for the first letter and the induction hypothesis for the remaining word.

Induct on the word \(\mathbf i\). For the empty word the identity is \(P_uF_u^{\mathbf j}=F_u^{\mathbf j}\), which follows from the idempotent part of the orthogonal-projection hypothesis. For a nonempty word, remove the first letter and apply the induction hypothesis at the next cyclic sector.

Induct on \(k\). The base case \(k=1\) is the hypothesis. For the inductive step split \(W\) into its first length-\(m\) block and the remaining length-\(km\) word, apply the concatenation law, and use that the junction shift \(m\cdot 1=0\) returns to the base sector before applying the block identity and the induction hypothesis.

Expand each \(A_k^{\sigma _k}X_k =\sum _{\mathbf j}(\Omega _{k+1}(X_k))_{\mathbf j} A_k^{\sigma _k}F_{k+1}^{\mathbf j}\) using the right inverse, apply the noncommutative distributivity identity

and pull the scalar coefficients \(\prod _k(\Omega _{k+1}(X_k))_{\rho _k}\) out of each ordered product.

Apply the grading at sector \(-(k+1)\) and simplify the cyclic index.

Induct on the word. The head junction \(P_uA^iP_{u+1}=P_uA^i\) collapses by the shift and idempotency; the induction hypothesis handles the tail, and the accumulated offset realigns.

A blocked letter is a word of length exactly \(m\); the accumulated offset \(m\cdot 1=0\) vanishes, so the telescoping gives the diagonal corner.

The decoding bijection identifies a word-indexing function with its ordered list of letters, and the iterated blocking bijection concatenates the decoded length-\(m\) blocks in order.

Induct on \(W\). The head blocked letter collapses by the blocked-letter diagonal corner to \(\varphi _k(\bar A_k^I)\), and multiplicativity of \(\varphi \) combines the letters:

The empty word is excluded so the induction stays inside the corner.

Set \(\hat\Omega _k(Y)=\Omega _k(\varphi _k^{-1}(Y))\) for corner-supported \(Y\). Applying \(\varphi _k\) to the sector recovery and using the corner product of a flattened word,

Apply the right-inverse contraction with the ambient corner right inverse in place of the abstract right inverse.

Choose the projection family contained in the cyclic sector decomposition. It satisfies

Apply Theorem 22.2.3.6.

Choose the projection family contained in the cyclic sector decomposition. It satisfies

Apply Theorem 22.2.3.7.

This is the cyclic-sector existence part of Lemma bdcf, translated into the inverse indexing convention of Definition 22.2.3.5.

Let \(U\) diagonalize \(P_k\), so that

Write \(B^i=U^\dagger (A^{(m)})^iU\), and let \(B_{kk}^i\) be the corner block on the support of \(P_k\). The sector tensor is \(C_k^i=V_k^\dagger B_{kk}^iV_k\), where \(V_k\) identifies the support with \(\mathbb {C}^{\dim C_k}\). The linear corner isomorphism is the inverse transport

Hence

Choose the projection families \(P_u\) and \(Q_v\) from the two cyclic sector decompositions. The single-site relations \(P_{u+1}A^i=A^iP_u\) and \(Q_{v+1}B^i=B^iQ_v\) give the following trace-sum identity, where \(T\) is the one-site rotation of physical words:

The first and last equalities in (2) are the cyclic-sector trace formula after reindexing blocked words as physical words of length \(Nm\).

Since the peripheral spectrum of \(\mathcal{E}_A\) consists of \(m\)-th roots of unity, the corresponding lemma in the companion quantum-channel volume  [ 216 , “Powering collapses peripheral eigenvalues” ] gives \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_A^m)=\{ 1\} \), hence \(\zeta =1\). The cyclic trace gives \(\operatorname{tr}U=\operatorname{tr}(P_uUP_v)=\operatorname{tr}(P_vP_uU)=0\).

Put \(W=\sum _{t=0}^{m-1}\mathcal{E}_A^t(U)\). The equation \(\mathcal{E}_A^m(U)=U\) gives \(\mathcal{E}_A(W)=W\), and trace preservation gives \(\operatorname{tr}W=0\). Irreducibility of \(A\) therefore gives \(W=0\). The shift relation \(P_{k+1}A^i=A^iP_k\) moves \(\mathcal{E}_A^t(U)\) into the \((u+t,v+t)\) corner, so only the \(t=0\) term contributes after multiplying by \(P_u\) and \(P_v\). Thus \(U=P_uWP_v=0\).

