Hockey stick $f$-divergences
Source: arXiv:2607.08760 · Published 2026-07-09 · By Fumio Hiai, Milán Mosonyi, Marco Tomamichel
TL;DR
This paper provides a comprehensive and unified treatment of a recently introduced class of quantum divergences known as hockey stick f-divergences, which extend classical f-divergences to the quantum domain via hockey stick divergences. The authors significantly generalize previous studies by considering non-normalized quantum states, a broader family of quantum hockey stick divergences, and an integral representation parameterized by an additional real number. They also extend the theory beyond finite-dimensional Hilbert spaces to general von Neumann algebras, thereby encompassing infinite-dimensional quantum systems and classical settings in one framework. Central to their contributions are novel representations of hockey stick f-divergences using Neyman-Pearson error probabilities, extensions of results concerning reversibility detection of quantum channels, and establishing that regularized hockey stick Rényi divergences coincide with the widely studied Petz-type divergences for α ∈ (0,1) and sandwiched Rényi divergences for α > 1. The paper also provides partial characterizations of when differing definitions of quantum f-divergences yield the same values, advancing the unification and understanding of quantum distinguishability measures.
Key findings
- Hockey stick f-divergences admit an integral representation in terms of hockey stick divergences parametrized by a real parameter a > 0, extending classical decompositions to the quantum setting (Corollary III.15, III.16).
- The hockey stick f-divergences can be represented explicitly via Neyman-Pearson error probabilities on pairs of quantum states, generalizing earlier finite-dimensional results to von Neumann algebra settings (Sections IV.C, V.C).
- Regularized hockey stick Rényi α-divergences coincide exactly with Petz-type Rényi divergences for α in (0,1) and with sandwiched Rényi divergences for α > 1, confirming a unifying link between these important quantum divergences (Sections VI.B–VI.D).
- An extension of Jenčová's criterion for detecting reversibility of quantum channels is proven using hockey stick divergences on state pairs, linking channel properties to distinguishability measures (Section V.E).
- New results characterize conditions under which various quantum f-divergences—including measured, maximal, and hockey stick variants—yield identical values on pairs of quantum states, partially unifying diverse quantum f-divergence definitions (Sections VII.A–VII.C).
- The theory extends to non-normalized states, allowing for more general applications beyond density operator pairs.
- Earlier integral decompositions of the Umegaki relative entropy and classical f-divergences are unified and generalized with more flexible parameterizations and operator algebra contexts.
- The integral framework includes handling of differentiability, joint lower semicontinuity, and martingale convergence of measured hockey stick divergences in von Neumann algebras.
Threat model
The adversary considered is an abstract quantum hypothesis tester attempting to distinguish between two quantum states (or non-normalized positive operators) within a von Neumann algebraic framework. The adversary's capability is bounded by quantum measurement theory, described via POVMs and Neyman-Pearson tests, with error probabilities capturing performance. The adversary cannot circumvent the operator-algebraic constraints such as positivity or state normalization, nor perform operations beyond the algebra on which the states are defined.
Methodology — deep read
The paper's methodology is primarily theoretical and mathematical, rooted in functional analysis, operator algebras, and quantum information theory. The authors start by setting a threat model that considers pairs of quantum states (possibly non-normalized) in finite-dimensional Hilbert spaces and generalized von Neumann algebra contexts—allowing infinite dimension and non-commutative algebras—which are natural generalizations of classical probability measures. The adversary model corresponds to distinguishing these states via quantum hypothesis testing, where Neyman-Pearson tests and error probabilities serve as operational quantities expressing distinguishability.
They derive classical f-divergences via the perspective function of convex functions f on (0,∞), extended by integral decompositions involving hockey stick functions defined as (id - t)^+ and (id - t)^-. These classical hockey stick divergences link to Neyman-Pearson tests and error probabilities (Section III). This setting is then quantized by replacing probability measures with density operators or positive operators in von Neumann algebras. The authors construct quantum hockey stick divergences by considering measured versions (optimized over POVMs), maximal versions, and maximal hockey stick versions of these classical quantities.
The main technical contribution is expressing the quantum hockey stick f-divergences as integrals over parameter t of hockey stick divergences weighted by the Lebesgue-Stieltjes measure induced by the derivative of f'. This involves detailed manipulations of operator inequalities and spectral decompositions, as well as the use of integral representations adapted to von Neumann algebra settings, leveraging Haagerup’s reduction techniques.
