Quantum Rényi and $f$-divergences from integral representations

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 inequalities with applications in differential privacy and explore various other applications of the new divergences.