Integral formula for quantum relative entropy implies data processing inequality

Péter E. Frenkel

Eötvös University, Institute of Mathematics, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary
Rényi Institute, Budapest, Reáltanoda u. 13-15, 1053 Hungary

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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.

Umegaki's quantum relative entropy, introduced in 1959, is a fundamental measure of dissimilarity of two quantum states. The main result of the paper is a new integral formula relating the quantum relative entropy to the trace norms of linear combinations of the two states. This leads to integral formulas for the higher order directional derivatives of von Neumann entropy and to a better understanding of data processing inequalities. It also has applications to the general probabilistic model of information theory.

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].

► BibTeX data

► References

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Cited by

[1] Anna Jenčová, "Recoverability of quantum channels via hypothesis testing", Letters in Mathematical Physics 114 1, 31 (2024).

[2] Milán Mosonyi, Gergely Bunth, and Péter Vrana, "Geometric relative entropies and barycentric Rényi divergences", Linear Algebra and its Applications (2024).

[3] Christoph Hirche and Marco Tomamichel, "Quantum Rényi and $f$-divergences from integral representations", arXiv:2306.12343, (2023).

[4] Milán Mosonyi, Gergely Bunth, and Péter Vrana, "Geometric relative entropies and barycentric Rényi divergences", arXiv:2207.14282, (2022).

[5] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li, "Relative entropy decay and complete positivity mixing time", arXiv:2209.11684, (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 07:07:50) and SAO/NASA ADS (last updated successfully 2024-06-22 07:07:51). The list may be incomplete as not all publishers provide suitable and complete citation data.