They investigate differentiability properties of f, derive representation formulas for the divergences via Neyman-Pearson error probabilities, and extend these concepts rigorously to general von Neumann algebras beyond finite-dimensional matrices. They use operator algebraic tools to ensure measurability and lower semicontinuity of the divergences and establish convergence properties under martingales.
For regularization, they consider infinite iterations (tensor powers) of quantum states and prove that the regularized hockey stick Rényi α-divergences match known Petz or sandwiched Rényi divergences depending on the order α. This requires careful analysis of the asymptotic behavior and continuity under tensor products.
They also study equality conditions: when different quantum f-divergences coincide on given pairs of states, using algebraic and functional analytic criteria.
The paper does not present empirical data or experiments, as it focuses on analytical formula derivations and proofs. The rigorous step-by-step development includes identifying the functional analytical definitions, constructing integral decompositions, proving operator inequalities, and extending finite-dimensional results to infinite-dimensional von Neumann algebras. They leverage prior results from Frenkel, Jenčová, and others and integrate them into this broader framework. The full proofs span multiple sections and appendices, many relying on spectral calculus, properties of operator convex functions, and Radon-Nikodym derivatives in operator algebras. Code and numerical reproducibility are not applicable due to the theoretical nature.
Technical innovations
- Generalization of quantum hockey stick f-divergences to non-normalized states and to the setting of general von Neumann algebras beyond finite dimensions.
- Integral decomposition of quantum f-divergences using an additional real parameter to unify and extend classical and quantum divergence representations.
- Representation of quantum hockey stick f-divergences in terms of Neyman-Pearson error probabilities within von Neumann algebra contexts, expanding prior finite-dimensional results.
- Proof that regularized hockey stick Rényi α-divergences exactly coincide with Petz-type divergences for α ∈ (0,1) and sandwiched Rényi divergences for α > 1, linking multiple quantum divergence families.
Limitations
- The results are mostly analytical and abstract, lacking explicit numerical examples or empirical validation.
- The paper does not address robustness or behavior of these divergences under adversarial perturbations or noisy quantum channels.
- Extension to non-von Neumann algebraic forms or infinite measures outside standard operator algebra settings is not discussed.
- No algorithmic or computational aspects of estimating these divergences in practice are provided.
- The equality characterizations of when different quantum f-divergences coincide remain partial and incomplete in some regimes.
- Some technical assumptions (e.g., differentiability and convexity conditions on f) limit the generality of the integral representations.
Open questions / follow-ons
- Can the partial equality characterizations between different quantum f-divergences be completed or refined for broader function classes and state pairs?
- How can hockey stick f-divergences be effectively computed or approximated for high-dimensional or infinite-dimensional quantum states in practice?
- What is the operational significance of hockey stick f-divergences beyond hypothesis testing, e.g., in quantum communication or cryptography?
- How robust are hockey stick f-divergences and their characterizations under noise, adversarial perturbations, or imperfect measurements?
Why it matters for bot defense
While this paper is highly theoretical and focused on quantum information theory and operator algebras, its core insights about f-divergences and their integral decompositions via hockey stick divergences speak broadly to the theory of statistical distinguishability measures. For bot-defense and CAPTCHA practitioners, these advanced notions contribute foundational understanding of how to quantify distinguishability under complex non-commutative settings that could arise in secure multi-party or quantum-enhanced bot detection systems. The connection to Neyman-Pearson error probabilities aligns with the use of statistical tests to separate benign human behavior from adversarial bot behavior. However, the direct applicability is limited because the divergences studied here operate in infinite-dimensional quantum spaces rather than classical or practical CAPTCHA models. Nonetheless, the novel integral representations and connections to error probabilities provide theoretical inspiration for designing new, possibly more powerful f-divergence-based distinguishers for bot detection that incorporate operator-algebraic or high-dimensional features. Bot-defense engineers could monitor this line of work to inform future developments in adversarial distinguishability measures beyond classical IP or behavioral metrics.
Cite
@article{arxiv2607_08760,
title={ Hockey stick $f$-divergences },
author={ Fumio Hiai and Milán Mosonyi and Marco Tomamichel },
journal={arXiv preprint arXiv:2607.08760},
year={ 2026 },
url={https://arxiv.org/abs/2607.08760}
}