Continuity of Entropies via Integral Representations

Abstract

We show that Frenkel’s integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: 1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; 2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; 3) better estimates on the quantum capacity of approximately degradable channels; 4) an improved continuity relation for the entanglement cost; 5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and 6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of ‘filtered’ relative entropy distances.