A two-parameter entropy and its fundamental properties

Partha Guha; Shigeru Furuichi; Supriyo Dutta

REVIEW 4 major objections 4 minor 29 references

A two-parameter entropy and its fundamental properties

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A two-parameter deformed logarithm is claimed to yield a generalized Tsallis entropy with a full information-theoretic toolkit.

desk verdict The paper's own algebra cancels the r parameter from its entropy and divergence, so the advertised two-parameter family is Tsallis entropy with q=2k+1; the remaining one-parameter proofs are largely correct but standard. read the letter →

arxiv 1908.01696 v3 pith:6GS5J4KB submitted 2019-08-05 math-ph cs.ITmath.ITmath.MP

Supriyo Dutta , Shigeru Furuichi , Partha Guha This is my paper · ORCID

classification math-phcs.ITmath.ITmath.MP MSC 94A1794A15

keywords two-parameterentropydeformedlogarithmTsallisrelativechainrulesub-additivityinformationmonotonicityHessianmetric

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 sets out to build a two-parameter generalization of Tsallis entropy and relative divergence from a deformed logarithm $\ln_{k,r}(x) = (x^k - x^{-k})/(2k x^r)$. Its aim is to give this entropy the same working toolkit that Shannon and Tsallis entropies have: a chain rule, sub-additivity, strong sub-additivity, joint convexity, pseudo-additivity for independent systems, and information monotonicity under coarse-graining. The authors choose the deformed logarithm so that the weighted expression $x^{r+k}\ln_{k,r}(x)$ simplifies to $(x^{2k}-1)/(2k)$, which makes a product rule hold and lets all these properties be derived by convexity arguments. If the construction works as claimed, it would place the Sharma-Mittal entropy family inside classical information theory and connect it to an information-geometric Hessian metric, with Tsallis and Shannon entropies recovered at special parameter values.

What carries the argument

The load-bearing object is the two-parameter deformed logarithm $\ln_{k,r}(x) = (x^k - x^{-k})/(2k x^r)$, with $x^r$ placed in the denominator and $0<k\le \tfrac12$. This placement makes $x^{r+k}\ln_{k,r}(x) = (x^{2k}-1)/(2k)$, a simple power, and it is exactly what makes the product rule of Lemma 1 hold. That product rule is the mechanism: it turns the deformed logarithm into a calculus in which joint entropies decompose into marginals and conditionals, so the chain rule, sub-additivity, strong sub-additivity, joint convexity, and information monotonicity all follow from Jensen-type convexity arguments applied to that one identity.

What would settle it

Take $P=(1/2,1/3,1/6)$ and compute $S_{1/4,1}(P)$ and $S_{1/4,2}(P)$ directly from Definition 2: because $S_{k,r} = (1-\sum_x p^{2k+1})/(2k)$, both numbers are identical, which would contradict the claim that $S_{k,r}$ depends on two parameters; the same check on $D_{1/4,1}(P\|Q)$ and $D_{1/4,2}(P\|Q)$ gives equal values.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the pair $S_{k,r}(X) = -\sum_x p(x)^{r+k+1} \ln_{k,r}(p(x))$ and $D_{k,r}(P\|Q) = \sum_x p(x)(p(x)/q(x))^{r-k} \ln_{k,r}(p(x)/q(x))$ constitutes a genuine two-parameter generalized Tsallis entropy and divergence. The engine is the product rule $(xy)^{r+k}\ln_{k,r}(xy) = x^{r+k}\ln_{k,r}(x) + y^{r+k}\ln_{k,r}(y) + 2k x^{r+k} y^{r+k} \ln_{k,r}(x) \ln_{k,r}(y)$, which makes the joint entropy split into a marginal plus a conditional term. From this product rule the paper derives the chain rule, sub-additivity, strong sub-additivity, pseudo-additivity, joint convexity, and information monotonicity for the divergence, and it identifies parameter choices that recover Tsallis entropy and Tsallis relative entropy. The main asserted payoff is not any single formula but the infrastructure: conditional entropies satisfy the same inequalities as classical ones, and the divergence induces a Hessian metric on the probability simplex.

Load-bearing premise

Everything rests on the choice to put $x^r$ in the denominator of the deformed logarithm, $\ln_{k,r}(x) = (x^k - x^{-k})/(2k x^r)$, because that placement is what makes the product rule of Lemma 1 hold, and the chain rule, sub-additivity, and all later properties are derived from that product rule alone.

Editorial extensions

If this is right

If correct, the chain rule $S_{k,r}(X,Y) = S_{k,r}(X) + S_{k,r}(Y|X)$ holds, and with it sub-additivity $S_{k,r}(X,Y) \le S_{k,r}(X) + S_{k,r}(Y)$ for arbitrary random variables.
The divergence $D_{k,r}$ is nonnegative, vanishes only for $P=Q$, is permutation invariant, admits a zero-probability extension, and is pseudo-additive under tensor products.
$D_{k,r}$ is jointly convex in $(P,Q)$ and monotone under stochastic maps, so it qualifies as an information-theoretic divergence and can support data-processing-type inequalities.
The divergence induces a Hessian metric on the probability simplex, giving the family a dually flat information-geometric structure analogous to that of the Kullback-Leibler divergence.
The special parameter limits recover Tsallis entropy and divergence, with Shannon entropy reached in the $k\to 0$ limit, making the construction a potential interpolation tool.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Direct algebra from Definition 2 shows the paper's $r$ drops out: $S_{k,r}(X) = (1-\sum_x p(x)^{2k+1})/(2k)$ and $D_{k,r}(P\|Q) = (1-\sum_x p(x)^{1-2k}q(x)^{2k})/(2k)$, so the two-parameter label is not supported by the definitions; the family is the one-parameter Tsallis family reindexed with $q=2k+1$.
The chain rule, sub-additivity, and convexity proved in the paper are therefore properties of that one-parameter family; a genuine two-parameter theory needs a deformation in which the second parameter survives the entropy weighting.
One testable route to a real second parameter would be to weight the deformed logarithm by $p^\alpha$ with an exponent independent of the one inside the logarithm, so that no telescoping cancellation occurs; the resulting product rule would be more complex but would keep $r$ visible.
The paper's Hessian calculation gives $g_{ii} = (1-2k+4r)/p_i$, which depends on $r$, even though the divergence it comes from does not; recomputing the second derivative from the simplified divergence suggests that $r$-dependence is an artifact of differentiating the un-simplified form.

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.

Desk Editor's Note

read the letter

Bottom line: the headline claim is false. Substituting Definition 1 into Definition 2 gives S_{k,r}(X) = (1 - Σ p^{2k+1})/(2k), and D_{k,r}(P||Q) = (1 - Σ p^{1-2k} q^{2k})/(2k). Both are independent of r; the first is Tsallis entropy with q=2k+1. The paper's own equations cancel the parameter it advertises.

There is real work here, and I want to credit it. The product rule in Lemma 1 is a clean algebraic identity, and it genuinely drives the chain rule, sub-additivity, strong sub-additivity, joint convexity, and information monotonicity proofs. Those proofs are self-contained and mostly correct for the surviving one-parameter family on 0

The soft spots are not minor. The claimed reduction to Furuichi's (α,β)-entropy, equation (19), is only correct when r=k; for general r there is a missing factor p^{r-k}. The Fisher metric calculation in Section 5 cannot be right because D does not depend on r; direct differentiation gives g_ii=(1-2k)/p_i, not (-2k+4r+1)/p_i. Worse, at k=1/2, which is inside the stated parameter range, the divergence is identically zero, so the metric degenerates. There is also a notational whiplash in Section 2: the original Sharma-Mittal logarithm uses x^r in the numerator and r<0, then Definition 1 moves x^r to the denominator and requires r>0. That switch is what makes the product rule work, but it is chosen for exactly that reason and has no external justification.

The citations are fine: some self-citations appear, but nothing load-bearing depends on them.

Who is this for? Someone who wants a self-contained re-derivation of standard Tsallis entropy properties with a particular deformed logarithm might extract some pedagogical value from the proof sections. But the paper as written does not deliver the two-parameter generalization it promises. I would not send this to peer review in its current form. The honest recommendation is to reject and invite the authors to resubmit a much shorter note, stripped of the spurious r parameter, with the Hessian corrected.

Referee Report

4 major / 4 minor

Summary. The paper proposes a two-parameter deformed logarithm ln_{k,r}(x) and uses it to define a generalized Tsallis entropy S_{k,r}(X) and a generalized Tsallis relative entropy D_{k,r}(P||Q). It derives pseudo-additivity, chain rules, sub-additivity, strong sub-additivity, joint convexity, information monotonicity, and a Hessian information-geometric structure for these objects. The central claim is that S_{k,r} and D_{k,r} form a genuine two-parameter extension of Tsallis and Shannon entropy.

