Hockey stick $f$-divergences

Abstract

In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025. In particular, we consider non-normalized states and hockey stick $f$-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from HircheTomamichel_integral,LiuHircheCheng2025 to this setting. Our main results here are the representation of the hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in LiuHircheCheng2025, an extension of Jen cová's result Jencova2023 on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in HircheTomamichel_integral showing that the regularized hockey stick Rényi $α$