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 \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.

• The paper introduces quantum hockey stick f-divergences, a novel measure generalizing classical f-divergences to handle non-commutative and non-normalized settings.
• It establishes integral representation theorems that connect these divergences to Neyman-Pearson error probabilities in quantum hypothesis testing.
• The framework bridges multiple quantum divergence types by linking Petz-type, maximal, and sandwiched Rényi divergences and characterizing quantum channel reversibility.

Hockey Stick $f$-Divergences: A Technical Overview

Introduction

The paper "Hockey stick $f$-divergences" (2607.08760) systematically develops and extends a novel class of quantum $f$-divergences, constructed using quantum analogs of classical 'hockey stick' divergences. These divergences generalize operationally relevant statistical distance measures between quantum states and are central to quantum information theory, particularly in the context of hypothesis testing, entropy, and channel discrimination. The work provides a unified framework for these divergences, including non-normalized states, extensions to general von Neumann algebras, and integral representations with additional parameters, and establishes strong connections to Neyman-Pearson theory and reversibility of quantum channels.

Theoretical Foundation and Definitions

The starting point is the classical $f$-divergence, which, for probability measures $P$ and $Q$ and a convex function $f : (0, \infty) \to \mathbb{R}$, takes the form

$$D_f(P \| Q) = \int f\left( \frac{dP}{dQ} \right) dQ.$$

This framework encompasses measures such as Kullback-Leibler divergence and Rényi divergences. The hockey stick divergence corresponds to choosing piecewise linear $f$, e.g., $(x-t)_+$, yielding

$f$0

which operationally connects to error exponents in hypothesis testing.

Quantum extensions are nontrivial due to noncommutativity. The Petz-type, maximal, and minimal (measured) quantum $f$1-divergences represent prominent quantum analogs. The paper encapsulates these within a general formalism using quantum hockey stick divergences. The 'quantum hockey stick' is defined through

$f$2

with analogous definitions for the negative part.

The unifying construction is the hockey stick $f$3-divergence:

$f$4

Key results show independence from the choice of $f$5, generalizing the classical integral representation to the quantum case.

Main Results

Extensions and Representation Theorems

The authors extend the theory in three key directions:

• Non-normalized States and General von Neumann Algebras: Quantum $f$6-divergences are constructed not only on normalized density operators but also for arbitrary positive trace class operators and, more generally, normal functionals on von Neumann algebras. This functional-analytic generality makes the results applicable well beyond finite dimensions.
• Integral Representation and Parameterization: The paper introduces a more general integral decomposition parameterized by an additional real variable, generalizing previous constructions that were restricted to normalized states or specific normalization conventions.
• Equivalence and Limit Relations: Notably, it is shown that for specific parameter ranges, the regularized 'hockey stick' Rényi divergences coincide with the Petz-type for $f$7 and with sandwiched (a.k.a. maximal) Rényi divergences for $f$8. This provides a new perspective on the operational meaning and mathematical structure of these widely-used quantum divergences.

Operationally-Relevant Representations

A central technical achievement is the representation of hockey stick $f$9-divergences in terms of Neyman-Pearson error probabilities. Explicitly,

$f$0

where $f$1 captures the type II error under the Neyman-Pearson test at threshold $f$2. The mapping between error exponents and divergence integrals provides a direct link between mathematical divergence properties and operational discrimination tasks in quantum hypothesis testing.

Channel Reversibility and Sufficiency

Extending results of Jenčová, the authors characterize reversibility (sufficiency) of a quantum channel on a pair of states via the hockey stick divergences. If equality holds in the data processing inequality for these divergences for all relevant $f$3 or for a sufficiently rich family (specified by support of $f$4), then the channel is reversible (i.e., preserves the distinguishability structure given by $f$5-divergences).

Characterization of Equality Cases

Partial results are provided on the characterization of when different quantum $f$6-divergences coincide on a pair of (possibly non-normalized) quantum states. The interplay of commutativity, extremality of quantum operations (measurements, channels), and the structure of reverse tests is elucidated.

Strong Results and Novel Claims

• Unified Approach: The central claim is that the new family of hockey stick $f$7-divergences is not equivalent to any previously studied quantum $f$8-divergence (including Petz, maximal, or measured types), except in classical or commuting limits.
• Operational Characterization: The representation in terms of Neyman-Pearson probabilities is claimed to extend not only to finite-dimensional cases but to the general von Neumann algebraic setting, including infinite dimensions.
• Regularized Limits: The regularized hockey stick Rényi divergences interpolate sharply between Petz and sandwiched Rényi divergences, up to "almost everywhere" identification in the limit of tensor powers ("i.i.d. regularization") for $f$9 outside 1.

Implications and Potential Impact

Theoretical Implications

This work bridges the gap between operational distinguishability (via hypothesis-testing error exponents) and mathematically natural divergence functionals in the quantum regime. The integral representation unifies and generalizes known constructions, offering a potentially canonical quantization of classical $f$0-divergences for arbitrary operator-algebraic contexts.

The equivalence of regularized hockey stick and sandwiched Rényi divergences, and their link to Petz-type in different $f$1 ranges, puts the operational meaning of these quantities on firmer foundations.

Further, channel reversibility criteria grounded in these divergences offer new tools for quantum Markov chain theory, resource theory structure, and the algebraic study of quantum sufficiency.

Practical and Future Directions

The explicit connection to hypothesis-testing suggests possible applications in quantum information tasks where strong error exponents or finite blocklength effects are significant (e.g., quantum channel coding, cryptography, or metrology).

Extensions to generalized resource theories and non-i.i.d. settings seem plausible, especially leveraging the general von Neumann algebraic formalism.

It is likely that these divergences will appear in characterizations of non-asymptotic rates and strong converse properties in quantum Shannon theory, and possibly in understanding asymptotic spectral rates in quantum statistical mechanics.

Future research directions include:

• Full classification of equality cases for all types of quantum $f$2-divergences and deeper analysis of their operational significance
• Extension to quantum dynamic resource theories (resource non-generating channels, etc.)
• Development of efficient numerical methods for computing hockey stick divergences in large-dimensional systems
• Exploration of connections to other generalized divergences, such as monotone metrics and geodesics in quantum information geometry

Conclusion

"Hockey stick $f$3-divergences" (2607.08760) delivers a rigorous and comprehensive extension of $f$4-divergence theory into the quantum domain. It provides both foundational mathematical results—encompassing generalizations to the von Neumann algebraic setting and integral representations—and operational links to hypothesis testing and channel reversibility. The paper's results are positioned to be influential in both quantum information theory and mathematical physics, particularly in areas where operational quantum distinguishability and symmetry properties are of fundamental importance.