Significance. If the central claim were correct, the paper would provide a useful two-parameter entropy/divergence family with the standard information-theoretic toolbox. The manuscript is self-contained, with explicit definitions and mostly checkable algebraic proofs. However, direct substitution shows that the parameter r cancels identically from both S_{k,r} and D_{k,r}, so the proposed objects are actually one-parameter families. Consequently, the main contribution is not realized, and several derived statements are internally inconsistent with the paper's own definitions. The explicit, machine-checkable algebra is a strength, but it is used here to disprove the paper's central claim.

major comments (4)

[Section 3, Definition 2 and Eq. (19)] The parameter r cancels identically from the proposed entropy. Substituting ln_{k,r}(x)=(x^{2k}-1)/(2k x^{r+k}) into Definition 2 gives S_{k,r}(X)=Σ_x [p(x)-p(x)^{2k+1}]/(2k)=(1-Σ_x p(x)^{2k+1})/(2k), which is independent of r and is the Tsallis entropy with q=2k+1. The assertion that α=1-k+r and β=1+k+r in Eq. (19) recover Definition 2 is incorrect: Eq. (19) then gives Σ_x (p(x)^{1-k+r}-p(x)^{1+k+r})/(2k), which equals the Definition 2 expression only when r=k. The two-parameter claim is therefore false by construction.
[Section 4, Definition 5] The same cancellation occurs in the divergence. Substituting ln_{k,r} into D_{k,r}(P||Q)=Σ_x p(x)(p(x)/q(x))^{r-k} ln_{k,r}(p(x)/q(x)) gives D_{k,r}(P||Q)=Σ_x [p(x)-p(x)^{1-2k} q(x)^{2k}]/(2k), which is independent of r. Hence Theorems 6-8, Lemma 9, and the other divergence properties are properties of a one-parameter family, not of the claimed two-parameter generalization. The parameter r enters Definition 5 only through the deformed logarithm and cancels before any information-theoretic statement is made.
[Section 5, Eq. (85) and Theorem 9] The Hessian computation is internally inconsistent. Since D_{k,r} is r-independent, the induced metric cannot depend on r. Direct differentiation of the simplified expression D_{k,r}(P||Q)=Σ_i [p_i-p_i^{1-2k}q_i^{2k}]/(2k) gives ∂²D/∂p_i²|_{Q=P}=(1-2k)/p_i, not the reported (-2k+4r+1)/p_i. Thus Eq. (85), the metric in Eq. (86), and the Hessian-manifold claim in Theorem 9 are not supported by the definitions. Moreover, at k=1/2, which lies in the stated range 0<k≤1/2, the metric degenerates.
[Section 4, Lemma 9] At the boundary k=1/2 of the allowed parameter range, the divergence collapses identically: D_{1/2,r}(P||Q)=Σ_x [p(x)-q(x)]/(1)=0 for all probability distributions P and Q. This contradicts the positivity claim in Lemma 9, which states that equality holds only for P=Q, and it violates the divergence axioms listed in Section 5. The one-parameter object itself therefore has a genuine internal consistency problem at an admissible parameter value.

minor comments (4)

[Theorem 9 and Conclusion] The term 'Hassian' should be 'Hessian' in Theorem 9 and in the concluding discussion; this is a typographical issue but appears repeatedly.
[Eq. (8) and Definition 1] Equation (8) defines Ln_{k,r}(x) with x^r in the numerator, while Definition 1 defines ln_{k,r}(x) with x^r in the denominator. The two functions are different, and the notation is easy to confuse; an explicit remark distinguishing them would improve clarity.
[Introduction, Eq. (4)] The displayed joint convexity inequality in Eq. (4) appears to contain a typo: the right-hand side should be (1-λ)D_{k,r}(P^{(1)}||Q^{(1)})+λD_{k,r}(P^{(2)}||Q^{(2)}), not D_{k,r}(P^{(1)}||Q^{(1)})+λD_{k,r}(P^{(1)}||Q^{(1)}).
[Theorem 1] The proof relies on a product-of-convex-functions fact with a citation but does not state the required hypotheses precisely; adding a short proof or an exact reference statement would make the argument fully self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivations follow from the stated definitions; the r-cancellation is a correctness flaw, not a circular one.

