Pith. sign in

REVIEW 9 minor 55 references

Hockey stick $f$-divergences

T0 review · 0 major / 9 minor · reviewed 2026-07-10 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Hockey stick divergences unify quantum Rényi theory in von Neumann algebras

desk verdict Solid unification and extension of hockey stick f-divergence theory to von Neumann algebras; one acknowledged coverage gap in the α>1 regularization read the letter →

arxiv 2607.08760 v1 pith:NXWWGEOE submitted 2026-07-09 quant-ph cs.ITmath-phmath.ITmath.MP

Fumio Hiai , Milán Mosonyi , Marco Tomamichel This is my paper · ORCID
classification quant-phcs.ITmath-phmath.ITmath.MP
keywords divergenceshockeystickquantumciteintegralresultsalpha
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper builds a unified theory of quantum f-divergences constructed by integrating families of quantum hockey stick divergences—quantities measuring how much one quantum state exceeds a scaled copy of another. The central construction takes a convex function f, decomposes it into a linear part plus an integral over its second derivative measure, and replaces each classical hockey stick divergence in that integral with a chosen quantum analogue. The resulting hockey stick f-divergence is shown to be independent of the auxiliary parameter a appearing in the decomposition, and it recovers the Petz-type f-divergence when the Petz-type hockey stick divergences are used. The paper extends the entire framework from finite-dimensional matrix algebras to general von Neumann algebras, proves that the measured hockey stick f-divergences admit a representation in terms of Neyman-Pearson error probabilities, and establishes that the regularized measured hockey stick Rényi α-divergences converge to the Petz-type Rényi divergence for α in (0,1) and to the sandwiched Rényi divergence for α greater than 1. Partial results characterize when different quantum f-divergence notions coincide on a pair of states, connecting equality cases to commutativity of the states.

What carries the argument

The perspective function Pf(x,y) = y·f(x/y) and its integral decomposition via Lebesgue-Stieltjes integration against df′; the quantum hockey stick divergences D^(q)_{(id−t)+} and their measured, Petz-type, and maximal variants; the Neyman-Pearson test N_t(ϱ∥σ) = {ϱ − tσ > 0} and associated error probabilities β_{0,t}, β_{1,t}; Haagerup's reduction theorem for passing from general von Neumann algebras to finite approximants; and the regularized (asymptotic) quantities D̄_α = lim_{n→∞} (1/n) D(ϱ^⊗n ∥ σ^⊗n).

What would settle it

A pair of states (ϱ, σ) in a von Neumann algebra for which the sandwiched Rényi divergence e^{D_α(ϱ∥σ)} is finite at some α > 1 but infinite for all β > α, and for which the regularized measured hockey stick Rényi divergence either does not exist or differs from e^{D_α(ϱ∥σ)}, would contradict the paper's central regularized identity. Additionally, a non-commuting pair of states in an injective von Neumann algebra for which the measured and measured-hockey-stick f-divergences coincide for some f with full support of df′ would refute the equality-case characterization in Theorem VII.1.

Watch

Extended reading notes

Core claim

The load-bearing mechanism is the integral decomposition identity: any convex function f can be written as a linear part plus an integral of hockey stick functions (id − t)_+ and (id − t)_− weighted by the second-derivative measure df′. By substituting quantum hockey stick divergences for the classical ones in this integral, one obtains a quantum f-divergence that is well-defined (independent of the auxiliary parameter a) whenever the hockey stick family is bounded, measurable, and non-negative. The paper proves that this construction, when specialized to the measured hockey stick divergences in general von Neumann algebras, yields a representation purely in terms of the Neyman-Pearson typeI

Load-bearing premise

For α > 1, the proof that the regularized hockey stick Rényi divergence equals the sandwiched Rényi divergence requires that either the sandwiched Rényi divergence is infinite or it is finite in a neighborhood of α. The argument breaks down when the divergence is finite at α but infinite for all larger exponents, leaving a boundary case unresolved.

