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arXiv · 2607.24687

Sharp continuity of quantum conditional entropy

Mario Berta·Pablo Costa Rico·Gereon Kossmann·Ludovico Lami·Julius A. Zeiss
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Abstract

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.

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Mario Berta, Pablo Costa Rico, Gereon Kossmann, Ludovico Lami, Julius A. Zeiss. 2026-07-27. Sharp continuity of quantum conditional entropy. https://arxiv.org/abs/2607.24687

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