If \(D_A\ne D_B\), use Theorem 22.2.3.41. Assume \(D_A=D_B\) and \(A\sim _{\mathrm{gp}}B\), say \(B^i=\zeta XA^iX^{-1}\) with \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \ne 0\). The transfer maps satisfy

With \(\mathcal{E}_A(\rho _A)=\rho _A\), put \(\sigma =X\rho _AX^*\). Then \(\mathcal{E}_B(\sigma )=|\zeta |^2\sigma \), while \(\mathcal{E}_B(\rho _B)=\rho _B\) for a nonzero positive fixed point \(\rho _B\). Perron–Frobenius eigenvalue uniqueness [ 244 , Theorem 6.3 ] gives \(|\zeta |^2=1\). Hence \(\mathcal{E}_B\) is conjugate to \(\mathcal{E}_A\), so \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_A)=\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_B)\). Since

equality of the two finite sets in (3) gives \(m_A=m_B\), a contradiction. Thus \(A\not\sim _{\mathrm{gp}}B\), and the irreducible trace-preserving overlap dichotomy gives \(\lim _{N\to \infty }O_{AB}(N)=0\).

A hypothetical gauge-phase equivalence produces a nonzero peripheral eigenvector of the blocked transfer map supported between distinct cyclic corners. By Lemma 22.2.3.28, periodicity of the original irreducible tensor forces this vector to vanish. Orthogonality of the corners alone is insufficient.

Write \(\bar A\sim \bigoplus _{u\in \mathbb {Z}_m}A_u\) and \(\bar B\sim \bigoplus _{v\in \mathbb {Z}_m}B_v\). For every \(N\),

For each sector pair \((u,v)\),

The finite double sum (4) therefore gives

If \(A\sim _{\mathrm{gp}}B\), then \(\bar A\sim _{\mathrm{gp}}\bar B\), so for some \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \ne 0\), \((\bar B)^\alpha =\zeta X(\bar A)^\alpha X^{-1}\). Thus \(V^{(N)}(\bar B)_\sigma =\zeta ^N V^{(N)}(\bar A)_\sigma \), and

Since \(\lim _{N\to \infty }O_{\bar A\bar A}(N) =\lim _{N\to \infty }O_{\bar B\bar B}(N)=m{\gt}0\),

hence \(|\zeta |=1\). Therefore

contradicting \(\lim _{N\to \infty }O_{\bar A\bar B}(N)=0\). Thus \(A\not\sim _{\mathrm{gp}}B\), and the irreducible trace-preserving overlap dichotomy gives \(\lim _{N\to \infty }O_{AB}(N)=0\).

Set \(\varphi _k=(\kappa _0\cdots \kappa _{k-1})^{-1}\), with the empty product equal to \(1\). For \(k{\lt}n\),

Separating the final factor in the product-one hypothesis gives \((\kappa _0\cdots \kappa _{n-1})\kappa _n=1\), which gives

The corner relation \(U_v=P_{v-q}U_vQ_v\) gives \(P_uU_{u+q}Q_{u+q}=U_{u+q}\); reindexing the sum by \(v=u+q\) turns \(\sum _uP_uU_{u+q}Q_{u+q}\) into \(\sum _vU_v\).

Since \(|\zeta |=1\), the scalar is nonzero. Multiplication by \(X^{-1}\) on the left and by \(X\) on the right gives \(X^{-1}B^iX=\zeta A^i\); multiplication by \(\zeta ^{-1}\) proves the claim.

The corner relation \(U_v=P_{v-q}U_vQ_v\) gives \(P_uU_{u+q}Q_{u+q}=U_{u+q}\), so \(U=\sum _vU_v\) after reindexing. Distinct projections are orthogonal, \(Q_vQ_w=0\) and \(P_{v-q}P_{w-q}=0\) for \(v\ne w\), which eliminates the cross terms in \(UU^\dagger \) and \(U^\dagger U\); the diagonal terms are \(U_vU_v^\dagger =P_{v-q}\) and \(U_v^\dagger U_v=Q_v\), so

The off-diagonal structure of \(B\) makes \(Q_vB^iQ_w\) vanish unless \(w=v+1\), so \(UB^iU^\dagger =\sum _vU_vQ_vB^iQ_{v+1}U_{v+1}^\dagger \). Summing the per-site proportionality over \(u\) and reindexing by \(v=u+q\) gives

The last equality is the off-diagonal structure of \(A\). The unit-modulus scalar and the unitary \(U\) exhibit \(A\) and \(B\) as repeated blocks.

If \(m_A\ne m_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\) by Theorem 22.2.3.29. Assume \(m_A=m_B=m\). If \(D_A\ne D_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\) by Theorem 22.2.3.41. Assume \(D_A=D_B\).