Editorial extensions

If this is right

  • The regularized measured hockey stick Rényi divergences provide a single construction that recovers both the Petz-type (α < 1) and sandwiched (α > 1) Rényi divergences, suggesting these are not ad hoc choices but natural endpoints of one integral family.
  • The Neyman-Pearson representation of measured hockey stick f-divergences in von Neumann algebras gives a direct operational meaning: these divergences are built entirely from optimal hypothesis-testing error probabilities.
  • The sufficiency theorem (Theorem V.21) shows that reversibility of a quantum channel for a pair of states can be detected by equality of hockey stick divergences for all t, or even for a single f with full support of df′, extending classical statistical sufficiency to the quantum setting.
  • The equality-case analysis shows that when the measured and measured-hockey-stick f-divergences coincide, the states must commute—at least in injective von Neumann algebras—tying the gap between different quantum f-divergence notions directly to noncommutativity.
  • Frenkel's integral representation of quantum relative entropy (D = D^{meas,hs}_{x log x}) is extended to injective von Neumann algebras via martingale convergence.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 9 minor

Summary. This paper provides a systematic and unified treatment of quantum hockey stick $f$-divergences, extending recent results from [7, 23, 30] in several directions. The main contributions are: (1) a general framework allowing non-normalized states, more general notions of quantum hockey stick divergences, and an integral representation with an additional real parameter $a$; (2) extension of the theory to general von Neumann algebras, including a representation of measured hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities (Theorem V.11), a differentiability result for measured hockey stick divergences in the von Neumann algebra setting (Theorem V.3), and a reversibility/sufficiency theorem (Theorem V.21); (3) proofs that the regularized measured hockey stick Rényi $α$-divergences coincide with the sandwiched Rényi divergences for $α>1$ (Theorem VI.2) and with the Petz-type Rényi divergences for $α∈(0,1)$ (Proposition VI.13), both extended to general von Neumann algebras; and (4) partial results characterizing when different notions of quantum $f$-divergences coincide (Section VII). The paper is 68 pages with detailed proofs throughout.

Significance. The paper makes a substantial contribution to the theory of quantum divergences in von Neumann algebras. The extension of the Neyman-Pearson error probability representation (Theorem V.11) and the differentiability theorem (Theorem V.3) to general von Neumann algebras are technically demanding results requiring non-trivial machinery (Kosaki's norm continuity, Takesaki's structure theorem, Haagerup's $L^p$-spaces). The regularized identities (Theorem VI.2 and Proposition VI.13) are operationally significant, establishing that the regularized measured hockey stick Rényi divergences recover the standard sandwiched and Petz-type Rényi divergences. The reversibility theorem (Theorem V.21) extends Jenčová's finite-dimensional result to injective von Neumann algebras. The equality-case analysis in Section VII, while partial, identifies interesting open problems. The concurrent works [45, 53] noted by the authors indicate active interest in this area.

