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 (CommunMath 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 quantumclassical states and another about fully quantum bipartite states.