Choose period-blocked sector decompositions

If no pair \((u,v)\) satisfies \(d_u=e_v\) and \(A_u\sim _{\mathrm{gp}}B_v\), then Theorem 22.2.3.32 gives \(\lim _{N\to \infty }O_{AB}(N)=0\).

Otherwise there are \(u_0,v_0\in \mathbb Z_m\) such that \(d_{u_0}=e_{v_0}\) and \(A_{u_0}\sim _{\mathrm{gp}}B_{v_0}\). Theorem 22.2.3.40 then gives repeated-block equivalence between \(A\) and \(B\).

The Gram matrix converges to the diagonal matrix whose \(j\)-th diagonal entry is the positive period of \(A_j\). Its determinant therefore has a nonzero limit, so the Gram matrix is invertible for every sufficiently large \(N\).

Periodicity makes every bond dimension positive. For distinct \(i,j\), the repeated-block alternative in Theorem 22.2.3.42 contradicts pairwise non-repetition, so the cross overlap tends to zero. Theorem 22.2.3.43 now gives the claimed independence along the common-period subsequence.

There are finitely many subfamilies. For each subfamily \(S\subseteq J\) apply Theorem 22.2.3.43 at the least common multiple \(p_S\) of the periods occurring in \(S\), the cross-overlap decay coming as before from Theorem 22.2.3.42 and pairwise non-repetition; this gives a threshold \(M_S\). Let \(L\) be the least common multiple of all the periods and take \(N_0\) to be the largest of the numbers \(L\, M_S\). Given \(N\ge N_0\), let \(S\) be the set of indices whose period divides \(N\). Every period occurring in \(S\) divides \(N\), so \(p_S\mid N\); writing \(N=p_Sn\) and using \(p_S\le L\) gives \(L\, M_S\le N\le Ln\), hence \(n\ge M_S\) and the subfamily is independent at the length \(N=p_Sn\).

The sector decomposition gives

Terms whose state vector vanishes contribute zero. Every remaining term belongs to the span of the nonzero state vectors, as does their sum.

By Theorem 22.2.3.42, either the overlap \(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \) tends to \(0\) or the bond dimensions agree and \(A\) and \(B\) are repeated blocks. In the decay case, equality of the MPV families gives, for every \(N{\gt}0\),

Along multiples of the period \(m_a\), (7) would imply

contradicting \(m_a{\gt}0\).

Invert the pure gauge and take the repeated-block phase to be one.

Write \(A^i=\zeta YB^iY^{-1}\) with \(|\zeta |=1\). A closed chain of length \(N\) is the trace of a product of \(N\) such factors. The neighbouring similarities cancel and the trace is invariant under the outer conjugation, while each site contributes one factor \(\zeta \).

Take \(\xi \) to be the reciprocal of the phase from Theorem 22.2.3.50, which is nonzero because it has modulus one. Rescaling every matrix of a block by \(\xi \) multiplies its length-\(N\) vector by \(\xi ^{N}\), and the two powers cancel.

Scaling every tensor letter by \(c\) multiplies the transfer map by \(c\overline c=|c|^2=1\).

Write \(A^i=\xi YB^iY^{-1}\) with \(|\xi |=1\). The transfer map of \(\xi C\) is \(|\xi |^2\) times the transfer map of \(C\), hence equals it, and the transfer map of \(YB^iY^{-1}\) is the congruence conjugate \(Y\mathcal{E}_B(Y^{-1}\cdot Y^{-\dagger })Y^{\dagger }\) of the transfer map of \(B\). Conjugation by an invertible linear map leaves the spectrum unchanged.

A primitive \(m\)-th root belongs to the second set, so \(m\) divides \(n\). Interchanging \(m\) and \(n\) gives the reverse divisibility.

The two spectra are respectively the sets of \(m\)-th and \(n\)-th roots of unity, so Theorem 22.2.3.54 applies.

Repeated blocks have the same peripheral spectrum by Theorem 22.2.3.53. Theorem 22.2.3.55 then identifies their periods.

Cyclicity of the trace makes every closed-chain coefficient, and hence every mixed overlap, invariant under the two similarities. Thus decay and non-decay are unchanged. For a non-decaying pair, compose its primed repeated-block relation with the two pure similarities.

Fix \(j_0\in J\) and suppose that its overlap with every \(B_k\) tends to zero. Let \(T\subseteq J\) be the set of all indices with this property. For every \(j\notin T\), choose \(k(j)\) for which the mixed overlap does not decay. The periodic overlap dichotomy identifies \(A_j\) with \(B_{k(j)}\) up to a gauge transformation and a phase; consequently their length-\(N\) vectors are scalar multiples of one another.

