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REVIEW 2 major objections 1 minor 28 references

The r-deformed α-z-Rényi relative entropy upper-bounds the Tsallis relative entropy more tightly than an earlier bound when applied to density operators.

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2026-07-03 04:36 UTC pith:KV6TO3UP

load-bearing objection A clean three-parameter extension of the α-z Rényi family via r-logarithm, with axiom and DPI checks, but the tighter Tsallis bound rests on an observation whose scope is unclear from the abstract. the 2 major comments →

arxiv 2607.01805 v1 pith:KV6TO3UP submitted 2026-07-02 math-ph cs.ITmath.FAmath.ITmath.MPmath.OA

r-deformed α-z-R\'enyi relative entropy

Srikrishna Maity , Shigeru Furuichi , Supriyo Dutta This is my paper
classification math-ph cs.ITmath.FAmath.ITmath.MPmath.OA
keywords Rényi relative entropyTsallis relative entropydata processing inequalityquantum divergencesdeformed logarithmsrelative entropy bounds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper defines a three-parameter family of relative entropies by inserting the r-logarithm into the α-z-Rényi relative entropy. All members of the family satisfy the axioms required of a divergence. The authors identify ranges of α, z, and r for which the data processing inequality holds. They prove that the new family upper-bounds the Tsallis relative entropy and show that this bound is strictly tighter than the previously known upper bound whenever the arguments are density operators.

Core claim

The r-deformed α-z-Rényi relative entropy is an upper bound of the Tsallis relative entropy. When the arguments are density operators the new bound is tighter than the upper bound already present in the literature. The family is constructed so that every member satisfies the axioms of a divergence, and the data processing inequality holds for explicitly stated ranges of the three parameters.

What carries the argument

The r-deformed α-z-Rényi relative entropy, obtained by replacing the ordinary logarithm with the r-logarithm inside the definition of the α-z-Rényi relative entropy.

Load-bearing premise

The r-logarithm deformation produces quantities that obey the standard axioms of a divergence inside the parameter ranges claimed by the paper.

What would settle it

A pair of density operators for which the r-deformed quantity exceeds the literature upper bound while still being asserted to bound the Tsallis relative entropy from above would falsify the tightness claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The new family satisfies the data processing inequality inside the exposed parameter ranges.
  • It furnishes an upper bound on the Tsallis relative entropy for any valid arguments.
  • For density operators the numerical value of the new bound lies below the value of the earlier literature bound.
  • The order relation between the two upper bounds can be checked directly on any chosen pair of density operators.

Where Pith is reading between the lines

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

  • The same deformation technique could be applied to other parameterized families of entropies to generate additional families of bounds.
  • The three-parameter freedom may allow optimization of the bound for specific quantum information tasks such as channel discrimination.
  • Whether the tightness advantage persists for non-density-operator arguments remains open and could be tested on positive operators with unequal trace.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript defines a three-parameter family of r-deformed α-z-Rényi relative entropies via the r-logarithm as a generalization of prior α-z versions. It asserts that all members satisfy the axioms of a divergence, identifies ranges of α, z, r where the data processing inequality holds, establishes that the new quantity is an upper bound on the Tsallis relative entropy, and states that this bound is tighter than a previously discussed upper bound when applied to density operators.

Significance. If the upper-bound ordering and DPI ranges are established rigorously for the claimed parameter regimes, the construction would supply a new family of divergences with potentially sharper bounding properties for Tsallis-based quantities in quantum information.

major comments (2)
  1. [Abstract] Abstract: the statement that the new quantity 'is more tighter when applicable to the density operators' is presented only as an observation; it is unclear whether the ordering between the two upper bounds is proven for all density operators (once α, z, r lie in the ranges where both quantities are defined and DPI holds) or verified only on selected examples. This directly affects the central comparison claim.
  2. [Abstract] Abstract: the ranges of α, z, r for which the divergence axioms hold independently of the definition and for which DPI is valid are said to be 'exposed,' yet no explicit conditions appear; without these the independence of the axioms from the r-deformation cannot be assessed.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'more tighter' is grammatically incorrect and should be replaced by 'tighter'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and will revise the abstract to improve clarity on both issues raised.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the new quantity 'is more tighter when applicable to the density operators' is presented only as an observation; it is unclear whether the ordering between the two upper bounds is proven for all density operators (once α, z, r lie in the ranges where both quantities are defined and DPI holds) or verified only on selected examples. This directly affects the central comparison claim.

