Quantum Rényi and f-divergences from integral representations
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Smooth Csiszár 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 Rényi divergences defined via our new quantum f-divergences are not additive in general, but that their regularisations surprisingly yield the Petz Rényi divergence for α < 1 and the sandwiched Rényi divergence for α > 1, unifying these two important families of quantum Rényi 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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