Choose a common positive multiple \(p\) of all periods. Along the lengths \(pN\), define the coproduct-indexed family \(w_N\colon T\amalg K\to (\mathbb C^d)^{\otimes pN}\) by \(w_N(\operatorname {inl}(j))=|V^{(pN)}(A_j)\rangle \) and \(w_N(\operatorname {inr}(k))=|V^{(pN)}(B_k)\rangle \). This family is eventually linearly independent. Indeed, distinct vectors within either summand have decaying overlaps by non-repetition and the overlap dichotomy, while the mixed overlaps vanish by the definition of \(T\).

Expand the proportionality relation at length \(pN\) using the grouped coefficients

Replace each term indexed by \(j\notin T\) by its scalar multiple of the corresponding \(B_{k(j)}\) term, and collect terms with the same value of \(k(j)\). Linear independence forces \(\operatorname {tr}(R_{j_0}^{pN})=0\) for every sufficiently large \(N\). Since each \(\mu _{j_0,q}^p\) is nonzero, geometric extrapolation gives the same identity at exponent zero. This says \(r_{j_0}=0\), contradicting the positive size of \(R_{j_0}\). Hence \(j_0\) has a non-decaying partner. Interchanging \(A\) and \(B\), inverting the nonzero scalars \(c_N\), and using conjugate symmetry of the mixed overlap gives the converse assertion. For any non-decaying pair, the periodic overlap dichotomy excludes its decay alternative and supplies the repeated-block relation.

Normalize all representatives by their pure Perron gauges and combine the gauges block diagonally. The multiplicity matrices and their grouped coefficients remain unchanged. Pure similarity preserves the total positive-length proportionality and pairwise non-repetition, so Theorem 22.2.3.58 applies to the normalized families. Finally apply Lemma 22.2.3.57.

Repeat the argument of Theorem 22.2.3.58 with the singleton coefficients \(\mu _j^N\) and \(\nu _k^N\). Their nonvanishing at every length replaces the geometric-extrapolation step. Equality with the two scalar-weight direct sums transports the positive-length proportionality relation to the corresponding block-diagonal tensors.

Choose a non-decaying partner \(f(j)\in K\) for every \(j\in J\). If \(f(j_1)=f(j_2)\), the two repeated-block relations compose to show that \(A_{j_1}\) and \(A_{j_2}\) are repeated, so pairwise non-repetition makes \(f\) injective. The symmetric construction gives an injection \(K\to J\). Therefore \(|J|=|K|\), and the first injection is the required bijection.

Theorem 22.1.2 produces a positive period for each block of either family, and Theorem 22.2.3.63 applied to those periods gives the matching.

Theorem 22.2.3.59 supplies non-decaying partners in both directions and classifies every such pair as repeated. Applying Theorem 22.2.3.61 gives a bijection \(\pi \colon J\to K\) of repeated pairs. Finally, Theorem 22.2.3.56 gives \(m_j=n_{\pi (j)}\) for every matched pair.

Regard each tensor as a one-block decomposition with one copy of weight one. Theorem 22.2.3.63 matches the two blocks. A left-canonical tensor defines a trace-preserving completely positive map, whose spectral radius is one.

Theorem 22.2.3.64 makes each member repeated with its representative, giving equal dimensions and periods. Lemma 9.4.1.2 replaces the similarity by a unitary matrix while keeping its phase. Repeated representatives would have the same matrix-product-vector phase class, contrary to their choice.

Let \(\Pi \) be the permutation matrix carrying the summand \(t=e(k)\) of \(\bigoplus _t\) to the summand \(k\) of \(\bigoplus _k\), and set \(Y=\Pi \bigl(\bigoplus _tX_t\bigr)\) and \(Y'=\bigl(\bigoplus _tX_t^{-1}\bigr)\Pi ^\dagger \). Then \(YY'=Y'Y=\mathbb {1}\), and

For unitary \(X_t\), \(X_t^{-1}=X_t^\dagger \), so \(Y'=Y^\dagger \). Multiplying the blockwise relation for \(k\) by \(c_k\) gives the same relation for the weights \(c_k\mu _k\) and \(c_{e^{-1}(t)}\mu '_t\), with the same \(X_t\), hence the same \(Y\) and \(Y'\).