minor comments (9)
  1. Theorem VI.2 (§VI.C): The assumption that either $eD_β(ϱ∥σ)<+∞$ for some $β∈(α,+∞)$ or $eD_α(ϱ∥σ)=+∞$ leaves a gap when $eD_α(ϱ∥σ)<+∞$ but $eD_β(ϱ∥σ)=+∞$ for all $β>α$. Remark VI.3 acknowledges this honestly. The gap is in coverage, not correctness: the lower bound (VI.168) holds unconditionally, and the result is complete in finite dimensions (where $ϱ_0≤σ_0$ implies $ϱ≤cσ$). The authors might consider noting whether the truncation technique ($σ↦σ+εϱ$) used in Corollary VI.4's proof could close this gap, even if they do not pursue it here.
  2. Theorem V.21 (§V.F): The reversibility theorem assumes $M$ and $N$ are injective von Neumann algebras, with Remark V.22 noting this assumption can be removed due to [45]. Since [45] is concurrent, it would help the reader to briefly indicate which specific result from [45] removes the injectivity assumption (e.g., Frenkel's representation for general von Neumann algebras).
  3. Section VII.C: The equality case $D^{meas,hs}_f = D^{max,hs}_f$ is left open. Example VII.8 shows that condition (v) of Proposition VII.7 (namely $ϱ-(ϱ-tσ)^+ ≥ 0$ for all $t>0$) does not imply commutativity, even for qubits. The authors note the mechanism is 'metric rather than algebraic.' It would strengthen the paper to state explicitly whether the qubit analysis can be pushed further, or whether this is genuinely open even in dimension 2.
  4. Notation: The paper uses $D^q_{(id-t)_+}$, $D^{q,hs}_f$, $Q^{q,hs}_α$, etc. with superscripts $q∈{P, meas, max}$. While defined, the density of superscripts can make some equations hard to parse (e.g., the chain of inequalities in (IV.95)). Consider adding a notation summary table.
  5. Lemma V.5 proof (§V.B): The proof uses Takesaki's crossed product structure theorem and Haagerup's $L^1$-space construction inside $e_N$ (the $τ$-measurable operators affiliated with $N$). A brief remark recalling why $L^1(M)⊂ e_N$ holds in this setting (with a precise reference to [14, Chap. 9] or [51]) would aid readers less familiar with this construction.
  6. Proposition VI.13 (§VI.D): The condition $ϱ(1)≤σ(1)$ is assumed. It would be worth noting whether this is a normalization convention or a genuine restriction, and whether the result extends to general $ϱ, σ∈ M^*_{≥0}$ by rescaling.
  7. References: The paper cites [45] (arXiv:2607.05195) and [53] (arXiv:2604.08380) as concurrent works. Since these are very recent, the authors should verify that the final published versions do not affect any claims of priority or novelty.
  8. Typo in §VI.D, line after (VI.177): 'If If' should be 'If'.
  9. Remark IV.49 (§IV.F): The statement that $D^P_{(id-t)_+}$ is not monotone under CPTP maps for any $t∈(0,+∞)$ is attributed to 'a simple argument' but not shown. A one-line justification or reference would be helpful.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; self-citations are to starting points for genuine extensions, not load-bearing uniqueness claims

full rationale

The paper extends hockey stick f-divergence theory from finite dimensions to general von Neumann algebras. The main results—Theorem V.11 (Neyman-Pearson representation), Theorem V.21 (reversibility detection), Theorem VI.2 (regularized identity for α>1), Proposition VI.13 (regularized identity for α∈(0,1))—each require genuinely new mathematical arguments: Theorem V.3 on differentiability uses Kosaki's norm continuity and Takesaki's crossed product structure; Proposition V.9 (dF/dG=x) is proved from scratch handling both atomic and continuous parts; the α∈(0,1) upper bound uses Nussbaum-Szkola distributions in finite von Neumann algebras (Lemma VI.7, proved here) combined with Haagerup's reduction theorem. The self-citations to [23] (Hirche-Tomamichel) and [7] (Beigi-Hirche-Tomamichel) serve as starting points: the finite-dimensional versions are extended, not merely reproduced. The definition of hockey stick f-divergences (Lemma IV.30/V.126) is an integral decomposition of f into hockey stick divergences, which is a construction, not a circular derivation. The consistency check D^{P,hs}_f = D^P_f (Proposition V.2) is verified by direct spectral computation, not assumed. The regularized identities use external results (Lemma VI.1 from [39] by different authors, Lemma VI.5 from [40] by Ogata) for their lower bounds. The only minor self-citation burden is Theorem VII.1 and VII.3 citing [15] (Hiai's monograph) for the commutativity criterion via relative entropy, but this is a standard reference to an established result, not a load-bearing uniqueness theorem that would force the conclusion. Score 1 reflects this minor self-citation pattern that is not load-bearing for the central claims.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or postulated objects. All mathematical objects (hockey stick divergences, f-divergences, relative modular operators, Haagerup L^p-spaces) are standard in the literature. The 'hockey stick f-divergence' itself was introduced in prior work [23]; this paper extends it. The free parameter a is proven to be gauge (independent of choice), not a fitted parameter.

