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arXiv:2607.08760

Hockey stick fff-divergences

Fumio Hiai, Milán Mosonyi, Marco Tomamichel

quant-phcs.ITmath-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations
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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.

Comments
See the concurrent and independent work https://doi.org/10.48550/arXiv.2607.05195 for the extension of hockey stick $f$-divergences to von Neumann algebras

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In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum fff-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 fff-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 fff-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 ααα-divergences coincide with the Petz-type Rényi divergences for α∈(0,1)α\in(0,1)α∈(0,1) and with the sandwiched Rényi divergences for α>1α>1α>1. Moreover, we give some partial results on the characterization of when different notions of quantum fff-divergences give the same value on a pair of quantum states.
We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero--one testing law, and the subchannel asymptotic equipartition property.
Integral representations of fff-divergences for general von Neumann algebras
We define and analyze hockeystick divergences and fff-divergences for normal positive functionals on general von Neumann algebras, generalizing and unifying previous work in classical probability and finite-dimensional von Neumann algebras. All the main properties of these state distinguishability measures (including in particular monotonicity, convexity, semicontinuity, bounds, state discrimination, data processing inequality) are derived from properties of the Jordan decomposition of selfadjoint normal functionals. This is done by representing the fff-divergences as integrals over hockeystick divergences, and their significance in quantum hypothesis testing is reviewed. The f0f_0f0​-divergence given by the information function f0(t)=tln⁡tf_0(t) = t \ln tf0​(t)=tlnt is shown to coincide with Araki's relative entropy, extending results of Frenkel to general von Neumann algebras.
The hockey-stick divergence is a fundamental quantity characterizing several statistical privacy frameworks that ensure privacy for classical and quantum data. In such quantum privacy frameworks, the adversary is allowed to perform all possible measurements. However, in practice, there are typically limitations to the set of measurements that can be performed. To this end, here, we comprehensively analyze the measured hockeystick divergence under several classes of practically relevant measurement classes. We prove several of its properties, including data processing and convexity. We show that it is efficiently computable by semi-definite programming for some classes of measurements and can be analytically evaluated for Werner and isotropic states. Notably, we show that the measured hockeystick divergence characterizes optimal privacy parameters in the quantum pufferfish privacy framework. With this connection and the developed technical tools, we enable methods to quantify and audit privacy for several practically relevant settings. Lastly, we introduce the measured hockey-stick divergence of channels and explore its applications in ensuring privacy for channels.
We prove a generalized quantum Stein lemma for i.i.d. normal states against convex, tensor-stable families on arbitrary von Neumann algebras. Assuming the existence of an alternative state with finite relative entropy from the null state, we show that, at every type-I error tolerance ε∈(0,1)\varepsilon\in (0,1)ε∈(0,1), the worst case type-II error exponent is achieved with the regularized relative entropy with a strong converse. The proof combines the integral representation of relative entropy by hockey-stick divergences, a modular testing bound, and convex minimax argument.
Maximal quantum fff-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum fff-divergences along unitary orbits of two quantum states. For any operator convex function f:(0,+∞)→Rf:(0,+\infty)\to\mathbb{R}f:(0,+∞)→R, we determine the exact minimum and maximum of U⟼S^f(ρ∥U∗σU) U \longmapsto \widehat S_f(ρ\|U^*σU) U⟼Sf​(ρ∥U∗σU) over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of ρρρ and σσσ, while the maximum corresponds to pairing the decreasing eigenvalues of ρρρ with the increasing eigenvalues of σσσ. Hence, the range of the maximal quantum fff-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum fff-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum fff-divergences.
Data-processing is a desired property of classical and quantum divergences and information measures. In information theory, the contraction coefficient measures how much the distinguishability of quantum states decreases when they are transmitted through a quantum channel, establishing linear strong data-processing inequalities (SDPI). However, these linear SDPI are not always tight and can be improved in most of the cases. In this work, we establish non-linear SDPI for quantum hockey-stick divergence for noisy channels that satisfy a certain noise criterion. We also note that our results improve upon existing linear SDPI for quantum hockey-stick divergences and also non-linear SDPI for classical hockey-stick divergence. We define FγF_γFγ​ curves generalizing Dobrushin curves for the quantum setting while characterizing SDPI for the sequential composition of heterogeneous channels. In addition, we derive reverse-Pinsker type inequalities for fff-divergences with additional constraints on hockey-stick divergences. We show that these non-linear SDPI can establish tighter finite mixing times that cannot be achieved through linear SDPI. Furthermore, we find applications of these in establishing stronger privacy guarantees for the composition of sequential private quantum channels when privacy is quantified by quantum local differential privacy.