Diagonalize \(P\) by a unitary \(U\). Since \(P^2=P\), the diagonal form of \(P\) has only \(0\)- and \(1\)-eigenvalues, so the basis splits as \(n+m=D\). Conjugating \(A\) by \(U\) preserves the MPV family by Theorem E.1.1. In this basis the relation \((\mathbb {1}-P)A^iP=0\) makes each letter upper block triangular. Write the conjugated letters as

Since \(B^{i_1}\cdots B^{i_N}\) is again upper block triangular, its trace satisfies

Hence the block-diagonal tensor \(A_1\oplus A_2\) has the same MPV coefficients as the conjugated tensor.

From \(\mathcal{E}_A(\rho )=\rho \) and positivity of each summand \(A^i\rho (A^i)^\dagger \), one gets \((\mathbb {1}-P)A^i\rho (A^i)^\dagger (\mathbb {1}-P)=0\) for each \(i\). Since \(\rho \) is supported on the range of \(P\), this forces \((\mathbb {1}-P)A^iP=0\).

This is the finite-dimensional case of [ Wol12 , Proposition 6.10 ] : the support projection of any stationary state is sub-harmonic. The equivalence between irreducibility and absence of nontrivial invariant projections is [ Wol12 , Theorem 6.2 ] ; see also  [ CPGSV21 , Section IV.A.1 ] .

The support projection \(P\) of \(\rho \) is neither \(0\) nor \(\mathbb {1}\), since \(\rho \neq 0\) and \(\rho \) is not positive definite. Theorem 9.1.4 gives \((\mathbb {1}-P)A^iP=0\), so a unitary change of basis diagonalizing \(P\) puts \(A^i\) in upper-triangular block form with diagonal blocks \(B_{11}^i,B_{22}^i\). Theorem E.1.1 preserves the MPV under the conjugation, and the trace identity (??) shows that deleting the off-diagonal blocks preserves all MPV coefficients.

By strong induction on \(D\). If \(A\) is irreducible, take \(r=1\). Otherwise a nontrivial invariant projection \(P\) gives the upper-triangular splitting of Theorem 9.1.5; deleting the off-diagonal part preserves the MPV family and yields two blocks of strictly smaller bond dimension. Apply the induction hypothesis to each block.

Both  [ PGVWC07 , Theorem 4 ] and  [ CPGSV21 , Section IV.A.1 ] obtain the block decomposition by the same invariant-projection splitting.

The Perron–Frobenius TP-gauge construction is applied to each irreducible block using the gauge (??). Gauge equivalence preserves irreducibility and the MPV family, and the positive scaling factors are absorbed into the weights \(\mu _k\).

Theorem E.1.3 supplies the nonzero irreducible blocks \((C_k)_{k=1}^r\), with \(V^{(N)}(A)_\sigma =V^{(N)}(\bigoplus _{k=1}^rC_k)_\sigma \) for every \(N\ge 1\) and every \(\sigma \). Applying the Perron–Frobenius TP gauge (??) to each \(C_k\) gives a left-canonical block \(B_k\) with weight \(\mu _k\), and

The gauge preserves every block dimension, so the structural bound \(\sum _kD_k\le D\) is unchanged.

The cyclic decomposition of the adjoint transfer map gives projections \((P_k)\) with \(\mathcal{E}_{A^\dagger }(P_{k+1})=P_k\). Iterating the cyclic relation \(m\) times gives \((\mathcal{E}_{A^\dagger })^m(P_k)=P_k\), and the blocked-transfer identity \(\mathcal{E}_{(A^{[m]})^\dagger }=(\mathcal{E}_{A^\dagger })^m\) identifies these with the adjoint transfer map of \(A^{[m]}\). Apply the block decomposition from adjoint-fixed projections.

Theorem 9.4.1 gives irreducibility of \((\mathcal{E}_A^\dagger )^m\) on each corner \(P_u\). The compression equivalences in (??) intertwine the adjoint transfer map of \(C_u\) with the corresponding corner restriction of \((\mathcal{E}_A^\dagger )^m\). The peripheral spectrum of the blocked adjoint map is

because \(\gamma \) is a primitive \(m\)th root of unity. Hence each nonzero irreducible corner restriction is primitive. Primitivity and irreducibility are preserved across \(\varphi _u\); then use Theorem E.4.3 to pass from the adjoint blocked map to the primal transfer map of \(C_u\).