Choose the phases and unitaries of Theorem 22.2.3.65. Multiplying (8) by \(\mu _{e(j,q)}\) and using Definition 22.2.3.67 gives

The enumeration is a bijection between grouped copies and original blocks. Let \(\Pi \) be the permutation matrix sending the copy \((j,q)\) to the block \(e(j,q)\), and set \(Y=\Pi \bigl(\bigoplus _{j,q}U_{j,q}\bigr)\). Then

and likewise \(Y^\dagger Y=\mathbb {1}\). Replacing \(\mu _{e(j,q)}\) by \(c_{j,q}\mu _{e(j,q)}\) and \(\mu '_{j,q}\) by \(c_{j,q}\mu '_{j,q}\) leaves the blockwise relations and \(Y\) unchanged; since \(Z_cP^i\) is the block sum with weights \(c_{j,q}\mu '_{j,q}\), this gives the transport of \(Z_c\). The same bijection and the equality of matched block dimensions give equality of total dimensions. The phases have modulus one, so the individual weight moduli are unchanged. For \(N{\gt}0\), the factors \(Y^\dagger Y=I\) between adjacent letters cancel, and cyclicity of the trace removes the outer \(Y\) and \(Y^\dagger \):

Group the blocks of \(T\) as in Theorem 22.2.3.68. For every \(N{\gt}0\), the resulting decomposition \(P\) satisfies \(V^{(N)}(P)=V^{(N)}(T)=V^{(N)}(Q)\). Theorem 22.2.2.15 matches the multiplicities of \(P\) and \(Q\) up to unit-modulus factors. Hence, for every grouped copy,

The first equality is the modulus statement of Theorem 22.2.3.68. Since \(e\) enumerates every original block, \(|\mu _k|=1\) for every \(k\).

Theorem 22.2.3.68 groups both sums into multiplicity forms \(P\) and \(Q\) over bases of periodic tensors without repetition, with unitaries \(Y_A\) and \(Y_B\). The two forms generate the same vectors at every positive length, so Theorem 22.2.2.14 gives multiplicity scalars \(z_{j,q}\) with \(z_{j,q}^{m_{\rho _j}}=1\) and a unitary \(Y\) with \(Z_zP^i=YQ^iY^\dagger \). Transporting \(Z_z\) back with \(Y_A\) as in Theorem 22.2.3.68 gives the scalars \(c_{e(j,q)}=z_{j,q}\) and \(W=Y_AYY_B^\dagger \). Each original block has the period of its representative, so \(c_k^{m_k}=1\).

Theorem 22.1.2 gives the periods \(m_j\). Mixing the physical index commutes with the direct sum: \(\sum _iu^{i',i}A^i=\bigoplus _j\mu _jC_j^{i'}\) with \(C_j^{i'}=\sum _iu^{i',i}A_j^i\). Since \(u\) is unitary, \(\mathcal{E}_{C_j}=\mathcal{E}_{A_j}\), so \(C_j\) is periodic with the same period. By hypothesis \(\bigoplus _j\mu _jC_j\) and \(A\) generate the same vectors at every positive length, so Theorem 22.3.1 gives \(c_j\) with \(c_j^{m_j}=1\) and a unitary \(U\) with \(\bigoplus _jc_j\mu _jC_j^{i'}=UA^{i'}U^\dagger \). With \(z_j=c_j^{-1}\) and \(Z=\bigoplus _jz_j\mathbb {1}\) this is \(\sum _iu^{i',i}A^i=ZUA^{i'}U^\dagger \). The matrix \(Z\) is diagonal with entries of modulus one and commutes with every \(A^i\). Finally, \(V_N(ZA)=\sum _jz_j^N\mu _j^NV_N(A_j)\): if \(m_j\mid N\) then \(z_j^N=1\), and otherwise \(V_N(A_j)=0\) (Theorem 7.4). Hence \(\mathcal{V}(A)=\mathcal{V}(ZA)\).

The twisted tensor \(\widetilde{A}_g\) coincides with the physical-index rotation \(A_{U(g)}\) of Definition G.1.1.1. By Theorem 22.1.22 the rotated tensor remains in irreducible form II, and the on-site symmetry hypothesis supplies the equality \(\mathcal{V}(A) = \mathcal{V}(A_{U(g)})\). Apply the periodic equal-case Fundamental Theorem (Theorem 22.2.2.1) to obtain the asserted \(\mathbb {Z}_{m_g}\)-gauge equivalence.

By Definition 22.3.4, let \((Y, u)\) be the virtual gauge family and factor system. Set \(X(g) := Y(g^{-1})\) and \(\omega (g, h) := u(h^{-1}, g^{-1})\). The projective multiplication law \(X(g)X(h) = \omega (g,h)X(gh)\) is then exactly the \((h^{-1}, g^{-1})\)-instance of the factor-system identity, using \((gh)^{-1} = h^{-1}g^{-1}\). The intertwining relation between \(A\) and \(\widetilde{A}_g\) transports unchanged.

