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arxiv: 2604.06908 · v1 · submitted 2026-04-08 · 🪐 quant-ph · cs.IT· hep-th· math-ph· math.IT· math.MP

Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective

Sayantan Roy , Atin Gayen , Aditi Kar Gangopadhyay , Sugata Gangopadhyay This is my paper

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classification 🪐 quant-ph cs.IThep-thmath-phmath.ITmath.MP
keywords quantum relative-alpha-entropyUmegaki relative entropyf-divergencenonlinear convexityPetz-Renyi divergenceNussbaum-Szkola distributionsquantum distinguishabilitygeometric quantum information
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The pith

A quantum relative-alpha-entropy extends Umegaki's relative entropy outside the f-divergence class while showing nonlinear convexity.

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

The paper defines a quantum relative-alpha-entropy to generalize Umegaki's relative entropy in a way that escapes the usual f-divergence structure. This definition brings a nonlinear convexity property that produces a corresponding convexity statement for the Petz-Renyi divergence when alpha is larger than one. The new divergence stays the same under unitary changes and adds up for independent systems. It matches the classical relative-alpha-entropy precisely when the states are replaced by their Nussbaum-Szkola probability distributions. A reader would care because this points to a view of quantum distinguishability that depends only on the relative positions of states rather than their individual sizes.

Core claim

We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-

What carries the argument

quantum relative-alpha-entropy, a divergence extending Umegaki's relative entropy outside f-divergences and carrying nonlinear convexity together with unitary invariance and tensor additivity

If this is right

  • The nonlinear convexity supplies a generalized convexity result for the Petz-Renyi divergence when alpha exceeds one.
  • The divergence remains invariant under unitary transformations applied to the states.
  • It is additive when the underlying states are replaced by their tensor products.
  • It depends only on the relative geometry between the states and not on their absolute magnitudes.
  • It coincides exactly with the classical relative-alpha-entropy when evaluated on Nussbaum-Szkola distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This geometric divergence could supply new bounds in quantum hypothesis testing or state discrimination problems that rely on distinguishability measures.
  • Analogous constructions might be attempted for other families of quantum entropies or divergences.
  • Numerical checks on low-dimensional systems such as pairs of qubit states would provide a direct test of the nonlinear convexity inequality.
  • The emphasis on relative geometry may connect to differential-geometric quantities already defined on the manifold of quantum states.

Load-bearing premise

The specific construction of the quantum relative-alpha-entropy satisfies both the nonlinear convexity property and the exact reduction to classical relative-alpha-entropy on Nussbaum-Szkola distributions.

What would settle it

Direct computation of the proposed divergence between two chosen non-commuting density operators, followed by comparison to the classical relative-alpha-entropy of their associated Nussbaum-Szkola probability distributions, would confirm or refute the claimed exact correspondence.

Figures

Figures reproduced from arXiv: 2604.06908 by Aditi Kar Gangopadhyay, Atin Gayen, Sayantan Roy, Sugata Gangopadhyay.

Figure 1
Figure 1. Figure 1: The Quantum Relative α-Entropy as a function of its order for three different sets of quantum states. C. A Nonlinear Convexity Framework for Quantum Divergences A real-valued function f : D → R, where D ⊆ R, is said to be convex if for all x, y ∈ D and t ∈ [0, 1], f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y). The set D ⊆ R itself is called convex if tx + (1 − t)y ∈ D for all x, y ∈ D and t ∈ [0, 1]. Analogously,… view at source ↗
Figure 2
Figure 2. Figure 2: The Quantum Relative α-Entropy vs Petz-Renyi- ´ α-Relative Entropy as functions of the order α. Let ρ =   0.5 0 0 0 0.25 0 0 0 0.25   and σ =   0.7 0 0 0 0.2 0 0 0 0.1   . Then S2(ρ∥σ) = (−2) log(0.425) + log(0.375) + log(0.54) ≈ 0.1649. Let Φ2 be a quantum channel from the system HA to HB, where BB = {|0⟩, |1⟩} is the basis for HB. We define the quantum channel Φ2 as Φ2(ρ) = P3 i=1 Kiρ… view at source ↗
read the original abstract

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript introduces a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. It establishes additivity under tensor products, unitary invariance, dependence solely on relative geometry of states, an exact correspondence to the classical relative-alpha-entropy via Nussbaum-Szkola distributions, and a nonlinear convexity property that yields a generalized convexity result for the Petz-Rényi divergence when alpha > 1.

Significance. If the results hold, the work supplies a new geometric notion of quantum distinguishability not captured by existing divergence frameworks and complements known convexity statements for Petz-Rényi quantities. The explicit definition together with direct proofs of the listed properties, the parameter-free construction, and the exact classical correspondence via Nussbaum-Szkola distributions are clear strengths that support the central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary correctly identifies the key contributions, including the definition outside the f-divergence class, additivity, unitary invariance, dependence on relative geometry, the exact Nussbaum-Szkola correspondence, and the nonlinear convexity property that generalizes convexity results for the Petz-Rényi divergence when alpha > 1.

