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 $f$-散度可表示为所谓冰球棍散度的积分形式。这一性质自然启发我们通过量子冰球棍散度构建相应的量子推广。根据这一构造方法，Kullback-Leibler散度可泛化为Umegaki相对熵，其积分形式与Frenkel近期发现的结果一致。研究表明，基于新量子$f$-散度定义的Rényi散度通常不满足可加性，但其正则化形式出人意料地统一了量子Rényi散度的两大重要分支：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，所有算子凸函数$f$对应的新量子$f$-散度的压缩系数均出现坍缩现象，这一特性复现了经典情形并解决了Lesniewski与Ruskai提出的长期猜想。本文推导了包括新型逆向Pinsker不等式在内的多种不等式，揭示了其在差分隐私中的应用价值，同时探讨了新散度的其他潜在应用场景。