4 Perron–Frobenius Theory for Channels and Transfer Maps

Quantum Perron–Frobenius theory governs the fixed points and spectral structure of positive and completely positive maps. For injective and irreducible transfer maps, positive fixed points are unique up to scaling and strictly positive definite, determining the canonical gauges. The ergodicity of irreducible channels and the spectral characterization of irreducibility are treated in the companion volume [ 216 , “Cesàro convergence for irreducible channels” and “Irreducibility iff the restricted CP spectral properties hold” ] , following Wolf [ 244 , Chapter 6 ] and Evans [ 193 ] .

4.1 Positive definiteness

Theorem 4.1.1 Transfer-map PSD fixed point — irreducible version

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Let \(A\) be an MPS tensor whose transfer map \(\mathcal{E}_A\) is irreducible. If \(\rho \geq 0\), \(\rho \neq 0\), and \(\mathcal{E}_A(\rho )=\rho \), then \(\rho \) is positive definite.

Proof ▶

The transfer map is completely positive, so Theorem [ 216 , “PSD fixed point is positive definite — irreducible version” ] applies.

Theorem 4.1.2 PSD fixed point is positive definite

✓

Let \(A\) be an injective MPS tensor and let \(\rho \ge 0\), \(\rho \neq 0\), be a fixed point of \(\mathcal{E}_A\). Then \(\rho \) is positive definite.

Proof ▶

Injectivity implies irreducibility of the transfer map. Apply Theorem 4.1.1.

4.2 Uniqueness

Theorem 4.2.1 Irreducible fixed-point uniqueness

✓

Let \(A\) be an MPS tensor whose transfer map is irreducible. If \(\rho ,\sigma \geq 0\) are fixed points of \(\mathcal{E}_A\) and \(\rho \neq 0\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Proof ▶

The transfer map is completely positive, so Theorem [ 216 , “Irreducible fixed-point uniqueness — CP version” ] applies.

Theorem 4.2.2 Uniqueness of PSD fixed point

✓

Let \(A\) be an injective MPS tensor. If \(\rho ,\sigma \geq 0\) are nonzero fixed points of \(\mathcal{E}_A\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Proof ▶

Injectivity implies irreducibility of the transfer map. Apply Theorem 4.2.1.

4.3 Existence and the Perron–Frobenius theorem

Theorem 4.3.1 Existence of PSD fixed point

✓

Assume \(D \ge 1\). Let \(A\) be an MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then there exists \(\rho \ge 0\), \(\rho \neq 0\), with \(\mathcal{E}_A(\rho ) = \rho \).

Proof ▶

Under the normalization \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), the transfer map is a quantum channel (Theorem 2.2.8). The Cesàro fixed-point theorem (the corresponding theorem in the companion quantum-channel volume [ 216 , “Cesàro fixed-point theorem” ] ) then provides the fixed point; injectivity is not needed for this step.

Theorem 4.3.2 Quantum Perron–Frobenius

✓

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

Proof ▶

Combine Theorems 4.3.1, 4.1.2, and 4.2.2.

Theorem 4.3.3 Quantum Perron–Frobenius for irreducible transfer maps

✓

Assume \(D\ge 1\). Let \(A\) be an MPS tensor such that its transfer map is irreducible and \(\sum _i (A^i)^\dagger A^i=\mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then \(\mathcal{E}_A\) has a unique positive definite fixed point, up to scalar multiple.

Proof ▶

Existence is the channel fixed-point theorem (Theorem 4.3.1). Irreducibility upgrades the fixed point to positive definite, and Theorem 4.2.1 gives uniqueness up to scalar multiple.

4.4 Right- and left-canonical gauges

Definition 4.4.1 Left-canonical normalization

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An MPS tensor \(A=(A^i)_i\) is left-canonical when

\begin{align} \sum _i (A^i)^\dagger A^i & =\mathbb {1}. \notag \end{align}

Equivalently, its Kraus map is trace-preserving.

Theorem 4.4.2 Left-canonical (trace-preserving) gauge

✓

Let \(\sigma \) be a fixed point of the adjoint transfer map \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \), and let \(S\) be invertible with \(S^\dagger S = \sigma \). Define \(A'^i = SA^iS^{-1}\). Then \(\sum _i (A'^i)^\dagger A'^i = \mathbb {1}\), i.e. the gauged Kraus map is trace-preserving. This is the left-canonical normalization.

Proof ▶

Conjugating each summand gives \((SA^iS^{-1})^\dagger (SA^iS^{-1}) =(S^\dagger )^{-1}(A^i)^\dagger \sigma A^iS^{-1}\). The adjoint fixed-point substitution leaves \((S^\dagger )^{-1}\sigma S^{-1}\), and \(\sigma =S^\dagger S\) cancels the outer factors. See Section A.1 for the full calculation.

Theorem 4.4.3 Square-root trace-preserving gauge from an adjoint fixed point

✓

Let \(\sigma \) be positive definite and satisfy \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \). Define

\begin{align} B^i & = \sigma ^{1/2}A^i\sigma ^{-1/2}. \label{eq:qpf_tp_gauge} \end{align}

Then \(B\) is trace-preserving: \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

Proof ▶

This is the specialization \(K_i = A^i\) of Theorem [ 216 , “Trace-preserving gauge from an adjoint fixed point, Kraus form” ] . Each conjugated summand is \((B^i)^\dagger B^i =\sigma ^{-1/2}(A^i)^\dagger \sigma A^i\sigma ^{-1/2}\). Substituting \(\sum _i(A^i)^\dagger \sigma A^i=\sigma \) and cancelling the outer inverse square roots gives \(\mathbb {1}\). The complete calculation appears in Section A.1.

