Integral formula for quantum relative entropy implies data processing inequality
Eötvös University, Institute of Mathematics, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary
Rényi Institute, Budapest, Reáltanoda u. 13-15, 1053 Hungary
| Published: | 2023-09-07, volume 7, page 1102 |
| Eprint: | arXiv:2208.12194v4 |
| Doi: | https://doi.org/10.22331/q-2023-09-07-1102 |
| Citation: | Quantum 7, 1102 (2023). |
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Abstract
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.
Popular summary
A binary reduction principle for generalized divergences is also presented, leading, in particular, to an improved Pinsker-style lower bound for the Holevo quantity of two quantum states in terms of their trace distance.
The paper is already cited by two preprints that apply the main result in essential ways:
[Anna Jencová, Recoverability of quantum channels via hypothesis testing, arXiv:2303.11707] and [Christoph Hirche, Marco Tomamichel, Quantum Rényi and $f$-divergences from integral representations, arXiv:2306.12343].
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► References
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[1] Kun Fang, Hamza Fawzi, and Omar Fawzi, "Uhlmann’s Theorem for Measured Divergences", IEEE Transactions on Information Theory 72 3, 1751 (2026).
[2] Giulio Chiribella and Kaumudibikash Goswami, "Maximum and Minimum Causal Effects of Physical Processes", PRX Quantum 6 2, 020335 (2025).
[3] Hayata Yamasaki, Kohdai Kuroiwa, Patrick Hayden, and Ludovico Lami, "Entanglement Cost for Infinite-Dimensional Physical Systems", Communications in Mathematical Physics 406 11, 277 (2025).
[4] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li, "Complete positivity order and relative entropy decay", Forum of Mathematics, Sigma 13, e31 (2025).
[5] Bartosz Regula, Ludovico Lami, and Nilanjana Datta, "Tight Relations and Equivalences Between Smooth Relative Entropies", IEEE Transactions on Information Theory 72 5, 3051 (2026).
[6] Gereon Koßmann and Mark M. Wilde, "Semidefinite Optimization of the Quantum Relative Entropy of Channels", IEEE Transactions on Information Theory 72 4, 2378 (2026).
[7] Pritam Sarkar, Diptiman Sen, and Arnab Sen, "Metric response of relative entropy: A universal indicator of quantum criticality", Physical Review B 113 18, 184415 (2026).
[8] Mario Berta, Ludovico Lami, and Marco Tomamichel, "Continuity of Entropies via Integral Representations", IEEE Transactions on Information Theory 71 3, 1896 (2025).
[9] Anna Jenčová, "Recoverability of quantum channels via hypothesis testing", Letters in Mathematical Physics 114 1, 31 (2024).
[10] Christoph Hirche and Marco Tomamichel, "Quantum Rényi and f-Divergences from Integral Representations", Communications in Mathematical Physics 405 9, 208 (2024).
[11] Paula Belzig, Li Gao, Graeme Smith, and Peixue Wu, "Reverse-Type Data Processing Inequality", Communications in Mathematical Physics 406 12, 295 (2025).
[12] Gereon Koßmann and René Schwonnek, "Optimising the relative entropy under semidefinite constraints", npj Quantum Information 12 1, 23 (2026).
[13] Haojian Li and Xiaojing Yan, "Unitary orbit optimization of quantum f-divergence", Quantum Information Processing 24 3, 70 (2025).
[14] Milán Mosonyi, Gergely Bunth, and Péter Vrana, "Geometric relative entropies and barycentric Rényi divergences", Linear Algebra and its Applications 699, 159 (2024).
[15] Patrick Andriolo, Esteban Vasquez, Elizabeth Agudelo, Max Riegler, Matej Pivoluska, and Gláucia Murta, "Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs", Brazilian Journal of Physics 56 4, 157 (2026).
[16] Sreejith Sreekumar and Mario Berta, "Limit Distribution Theory for Quantum Divergences", IEEE Transactions on Information Theory 71 1, 459 (2025).
[17] Gereon Koßmann and Mark M. Wilde, "Semidefinite optimization of the quantum relative entropy of channels", arXiv:2410.16362, (2024).
[18] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li, "Relative entropy decay and complete positivity mixing time", arXiv:2209.11684, (2022).
[19] Milán Mosonyi, Gergely Bunth, and Péter Vrana, "Geometric relative entropies and barycentric Rényi divergences", arXiv:2207.14282, (2022).
[20] Ruben Ibarrondo and Daniel Stilck França, "Average Contraction Coefficients of Quantum Channels", arXiv:2508.08214, (2025).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 23:47:16) and SAO/NASA ADS (last updated successfully 2026-08-17 23:47:17). The list may be incomplete as not all publishers provide suitable and complete citation data.
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