Associativity of matrix multiplication applied to \(X(g)X(h)X(k)\), combined with the projective multiplication law, gives

Since \((gh)k = g(hk)\), both sides of (10) are scalar multiples of the same matrix \(X((gh)k)\). Since \(X((gh)k) \in \mathrm{GL}_D(\mathbb {C})\) and \(D{\gt}0\), the matrix \(X((gh)k)\) is nonzero. Cancelling \(X((gh)k)\) from both sides of (10) yields \(\omega (g,h)\omega (gh,k) = \omega (g,hk)\omega (h,k)\) in \(\mathbb {C}^{\times }\).

Theorem 7.2.1.1 gives \(A\) the same positive-length matrix product vectors as \(\bigoplus _{k=1}^r\mu _kB_k\) with \(\mu _k{\gt}0\), \(\sum _kD_k\le D\), and each \(B_k\) irreducible and left-canonical, so \(\rho _{\operatorname{spec}}(\mathcal{E}_{B_k})=1\). By Theorem 22.1.6, each \(B_k\) is \(m_k\)-periodic, that is, \(\operatorname {Spec}_{\mathrm{per}}(\mathcal{E}_{B_k})=\{ \mu \in \mathbb {C}:\mu ^{m_k}=1\} \) for some \(m_k\ge 1\).

One-site blocking only permutes the physical labels. Use the associated permutation matrix for the refinement isometry. The channel assertion follows because the first power of a linear map is the map itself.

If two such expressions agree modulo \(m\), reduction modulo \(r\) identifies their representatives \(\alpha \). Cancelling that common representative gives \(pk=pk'\pmod m\), hence \(k=k'\pmod{m/r}\). Their bounds imply \(k=k'\). There are \(r(m/r)=m\) pairs of coordinates, so this injective map to the \(m\) residues is also surjective. Since \(m\) divides \(p(m/r)\), reducing \(k+1\) modulo \(m/r\) does not change \(\alpha +p(k+1)\) modulo \(m\).

The step coordinates partition the original projections, so the \(Q_\alpha \) are mutually orthogonal nonzero projections summing to \(\mathbb {1}_D\). Iterating \(P_uA^i=A^iP_{u+1}\) over a word of length \(p\) shifts the projector index by \(p\). Multiplying by the adjoint word and summing, left canonicality gives \(\mathcal{E}_{A^{[p]}}^\dagger (P_u)=P_{u+p}\). Iterating the shift relation gives \(P_u(A^{[p]})^I=(A^{[p]})^IP_{u+p}\). Since \(u\mapsto u+p\) permutes the orbit of \(\alpha \), summing over that orbit gives \(Q_\alpha (A^{[p]})^I=(A^{[p]})^IQ_\alpha \). Compress to the ranges of these projections. Their dimensions sum to \(D\), and the commuting projection resolution preserves both left canonicality and traces of every word. The empty-word trace gives the same dimension sum.

Take the cyclic spectral projections, with their indices negated to give \(P_uA^i=A^iP_{u+1}\). No projection vanishes: left canonicality and the shift relation give \(\mathcal{E}_A^\dagger (P_u)=P_{u+1}\), and \(\mathcal{E}_A^\dagger (0)=0\), so \(P_u=0\) would give \(P_{u+1}=0\) and hence, around the cycle, \(\mathbb {1}_D=\sum _vP_v=0\). Apply Theorem 22.4.7. For \(p{\gt}0\), Theorem 22.4.11 identifies each block’s peripheral spectrum, and Theorem 22.4.14 gives irreducibility. Together with left canonicality, these prove periodicity.

Spectral mapping for \(\mathcal{E}_A^p\) gives the first assertion: a complex number has unit modulus if and only if its positive \(p\)th power does. A primitive \(m\)th root raised to the \(p\)th power has order \(q\), giving (11). For compression, \((A^{[p]})^IV=VC^I\), so \(\mathcal{E}_{A^{[p]}}(VXV^\dagger )=V\mathcal{E}_C(X)V^\dagger \). The isometry makes \(X\mapsto VXV^\dagger \) injective; hence every eigenvalue of \(\mathcal{E}_C\) is an eigenvalue of \(\mathcal{E}_{A^{[p]}}\).

