Continuity of entropies via integral representations

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) a tighter continuity relation for quantum capacity; (4) better estimates on the quantum capacity of approximately degradable channels; (5) an improved continuity relation for the entanglement cost; (6) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (7) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.