full rationale

The paper's derivation chain is self-contained: Definition 1 fixes ln_{k,r}(x), Definition 2 defines S_{k,r}, and the product rule, chain rule, sub-additivity, joint convexity, and information monotonicity are proved from these definitions without assuming the conclusions. The cited works by the authors (Furuichi [7,8,9,15]) are not load-bearing: Eq. (19) is only a motivational comparison, and no theorem depends on a prior uniqueness or existence result. The serious defect that S_{k,r}(X) = (1 - Σ p^{2k+1})/(2k), so that r cancels and the claimed two-parameter entropy is just Tsallis entropy with q=2k+1, is an algebraic/correctness flaw rather than a circular step: the derivation does not use the two-parameter claim as a premise, and the stated reduction to Eq. (19) is incorrect for r≠k. There is no empirical fit, no prediction-from-fit, and no load-bearing self-citation, so the circularity burden is minimal.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The paper contributes a reparameterization of Tsallis entropy. The only hand-chosen inputs are the parameters k and r; r cancels out of both final formulas, so the effective free parameter is k. The axiomatic input is the ad hoc deformed logarithm of Definition 1, plus the restriction k<=1/2. No independent evidence supports the new objects as distinct from Tsallis entropy and standard f-divergences.

free parameters (2)

k
Hand-chosen parameter, 0<k<=1/2, appearing as the Tsallis parameter q=2k+1 in S_{k,r}; not fitted to data.
r
Hand-chosen parameter, r>0, declared to make a two-parameter family but cancels identically in S_{k,r} and D_{k,r}; a dummy parameter.

assumptions (5)

ad hoc to paper Definition 1: ln_{k,r}(x) = (x^k - x^{-k})/(2k x^r) is the deformed logarithm on which all subsequent constructions rest.
Introduced specifically to make the product rule (Lemma 1) and chain rule work; the original Sharma-Mittal logarithm of equation (8) does not. No external motivation is given for this particular deformation.
domain assumption Parameter range: 0<k<=1/2 and r>0.
Convexity of f(x)=x^{r-k+1}ln_{k,r}(x), lemmas 4, 5, 7 and the log-sum inequality all require k<=1/2. The paper restricts to this range without deriving it from any physical or mathematical principle.
standard math Convexity of x^{r-k+1}ln_{k,r}(x), used in Theorem 2 (log-sum inequality).
Follows from k<=1/2 after simplification; the proof uses Jensen's inequality.
standard math The divergence's Fisher metric is its second derivative at Q=P (equation 82).
Standard information-geometric definition; applied to D_{k,r} which is actually independent of r.
ad hoc to paper Unproved claim in Theorem 1: product of two positive convex functions with the same monotonicity is convex.
Stated without proof and used only for the preliminary old logarithm; generally not automatic for arbitrary convex functions.

invented entities (2)

Two-parameter generalized Tsallis entropy S_{k,r}
purpose: Claimed new entropy family with chain rule and sub-additivity.
Simplifies to (1 - sum p^{2k+1})/(2k), the Tsallis entropy S_q with q=2k+1, independent of r; it is not a new entity.
Two-parameter generalized Tsallis relative entropy D_{k,r}
purpose: Claimed new divergence with joint convexity and information monotonicity.
Simplifies to (1 - sum p^{1-2k} q^{2k})/(2k), a known f-divergence; at k=1/2 it is identically zero for all P,Q, so it fails to be a divergence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A two-parameter entropy and its fundamental properties." pith.science (2026). https://pith.science/paper/6GS5J4KB

@misc{pith2026190801696,
  author       = {Pith},
  title        = {Pith review of: A two-parameter entropy and its fundamental properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GS5J4KB}},
  note         = {Machine review of arXiv:1908.01696}
}

read the original abstract

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and the Shannon entropy for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an exposit investigation on the information-theoretic and information-geometric characteristics of the new generalized entropy and compare them with the properties of the Tsallis and the Shannon entropy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

[1]

Methods of information geometry , volume 191

Shun-ichi Amari and Hiroshi Nagaoka. Methods of information geometry , volume 191. American Mathematical Soc., 2007

work page 2007
[2]

Thermodynamic equilibrium and its stability for microcanonical systems described by the sharma-taneja-mit tal entropy

AM Scarfone and T Wada. Thermodynamic equilibrium and its stability for microcanonical systems described by the sharma-taneja-mit tal entropy. Physical Review E , 72(2):026123, 2005. 23

work page 2005
[3]

