On the problem of constraints in nonextensive formalism: A quantum mechanical treatment

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 one of Csiszar 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 ∈ (0, 1) and fidelity F smaller than 1 whereas the one obtained by using Csiszar 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.

[1]  D. Ion,et al.  Strong evidences for correlated nonextensive quantum statistics in hadronic scatterings , 2000 .

[2]  Włodarczyk,et al.  Interpretation of the nonextensivity parameter q in some applications of tsallis statistics and Levy distributions , 2000, Physical review letters.

[3]  Z. Wlodarczyk,et al.  Do we observe levy flights in cosmic rays , 1999 .

[4]  Boghosian.  Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  J. Naudts.  Generalized thermostatistics based on deformed exponential and logarithmic functions , 2003, cond-mat/0311438.

[6]  Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics , 2001, cond-mat/0303150.

[7]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[8]  C. Tsallis.  Extensive versus Nonextensive Physics , 1994 .

[9]  J. Naudts.  Generalized thermostatistics and mean-field theory , 2002, cond-mat/0211444.

[10]  S. Abe.  Quantum q-divergence , 2004 .

[11]  D. Stariolo.  The Langevin and Fokker-Planck equations in the framework of a generalized statistical mechanics , 1994 .

[12]  Remark on the escort distribution representation of nonextensive statistical mechanics , 2000, cond-mat/0006053.

[13]  T. Frank.  Nonlinear Fokker-Planck Equations: Fundamentals and Applications , 2004 .

[14]  Rodney W. Johnson,et al.  Comments on and correction to 'Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy' (Jan 80 26-37) , 1983, IEEE Trans. Inf. Theory.

[15]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[16]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[17]  C. Tsallis,et al.  Generalized statistical mechanics : connection with thermodynamics , 1991 .

[18]  J. Naudts.  CONTINUITY OF A CLASS OF ENTROPIES AND RELATIVE ENTROPIES , 2002, math-ph/0208038.

[19]  T. D. Frank,et al.  Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics , 2002 .

[20]  Ernesto P. Borges.  On a q -generalization of circular and hyperbolic functions , 1998 .

[22]  Andreas Daffertshofer,et al.  H-theorem for nonlinear Fokker–Planck equations related to generalized thermostatistics , 2001 .

[23]  A. R. Plastino,et al.  General thermostatistical formalisms, invariance under uniform spectrum translations, and Tsallis q-additivity , 1999 .

[24]  NON-EXTENSIVE STUDY OF RIGID AND NON-RIGID ROTATORS , 2003, cond-mat/0303523.

[25]  R. Johnson,et al.  Properties of cross-entropy minimization , 1981, IEEE Trans. Inf. Theory.

[26]  A. R. Plastino,et al.  Stellar polytropes and Tsallis' entropy , 1993 .

[27]  I. Csiszár.  A class of measures of informativity of observation channels , 1972 .

[28]  A. M. Scarfone,et al.  A new one-parameter deformation of the exponential function , 2002 .

[29]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[30]  C. Tsallis,et al.  The role of constraints within generalized nonextensive statistics , 1998 .

[31]  J. Naudts.  Deformed exponentials and logarithms in generalized thermostatistics , 2002, cond-mat/0203489.

[32]  S. Abe,et al.  Necessity of q-expectation value in nonextensive statistical mechanics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  C. Tsallis.  Some comments on Boltzmann-Gibbs statistical mechanics , 1995 .