    Authors: We acknowledge the referee's point that the abstract phrasing ('we observe') leaves open whether the tighter bound holds generally or only on examples. The manuscript investigates the order relationship analytically between the two upper bounds on the Tsallis relative entropy and concludes the new bound is tighter for density operators in the relevant regimes. To eliminate ambiguity, we will revise the abstract to state explicitly that the new upper bound is tighter (as established by the comparison in the main text) rather than using 'observe'. revision: yes

  2. Referee: [Abstract] Abstract: the ranges of α, z, r for which the divergence axioms hold independently of the definition and for which DPI is valid are said to be 'exposed,' yet no explicit conditions appear; without these the independence of the axioms from the r-deformation cannot be assessed.

    Authors: The explicit ranges of α, z, and r for which the divergence axioms hold (independent of the r-deformation) and for which the data processing inequality holds are derived and stated in the main body of the manuscript. The abstract summarizes this derivation with the word 'exposed'. We agree that the abstract would be clearer with the conditions listed, and we will revise it to include the specific parameter ranges for the axioms and for DPI. revision: yes

Circularity Check

0 steps flagged

No circularity; new definition yields independent properties

full rationale

The paper defines the r-deformed α-z-Rényi relative entropy via the r-logarithm and derives its divergence axioms, DPI ranges, and upper-bound relation to Tsallis entropy directly from that definition. The comparison to the literature bound is stated as an observation on density operators without any reduction of a claimed prediction to a fitted parameter or self-citation that bears the central load. All steps remain self-contained against external benchmarks and do not collapse by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the r-logarithm deformation preserves the standard divergence axioms and that the data-processing and upper-bound properties follow from the definition for the stated parameter ranges. No fitted numerical constants or new postulated entities appear in the abstract.

axioms (2)
  • domain assumption The defined quantity must be non-negative, zero if and only if the two states coincide, and satisfy other standard divergence axioms.
    Abstract states that all members satisfy the necessary axioms to be a divergence.
  • domain assumption Data processing inequality holds for the exposed ranges of α, z, r.
    Abstract claims the ranges are exposed but provides no explicit conditions.

pith-pipeline@v0.9.1-grok · 5752 in / 1230 out tokens · 31000 ms · 2026-07-03T04:36:19.359330+00:00 · methodology

0 comments
read the original abstract

In this article, we consider the $r$-logarithm for defining three-parameter family of R\'{e}nyi relative entropies that are generalization of the $\alpha$-$z$-R\'{e}nyi relative entropies. All the members of $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropies satisfy the necessary axioms to be a divergence. We expose the range of parameters $\alpha$, $z$ and $r$ for which the data processing inequality holds. We also establish that $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy is an upper bound of the Tsallis relative entropy. Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $\alpha$-$z$-R\'{e}nyi relative entropy and the other one, which is discussed in literature. We investigate the order relationship between these two upper bounds of the Tsallis relative entropy. We observe that our new upper bound is more tighter when applicable to the density operators.

Figures

Figures reproduced from arXiv: 2607.01805 by Shigeru Furuichi, Srikrishna Maity, Supriyo Dutta.

Figure 1
Figure 1. Figure 1: For different choices of ρ and σ the bounds of Tsallis relative entropy behaves differently, when Tr(ρ) = 1 and 0 < Tr(σ) < 1. q = 1.5 we have B1(ρ||σ) = 0.252861 and B2(ρ||σ) = 0.264951 that is B2(ρ||σ) > B1(ρ||σ). We plot the values of B1(ρ||σ) and B2(ρ||σ) with respect to α ∈ (0, 1) considering r = 0.7, z = 2, q = 1.5, in Figure 1a. Consider another examples. We consider ρ and σ as ρ =  0.2197 0.3956 0… view at source ↗
Figure 2
Figure 2. Figure 2: When ρ and σ are non-commutative density operators, B1(ρ||σ) acts as a bound of Tsallis relative entropy, which is tighter than B2(ρ||σ). References [1] Tim Van Erven and Peter Harremos. R´enyi divergence and Kullback-Leibler divergence. IEEE Transactions on Information Theory, 60(7):3797–3820, 2014. [2] Lian-He Shao, Yong-Ming Li, Yu Luo, and Zheng-Jun Xi. Quantum Coherence quantifiers based on R´enyi α-r… view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 3 internal anchors

  1. [1]

    R´ enyi divergence and Kullback-Leibler divergence.IEEE Transactions on Information Theory, 60(7):3797–3820, 2014

    Tim Van Erven and Peter Harremos. R´ enyi divergence and Kullback-Leibler divergence.IEEE Transactions on Information Theory, 60(7):3797–3820, 2014

    work page 2014

  2. [2]

    Quantum Coherence quantifiers based on R´ enyi α-relative entropy.Communications in Theoretical Physics, 67(6):631, 2017

    Lian-He Shao, Yong-Ming Li, Yu Luo, and Zheng-Jun Xi. Quantum Coherence quantifiers based on R´ enyi α-relative entropy.Communications in Theoretical Physics, 67(6):631, 2017

    work page 2017

  3. [3]

    Koenraad MR Audenaert and Nilanjana Datta.α-z-R´ enyi relative entropies.Journal of Mathematical Physics, 56(2), 2015

    work page 2015

  4. [4]

    Springer, 2002

    Victor G Kac and Pokman Cheung.Quantum calculus, volume 113. Springer, 2002

    work page 2002

  5. [5]

    John Wiley & Sons, 1999

    Thomas M Cover.Elements of information theory. John Wiley & Sons, 1999

    work page 1999

  6. [6]

    Deformed logarithms and entropies.Physica A: Statistical Mechanics and its Applications, 340(1-3):41–49, 2004

    Giorgio Kaniadakis, M Lissia, and AM Scarfone. Deformed logarithms and entropies.Physica A: Statistical Mechanics and its Applications, 340(1-3):41–49, 2004

    work page 2004

  7. [7]

    Relative entropies and their use in quantum information theory

    Felix Leditzky. Relative entropies and their use in quantum information theory.arXiv preprint arXiv:1611.08802, 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  8. [8]

    Beyond Boltzmann–Gibbs–Shannon in physics and elsewhere.Entropy, 21(7):696, 2019

    Constantino Tsallis. Beyond Boltzmann–Gibbs–Shannon in physics and elsewhere.Entropy, 21(7):696, 2019

    work page 2019

  9. [9]

    Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years

    Constantino Tsallis. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Brazilian Journal of Physics, 39:337–356, 2009

    work page 2009

  10. [10]

    A review of entropy measures for uncertainty quantification of stochastic processes.Advances in Mechanical Engineering, 11(6):1687814019857350, 2019

    Alireza Namdari and Zhaojun Li. A review of entropy measures for uncertainty quantification of stochastic processes.Advances in Mechanical Engineering, 11(6):1687814019857350, 2019

    work page 2019

  11. [11]

    A two-parameter entropy and its fundamental properties

    Supriyo Dutta, Shigeru Furuichi, and Partha Guha. A two-parameter entropy and its fundamental properties. Reviews in Mathematical Physics, 33(04):2130003, 2021

    work page 2021

  12. [12]

    Fundamental properties of Tsallis relative entropy

    Shigeru Furuichi, Kenjiro Yanagi, and Ken Kuriyama. Fundamental properties of Tsallis relative entropy. Journal of Mathematical Physics, 45(12):4868–4877, 2004

    work page 2004

  13. [13]

    Matrix trace inequalities related to the Tsallis relative entropies of real order

    Masatoshi Fujii and Yuki Seo. Matrix trace inequalities related to the Tsallis relative entropies of real order. Journal of Mathematical Analysis and Applications, 498(1):124877, 2021

    work page 2021

  14. [14]

    Matrix trace inequalities on the Tsallis entropies

    Shigeru Furuichi. Matrix trace inequalities on the Tsallis entropies.arXiv preprint arXiv:1001.1388, 2010

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  15. [15]

    Matrix trace inequalities on Tsallis relative entropy of negative order.Journal of Mathematical Analysis and Applications, 472(2):1499–1508, 2019

    Yuki Seo. Matrix trace inequalities on Tsallis relative entropy of negative order.Journal of Mathematical Analysis and Applications, 472(2):1499–1508, 2019. 13

    work page 2019

  16. [16]

    H¨ older’s inequal- ity: some recent and unexpected applications.Bulletin of the Belgian Mathematical Society-Simon Stevin, 24(2):199–225, 2017

    Nacib Albuquerque, Gustavo Ara´ ujo, Daniel Pellegrino, and Juan Benigno Seoane-Sep´ ulveda. H¨ older’s inequal- ity: some recent and unexpected applications.Bulletin of the Belgian Mathematical Society-Simon Stevin, 24(2):199–225, 2017

    work page 2017

  17. [17]

    Cambridge university press, 1994

    Roger A Horn and Charles R Johnson.Topics in matrix analysis. Cambridge university press, 1994

    work page 1994

  18. [18]

    Springer Science & Business Media, 2013

    Rajendra Bhatia.Matrix analysis, volume 169. Springer Science & Business Media, 2013

    work page 2013

  19. [19]

    Concavity of certain matrix trace and norm functions.Linear algebra and its applications, 439(5):1568–1589, 2013

    Fumio Hiai. Concavity of certain matrix trace and norm functions.Linear algebra and its applications, 439(5):1568–1589, 2013

    work page 2013

  20. [20]

    Concavity of certain maps on positive definite matrices and applications to Hadamard products

    Tsuyoshi Ando. Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear algebra and its applications, 26:203–241, 1979

    work page 1979

  21. [21]

    Monotonicity of a relative R´ enyi entropy.Journal of Mathematical Physics, 54(12), 2013

    Rupert L Frank and Elliott H Lieb. Monotonicity of a relative R´ enyi entropy.Journal of Mathematical Physics, 54(12), 2013

    work page 2013

  22. [22]

    Sandwiched R´ enyi divergence satisfies data processing inequality.Journal of Mathematical Physics, 54(12), 2013

    Salman Beigi. Sandwiched R´ enyi divergence satisfies data processing inequality.Journal of Mathematical Physics, 54(12), 2013

    work page 2013

  23. [23]

    Some operator and trace function convexity theorems

    Eric A Carlen, Rupert L Frank, and Elliott H Lieb. Some operator and trace function convexity theorems. Linear algebra and its applications, 490:174–185, 2016

    work page 2016

  24. [24]

    Completely positive linear maps on complex matrices.Linear algebra and its applications, 10(3):285–290, 1975

    Man-Duen Choi. Completely positive linear maps on complex matrices.Linear algebra and its applications, 10(3):285–290, 1975

    work page 1975

  25. [25]

    An analysis of completely-positive trace-preserving maps on M2.Linear algebra and its applications, 347(1-3):159–187, 2002

    Mary Beth Ruskai, Stanislaw Szarek, and Elisabeth Werner. An analysis of completely-positive trace-preserving maps on M2.Linear algebra and its applications, 347(1-3):159–187, 2002

    work page 2002

  26. [26]

    Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states.arXiv preprint arXiv:2304.00161, 2023

    Thomas Barthel and Qiang Miao. Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states.arXiv preprint arXiv:2304.00161, 2023

    work page arXiv 2023

  27. [27]

    Cambridge university press, 2013

    Mark M Wilde.Quantum information theory. Cambridge university press, 2013

    work page 2013

  28. [28]

    On the Araki-Lieb-Thirring inequality

    Koenraad MR Audenaert. On the araki-lieb-thirring inequality.arXiv preprint math/0701129, 2007. 14

    work page internal anchor Pith review Pith/arXiv arXiv 2007