Sharp continuity of quantum conditional entropy

Abstract

We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $\hat{I}´$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(\hat{I}´)+\hat{I}´\log (d^2-1)$ up to $\hat{I}´=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$, this bound is tight for every $\hat{I}´\in [0,1]$. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.

4 pages + 2 pages appendix