Left-canonicality makes \(K_i:=(A^i)^\dagger \) unital, irreducibility passes to the adjoint map, and the quantum Perron–Frobenius theorem provides a positive-definite fixed point; the cyclic peripheral-spectrum theorem and Theorem 9.3.1.1 then produce the cyclic-sector decomposition. Keep the projections and compression equivalences, and apply Theorem 9.3.1.2 to the sector blocks.

Start from Theorem E.5.4. With \(p_A,p_B\) the least common multiples of the period-removal lengths on the two sides, use \(p=\operatorname{lcm}(p_A,p_B)\). The prescribed-common-multiple lemma constructs both one-sided sector families at this length, and the positive-length blocking lemmas carry the one-sided tensor/nonzero-part equalities over to the common alphabet, giving (??) and (??). The two-sided blocking lemma applied to (??) gives (??).

Follow [ PGVWC07 , Theorem 4 ] . The nonzero irreducible decomposition is obtained by strict dimension induction. On a nonzero current block, Perron–Frobenius gives a positive semidefinite eigenvector at a positive spectral radius. If it is singular, positivity makes its support invariant: \((\mathbb {1}-P)A^iP=0\). After diagonalizing \(P\), every letter is upper block triangular and the finite-ring trace splits as

Discard all-zero blocks and repeat on the two strictly smaller diagonal blocks. This gives the finite nonzero irreducible family of Theorem E.1.3.

For each retained irreducible block, divide by the square root of its positive spectral radius and conjugate by the square root of its positive-definite Perron eigenvector. Theorem 6.7.5 gives the unital identity

Nonzero scalar rescaling preserves irreducibility. The proof passes from tensor irreducibility to irreducibility of the matrix action, carries invariant subspaces through the gauge similarity, and returns to tensor irreducibility using Theorem 8.7.3.4 and Lemma D.2.2. Hence Theorem 9.8.1 makes every fixed point scalar. Theorem 9.10.2 then diagonalizes a positive-definite dual fixed point by a unitary without changing unitality.

Apply this single-block construction to every member of the finite family. The square roots of the block spectral radii become positive weights, gauge invariance preserves every positive-length MPV coefficient, and the block dimensions are unchanged. Combining this blockwise result with the initial decomposition yields the arbitrary-input unnormalized form and its bound \(\sum _k\dim (A_k)\le D\). Together these conclusions give the structured witness.

If the witness had no blocks, its weighted block tensor would have zero MPV coefficients at every positive length, contrary to the assumed nonzero coefficient. Thus \(r{\gt}0\). Since the finite family \((a_k)\) is nonempty and every \(a_k{\gt}0\), its maximum \(s\) is positive. The definition \(\nu _k=a_k/s\) gives \(0{\lt}\nu _k\le 1\), and any maximizing index has \(\nu _k=1\). Finally, \(a_k=s\nu _k\) for every \(k\), so homogeneity of a length-\(N\) matrix product vector gives (??).

Choose the structured witness from Theorem 9.5.2. The assumed nonzero coefficient makes its block family nonempty, so Theorem 9.5.3 supplies \(s{\gt}0\) and the normalized positive weights. Equation (??) and homogeneity give

The witness satisfies all required block conditions and the bond-dimension bound.

Split according to whether there exist \(N{\gt}0\) and \(\sigma \) with \(V^{(N)}(A)_\sigma \neq 0\). In that case apply Theorem 9.5.4. If no such pair exists, every positive-length coefficient vanishes.

Apply Theorem 9.5.5. If every positive-length coefficient vanishes, take \(s=1\) and the empty retained family; the block conditions and bond-dimension bound are vacuous, and the MPV identity is the zero identity. In the nonzero branch, use the nonempty normalized family supplied by the second alternative.

Algebraic normality together with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) makes the transfer map strongly irreducible. It consequently has a positive-definite fixed point, no nontrivial invariant projection, and peripheral spectrum \(\{ 1\} \). The fixed-point eigenvalue and left-canonical normalization give \(r(\mathcal{E}_A)=1\). Thus no scalar rescaling remains, and the two conditions in Definition 9.6.1 hold for \(A\) itself.

Apply Theorem 9.11.1.3 to \(A\) and itself. It gives \(p\ge 1\), weights \(\mu _k\), and left-canonical, primitive, tensor-irreducible blocks \(C_k\) such that, for every \(N\ge 1\),

First, Theorem 9.11.1.1 makes each \(C_k\) algebraically normal. Second, Theorem 9.6.3 combines this conclusion with the left-canonical identity to make each \(C_k\) a CPSV normal tensor. Hence

satisfies the literal block-diagonal equation ??, so \(B\in \mathrm{CF}\). Equation ?? gives the second part of ??.