Circularity Check

0 steps flagged

No significant circularity; new definition with independent proofs

full rationale

The manuscript introduces an explicit new definition of quantum relative-alpha-entropy (distinct from Umegaki relative entropy and f-divergences) and supplies direct proofs of its listed properties: additivity under tensor products, unitary invariance, exact Nussbaum-Szkola correspondence to the classical relative-alpha-entropy, and nonlinear convexity that induces a generalized convexity statement for Petz-Rényi divergences when alpha > 1. None of these steps reduce by construction to the definition itself, to fitted parameters renamed as predictions, or to load-bearing self-citations whose content is unverified. The geometric interpretation follows from the stated axioms and the correspondence theorem rather than from any renaming or smuggling of prior ansatzes. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Central claim rests on a new definition of the divergence and standard quantum mechanics axioms for states and operators. The main addition is the invented divergence entity itself.

axioms (1)
  • standard math Standard axioms of quantum mechanics and operator algebras for defining relative entropies
    Invoked implicitly to extend Umegaki's relative entropy and establish properties.
invented entities (1)
  • quantum relative-alpha-entropy no independent evidence
    purpose: New divergence measuring quantum state distinguishability with nonlinear convexity and geometric focus
    Newly introduced construction outside existing f-divergence class.

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Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    Testing statistical hypotheses based on the density power divergence,

    N. M. A. Basu, A. Mandal and L. Pardo, “Testing statistical hypotheses based on the density power divergence,”Annals of the Institute of Statistical Mathematics, vol. 65, pp. 319–348, 2013

    work page 2013

  2. [2]

    Quantum f-divergences via nussbaum–szkoła distributions and applications to f-divergence inequalities,

    G. Androulakis and T. C. John, “Quantum f-divergences via nussbaum–szkoła distributions and applications to f-divergence inequalities,”Reviews in Mathematical Physics, vol. 36, p. 2360002, 2024

    work page 2024

  3. [3]

    Relative entropy of states of von neumann algebras,

    H. Araki, “Relative entropy of states of von neumann algebras,”Publications of The Research Institute for Mathematical Sciences, vol. 11, pp. 809–833, 1975

    work page 1975

  4. [4]

    Relative entropy for states of von neumann algebras ii,

    ——, “Relative entropy for states of von neumann algebras ii,”Publications of The Research Institute for Mathematical Sciences, 2005

    work page 2005

  5. [5]

    α−z-relative renyi entropies,

    K. M. R. Audenaert and N. Datta, “α−z-relative renyi entropies,”Journal of Mathematical Physics, vol. 56, p. 022202, 2015

    work page 2015

  6. [6]

    Robust minimum divergence procedures for count data models,

    A. Basu, S. Basu, and G. Chaudhury, “Robust minimum divergence procedures for count data models,”Sankhya: The Indian Journal of Statistic, vol. 59, pp. 11–27, 1997

    work page 1997

  7. [7]

    Robust and efficient estimation by minimizing a density power divergence,

    A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones, “Robust and efficient estimation by minimizing a density power divergence,”Biometrika, vol. 85, pp. 549–559, 1998

    work page 1998

  8. [8]

    Robust and efficient estimation by minimising a density power divergence,

    A. Basu, I. R. Harris, N. L. Hjort, and M. Jones, “Robust and efficient estimation by minimising a density power divergence,” Biometrika, vol. 85, pp. 549–559, 1998

    work page 1998

  9. [9]

    A. Basu, H. Shioya, and C. Park,Statistical Inference: The Minimum Distance Approach. Chapman & Hall/ CRC Monographs on Statistics and Applied Probability 120, 2011

    work page 2011

  10. [10]

    Continuity of quantum entropic quantities via almost convexity,

    A. Bluhm, ´A. Capel, P. Gondolf, and A. P. Hern ´andez, “Continuity of quantum entropic quantities via almost convexity,” IEEE Transactions on Information Theory, vol. 69, pp. 5869–5901, 2022

    work page 2022

  11. [11]

    Multinomial goodness-of-fit tests,

    N. Cressie and T. R. C. Read, “Multinomial goodness-of-fit tests,”J. R. Stat. Soc. Ser. B. Stat. Methodol., vol. 46, pp. 440–464, 1984

    work page 1984

  12. [12]

    Csisz ´ar and P

    I. Csisz ´ar and P. C. Shields,Information Theory and Statistics: A Tutorial. Hanover: Foundations and Trends in Communications and Information Theory, 2004

    work page 2004

  13. [13]

    Information-type measures of difference of probability distributions and indirect observation,

    I. Csisz ´ar, “Information-type measures of difference of probability distributions and indirect observation,”Studia Scien- tiarum Mathematicarum Hungarica, vol. 2, pp. 229–318, 1967

    work page 1967

  14. [14]

    Min- and max-relative entropies and a new entanglement monotone,

    N. Datta, “Min- and max-relative entropies and a new entanglement monotone,”IEEE Transactions on Information Theory, vol. 55, p. 2816–2826, 2009

    work page 2009

  15. [15]

    R ´enyi divergence and kullback-leibler divergence,

    T. V . Erven and P. Harremos, “R ´enyi divergence and kullback-leibler divergence,”IEEE Transactions on Information Theory, vol. 60, pp. 3797–3820, 2014

    work page 2014

  16. [16]

    Projection theorems and estimating equations for power-law models,

    A. Gayen and M. A. Kumar, “Projection theorems and estimating equations for power-law models,”Journal of Multivariate Analysis, vol. 184, p. 104734, 2021. 31

    work page 2021

  17. [17]

    Generalized Fisher-Darmois-Koopman-Pitman theorem and Rao-Blackwell type estimators for power-law distribu- tions,

    ——, “Generalized Fisher-Darmois-Koopman-Pitman theorem and Rao-Blackwell type estimators for power-law distribu- tions,”IEEE Transactions on Information Theory, vol. 69, pp. 7565–7583, 2023

    work page 2023

  18. [18]

    A unified approach to the pythagorean identity and projection theorem for a class of divergences based on m-estimations,

    A. Gayen, S. Roy, and A. K. Gangopadhyay, “A unified approach to the pythagorean identity and projection theorem for a class of divergences based on m-estimations,”Statistics, vol. 58, pp. 842–880, 2024

    work page 2024

  19. [19]

    Quantumf-divergences in von neumann algebras. i. standardf-divergences,

    F. Hiai, “Quantumf-divergences in von neumann algebras. i. standardf-divergences,”Journal of Mathematical Physics, vol. 59, p. 102202, 2018

    work page 2018

  20. [20]

    Different quantum f-divergences and the reversibility of quantum operations,

    F. Hiai and M. Mosonyi, “Different quantum f-divergences and the reversibility of quantum operations,”Reviews in Mathematical Physics, vol. 29, p. 1750023, 2017

    work page 2017

  21. [21]

    A comparison of related density based minimum divergence estimators,

    M. C. Jones, N. L. Hjort, I. R. Harris, and A. Basu, “A comparison of related density based minimum divergence estimators,” Biometrika, vol. 88, pp. 865–873, 2001

    work page 2001

  22. [22]

    Fidelity for mixed quantum states,

    R. Jozsa, “Fidelity for mixed quantum states,”Journal of Modern Optics, vol. 41, pp. 2315–2323, 1994

    work page 1994

  23. [23]

    Scale-invariant divergences for density functions,

    T. Kanamori, “Scale-invariant divergences for density functions,”Entropy, vol. 16, pp. 2611–2628, 2014

    work page 2014

  24. [24]

    On Information and Sufficiency,

    S. Kullback and R. A. Leibler, “On Information and Sufficiency,”The Annals of Mathematical Statistics, vol. 22, pp. 79–86, 1951

    work page 1951

  25. [25]

    Cram ´er–Rao lower bounds arising from generalized Csisz ´ar divergences,

    M. A. Kumar and K. V . Mishra, “Cram ´er–Rao lower bounds arising from generalized Csisz ´ar divergences,”Info. Geo., vol. 3, pp. 33–59, 2020

    work page 2020

  26. [26]

    Projection theorems for the r ´enyi divergence on alpha-convex sets,

    M. A. Kumar and I. Sason, “Projection theorems for the r ´enyi divergence on alpha-convex sets,”IEEE Transactions on Information Theory, vol. 62, pp. 4924–4935, 2016

    work page 2016

  27. [27]

    Minimization problems based on relativeα-entropy i: Forward projection,

    M. A. Kumar and R. Sundaresan, “Minimization problems based on relativeα-entropy i: Forward projection,”IEEE Transactions on Information Theory, vol. 61, pp. 5063–5080, 2015

    work page 2015

  28. [28]

    Minimization problems based on relativeα-entropy ii: Reverse projection,

    ——, “Minimization problems based on relativeα-entropy ii: Reverse projection,”IEEE Transactions on Information Theory, vol. 61, pp. 5081–5095, 2015

    work page 2015

  29. [29]

    Robust regression with density power divergence: Theory, comparisons, and data analysis,

    A. C. M. Riani, A. C. Atkinson and D. Perrotta, “Robust regression with density power divergence: Theory, comparisons, and data analysis,”Entropy, vol. 22, pp. 1099–4300, 2020

    work page 2020

  30. [30]

