Pré-Publication, Document De Travail Année : 2024

Continuity bounds for quantum entropies arising from a fundamental entropic inequality

Résumé

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $ρ_1$ and $ρ_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $(ρ_1 - ρ_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

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Dates et versions

hal-05022361 , version 1 (06-04-2025)
hal-05022361 , version 2 (03-08-2025)
hal-05022361 , version 3 (25-08-2025)

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Koenraad Audenaert, Bjarne Bergh, Nilanjana Datta, Michael G Jabbour, Ángela Capel, et al.. Continuity bounds for quantum entropies arising from a fundamental entropic inequality. 2025. ⟨hal-05022361v1⟩
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