Induction on a nonempty physical word \(w\) of length \(N\), using (??) at each letter and \(UU^\dagger =\mathbb {1}_S\) between adjacent letters, gives

Applying cyclicity of the trace and \(UU^\dagger =\mathbb {1}_S\) to (??) gives

Applying Theorem 2.5.9 to the right-hand side of (??) yields \(\operatorname{tr}(A^w)=\sum _{k=1}^{r}\mu _k^N\operatorname{tr}(A_k^w)\), which is (??).

The canonical-form data supply normality of every \(A_k\) and the positive-length identity \(|V^{(N)}(A)\rangle =\sum _k\mu _k^N|V^{(N)}(A_k)\rangle \). Substitution of these two facts into Theorem 10.1.4 gives (??).

Proceed by strong induction on the bond dimension. If \(A\) is irreducible, the single corner \(V=\mathbb {1}\) suffices. Otherwise, \(A\) has a nontrivial invariant projection \(P\), which commutes with every \(A^i\) by Theorem E.2.1. Factoring \(P=V_1V_1^\dagger \) and \(\mathbb {1}-P=V_2V_2^\dagger \) through support isometries by Lemma E.2.3 splits \(A\) into two corner tensors of strictly smaller bond dimension. For \(j=1,2\), commutation gives \(A^iV_j=V_j(V_j^\dagger A^iV_j)\), and each corner inherits invariant-projector closure by Lemma E.2.4. The two recursively obtained families are combined. Ranges from different corners are orthogonal because

Equality of matrix product vectors follows from Lemma E.2.6.

Decompose \(A\) into irreducible corners \(B_k\) along isometries \(V_k\) by Theorem 9.7.2; every matrix product vector coefficient of \(A\) is the sum of those of the corners. A corner whose matrices all vanish contributes nothing at positive length and is dropped. Every remaining corner has a Perron eigenpair \(\mathcal{E}_{B_k}(\rho _k)=r_k\rho _k\) by Lemma E.3.1. The absence of \(p\)-periodic vectors, applied to the corner \(B_k\), gives that every eigenvalue of \(\mathcal{E}_{B_k}\) of modulus \(r_k\) equals \(r_k\), so \(A_k:=r_k^{-1/2}B_k\) is normal by Lemma E.3.2. Setting \(\mu _k:=r_k^{1/2}\) gives \(\mu _kA_k=B_k\) at the level of matrices, so for every word \(w\) of positive length \(N\) the weight cancels:

Summing over \(k\) gives (??), while the corner reconstruction gives (??). The retained dimensions satisfy \(\sum _kD_k\le \sum _k\operatorname{tr}(V_kV_k^\dagger )=\operatorname{tr}(\mathbb {1})=D\).

Let \(\lambda _{\max }\) be the largest eigenvalue of \(X\). Then \(\rho :=\lambda _{\max }\mathbb {1}-X\) is positive semidefinite, singular, and nonzero because \(X\) is not scalar. Unitality and the fixed-point equation for \(X\) give \(\mathcal{E}_A(\rho )=\rho \). Apply Theorem 9.1.5. The non-scalar hypothesis is needed because scalar multiples of \(\mathbb {1}\) are fixed by every unital map and yield no nontrivial support projection.

Tensor irreducibility gives irreducibility of the completely positive transfer map. By Wolf’s Theorem 6.6 for irreducible unital Kraus maps, every fixed point is scalar.

The positive square root of \(\rho \) is invertible. Substituting \(B^i=r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}\) and using \(\mathcal{E}_A(\rho )=r\rho \) gives

which is (??). For \(r=1\), the change from \(A\) to \(B\) is a similarity, so cyclicity of the trace gives equality of all MPV coefficients.

Write the retained part of the canonical-form decomposition as \(C^i=\bigoplus _k\mu _kA_k^i\), and let \(U\in \mathbb {C}^{S\times D}\) satisfy \(UU^\dagger =\mathbb {1}_S\) and \(A^i=U^\dagger C^iU\). For each normal block, first choose its pure trace-preserving Perron gauge and then diagonalize its positive definite fixed point by a unitary. This gives \(X_k\in \mathrm{GL}(D_k,\mathbb {C})\) and blocks \(B_k\) satisfying the two canonical-form-II identities. Put \(X=\bigoplus _kX_k\) and extend it to the omitted zero coordinates by

Its inverse is

The identity \(UU^\dagger =\mathbb {1}_S\) gives \(GG^{-1}=G^{-1}G=\mathbb {1}_D\). It also gives

and

Therefore

which has the same ambient reconstruction and satisfies canonical form II.

