B Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
This appendix supplies the compactness, gauge, similarity, and finite-sum calculations used in Chapter 6.
B.1 Density matrices, Brouwer’s theorem, and Cesàro limits
Write \(\mathcal{D}_D\) for the density matrices of Definition 4.4.6. The next results provide the compact convex domain and the limit argument used for channel fixed points.
\(\mathcal{D}_D\) is compact in the entrywise topology on \(M_{D}(\mathbb {C})\).
The positive semidefinite cone is closed, being the preimage of the closed non-negative cone under continuous quadratic forms. If \(\rho \ge 0\) with \(\operatorname{tr}(\rho ) = 1\), then each entry satisfies \(|\rho _{ij}|^2 \le \rho _{ii}\rho _{jj} \le 1\): the diagonal entries are non-negative and sum to \(1\), and the off-diagonal bound is the positive semidefinite Cauchy–Schwarz inequality. Hence the matrix entries are uniformly bounded, and Heine–Borel gives compactness.
\(\mathcal{D}_D\) is convex.
A convex combination of positive semidefinite matrices is positive semidefinite, and the trace is linear.
For \(D \ge 1\), \(\mathcal{D}_D \neq \varnothing \).
\(\frac{1}{D}\mathbb {1}_D\) is a density matrix.
If \(E\) is a quantum channel, then \(E(\mathcal{D}_D) \subseteq \mathcal{D}_D\).
Complete positivity implies positivity by Theorem 4.2.3, so \(E(\rho ) \ge 0\); trace preservation gives \(\operatorname{tr}(E(\rho )) = \operatorname{tr}(\rho ) = 1\).
Let \(D {\gt} 0\). Every continuous map from the compact convex set of density matrices in \(M_{D}(\mathbb {C})\) to itself has a fixed point.
The proof starts from Brouwer’s theorem for products of simplices, transfers it to closed cubes, and then to compact retracts of finite-dimensional real normed spaces. The density-matrix set is such a compact retract, using the explicit density retraction.
Let \(E\) be a quantum channel and \(\rho \in \mathcal{D}_D\). Then, for every \(N\geq 0\),
Every iterate \(E^n(\rho )\) belongs to \(\mathcal{D}_D\) by Theorem B.1.4. Equation (??) is their convex average: positivity is preserved by sums and multiplication by \((N+1)^{-1}\), while its trace is \((N+1)^{-1}\sum _{n=0}^{N}1=1\).
Let \(E\) be a quantum channel, \(\rho \in \mathcal{D}_D\), and \(\psi :\mathbb {N}\to \mathbb {N}\) satisfy \(\psi (k)\to \infty \). If
then \(\sigma \in \mathcal{D}_D\) and \(E(\sigma )=\sigma \).
Lemma B.1.6 and closedness of the compact set \(\mathcal{D}_D\) give \(\sigma \in \mathcal{D}_D\). The telescope identity yields
Both \(E^{\psi (k)+1}(\rho )\) and \(\rho \) lie in the compact set \(\mathcal{D}_D\), so their difference is uniformly bounded. The right-hand side tends to zero. Continuity of \(E\) and convergence of the subsequence therefore give \(E(\sigma )-\sigma =0\).
B.2 Canonical-gauge algebra
For the right-canonical gauge in Theorem 6.5.1, each summand becomes
Hence
For the left-canonical gauge in Theorem 6.5.3,
and therefore
Finally, for the square-root gauge (??), self-adjointness of \(\sigma ^{1/2}\) gives
Summing, substituting the adjoint fixed-point equation, and cancelling the outer square-root factors yields
If \(\sigma \) is positive definite and \(B\) is defined by (??), then \(A\) and \(B\) are gauge equivalent.
The gauge matrix is \(\sigma ^{1/2}\), which is invertible because \(\sigma \) is positive definite. Thus (??) has the form of the gauge relation (??).
B.3 Similarity bookkeeping
For any invertible \(C \in M_{D}(\mathbb {C})\) and any linear map \(E\) on \(M_{D}(\mathbb {C})\), write \(S_C(E)(X)=C^{-1}E(CXC^\dagger )(C^\dagger )^{-1}\) for the similarity transform by \(C\). The transforms by \(C\) and \(C^{-1}\) compose to the identity:
Direct computation: the inner conjugation \(C(C^{-1}X(C^\dagger )^{-1})C^\dagger \) collapses to \(X\) using \(CC^{-1} = \mathbb {1}\) and \((C^\dagger )^{-1}C^\dagger = \mathbb {1}\), and the outer conjugation by \(C(\, \cdot \, )C^\dagger \) then cancels the inner \(C^{-1}(\, \cdot \, )(C^\dagger )^{-1}\) on \(E(X)\) for the same reason.
For any invertible \(C \in M_{D}(\mathbb {C})\) and any linear map \(E\) on \(M_{D}(\mathbb {C})\), the similarity transform \(X \mapsto C^{-1}E(CXC^\dagger )(C^\dagger )^{-1}\) is irreducible if and only if \(E\) is.
The forward direction follows from Lemma 6.6.1 applied with the inverse \(C^{-1}\), whose determinant is also nonzero: if the similarity transform of \(E\) by \(C\) is irreducible, then so is its similarity transform by \(C^{-1}\), which equals \(E\) by (??). The reverse direction is Lemma 6.6.1 itself with \(c = 1\).
B.4 Auxiliary Perron reductions
If some Kraus operator \(A^i \neq 0\), then the adjoint transfer map \(\mathcal{E}_A^\dagger (X) = \sum _i (A^i)^\dagger X A^i\) is nonzero.
If \(\mathcal{E}_A^\dagger = 0\) then \(\mathcal{E}_A^\dagger (\mathbb {1}) = 0\), so \(\sum _i (A^i)^\dagger A^i = 0\). Since each summand \((A^i)^\dagger A^i\) is positive semidefinite, each \((A^i)^\dagger A^i = 0\) and hence each \(A^i = 0\).
Under the hypotheses of Theorem 6.10.1, the real-valued spectral radius of \(E\) is also equal to \(r\).
This is the real-valued reformulation of Theorem 6.10.1.
B.5 Exponential truncation and scalar reformulations
Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\), let \(A \ge 0\) be nonzero, and let \(t {\gt} 0\). Then the finite exponential truncation satisfies
This is the finite-sum core of the completely positive specialization of [ Wol12 , Theorem 6.2(3) ] .
If a nonzero vector \(v\) annihilated the quadratic form in (??), positivity of every summand would make \(E^k(A)v=0\) for \(0\leq k\leq D-1\). Expanding \((\mathbb {1}+E)^{D-1}(A)\) by the binomial theorem would then make it annihilate \(v\), contrary to Theorem 6.2.1.