Theorem 4.4.4 Square-root trace-preserving gauge from an adjoint eigenvector

✓

Let \(\sigma \) be positive definite and satisfy \(\sum _i (A^i)^\dagger \sigma A^i = r\sigma \) for \(r{\gt}0\). Define

\begin{align} B^i & = \sigma ^{1/2}(r^{-1/2}A^i)\sigma ^{-1/2}. \label{eq:qpf_tp_gauge_eigenvector} \end{align}

Then \(B\) is trace-preserving: \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

Proof ▶

Rescale \(A\) by \(r^{-1/2}\). The adjoint transfer map is quadratic in the tensor, so the eigenvector equation becomes a fixed-point equation. Theorem 4.4.3 then gives the result.

Remark 4.4.5

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The right-canonical (unital) gauge and the left-canonical (trace-preserving) gauge arise from different similarity transformations: the former uses a fixed point of the transfer map \(\mathcal{E}_A\), whereas the latter uses a fixed point of the adjoint map \(\mathcal{E}_A^\dagger \). In general, a transfer map cannot be brought into simultaneously unital and trace-preserving form by a single similarity transform.

4.5 Perron–Frobenius eigenvector existence

Every irreducible MPS tensor admits a positive definite eigenvector for its adjoint transfer map with positive eigenvalue, which yields a trace-preserving (left-canonical) gauge. The underlying fixed-point and positive-definiteness theorems follow Wolf [ 244 , Theorems 6.3 and 6.5 ] and Evans [ 193 ] ; the gauge construction follows Cirac et al. [ 13 , Appendix A ] .

Theorem 4.5.1 Irreducibility of the transfer map of an irreducible tensor

✓

If an MPS tensor \(A\) is irreducible, then its transfer map \(\mathcal{E}_A\) is irreducible.

Proof ▶

If \(P\) is a nontrivial invariant projection for \(\mathcal{E}_A\), then

\begin{align} (\mathbb {1}-P)A^iP & = 0 \notag \end{align}

for every \(i\), which is exactly a nontrivial invariant projection for the family \(\{ A^i\} \), contradicting irreducibility of \(A\).

Theorem 4.5.2 Positive-definite adjoint eigenvector for irreducible tensors

✓

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive definite matrix \(\sigma \) and a positive real \(r {\gt} 0\) such that \(\mathcal{E}_A^\dagger (\sigma ) = r\sigma \).

This combines the eigenvector-existence part of [ 244 , Theorem 6.5 ] with the irreducible upgrade to positive definiteness ( [ 244 , Theorem 6.3(2)–(3) ] ); the application to the adjoint transfer map follows [ 13 , Appendix A ] .

Proof ▶

By Theorem 4.5.1, \(\mathcal{E}_A\) is irreducible, and since \(\mathcal K_A = \mathcal{E}_A\) (the corresponding theorem in the companion quantum-channel volume [ 216 , “Kraus-map form of the transfer map” ] ), its Kraus map \(\mathcal K_A\) is irreducible too (the corresponding theorem in the companion quantum-channel volume [ 216 , “Irreducibility in Kraus-map notation” ] ). Applying Theorem [ 216 , “Positive-definite adjoint eigenvector for irreducible Kraus families” ] to \(K_i = A^i\) gives a positive definite \(\sigma \) and a positive real \(r {\gt} 0\) with \(\sum _i (A^i)^\dagger \sigma A^i = r\sigma \), i.e. \(\mathcal{E}_A^\dagger (\sigma ) = r\sigma \).

Theorem 4.5.3 TP-gauge existence for irreducible tensors

✓

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive real \(r\), a positive definite matrix \(\sigma \), and a tensor \(B\) gauge-equivalent to \(r^{-1/2}A\) such that

\begin{align} B^i & = \sigma ^{1/2}(r^{-1/2}A^i)\sigma ^{-1/2}, \label{eq:qpf_irred_tp_gauge} \end{align}

and \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

This corresponds to the spectral rescaling and trace-preserving gauge construction of [ 13 , Appendix A ] .

Proof ▶

By Theorem 4.5.2, obtain \(\sigma \) positive definite and \(r {\gt} 0\) with \(\mathcal{E}_A^\dagger (\sigma ) = r\sigma \). Set \(c = r^{-1/2}\) and \(A'^i = cA^i\). Then \(\mathcal{E}_{A'}^\dagger (\sigma ) = \sigma \). By the square-root trace-preserving eigenvector gauge construction (Theorem 4.4.4), the tensor defined by (3) satisfies \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\). Lemma A.1.1 gives the required gauge equivalence to \(r^{-1/2}A\).

Theorem 4.5.4 Unital-gauge existence for irreducible tensors

✓

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive real \(r\), a positive definite matrix \(\rho \), and a tensor \(B\) gauge-equivalent to \(r^{-1/2}A\) such that

\begin{align} B^i & = r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}, \label{eq:qpf_irred_unital_gauge} \end{align}

and \(\sum _i B^i(B^i)^\dagger = \mathbb {1}\).

This corresponds to the Perron–Frobenius unital gauge of [ 228 , Theorem 4 ] for an irreducible block.

Proof ▶

Apply Theorem 4.5.2 to the conjugate-transposed Kraus family. Irreducibility transfers to that family, and the nonzero Kraus hypothesis is preserved by taking adjoints. Translating the resulting adjoint eigenvector equation back gives \(\mathcal{E}_A(\rho )=r\rho \) with \(\rho \) positive definite and \(r{\gt}0\). The unital gauge theorem then gives the representative (4), and the same square-root similarity gives gauge equivalence to \(r^{-1/2}A\).