Orthogonality gives \(Q_\alpha P_u=P_uQ_\alpha =P_u\). Therefore \(V(V^\dagger P_uV)V^\dagger =P_u\). Expanding the compressed adjoint map yields \(\mathcal{E}_C^\dagger (Y)=V^\dagger \mathcal{E}_{A^{[p]}}^\dagger (VYV^\dagger )V\). Apply this identity to \(Y=V^\dagger P_uV\) and use \(\mathcal{E}_{A^{[p]}}^\dagger (P_u)=P_{u+p}\).

The ambient spectral bound gives one inclusion. For the other, put \(R_k=V^\dagger P_{[\alpha +pk]_m}V\). The compressed adjoint map satisfies \(\mathcal{E}_C^\dagger (R_k)=R_{k+1}\). If \(z^q=1\), the sum \(X_z=\sum _{k=0}^{q-1}z^kR_k\) therefore satisfies \(\mathcal{E}_C^\dagger (X_z)=z^{-1}X_z\). It is nonzero: since \(P_{[\alpha +pk]_m}\le Q_\alpha =VV^\dagger \) and distinct orbit projections are orthogonal,

Given \(\lambda ^q=1\), choose \(z=\overline{\lambda }^{-1}\). The adjoint has eigenvalue \(\overline{\lambda }\), so \(\mathcal{E}_C\) has eigenvalue \(\lambda \). Its modulus is one.

On \(S=\ker (F^p-1)\), the restriction of \(F\) is annihilated by the separable polynomial \(t^p-1\), and hence is diagonalizable. Each eigenvalue has modulus one and therefore is an \(m\)th root of unity. Thus \(F^m\) is the identity on \(S\). A point of periods \(m\) and \(p\) also has period \(r\); conversely, \(r\mid p\) gives the other implication.

The projections \(R_j=F^j(Q_\alpha )\), for \(0\le j{\lt}r\), are pairwise orthogonal and sum to \(\mathbb {1}_D\), since their cyclic-sector indices partition \(\mathbb {Z}_m\). The letter relation gives \(F(P_uY)=P_{u+1}F(Y)\), so \(R_jF^j(X)=F^j(X)\). By Theorem 22.4.12, \(F^r(X)=X\). Thus \(S=\sum _{j=0}^{r-1}F^j(X)\) is fixed by the irreducible unital map \(F\), and \(S=c\mathbb {1}_D\). Orthogonality gives \(Q_\alpha S=X=cQ_\alpha \).

Let \(R\) be an invariant orthogonal projection for \(\mathcal{E}_C^\dagger \). The projection \(L=VRV^\dagger \) has invariant corner under \(F^p\), where \(F=\mathcal{E}_A^\dagger \). The faithful invariant state of \(\mathcal{E}_A\) implies \(F^p(L)=L\). Since \(Q_\alpha L=L\), fixed-point rigidity gives \(L=cQ_\alpha \), and compression gives \(R=c\mathbb {1}\). Idempotence forces \(R=0\) or \(R=\mathbb {1}\). Since \(\mathcal{E}_C\) and \(\mathcal{E}_C^\dagger \) have the same invariant orthogonal projections, \(C\) is irreducible.

Put \(f_u=c_{\alpha _u}^{k_u}\) and \(b_u=f_{u+1}/f_u\). The coordinate shift and \(c_\alpha ^q=1\) give \(f_{u+p}=c_{\alpha _u}f_u\), including when \(k_u=q-1\). The product telescopes to \(f_{u+p}/f_u=c_{\alpha _u}\), proving (14).

Orthogonality gives \(P_u A'^i=c_uP_u A^i=c_uA^iP_{u+1}\). Induction on the word length gives the first identity in (15); summing it over \(u\) gives the second.

Put \(U=\sum _uc_uP_u\). Orthogonality gives \(U^\dagger U=\sum _u|c_u|^2P_u=\mathbb {1}\). Since \(A'^i=UA^i\), \(\sum _i(A'^i)^\dagger A'^i=\sum _i(A^i)^\dagger U^\dagger UA^i=\mathbb {1}\).

Since \(m/r{\gt}0\), each root \(c_\alpha \) has modulus one. Apply Theorem 22.4.15 in \(U(1)\) to obtain unit phases \(b_u\), and set \(A'^i=\sum _ub_uP_uA^i\). Theorem 22.4.18 gives left canonicality. Substituting (14) into (15) gives (16).

