Optimal uniform continuity bound for conditional entropy of classical--quantum states

TL;DR

This paper derives an optimal uniform continuity bound for quantum conditional entropy of classical-quantum states, improving previous bounds and raising open questions for broader classes of states.

Contribution

It extends a recent classical result to quantum states, establishing an optimal bound for quantum conditional entropy and its application to entanglement measures.

Findings

Established an optimal continuity bound for quantum conditional entropy.

Improved the continuity bound for entanglement of formation.

Identified open questions for other state classes regarding continuity bounds.

Abstract

In this short note, I show how a recent result of Alhejji and Smith [arXiv:1909.00787] regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical--quantum states. The bound is optimal in the sense that there always exists a pair of classical--quantum states saturating the bound, and so no further improvements are possible. An immediate application is a uniform continuity bound for entanglement of formation that improves upon the one previously given by Winter in [arXiv:1507.07775]. Two intriguing open questions are raised regarding other possible uniform continuity bounds for conditional entropy, one about quantum--classical states and another about fully quantum bipartite states.