Continuity bounds for quantum entropies arising from a fundamental entropic inequality
- URL: http://arxiv.org/abs/2408.15306v2
- Date: Fri, 4 Oct 2024 16:03:07 GMT
- Title: Continuity bounds for quantum entropies arising from a fundamental entropic inequality
- Authors: Koenraad Audenaert, Bjarne Bergh, Nilanjana Datta, Michael G. Jabbour, Ángela Capel, Paul Gondolf,
- Abstract summary: We establish a tight upper bound for the difference in von Neumann entropies between two quantum states.
This yields a novel entropic inequality that implies the well-known Audenaert-Fannes inequality.
- Score: 9.23607423080658
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $\rho_1$ and $\rho_2$. 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 $(\rho_1 - \rho_2)$. 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. Our proofs are largely based on majorization theory and convex optimization. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.
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