free parameters (1)
  • parameter a in integral representation = arbitrary in (0,+∞)
    The integral representation of hockey stick f-divergences includes a free real parameter a∈(0,+∞). Lemma IV.30 proves the representation is independent of a, so this is not a fitting parameter but a gauge freedom. Not a circularity concern.
assumptions (5)
  • standard math Haagerup's reduction theorem for σ-finite von Neumann algebras
    Used in Section VI.A to approximate general von Neumann algebras by finite ones with faithful tracial states. This is a published theorem from Takesaki, not introduced by this paper.
  • standard math Kosaki's norm continuity of the map ψ↦|ψ| on M*
    Used in the proof of Lemma V.5 (equation V.138) to establish lower semicontinuity of support projections. Standard result in von Neumann algebra theory.
  • standard math Existence of faithful normal semi-finite weights on von Neumann algebras for standard form representation
    Used in Section V.A to set up the relative modular operator and Haagerup L^p-space framework. Standard assumption in the theory.
  • domain assumption Finiteness condition eD_β(ϱ∥σ)<+∞ for some β>α in Theorem VI.2
    This is a load-bearing assumption for the α>1 regularization result. Without it, the upper bound proof fails. The paper acknowledges this limitation in Remark VI.3.
  • domain assumption Injectivity of von Neumann algebras M and N in Theorems V.21 and VII.1
    The reversibility theorem (Theorem V.21) and equality-case results (Theorem VII.1) assume injective von Neumann algebras to apply Corollary V.19 (Frenkel's representation). The paper notes this assumption can be removed using recent work [45].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hockey stick $f$-divergences." pith.science (2026). https://pith.science/paper/NXWWGEOE

@misc{pith2026260708760,
  author       = {Pith},
  title        = {Pith review of: Hockey stick $f$-divergences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXWWGEOE}},
  note         = {Machine review of arXiv:2607.08760}
}
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 cov\'a'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 R\'enyi $\alpha$-divergences coincide with the Petz-type R\'enyi divergences for $\alpha\in(0,1)$ and with the sandwiched R\'enyi divergences for $\alpha>1$. 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    2022-2.1.1- NL-2022-00004)

    MM was also partially supported by the National Research, Development and Innovation Office of Hungary (NKFIH) via the research grants K 146380 and EXCELLENCE 151342, and by the Ministry of Culture and Innovation and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary (Grant No. 2022-2.1.1...

    work page 2022

  2. [2]

    Accardi and C

    L. Accardi and C. Cecchini. Conditional expectations in von neumann algebras and a theorem of takesaki.J. Funct. Anal., 45:245–273, 1982

    work page 1982

  3. [3]

    S. M. Ali and S. D. Silvey. A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society: Series B (Methodological), 1966

    work page 1966

  4. [4]

    Quantum $f$-divergences via Nussbaum-Szko{\l}a Distributions in Semifinite von Neumann Algebras

    T. Anastasiadis and G. Androulakis. Quantumf-divergences via Nussbaum-Szko la distributions in semifinite von Neumann algebras. 2026. arXiv:2604.19853

    work page Pith review arXiv 2026

  5. [5]

    H. Araki. Relative entropy of states of von Neumann algebras ii.Publ. Res. Inst. Math. Sci., 13(3):173–192, 1977

    work page 1977

  6. [6]

    K. M. R. Audenaert, M. Nussbaum, A. Szkola, and F. Verstraete. Asymptotic error rates in quantum hypoth- esis testing.Communications in Mathematical Physics, 279:251–283, 2008

    work page 2008

  7. [7]

    Koenraad M. R. Audenaert and Nilanjana Datta.α-z-relative Renyi entropies.J. Math. Phys., 56:022202, 2015

    work page 2015

  8. [8]

    Some properties and applications of the new quantum $f$-divergences

    Salman Beigi, Christoph Hirche, and Marco Tomamichel. Some properties and applications of the new quantum f-divergences. arXiv:2501.03799, 2025

    work page Pith review arXiv 2025

Show all 55 references
  1. [9]

