Hockey stick $f$-divergences
2026-07-09 • Information Theory
Information Theory
AI summaryⓘ
The authors present a detailed study of quantum f-divergences, which are ways to measure differences between quantum states, extending previous work on a specific type called hockey stick divergences. They explore these concepts for more general quantum states and mathematical settings, including infinite-dimensional cases called von Neumann algebras. Key results include expressing these divergences through error probabilities in quantum hypothesis testing, and relating them to known quantum divergences like the Petz and sandwiched Rényi divergences. They also investigate conditions under which different definitions of f-divergences agree on given quantum states.
quantum f-divergenceshockey stick divergencesquantum statesvon Neumann algebrasNeyman-Pearson error probabilitiesquantum hypothesis testingPetz Rényi divergencessandwiched Rényi divergencesquantum channelsreversibility detection
Authors
Fumio Hiai, Milán Mosonyi, Marco Tomamichel
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 \cite{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 \cite{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 \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{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 \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$-divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$. Moreover, we give some partial results on the characterization of when different notions of quantum $f$-divergences give the same value on a pair of quantum states.