Continuity of certain quantum channel divergences

Abstract

The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities $0$ and $1$. The remedy is the Arveson--Radon--Nikodym derivative $T$ of one map with respect to the sum $\Omega$, whose transpose is the environment effect selecting that map in a Stinespring dilation of $\Omega$. In $T$, these quantities and all maximal $f$-divergences of operator convex $f$ are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most $2\Vert T_1-T_2{\Vert }_{\infty }^{1/2}$, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.