A Perron–Frobenius Theory for Channels and Transfer Maps: Supporting Results
Canonical-gauge calculations supporting Chapter 4 verify the normalization of the transformed Kraus operators directly.
A.1 Canonical-gauge algebra
For the right-canonical gauge in Theorem [ 216 , “Right-canonical (unital) gauge” ] , each summand becomes
\begin{align} (S^{-1}A^iS)(S^{-1}A^iS)^\dagger & =S^{-1}A^iSS^\dagger (A^i)^\dagger (S^\dagger )^{-1} \notag \\ & =S^{-1}A^i\rho (A^i)^\dagger (S^\dagger )^{-1}. \notag \end{align}
Hence
\begin{align} \sum _iA’^i(A’^i)^\dagger & =S^{-1}\mathcal{E}_A(\rho )(S^\dagger )^{-1} =S^{-1}\rho (S^\dagger )^{-1} =\mathbb {1}. \notag \end{align}
For the left-canonical gauge in Theorem 4.4.2,
\begin{align} (SA^iS^{-1})^\dagger (SA^iS^{-1}) & =(S^\dagger )^{-1}(A^i)^\dagger S^\dagger SA^iS^{-1} \notag \\ & =(S^\dagger )^{-1}(A^i)^\dagger \sigma A^iS^{-1}, \notag \end{align}
and therefore
\begin{align} \sum _i(A’^i)^\dagger A’^i & =(S^\dagger )^{-1} \left(\sum _i(A^i)^\dagger \sigma A^i\right)S^{-1} =(S^\dagger )^{-1}\sigma S^{-1} =\mathbb {1}. \notag \end{align}
Finally, for the square-root gauge (1), self-adjointness of \(\sigma ^{1/2}\) gives
\begin{align} (B^i)^\dagger B^i & =\sigma ^{-1/2}(A^i)^\dagger \sigma ^{1/2}\sigma ^{1/2}A^i\sigma ^{-1/2} \notag \\ & =\sigma ^{-1/2}(A^i)^\dagger \sigma A^i\sigma ^{-1/2}. \notag \end{align}
Summing, substituting the adjoint fixed-point equation, and cancelling the outer square-root factors yields
\begin{align} \sum _i(B^i)^\dagger B^i & =\sigma ^{-1/2} \left(\sum _i(A^i)^\dagger \sigma A^i\right)\sigma ^{-1/2} =\sigma ^{-1/2}\sigma \sigma ^{-1/2} =\mathbb {1}. \notag \end{align}
If \(\sigma \) is positive definite and \(B\) is defined by (1), then \(A\) and \(B\) are gauge equivalent.