Continuity of entropies via integral representations

TL;DR

This paper introduces a dimension-independent semi-continuity relation for quantum relative entropy using integral representations, leading to improved bounds and solutions for several quantum information measures.

Contribution

It provides a novel semi-continuity relation for quantum relative entropy that enhances understanding and bounds of various quantum information measures.

Findings

Resolved Wilde's conjecture for conditional entropy with equal marginals

Strengthened the Fannes-Audenaert inequality for quantum entropy

Improved estimates on quantum capacity of approximately degradable channels

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.