Hockey stick $f$-divergences

2026-07-09

AI Summary

This work aims to unify and extend the theory of quantum $f$-divergences based on the hockey-stick divergence, addressing key challenges concerning unnormalized states, general von Neumann algebraic settings, and characterizations of equivalence among divergences. By leveraging operator-theoretic methods, integral representations, and techniques from von Neumann algebras, the authors establish—for the first time—the hockey-stick $f$-divergence framework in infinite-dimensional general von Neumann algebras. They connect this divergence to Neyman–Pearson hypothesis testing error probabilities and unify regularized forms of both Petz-type and sandwiched Rényi divergences. This contribution not only generalizes existing finite-dimensional results to the infinite-dimensional setting but also yields a novel criterion for quantum channel reversibility and partially resolves the long-standing problem of characterizing equivalence among quantum $f$-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 \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.

Problem

• quantum f-divergences
• hockey stick divergences
• von Neumann algebras
• Rényi divergences
• reversibility detection

Innovation

• quantum f-divergences
• hockey stick divergence
• von Neumann algebras
• Rényi divergences
• Neyman-Pearson error probability