Sharp continuity of quantum conditional entropy

We prove the sharp uniform continuity bound for quantum conditional entropy.

If two bipartite states are at trace distance at most $\delta$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(\delta)+\delta\log(d^2-1)$ up to $\delta=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy.

When $\dim B\ge d$, this bound is tight for every $\delta\in[0,1]$.

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.