    V. P. Belavkin and P. Staszewski.C ∗-algebraic generalization of relative entropy and entropy.Ann. Inst. H. Poincar´ e Phys. Th´ eor., 37:51–58, 1982

    1982

  2. [10]

    D. S. Bernstein.Matrix mathematics: theory, facts, and formulas. Princeton University Press, 2009. 67

    2009

  3. [11]

    Scholz, and Marco Tomamichel

    Mario Berta, Volkher B. Scholz, and Marco Tomamichel. R´ enyi divergences as weighted non-commutative vector-valuedL p-spaces.Ann. Henri Poincar´ e, 19:1843–1867, 2018. arXiv:1608.05317

    2018 arXiv

  4. [12]

    Information type measure of difference of probability distributions and indirect observations

    Imre Csisz´ ar. Information type measure of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar., 2:299–318, 1967

    1967

  5. [13]

    Defining quantum divergences via convex optimization.Quantum, 5:387, January 2021

    Hamza Fawzi and Omar Fawzi. Defining quantum divergences via convex optimization.Quantum, 5:387, January 2021

    2021

  6. [14]

    P´ eter E. Frenkel. Integral formula for quantum relative entropy implies data processing inequality.Quantum, 7:1102, 2023. arXiv:2208.12194

    2023 arXiv

  7. [15]

    Hiai.Lectures on Selected Topics in von Neumann Algebras

    F. Hiai.Lectures on Selected Topics in von Neumann Algebras. EMS Press, Berlin, 2021

    2021

  8. [16]

    Hiai.Quantumf-Divergences in von Neumann Algebras: Reversibility of Quantum Operations

    F. Hiai.Quantumf-Divergences in von Neumann Algebras: Reversibility of Quantum Operations. Mathe- matical Physics Studies. Springer, Singapore, 2021

    2021

  9. [17]

    Hiai and H

    F. Hiai and H. Kosaki. Connections of unbounded operators and some related topics: von Neumann algebra case.Internat. J. Math., 32:2150024, 88 pp., 2021

    2021

  10. [18]

    Hiai and M

    F. Hiai and M. Tsukada. Strong martingale convergence of generalized conditional expectations on von neumann algebras.Trans. Amer. Math. Soc., 282:791–798, 1984

    1984

  11. [19]

    Quantumf-divergences in von Neumann algebras I

    Fumio Hiai. Quantumf-divergences in von Neumann algebras I. Standardf-divergences.J. Math. Phys., 59:102202, 2018

    2018

  12. [20]

    Quantumf-divergences in von Neumann algebras II

    Fumio Hiai. Quantumf-divergences in von Neumann algebras II. Maximalf-divergences.J. Math. Phys., 60:012203, 2019

    2019

  13. [21]

    Fumio Hiai and Anna Jenˇ cov´ a.α-z-R´ enyi divergences in von Neumann algebras: data processing inequality, reversibility, and monotonicity properties inα,z.Commun. Math. Phys., 405:271, 2024

    2024

  14. [22]

    Different quantumf-divergences and the reversibility of quantum operations

    Fumio Hiai and Mil´ an Mosonyi. Different quantumf-divergences and the reversibility of quantum operations. Reviews in Mathematical Physics, 29(7):1750023, 2017

    2017

  15. [23]

    Quantum R´ enyi divergences and the strong converse exponent of state discrimination in operator algebras.Annales Henri Poincar´ e, 24:1681–1724, 2023

    Fumio Hiai and Mil´ an Mosonyi. Quantum R´ enyi divergences and the strong converse exponent of state discrimination in operator algebras.Annales Henri Poincar´ e, 24:1681–1724, 2023. arXiv:2110.07320

    2023 arXiv

  16. [24]

    Quantum R´ enyi andf-divergences from integral representations

    Christoph Hirche and Marco Tomamichel. Quantum R´ enyi andf-divergences from integral representations. Commun. Math. Phys., 405(208), 2024. arXiv:2306.12343

    2024 arXiv

  17. [25]

    Jenˇ cov´ a

    A. Jenˇ cov´ a. R´ enyi relative entropies and noncommutativeLp-spaces.Ann. Henri Poincar´ e, 19:2513–2542,

  18. [26]

    Jenˇ cov´ a

    A. Jenˇ cov´ a. R´ enyi relative entropies and noncommutativeLp-spaces II.Annales Henri Poincar´ e, 22:3235– 3254, 2021. arXiv:1707.00047

    2021 arXiv

  19. [27]

    Recoverability of quantum channels via hypothesis testing.Letters in Mathematical Physics, 114(1), 2024

    Anna Jenˇ cov´ a. Recoverability of quantum channels via hypothesis testing.Letters in Mathematical Physics, 114(1), 2024. arXiv: 2303.11707

    2024 arXiv

  20. [28]

    R. V. Kadison. Strong continuity of operator functions.Pacific J. Math., 26:121–129, 1968

    1968

  21. [29]

    H. Kosaki. On the continuity of the mapφ/∫hortrightarrow|φ|from the predual of aW∗-algebra.J. Funct. Anal., 59:123–131, 1984

    1984

  22. [30]

    Device-independent quantum key distribution in the commuting operator framework

    Gereon Kossmann, Ren´ e Schwonnek, Po-Chieh Liu, and Hao-Chung Cheng. Device-independent quantum key distribution in the commuting operator framework. arXiv:2607.03579

    arXiv

  23. [31]

    Layer cake representations for quantum divergences

    Po-Chieh Liu, Christoph Hirche, and Hao-Chung Cheng. Layer cake representations for quantum divergences. arXiv:2507.07065, 2025

    2025 arXiv

  24. [32]

    Matsumoto

    K. Matsumoto. A new quantum version off-divergence. InNagoya Winter Workshop 2015: Reality and Measurement in Algebraic Quantum Theory, pages 229–273, 2018

    2015

  25. [33]

    Geometric relative entropies and barycentric R´ enyi diver- gences.Linear Algebra and its Applications, 699:159–276, 2024

    Mil´ an Mosonyi, Gergely Bunth, and P´ eter Vrana. Geometric relative entropies and barycentric R´ enyi diver- gences.Linear Algebra and its Applications, 699:159–276, 2024. arXiv:2207.14282

    2024 arXiv

  26. [34]

    Test-measured r´ enyi divergences.IEEE Transactions on Information Theory, 69(2):1074–1092, February 2023

    Mil´ an Mosonyi and Fumio Hiai. Test-measured r´ enyi divergences.IEEE Transactions on Information Theory, 69(2):1074–1092, February 2023. arXiv:2201.05477

    2023 arXiv

  27. [35]

    Quantum hypothesis testing and the operational interpretation of the quantum R´ enyi relative entropies.Communications in Mathematical Physics, 334(3):1617–1648, 2015

    Mil´ an Mosonyi and Tomohiro Ogawa. Quantum hypothesis testing and the operational interpretation of the quantum R´ enyi relative entropies.Communications in Mathematical Physics, 334(3):1617–1648, 2015

    2015

  28. [36]

    On quantum R´ enyi entropies: A new generalization and some properties.Journal of Mathematical Physics, 54(12):122203, December 2013

    Martin M¨ uller-Lennert, Fr´ ed´ eric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. On quantum R´ enyi entropies: A new generalization and some properties.Journal of Mathematical Physics, 54(12):122203, December 2013

    2013

  29. [37]

    An information-spectrum approach to classical and quantum hy- pothesis testing for simple hypotheses.IEEE Transactions on Information Theory, 53(2):534–549, February 2007

    Hiroshi Nagaoka and Masahito Hayashi. An information-spectrum approach to classical and quantum hy- pothesis testing for simple hypotheses.IEEE Transactions on Information Theory, 53(2):534–549, February 2007

    2007

  30. [38]