Inf ormation ge- ometry of κ-exponential families: Dually-ﬂat, hessian and legendre struc- tures

Antonio Scarfone, Hiroshi Matsuzoe, and Tatsuaki Wada. Inf ormation ge- ometry of κ-exponential families: Dually-ﬂat, hessian and legendre struc- tures. Entropy, 20(6):436, 2018

work page 2018
[4]

Legendre structure of the thermostatistics th eory based on the sharma–taneja–mittal entropy

AM Scarfone. Legendre structure of the thermostatistics th eory based on the sharma–taneja–mittal entropy. Physica A: Statistical Mechanics and its Applications, 365(1):63–70, 2006

work page 2006
[5]

Gauss la w of error revisited in the framework of sharma-taneja-mittal information m easure

Antonio M Scarfone, Hiroki Suyari, and Tatsuaki Wada. Gauss la w of error revisited in the framework of sharma-taneja-mittal information m easure. Central European Journal of Physics , 7(3):414–420, 2009

work page 2009
[6]

Possible generalization of boltzmann-gibbs st atistics

Constantino Tsallis. Possible generalization of boltzmann-gibbs st atistics. Journal of statistical physics , 52(1-2):479–487, 1988

work page 1988
[7]

Fundamenta l properties of tsallis relative entropy

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

work page 2004
[8]

Information theoretical properties of tsallis e ntropies

Shigeru Furuichi. Information theoretical properties of tsallis e ntropies. Journal of Mathematical Physics , 47(2):023302, 2006

work page 2006

Show all 29 references

[9]

On uniqueness theorems for tsallis entropy and tsallis relative entropy

Shigeru Furuichi. On uniqueness theorems for tsallis entropy and tsallis relative entropy. IEEE Transactions on Information Theory , 51(10):3638– 3645, 2005

2005
[10]

Entropy of type ( α , β ) and other generalized measures in information theory

Bhu D Sharma and Inder J Taneja. Entropy of type ( α , β ) and other generalized measures in information theory. Metrika, 22(1):205–215, 1975

1975
[11]

On some functional equations concerning entropy, d irected divergence and inaccuracy

DP Mittal. On some functional equations concerning entropy, d irected divergence and inaccuracy. Metrika, 22(1):35–45, 1975

1975
[12]

Deformed logarithms an d en- tropies

G Kaniadakis, M Lissia, and AM Scarfone. Deformed logarithms an d en- tropies. Physica A: Statistical Mechanics and its Applications , 340(1-3):41– 49, 2004

2004
[13]

Convex optimization

Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004

2004
[14]

Elements of information theory

Thomas M Cover and Joy A Thomas. Elements of information theory . John Wiley & Sons, 2012

2012
[15]

An axiomatic characterization of a two-param eter ex- tended relative entropy

Shigeru Furuichi. An axiomatic characterization of a two-param eter ex- tended relative entropy. Journal of Mathematical Physics , 51(12):123302, 2010

2010
[16]

Multivariable Calculus

James Stewart. Multivariable Calculus . Brooks/Cole, CA, 1995

1995
[17]

Information geometry of divergence functions

Shun-ichi Amari and Andrzej Cichocki. Information geometry of divergence functions. Bulletin of the Polish Academy of Sciences: Technical Scien ces, 58(1):183–195, 2010. 24

2010
[18]

Divergence function, information monotonicit y and in- formation geometry

Shun-ichi Amari. Divergence function, information monotonicit y and in- formation geometry. In Workshop on Information Theoretic Methods in Science and Engineering (WITMSE) . Citeseer, 2009

2009
[19]

Ihara Zeta Entropy

Supriyo Dutta and Partha Guha. Ihara Zeta Entropy. arXiv e-prints , page arXiv:1906.02514, Jun 2019

1906 arXiv
[20]

Formal groups and z-entropies

Piergiulio Tempesta. Formal groups and z-entropies. Proceedings of the Royal Society A: Mathematical, Physical and Engineerin g Sciences , 472(2195):20160143, 2016

2016
[21]

Image thresholding using tsallis entropy

M Portes De Albuquerque, Israel A Esquef, and AR Gesualdi Me llo. Image thresholding using tsallis entropy. Pattern Recognition Letters, 25(9):1059– 1065, 2004

2004
[22]

Optimal multi-level thresholding ba sed on maximum tsallis entropy via an artiﬁcial bee colony approach