    On quantum r ´enyi entropies: A new generalization and some properties,

    M. M ¨uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, “On quantum r ´enyi entropies: A new generalization and some properties,”Journal of Mathematical Physics, vol. 54, p. 122203, 2013

    work page 2013

  31. [31]

    Estimators, escort probabilities, andϕ-exponential families in statistical physics,

    J. Naudts, “Estimators, escort probabilities, andϕ-exponential families in statistical physics,”Journal of Inequalities in Pure & Applied Mathematics, vol. 5, p. 102, 2004

    work page 2004

  32. [32]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000

    work page 2000

  33. [33]

    The chernoff lower bound for symmetric quantum hypothesis testing,

    M. Nussbaum and A. Szkoła, “The chernoff lower bound for symmetric quantum hypothesis testing,”The Annals of Statistics, vol. 37, 2009

    work page 2009

  34. [34]

    Strong converse and stein’s lemma in quantum hypothesis testing,

    T. Ogawa and H. Nagaoka, “Strong converse and stein’s lemma in quantum hypothesis testing,”IEEE Transactions on Information Theory, vol. 46, pp. 2428–2433, 2000

    work page 2000

  35. [35]

    Quantum entropy and its applications to quantum communication and statistical physics,

    M. Ohya and N. Watanabe, “Quantum entropy and its applications to quantum communication and statistical physics,” Entropy, vol. 12, pp. 1194–1245, 2010

    work page 2010

  36. [36]

    Generalized quantum hellinger divergences generated by monotone functions,

    H. Osaka and H. Shudo, “Generalized quantum hellinger divergences generated by monotone functions,”Open Systems & Information Dynamics, vol. 32, p. 2550013, 2025

    work page 2025

  37. [37]

    Pardo,Statistical Inference Based on Divergence Measures

    L. Pardo,Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC, Taylor and Francis group, Boca Raton, Florida, USA, 2006

    work page 2006

  38. [38]

    K. R. Parthasarathy,Lectures on Quantum computation, quantum error correcting codes and information theory. New Delhi, India: Narosa Pub., 2006. 32

    work page 2006

  39. [39]

    Quasi-entropies for finite quantum systems,

    D. Petz, “Quasi-entropies for finite quantum systems,”Reports on Mathematical Physics, vol. 23, pp. 57–65, 1986

    work page 1986

  40. [40]

    Heidelberg: Springer, 2007

    ——,Quantum information theory and quantum statistics. Heidelberg: Springer, 2007

    work page 2007

  41. [41]

    On measures of entropy and information,

    A. R ´enyi, “On measures of entropy and information,” inProceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, USA, 1961, pp. 547–561

    work page 1961

  42. [42]

    A measure of discrimination and its geometric properties,

    R. Sundaresan, “A measure of discrimination and its geometric properties,”Proceedings IEEE International Symposium on Information Theory, pp. 264–, 2002

    work page 2002

  43. [43]

    Guessing under source uncertainty,

    ——, “Guessing under source uncertainty,”IEEE Transactions on Information Theory, vol. 53, p. 269–287, 2007

    work page 2007

  44. [44]

    Theχ 2-divergence and mixing times of quantum markov processes,

    K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, and F. Verstraete, “Theχ 2-divergence and mixing times of quantum markov processes,”Journal of Mathematical Physics, vol. 51, p. 122201, 2010

    work page 2010

  45. [45]

    Tomamichel,Quantum Information Processing with Finite Resources – Mathematical Foundations

    M. Tomamichel,Quantum Information Processing with Finite Resources – Mathematical Foundations. Cham: Springer Cham, 2015

    work page 2015

  46. [46]

    Possible generalization of bolzmann-gibbs statistics,

    C. Tsallis, “Possible generalization of bolzmann-gibbs statistics,”J. Stat. Phys., vol. 52, pp. 479–487, 1988

    work page 1988

  47. [47]

    The role of constraints within generalized non-extensive statistics,

    C. Tsallis, R. S. Mendes, and A. R. Plastino, “The role of constraints within generalized non-extensive statistics,”Phys. A., vol. 261, pp. 534–554, 1998

    work page 1998

  48. [48]

    Relative entropy and the wigner-yanase-dyson-lieb concavity in an interpolation theory,

    A. Uhlmann, “Relative entropy and the wigner-yanase-dyson-lieb concavity in an interpolation theory,”Communications in Mathematical Physics, vol. 54, p. 21–32, 1977

    work page 1977

  49. [49]

    Conditional expectation in an operator algebra. IV . Entropy and information,

    H. Umegaki, “Conditional expectation in an operator algebra. IV . Entropy and information,”Kodai Mathematical Seminar Reports, vol. 14, pp. 59–85, 1962

    work page 1962

  50. [50]

    Zhang,Matrix Theory: Basic Results and Techniques

    F. Zhang,Matrix Theory: Basic Results and Techniques. New York: Springer, 2011

    work page 2011