Sharp continuity of quantum conditional entropy

arXiv (Cornell University)

Abstract

We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $δ$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(δ)+δ\log(d^2-1)$ up to $δ=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$, this bound is tight for every $δ\in[0,1]$. 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.

Authors 5

1. Mario Berta Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

2. Pablo Costa Rico Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

3. Gereon Koßmann Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

4. Ludovico Lami

   Scuola Normale Superiore

   Affiliation as printed

   Scuola Normale Superiore , Pisa , Italy

5. Julius A. Zeiss Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany