On The Problem of Constraints In Nonextensive Formalism: A Quantum Mechanical Treatment

Altug Arda; Gokhan B. Bagci; Ramazan Sever

REVIEW

On The Problem of Constraints In Nonextensive Formalism: A Quantum Mechanical Treatment

Gokhan B. Bagci, Altug Arda, Ramazan Sever

Statistical Mechanics · Submitted 2 Mar 2006 · 32 references

Pith's verdict: Real Werner-state calculation, but the negative-values punchline is an artifact of an unregularized singular state; needs revision before it publishes.

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arXiv IDcond-mat/0603047 (version 3)

Pith IDLOYE5AV6, Pith's permanent record of this paper and its review

CitationAltug Arda, Gokhan B. Bagci, Ramazan Sever. "On The Problem of Constraints In Nonextensive Formalism: A Quantum Mechanical Treatment." arXiv:cond-mat/0603047 (2006). https://pith.science/paper/LOYE5AV6

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Claim paper

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Editor's note

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Short version: the Werner-state calculation is real and the no-go argument is worth stating, but the paper's own Eq. (26) undercuts the abstract's negative-values claim, and the constraint-mapping criterion is imported rather than proven.

What's new: Eq. (21), the explicit Bregman-type divergence for the Werner state, is not in the cited literature, and the perturbative regularization with epsilon is correct as far as it goes. The observation that the Bregman and Csiszár forms give different qualitative behavior in F, and the attempt to write a Bregman form compatible with the third choice, are legitimate extensions. The paper correctly credits Frank and Abe-Bagci for the free-energy-difference identifications; self-citation here is not padding because the relevant equations are rederived in the text.

Where it gets soft: the central advertised result is the negative values of D_q for q in (0,1), and that claim is an artifact. Eq. (21) is obtained by effectively treating sigma^{q-1} as 0 on the subspace orthogonal to |Psi->, but for q<1 that operator is unbounded on the singular pure state. The paper's own regularized calculation, Eq. (26), gives D_q -> +infinity as epsilon -> 0. So the 'negative values' in Eq. (21) do not survive the limit that the authors themselves introduce. The full-text abstract retreats to 'suitable for q>1,' but the headline abstract still advertises the negative-values argument. That is an internal inconsistency, not a matter of taste.

The second soft spot is the mapping from divergence type to constraint choice. The paper equates compatibility with a thermostatistical constraint to the ability to write the divergence as a free-energy difference under that constraint's maximum-entropy distribution. That criterion comes from prior work (Abe-Bagci) and is not proven here. If the criterion is not right, the whole 'first choice vs. third choice' story collapses. The no-go conclusion also only rules out the particular escort substitution p_i -> P_i = p_i^q / sum_j p_j^q with q -> 1/q, not all possible Bregman forms. The authors do flag the concavity problem, which is a real extra argument, but it is not enough to make the no-go fully general.

Bottom line: the algebra checks out and there is a small new result here. But the paper as it stands overstates its conclusion in the abstract, and the main advertised anomaly is not robust. It deserves a careful referee, not a desk rejection, because the contradiction is fixable and the explicit calculation is a useful data point. A referee should ask the authors to reconcile Eq. (21) with Eq. (26), drop the negative-values claim, and either prove or clearly caveat the constraint-mapping criterion.

Referee report

Significance

If the central claims were sound, the paper would provide a concrete quantum-mechanical argument for preferring the normalized-q-expectation-value constraint in nonextensive statistics. The algebraic computations are transparent and contain no fitted parameters; the authors also include a regularization calculation, which is a useful check. The free-energy-identification and the no-go discussion are potentially interesting. However, the main superiority argument is undermined by an internal inconsistency between the unregularized and regularized calculations, and the claimed qualitative behavior is not supported by the regularized expressions. The paper's value would depend on a substantial revision that removes or rigorously justifies the negative-values claim.