Yudong Zhang and Lenan Wu. Optimal multi-level thresholding ba sed on maximum tsallis entropy via an artiﬁcial bee colony approach. Entropy, 13(4):841–859, 2011

2011
[23]

Application of tsallis entropy to eeg: quantifying the prese nce of burst suppression after asphyxial cardiac arrest in rats

Dandan Zhang, Xiaofeng Jia, Haiyan Ding, Datian Ye, and Nitish V Thakor. Application of tsallis entropy to eeg: quantifying the prese nce of burst suppression after asphyxial cardiac arrest in rats. IEEE transactions on biomedical engineering , 57(4):867–874, 2009

2009
[24]

Tsallis entropy and cor ti- cal dynamics: the analysis of eeg signals

A Capurro, L Diambra, D Lorenzo, O Macadar, MT Martin, C Most accio, A Plastino, E Rofman, ME Torres, and J Velluti. Tsallis entropy and cor ti- cal dynamics: the analysis of eeg signals. Physica A: Statistical Mechanics and its Applications , 257(1-4):149–155, 1998

1998
[25]

Applications of entrop y in ﬁnance: A review

Rongxi Zhou, Ru Cai, and Guanqun Tong. Applications of entrop y in ﬁnance: A review. Entropy, 15(11):4909–4931, 2013

2013
[26]

Ts allis mutual information for document classiﬁcation

Marius Vila, Anton Bardera, Miquel Feixas, and Mateu Sbert. Ts allis mutual information for document classiﬁcation. Entropy, 13(9):1694–1707, 2011

2011
[27]

Quantum discord thr ough the generalized entropy in bipartite quantum states

Xi-Wen Hou, Zhi-Peng Huang, and Su Chen. Quantum discord thr ough the generalized entropy in bipartite quantum states. The European Physical Journal D , 68(4):87, 2014

2014
[28]

R´ enyi squashed entanglement, discord, and relative entropy diﬀerences

Kaushik P Seshadreesan, Mario Berta, and Mark M Wilde. R´ enyi squashed entanglement, discord, and relative entropy diﬀerences. Journal of Physics A: Mathematical and Theoretical , 48(39):395303, 2015

2015
[29]

Sharma–m ittal quan- tum discord

Souma Mazumdar, Supriyo Dutta, and Partha Guha. Sharma–m ittal quan- tum discord. Quantum Information Processing , 18(6):169, 2019. 25

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Methods of information geometry , volume 191

Shun-ichi Amari and Hiroshi Nagaoka. Methods of information geometry , volume 191. American Mathematical Soc., 2007

work page 2007

[2] [2]

Thermodynamic equilibrium and its stability for microcanonical systems described by the sharma-taneja-mit tal entropy

AM Scarfone and T Wada. Thermodynamic equilibrium and its stability for microcanonical systems described by the sharma-taneja-mit tal entropy. Physical Review E , 72(2):026123, 2005. 23

work page 2005

[3] [3]

Inf ormation ge- ometry of κ-exponential families: Dually-ﬂat, hessian and legendre struc- tures

Antonio Scarfone, Hiroshi Matsuzoe, and Tatsuaki Wada. Inf ormation ge- ometry of κ-exponential families: Dually-ﬂat, hessian and legendre struc- tures. Entropy, 20(6):436, 2018

work page 2018

[4] [4]

Legendre structure of the thermostatistics th eory based on the sharma–taneja–mittal entropy

AM Scarfone. Legendre structure of the thermostatistics th eory based on the sharma–taneja–mittal entropy. Physica A: Statistical Mechanics and its Applications, 365(1):63–70, 2006

work page 2006

[5] [5]

Gauss la w of error revisited in the framework of sharma-taneja-mittal information m easure

Antonio M Scarfone, Hiroki Suyari, and Tatsuaki Wada. Gauss la w of error revisited in the framework of sharma-taneja-mittal information m easure. Central European Journal of Physics , 7(3):414–420, 2009

work page 2009

[6] [6]

Possible generalization of boltzmann-gibbs st atistics

Constantino Tsallis. Possible generalization of boltzmann-gibbs st atistics. Journal of statistical physics , 52(1-2):479–487, 1988

work page 1988

[7] [7]

Fundamenta l properties of tsallis relative entropy

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

work page 2004

[8] [8]

Information theoretical properties of tsallis e ntropies

Shigeru Furuichi. Information theoretical properties of tsallis e ntropies. Journal of Mathematical Physics , 47(2):023302, 2006

work page 2006