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

TL;DR

This paper examines different quantum divergences in nonextensive formalism, highlighting how their differences impact physical measures like purity and fidelity, and discusses the compatibility of Bregman type relative entropy with third choice thermostatistics.

Contribution

It compares two forms of quantum divergences in nonextensive formalism, revealing their implications for physical measures and the limitations of Bregman type relative entropy.

Findings

Bregman type divergence leads to negative values for certain q

Csiszár type divergence avoids anomalies and is more consistent

Different divergences exhibit distinct qualitative behaviors with respect to fidelity

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.