Continuity of entropies via integral representations

• URL: http://arxiv.org/abs/2408.15226v2
• Date: Thu, 3 Oct 2024 17:36:48 GMT
• Title: Continuity of entropies via integral representations
• Authors: Mario Berta, Ludovico Lami, Marco Tomamichel,
• Abstract summary: We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. 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; and (3) better estimates on the quantum capacity of approximately degradable channels.
• Score: 16.044444452278064
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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