Major objections (4)

[Section II, Eqs. (21) and (26)] The negative-values argument that motivates the abstract's superiority claim is contradicted by the paper's own regularized calculation. Equation (21), which gives negative values for q∈(0,1), is obtained by effectively treating σ^{q-1} as bounded for the pure state σ=|Ψ−⟩⟨Ψ−|. But for q<1, q−1 is negative and the eigenvalues of σ orthogonal to |Ψ−⟩ are zero, so σ^{q-1} is unbounded. The regularized calculation in Eq. (24) leads to Eq. (26), D_q→+∞ as ε→0 for q∈(0,1), not the finite negative value of Eq. (21). The negative values are therefore an artifact of the singular pure state. The supplied abstract, which cites negative values for q∈(0,1) and F<1 as evidence of the superiority of the third choice, is also inconsistent with the full-text abstract and Section III, which state that the Bregman type is suitable only for q>1. This inconsistency directly undermines the claimed superiority of the third-choice thermostatistics.
[Section III and Eq. (28)] The statement that "the Bregman type decreases as fidelity F decreases" is not supported by the regularized expression. For q>1, which is the only regime in which D_q is finite for the singular σ, Eq. (28) increases as F decreases; for example, at q=2 it goes from 0 at F=1 to 3/4 at F=1/4. The claimed qualitative difference appears to rely on the invalid q∈(0,1) values from Eq. (21). If the authors intend a different regime, they need to reconcile it with Eq. (26).
[Section II, Eqs. (35)-(36)] The central mapping between divergence type and constraint choice rests on the premise that compatibility of a divergence with a given constraint is equivalent to the divergence being expressible as a free-energy difference under that constraint's maximum-entropy distribution. This premise is imported from Ref. [29] and is not proved in the present paper. If this criterion is not accepted, the conclusion that the Bregman type is related to the first choice and the Csiszár type to the third choice does not follow from the calculations.
[Abstract and Section III] The comparison underlying the "superiority" claim is not between two constraints at the same value of q. The Csiszár expression is evaluated for q∈(0,1), while the Bregman expression is regularized only for q>1. The observed difference in qualitative behavior may therefore be a q-regime effect rather than a constraint effect. Establishing superiority would require comparing the two constraints at a common q, or a physical argument for why the relevant q ranges should differ.

Minor objections (4)

[Section II, cross-references] The cross-references in the free-energy discussion are incorrect: "substitute ~p_i^ordinary given by Eq. (23)" should refer to the ordinary maximum-entropy distribution in Eq. (30), and "substitute ~p_i^normalized in Eq. (26)" should refer to Eq. (33). Similarly, "Eq. (29) shows us" should refer to Eq. (36).
[Eqs. (23) and (37)] Equation (23) has an unmatched parenthesis at the end ("− 1)]"), and Eq. (37) is missing the square bracket around the second trace difference, so it reads as −Tr(ρσ^{1/q−1}) − Tr(σ^{1/q}) instead of −[Tr(ρσ^{1/q−1}) − Tr(σ^{1/q})].
[Eqs. (30) and (33)] The quantities \tilde S_q and \tilde U with superscripts "ordinary" and "normalized" are used but not defined in the text; they should be defined as the Tsallis entropy and internal energy evaluated at the corresponding maximum-entropy distribution.
[Title and abstract] The term "purity" is misleading: the calculation is of a relative entropy (divergence) between the Werner state and a pure state, not of the purity Tr(ρ^2). Consider using "fidelity" or "relative entropy" consistently.

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The paper

The paper claims

Two quantum relative entropies disagree on Werner states because they encode different constraints in nonextensive thermostatistics.

The reading

The paper sets out to resolve an apparent inconsistency in Tsallis nonextensive statistics: two quantum relative entropies, the Bregman-type divergence and the Csiszár-type divergence, give different answers when used to measure the purity of a Werner state. It argues that the discrepancy is not a flaw but a fingerprint, because the Bregman divergence is the natural free-energy difference for thermostatistics built on ordinary expectation values (first choice), while the Csiszár divergence is the natural free-energy difference for thermostatistics built on normalized $q$-expectation values (third choice). For $q\in(0,1)$, the Csiszár expression is positive and well-behaved, whereas the Bregman expression becomes negative for fidelity $F<1$, which the paper reads as evidence that the third-choice constraint scheme is the consistent one. It closes with a no-go result: rewriting the Bregman divergence through escort probabilities and $q\to1/q$ does not yield a legitimate third-choice relative entropy, because the escort normalization is an identity rather than an independent constraint and the escort entropy is nonconcave in some intervals.

What carries the argument

The load-bearing objects are the two quantum relative entropies themselves. The Csiszár-type divergence is $I_q(\rho\|\sigma)=\frac{1}{q-1}[\mathrm{Tr}(\rho^q\sigma^{1-q})-1]$, and the Bregman-type divergence is $D_q(\rho\|\sigma)=\frac{1}{q-1}[\mathrm{Tr}(\rho^q)-\mathrm{Tr}(\rho\sigma^{q-1})]-[\mathrm{Tr}(\rho\sigma^{q-1})-\mathrm{Tr}(\sigma^q)]$. The argument uses the Werner state as the discriminating test system, the maximum-entropy variational problems of Eqs. (29) and (32) as the interpretive bridge, and the escort substitution $p_i\to P_i=p_i^q/\sum_jp_j^q$ with $q\to1/q$ as the route whose failure establishes the no-go conclusion. Substituting the maximum-entropy distributions back into the divergences is what turns each divergence into a free-energy difference, tying each to one constraint choice.

Extended reading notes

Core claim

The central claim is that the Bregman-type quantum relative entropy $D_q(\rho\|\sigma)$ and the Csiszár-type quantum relative entropy $I_q(\rho\|\sigma)$ are not rival definitions of the same quantity but expressions of two different thermostatistical constraint choices. On the Werner state $\rho_W$ with fidelity $F$, Csiszár gives $I_q=(1-F^q)/(1-q)$, which is positive for $q\in(0,1)$ and grows as $F$ falls, while Bregman gives $D_q=\frac{1}{q-1}[F^q+3(\frac{1-F}{3})^q-F]-(F-1)$, which is negative for $q\in(0,1)$ and $F<1$ and falls with $F$. The paper identifies the cause by substituting each divergence into its own maximum-entropy distribution: the Bregman form becomes the free-energy difference under ordinary expectation values (Eq. 35), and the Csiszár form becomes the free-energy difference under normalized $q$-expectation values (Eq. 36). It then considers whether the Bregman form can be adapted to the third choice by the escort transformation $p_i\to P_i=p_i^q/\sum_jp_j^q$ with $q\to1/q$ and concludes that it cannot: the escort normalization is an identity, and the escort entropy is not concave in some regimes.

Load-bearing premise

The argument depends on identifying a divergence's compatibility with a constraint with its being writable as a free-energy difference under that constraint's maximum-entropy distribution, and on treating the escort substitution as the only way to give the Bregman form a third-choice meaning.

Editorial extensions

If this is right

For $q\in(0,1)$, the Csiszár divergence on Werner states is positive and increases as fidelity $F$ decreases, whereas the Bregman divergence is negative and decreases with $F$; the two measures differ in sign and in qualitative trend.
Each divergence is a free-energy difference, but only when paired with its matching constraint: Bregman with ordinary expectation values and Csiszár with normalized $q$-expectation values.
The escort-probability rewriting of the Bregman divergence fails as a third-choice relative entropy because it treats an identity as a constraint and because the escort entropy is nonconcave in some intervals.
The quantum Werner-state calculation selects the constraint scheme by positivity alone, without needing the more abstract Shore–Johnson axioms used in the classical case.

Reading between the lines

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

If the constraint-diagnosis is right, the sign split between the two divergences should persist for other mixed states, not just Werner states; evaluating both on random two-qubit states across $q\in(0,1)$ would be a quick numerical check.
The no-go result is proved only for the usual escort substitution; a differently deformed relative entropy that realizes the third choice without imposing escort normalization as an independent constraint might evade the obstruction.
One could invert the logic and use positivity of a quantum divergence as the selection rule: find, for a given state and $q$, which constraint scheme keeps the divergence positive-definite.