    E. Nelson. Notes on non-commutative integration.J. Funct. Anal., 15:103–116, 1974

    1974

  31. [39]

    The Chernoff lower bound for symmetric quantum hypothesis testing

    Michael Nussbaum and Arleta Szko la. The Chernoff lower bound for symmetric quantum hypothesis testing. The Annals of Statistics, 37:1040–1057, 2009

    2009

  32. [40]

    L. Gao O. Fawzi and M. Rahaman. Asymptotic equipartition theorems in von Neumann algebras.Ann. Henri Poincar´ e, 27:95–141, 2026

    2026

  33. [41]

    A generalization of powers–størmer inequality.Lett

    Yoshiko Ogata. A generalization of powers–størmer inequality.Lett. Math. Phys., 97:339–346, 2011

    2011

  34. [42]

    D. Petz. Quasi-entropies for states of a von Neumann algebra.Publ. Res. Inst. Math. Sci., 21:781–800, 1985

    1985

  35. [43]

    Quasi-entropies for finite quantum systems.Reports in Mathematical Physics, 23:57–65, 1986

    D´ enes Petz. Quasi-entropies for finite quantum systems.Reports in Mathematical Physics, 23:57–65, 1986

    1986

  36. [44]

    Sufficient subalgebras and the relative entropy of states of a von Neumann algebra.Commun

    D´ enes Petz. Sufficient subalgebras and the relative entropy of states of a von Neumann algebra.Commun. Math. Phys., 105:123–131, 1986

    1986

  37. [45]

    Sufficiency of channels over von Neumann algebras.Quart

    D´ enes Petz. Sufficiency of channels over von Neumann algebras.Quart. J. Math. Oxford Ser. (2), 153:97–108, 1988

    1988

  38. [46]

    Lechner R.C

    G. Lechner R.C. da Silva, M.B. Fr¨ ob and L. Sangaletti. Integral representations off-divergences for general 68 von Neumann algebras. 2026. arXiv:2607.05195

    2026 arXiv

  39. [47]

    On measures of entropy and information

    Alfr´ ed R´ enyi. On measures of entropy and information. InProc. 4th Berkeley Sympos. Math. Statist. and Prob., volume I, pages 547–561. Univ. California Press, Berkeley, California, 1961

    1961

  40. [48]

    Igal Sason and Sergio Verd´ u.f-divergence inequalities.IEEE Transactions on Information Theory, 62(11):5973–6006, 2016

    2016

  41. [49]

    Str˘ atil˘ a.Modular Theory in Operator Algebras

    S. Str˘ atil˘ a.Modular Theory in Operator Algebras. Editura Academiei and Abacus Press, Tunbridge Wells, 1981

    1981

  42. [50]

    Takesaki

    M. Takesaki. Duality for crossed products and the structure of von Neumann algebras of type iii.Acta Math., 131:249–310, 1973

    1973

  43. [51]

    Takesaki.Theory of operator algebras I, volume 124 ofEncyclopaedia of Mathematical Sciences

    M. Takesaki.Theory of operator algebras I, volume 124 ofEncyclopaedia of Mathematical Sciences. Springer- Verlag, 2002

    2002

  44. [52]

    Terp.L p spaces associated with von Neumann algebras

    M. Terp.L p spaces associated with von Neumann algebras. Notes, Copenhagen University, 1981

    1981

  45. [53]

    H. Umegaki. Conditional expectation in an operator algebraalgebra, IV: Entropy and information.Kodai Math. Sem. Rep., 14:59–85, 1962

    1962

  46. [54]

    van Luijk and H

    L. van Luijk and H. Wilming. Sufficiency and Petz recovery for positive maps. 2026. arXiv:2604.08380

    2026 arXiv

  47. [55]

    Wilde, Andreas Winter, and Dong Yang

    Mark M. Wilde, Andreas Winter, and Dong Yang. Strong converse for the classical capacity of entanglement- breaking and Hadamard channels via a sandwiched R´ enyi relative entropy.Communications in Mathematical Physics, 331(2):593–622, October 2014

    2014

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.