From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912202814914560 |
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| author | Lanier, Dimitri Béguinot, Julien Rioul, Olivier |
| author_facet | Lanier, Dimitri Béguinot, Julien Rioul, Olivier |
| contents | Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between $χ^2$ and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum $f$-divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_14340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence Lanier, Dimitri Béguinot, Julien Rioul, Olivier Quantum Physics Information Theory Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our methodology is particularly simple as it does not require any elaborate matrix analysis machinery but only basic linear algebra. It is also effective, as illustrated by two examples improving existing bounds: (i)~an improved quantum Pinsker inequality is derived between $χ^2$ and trace norm, and leveraged to improve a bound in decoherence theory; (ii)~a new reverse quantum Pinsker inequality is derived for any quantum $f$-divergence, and compared to previous (Audenaert-Eisert and Hirche-Tomamichel) bounds. |
| title | From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence |
| topic | Quantum Physics Information Theory |
| url | https://arxiv.org/abs/2501.14340 |