PreprintCC BY 4.0arXiv
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 δ and d=dim A, the optimal dimension-only modulus of continuity is h2(δ)+δlog(d2−1) up to δ=1−d−2 and 2log d thereafter, where h2 denotes the binary entropy. When dim B≥ d, this bound is tight for every δ∈[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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BibTeX
@misc{berta2026sharp,
title = {{Sharp continuity of quantum conditional entropy}},
author = {Mario Berta and Pablo Costa Rico and Gereon Kossmann and Ludovico Lami and Julius A. Zeiss},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24687},
eprint = {2607.24687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24687},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp continuity of quantum conditional entropy
AU - Mario Berta
AU - Pablo Costa Rico
AU - Gereon Kossmann
AU - Ludovico Lami
AU - Julius A. Zeiss
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.24687
UR - https://arxiv.org/abs/2607.24687
AB - 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)+δlog(d^2−1) up to δ=1−d^−2 and 2log d thereafter, where h_2 denotes the binary entropy. When dim B≥ d, this bound is tight for every δ∈[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.
ER -