Quantum R\'enyi and $f$-divergences from integral representations

• URL: http://arxiv.org/abs/2306.12343v2
• Date: Wed, 16 Aug 2023 16:01:26 GMT
• Title: Quantum R\'enyi and $f$-divergences from integral representations
• Authors: Christoph Hirche, Marco Tomamichel
• Abstract summary: Smooth Csisz'ar $f$-divergences can be expressed as integrals over so-called hockey stick divergences. We find that the R'enyi divergences defined via our new quantum $f$-divergences are not additive in general. We derive various inequalities, including new reverse Pinsker inequalites with applications in differential privacy.
• Score: 13.869502085838452
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Smooth Csisz\'ar $f$-divergences can be expressed as integrals over so-called hockey stick divergences. This motivates a natural quantum generalization in terms of quantum Hockey stick divergences, which we explore here. Using this recipe, the Kullback-Leibler divergence generalises to the Umegaki relative entropy, in the integral form recently found by Frenkel. We find that the R\'enyi divergences defined via our new quantum $f$-divergences are not additive in general, but that their regularisations surprisingly yield the Petz R\'enyi divergence for $\alpha < 1$ and the sandwiched R\'enyi divergence for $\alpha > 1$, unifying these two important families of quantum R\'enyi divergences. Moreover, we find that the contraction coefficients for the new quantum $f$ divergences collapse for all $f$ that are operator convex, mimicking the classical behaviour and resolving some long-standing conjectures by Lesniewski and Ruskai. We derive various inequalities, including new reverse Pinsker inequalites with applications in differential privacy and also explore various other applications of the new divergences.

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