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.

翻译：光滑的Csiszár $f$-散度可以表示为所谓曲棍球棒散度的积分形式。这一特性自然地引出了基于量子曲棍球棒散度的量子推广形式，本文对此展开研究。通过该构造方法，Kullback-Leibler散度泛化为Umegaki相对熵，并呈现为Frenkel近期发现的积分形式。研究发现，基于新型量子$f$-散度定义的Rényi散度通常不满足可加性，但其正则化形式却能惊人地统一两个重要的量子Rényi散度家族：当$\alpha < 1$时退化为Petz Rényi散度，当$\alpha > 1$时退化为夹层Rényi散度。此外，我们发现所有算子凸函数$f$对应的新型量子$f$-散度的压缩系数均退化为经典情形，这解决了Lesniewski与Ruskai提出的若干长期猜想。本文推导了包括新型逆向Pinsker不等式在内的多种不等式，该类不等式在差分隐私领域具有应用价值，并进一步探讨了新型散度在其他场景中的应用。