Continuity bounds for quantum entropies arising from a fundamental entropic inequality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915462472794112 |
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| author | Audenaert, Koenraad Bergh, Bjarne Datta, Nilanjana Jabbour, Michael G. Capel, Ángela Gondolf, Paul |
| author_facet | Audenaert, Koenraad Bergh, Bjarne Datta, Nilanjana Jabbour, Michael G. Capel, Ángela Gondolf, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity bounds for quantum entropies arising from a fundamental entropic inequality Audenaert, Koenraad Bergh, Bjarne Datta, Nilanjana Jabbour, Michael G. Capel, Ángela Gondolf, Paul Quantum Physics Mathematical Physics 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. |
| title | Continuity bounds for quantum entropies arising from a fundamental entropic inequality |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2408.15306 |