The TP hypothesis makes \(\mathcal{E}_A\) a channel with a nonzero PSD fixed point. Irreducibility makes \(\mathcal{E}_A\) an irreducible CP map, and Theorem 6.2.4 upgrades the fixed point to positive definite. The spectral theorem diagonalizes it by a unitary conjugation, which preserves the TP normalization.

If the bond dimensions differ, Theorem 7.5.3 gives the decay. If the bond dimensions agree, the non-equivalence hypothesis excludes gauge-phase equivalence, so Theorem 7.6.3 gives the decay.

Injective blocks are normal. The block-diagonal decomposition theorem gives the MPV expansion of \(\bigoplus _k \mu _k A_k\) in terms of the block MPVs. The self-overlap limits are part of the hypotheses, and Theorem 9.9.1.3 supplies the off-diagonal decay. Therefore Lemma F.1.10 gives the eventual linear independence required in Definition 10.1.1.

The normal-canonical hypotheses already give the block-diagonal MPV expansion, the self-overlap normalization, and the eventual linear independence coming from irreducible trace-preserving overlap decay. Supplying algebraic normality for each block adds the remaining clause in Definition 10.1.1.

Pass to the conjugate-transposed Kraus family \(K_i:=(A^i)^\dagger \), which is unital with irreducible Kraus map. The TP fixed-point theorem for \(\mathcal{E}_A\) provides a positive definite fixed point for the adjoint of \(K\), so Theorem 4.10.2.2 shows that every peripheral eigenvalue is a root of unity. Since \(\sigma _\partial (\mathcal{E}_A)\) is finite (Theorem 4.10.2), Lemma 4.10.1.2 gives a common exponent \(p\) with \(\mu ^p=1\) for every \(\mu \in \sigma _\partial (\mathcal{E}_A)\), and Lemma 4.10.1.3 gives \(\sigma _\partial (\mathcal{E}_A^p)=\{ 1\} \). Blocking by \(p\) takes the \(p\)th power of the transfer map, so \(\mathcal{E}_{A^{[p]}}\) is primitive in the peripheral-spectrum sense of Definition 4.8.4. The blocked map need not remain irreducible: eigenvalue \(1\) may have several cyclic corners. The adjoint-fixed projections are therefore compressed before irreducibility and primitivity are asserted sector by sector in Theorem 9.3.1.3. This is the periodicity-removal step of [ CPGSV16 , Section 2.3 ] for an irreducible block already in TP gauge.

Theorem 9.8.5 provides the unitary and the diagonal positive-definite fixed point, and unitary conjugation preserves trace preservation and the MPV family (Theorem E.1.1).

Apply Theorem 9.10.1 to the conjugate-transposed family \(i\mapsto (A^i)^\dagger \): unitality of \(A\) is trace preservation for this family, whose transfer map is the dual of \(\mathcal{E}_A\). Translating back gives (??).

Tensor irreducibility first gives irreducibility of the transfer map. That map has a PSD fixed point by trace preservation, and irreducibility upgrades it to a positive-definite fixed point (Theorem 6.2.4). Theorem 8.7.4.2 then gives normality.

Theorem 9.2.1.1 supplies the left-canonical nontrivial blocks, the positive-length equality, and the bound \(\sum _kD_k\le D\). Apply Theorem E.4.2 directly to this finite irreducible family. Its proof first applies Theorem 9.9.2.1 to each block and then takes the least common multiple of the resulting periods through Theorem E.4.1. This gives one positive length \(p\) at which every blocked transfer map is primitive. Lemma E.4.12 keeps the blocked weights nonzero, and Lemma E.4.10 carries the positive-length equality through blocking, giving (??).

Apply Theorem 9.3.1.4 to choose the common blocking length and the two common cyclic-sector families. For either family, write \(G_k\) for its cyclic-sector blocks and \((\nu _x,C_x)\) for the weights and blocks of the flattened common-sector family from Lemma E.5.10. The common-alphabet nonzero-part identity gives, for every length \(N\) and blocked word \(\sigma \),

Applying this identity on the two sides of the comparison yields a positive length \(p\) and weighted primitive block families such that, for every \(N\ge 1\) and blocked word \(\sigma \),

The two weighted nonzero parts are equal at every positive length. The structural properties and the nonvanishing of the weights come from the common-sector families.