Choose \((A,W)\) from the refinement data. The definition of \(C\) gives \(\mathcal{E}_C = \mathcal{E}_B\) and equality of MPV families between \(C\) and \(A^{[p]}\). To see that \(C\) is still in irreducible form II, transport each periodic block \(B_j\) of \(B\) through the same physical isometry \(W\). The left-canonical normalization is preserved because \(W^\dagger W = \mathbb {1}\), while irreducibility and the peripheral spectrum are preserved because the transfer map of each transported block agrees with that of \(B_j\). Combining these transported blocks with the same weights gives an irreducible-form decomposition of \(C\).

Apply the \(Z\)-gauge extraction hypothesis to obtain a blocked \(\mathbb {Z}_m\)-gauge witness between \(C\) and \(A^{[p]}\), and then apply the construction of \(\widetilde A\) to that witness.

Apply Theorem 22.4.20 to the refinement witness of \(B\). This produces a blocked tensor \(C\) in irreducible form II with \(\mathcal{E}_C = \mathcal{E}_B\) and the same MPV family as \(A^{[p]}\). The blocked equal-case construction then gives a left-canonical tensor \(\widetilde A\) with \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\). Combining the two transfer-map identities yields \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\).

Combine Theorem 22.4.26 with the witness-level forward implication: once \(\mathcal{E}_B\) is identified with \(\mathcal{E}_{\widetilde A^{[p]}}\) for a left-canonical \(\widetilde A\), the blocking identity gives \(\mathcal{E}_B = (\mathcal{E}_{\widetilde A})^p\).

Combine the forward implication (Theorem 22.4.29) with the reverse implication (Theorem 22.4.35) into a single biconditional. The forward direction assumes the construction of \(\widetilde A\); the reverse direction assumes the transfer-map construction. Together they give the full equivalence under those hypotheses.

The construction supplies \(\widetilde A\) with \(\sum _i (\widetilde A^i)^\dagger \widetilde A^i = \mathbb {1}\) and \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\). The left-canonical property makes \(\mathcal{E}_{\widetilde A}\) a CPTP channel, and the blocking identity \(\mathcal{E}_{\widetilde A^{[p]}} = (\mathcal{E}_{\widetilde A})^p\) then identifies \(\mathcal{E}_B\) with the \(p\)-th power of the channel \(\mathcal{E}_{\widetilde A}\).

The block \((1)\) has identity transfer map and period one. Since \(A^{[2]}=B\), equality of the blocked and target transfer maps gives a refinement isometry. If \(\mathcal{E}_B=\mathcal F^2\) for a channel \(\mathcal F\), then \(\mathcal{E}_B\) preserves trace. This contradicts \(\operatorname{tr}\mathcal{E}_B(1)=16\ne 1\).

Choose a tensor \(A'\) whose matrix family realizes the root channel with at most \(d\) nonzero terms. Since this channel is trace-preserving, the matrices of \(A'\) satisfy \(\sum _i (A'^i)^\dagger A'^i = \mathbb {1}\). The identity \(\mathcal{E}_C = (\mathcal{E}_{A'})^p\) is then equivalent to \(\mathcal{E}_C = \mathcal{E}_{A'^{[p]}}\) by the blocking formula.

The two tensors give Kraus representations of the same completely positive map, of respective sizes \(d^p\) and \(d\). Since \(d\le d^p\), isometric freedom of Kraus representations gives \(A^{[p],\alpha }=\sum _j W_{\alpha j}B^j\) with \(W^*W=I\). Expanding products and taking traces gives the refinement identity.

From the reverse transfer-map root hypothesis, extract \(A\) with \(\mathcal{E}_B = \mathcal{E}_{A^{[p]}}\). This matches two finite Kraus families of the same completely positive map: the blocked family \(A^{[p]}\) with \(d^p\) operators and the original family \(B\) with \(d\) operators. The cardinality comparison \(d \le d^p\) holds whenever \(p \ge 1\), so [ 244 , Theorem 2.1(4) ] supplies an isometry \(V \in M_{d^p \times d}(\mathbb {C})\) with \(V^\dagger V = \mathbb {1}\) and \((A^{[p]})^\alpha = \sum _j V_{\alpha ,j}B^j\). Expanding the coefficient of \(A^{[p]}\) at the word \(\tau \) by linearity of matrix multiplication and of the trace, the contributions reorganise into the \(V\)-weighted sum over length-\(N\) words of \(B\) required by Definition 22.4.3, exhibiting \(V\) as the isometry \(W\) in the statement of \(p\)-refinability.

A subspace invariant under every \(A^i\) is invariant under every product of \(p\) letters. Equality of transfer maps preserves irreducibility. For the period assertion, blocking gives \(\gcd (m,p)\) nonzero orthogonal orbit projections, each commuting with every blocked letter. Irreducibility forces each projection to be the identity. Orthogonality then permits only one orbit.