Original abstract

Relative entropy (divergence) of Bregman type recently proposed by T. D. Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism. It has been observed that two different expressions arise due to two different forms of quantum divergences. It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas the one of Csisz\`{a}r type is related to the third choice thermostatistics. The superiority of the third choice thermostatistics to the first choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q$\in (0,1)$ and fidelity F smaller than 1 whereas the one obtained by using Csisz\`{a}r type is free from such anomalies. Moreover, it has been noted that these two measures show different qualitative behavior with respect to F. Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore-Johnson axioms. The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further. The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,009 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Relative entropy (divergence) of Bregman type recently proposed by T. D. Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism. It has been observed that two different expressions arise due to two different forms of quantum divergences. It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas the one of Csisz\`{a}r type is related to the third choice thermostatistics. The superiority of the third choice thermostatistics to the first choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q$\in (0,1)$ and fidelity F smaller than 1 whereas the one obtained by using Csisz\`{a}r type is free from such anomalies. Moreover, it has been noted that these two measures show different qualitative behavior with respect to F. Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore-Johnson axioms. The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further. The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Relative entropy (divergence) of Bregman type recently proposed by T. D.	background 0.79	0.09	0.02	unclear 0.77
s1	Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism.	contribution 0.58	0.80	0.06	asserted 1.00
s2	It has been observed that two different expressions arise due to two different forms of quantum divergences.	result 1.00	0.82	0.07	asserted 1.00
s3	It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas the one of Csisz\`{a}r type is related to the third choice thermostatistics.	interpretation 0.99	0.66	0.22	asserted 0.95
s4	The superiority of the third choice thermostatistics to the first choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q$\in (0,1)$ and fidelity F smaller than 1 whereas the one obtained by using Csisz\`{a}r type is free from such anomalies.	result 0.57	0.85	0.62	asserted 1.00
s5	Moreover, it has been noted that these two measures show different qualitative behavior with respect to F.	result 0.99	0.83	0.07	asserted 1.00
s6	Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore-Johnson axioms.	interpretation 0.91	0.65	0.27	asserted 1.00
s7	The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further.	contribution 0.82	0.81	0.06	hedged possibility 0.59
s8	The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.	result 0.96	0.86	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.87.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (32)

[29]

T. D. Frank, Physica A 310 (2002) 397

View on Pith 2002
[1]

C. Tsallis, J.Stat. Phys. 52 (1988) 479

View on Pith 1988
[2]

C. Tsallis, in: New Trends in Magnetism, Magnetic Materi als and their Applications, eds. J.L.Mor´ an-Lopez and J.M. S´ anchez (Plenum Press, New York, 1994), p.451

View on Pith 1994
[3]

C. Tsallis, Some comments on Boltzmann-Gibbs statistic al mechanics, Chaos, Solitons and Fractals 6 (1995) 539

View on Pith 1995
[4]

E.M.F. Curado and C. Tsallis, J. Phys. A 24 (1991) L69; Corrigenda: J. Phys. A 24 (1991) 3187; 25 (1992) 1019

View on Pith 1991
[5]

B. M. Boghosian, Phys. Rev. E 53, 4754 (1996)

View on Pith 1996
[6]

G. Wilk, Z. Wlodarcsyk, Nucl. Phys. B (Proc. Suppl.) 75A (1999) 191

View on Pith 1999
[7]

G. Wilk, Z. Wlodarcsyk, Phys. Rev. Lett. 84 (2000) 2770

View on Pith 2000
[8]

I. Bediaga, E. M. F. Curado, J. Miranda, Physica A 286 (2000) 156

View on Pith 2000
[9]

M. L. D. Ion, D. B. Ion, Phys. Lett. B 482 (2000) 57

View on Pith 2000
[10]

O. Kayacan, F. Buyukkilic, D. Demirhan, Physica A 301 (2001) 255

View on Pith 2001
[11]

A. R. Plastino, A. Plastino, Phys. Lett. A 174 (1993) 384

View on Pith 1993
[12]

G. B.Bagci, R. Sever, C. Tezcan, Mod. Phys. Lett. 18 (2004) 467

View on Pith 2004
[13]

D.A. Stariolo, Phys. Lett. A 185 (1993) 262

View on Pith 1993
[14]

T. D. Frank, A. Daﬀertshofer, Physica A 295 (2001) 455

View on Pith 2001
[15]

Ernesto P. Borges, J. of Phys. A 31 (1998) 5281

View on Pith 1998
[16]

J. Naudts, Physica A 316 (2002) 323. 7

View on Pith 2002
[17]

J. Naudts, Physica A 340 (2004) 32

View on Pith 2004
[18]

J. Naudts, Rev. Math. Phys. 16 (2004) 809

View on Pith 2004
[19]

J. Naudts, Physica A 332 (2004) 279

View on Pith 2004
[20]

G. Kaniadakis, Physica A 296 (2001) 405; Phys. Rev. E 66 (2002) 056125; Phys. Rev. E 72 (2005) 036108

View on Pith 2001
[21]

S. Abe, A. K. Rajagopal Phys. Rev. Lett. 91 (2003) 120601

View on Pith 2003
[22]

I. Csisz´ ar, Per. Math. Hung. 2 (1972) 191

View on Pith 1972
[23]

T. D. Frank, Nonlinear Fokker-Planck Equations: Funda mentals and Applications, Springer Verlag, Berlin, 2005

View on Pith 2005
[24]

which reads Dq(ρ ‖ σ) = 1 q − 1 [T r(ρq) − T r(ρσq−1)] − [T r(ρσq−1) − T r(σq)]. (12) S. Abe [25] made use of Eq. (11) in order to calculate the purity of We rner state in the nonextensive framework. For this purpose, he used Werner state [26] which is given by the den sity matrix ρW = F | Ψ −⟩⟨Ψ − | + 1 − F 3 (| Ψ +⟩⟨Ψ + | + | Φ +⟩⟨Φ + | + | Φ −⟩⟨Φ − |),...

View on Pith
[25]

L. M. Bregman, USSR comp. math. math. phys. 7 (1967) 200

View on Pith 1967
[26]

Sumiyoshi Abe, Physica A 344 (2004) 359

View on Pith 2004
[27]

R. F. Werner, Phys. Rev. A 40 (1989) 4277

View on Pith 1989
[28]

C. Tsallis, R. S. Mendes, A. R. Plastino, Physica A 261 (1998) 534

View on Pith 1998
[30]

S. Abe, G. B. Bagci, Phys. Rev. E 71 (2005) 016139

View on Pith 2005
[31]

S. Abe, Phys. Lett. A 275 (2000) 250

View on Pith 2000
[32]

R. P. Di Sisto, S. Martinez, R. B. Orellana, A. R. Plastin o, A. Plastino, Physica A 265 (1999) 590

View on Pith 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [29]

T. D. Frank, Physica A 310 (2002) 397

View on Pith 2002

[2] [1]

C. Tsallis, J.Stat. Phys. 52 (1988) 479

View on Pith 1988

[3] [2]

C. Tsallis, in: New Trends in Magnetism, Magnetic Materi als and their Applications, eds. J.L.Mor´ an-Lopez and J.M. S´ anchez (Plenum Press, New York, 1994), p.451

View on Pith 1994

[4] [3]

C. Tsallis, Some comments on Boltzmann-Gibbs statistic al mechanics, Chaos, Solitons and Fractals 6 (1995) 539

View on Pith 1995

[5] [4]

E.M.F. Curado and C. Tsallis, J. Phys. A 24 (1991) L69; Corrigenda: J. Phys. A 24 (1991) 3187; 25 (1992) 1019

View on Pith 1991

[6] [5]

B. M. Boghosian, Phys. Rev. E 53, 4754 (1996)

View on Pith 1996

[7] [6]

G. Wilk, Z. Wlodarcsyk, Nucl. Phys. B (Proc. Suppl.) 75A (1999) 191

View on Pith 1999

[8] [7]

G. Wilk, Z. Wlodarcsyk, Phys. Rev. Lett. 84 (2000) 2770

View on Pith 2000

[9] [8]

I. Bediaga, E. M. F. Curado, J. Miranda, Physica A 286 (2000) 156

View on Pith 2000

[10] [9]

M. L. D. Ion, D. B. Ion, Phys. Lett. B 482 (2000) 57

View on Pith 2000

[11] [10]

O. Kayacan, F. Buyukkilic, D. Demirhan, Physica A 301 (2001) 255

View on Pith 2001

[12] [11]

A. R. Plastino, A. Plastino, Phys. Lett. A 174 (1993) 384

View on Pith 1993

[13] [12]

G. B.Bagci, R. Sever, C. Tezcan, Mod. Phys. Lett. 18 (2004) 467

View on Pith 2004

[14] [13]

D.A. Stariolo, Phys. Lett. A 185 (1993) 262

View on Pith 1993

[15] [14]

T. D. Frank, A. Daﬀertshofer, Physica A 295 (2001) 455

View on Pith 2001

[16] [15]

Ernesto P. Borges, J. of Phys. A 31 (1998) 5281

View on Pith 1998

[17] [16]

J. Naudts, Physica A 316 (2002) 323. 7

View on Pith 2002

[18] [17]

J. Naudts, Physica A 340 (2004) 32

View on Pith 2004

[19] [18]

J. Naudts, Rev. Math. Phys. 16 (2004) 809

View on Pith 2004

[20] [19]

J. Naudts, Physica A 332 (2004) 279

View on Pith 2004

[21] [20]

G. Kaniadakis, Physica A 296 (2001) 405; Phys. Rev. E 66 (2002) 056125; Phys. Rev. E 72 (2005) 036108

View on Pith 2001

[22] [21]

S. Abe, A. K. Rajagopal Phys. Rev. Lett. 91 (2003) 120601

View on Pith 2003

[23] [22]

I. Csisz´ ar, Per. Math. Hung. 2 (1972) 191

View on Pith 1972

[24] [23]

T. D. Frank, Nonlinear Fokker-Planck Equations: Funda mentals and Applications, Springer Verlag, Berlin, 2005

View on Pith 2005

[25] [24]

which reads Dq(ρ ‖ σ) = 1 q − 1 [T r(ρq) − T r(ρσq−1)] − [T r(ρσq−1) − T r(σq)]. (12) S. Abe [25] made use of Eq. (11) in order to calculate the purity of We rner state in the nonextensive framework. For this purpose, he used Werner state [26] which is given by the den sity matrix ρW = F | Ψ −⟩⟨Ψ − | + 1 − F 3 (| Ψ +⟩⟨Ψ + | + | Φ +⟩⟨Φ + | + | Φ −⟩⟨Φ − |),...

View on Pith

[26] [25]

L. M. Bregman, USSR comp. math. math. phys. 7 (1967) 200

View on Pith 1967

[27] [26]

Sumiyoshi Abe, Physica A 344 (2004) 359

View on Pith 2004

[28] [27]

R. F. Werner, Phys. Rev. A 40 (1989) 4277

View on Pith 1989

[29] [28]

C. Tsallis, R. S. Mendes, A. R. Plastino, Physica A 261 (1998) 534

View on Pith 1998

[30] [30]

S. Abe, G. B. Bagci, Phys. Rev. E 71 (2005) 016139

View on Pith 2005

[31] [31]

S. Abe, Phys. Lett. A 275 (2000) 250

View on Pith 2000

[32] [32]

R. P. Di Sisto, S. Martinez, R. B. Orellana, A. R. Plastin o, A. Plastino, Physica A 265 (1999) 590

View on Pith 1999