Continuity of entropies via integral representations
<|reference_start|>Continuity of entropies via integral representations: We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.<|reference_end|>
@article{berta2024continuity,
    title={Continuity of entropies via integral representations},
    author={Mario Berta and Ludovico Lami and Marco Tomamichel},
    journal={arXiv preprint arXiv:2408.15226},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.15226},
    primaryClass={quant-ph cs.IT math-ph math.IT math.MP}
}