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

In this short note, I show how a recent result of Alhejji and Smith (A tight uniform continuity bound for equivocation, 2019. arXiv:1909.00787v1) 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 the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. 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.