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International Journal of Theoretical Physics

Generalized Position and Momentum Tsallis Entropies

Angsula Ghosh, Puspitapallab Chaudhuri

Published Oct 1, 2000• 20 citations
Publisher
DOI
10.1023/a:1026432919521

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arXiv:1006.4699•Journal of Physics A Mathematical and Theoretical•Feb 9, 2011•42 citations
In this study, the information‐theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy, and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava–Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also, they are obtained in the momentum space in terms of the multivariate Bell polynomials of Combinatorics. We observed that the Fisher information increases with n in both the position and momentum spaces, but decreases with for all the diatomic molecules considered. The Shannon entropy also increases with increasing n in the position space and decreases with increasing . The variations of the Renyi and Tsallis entropies with are also discussed. The exact and numerical values of the Onicescu information energy are also obtained, after which the ratio of information‐theoretic impetuses to lengths for Fisher, Shannon, and Renyi are obtained. © 2015 Wiley Periodicals, Inc.
A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For a given input state, different effects of each unraveling result in some probability distribution at the output. As is shown, all Tsallis' entropies of positive order as well as some of Rényi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Rényi and Tsallis entropies. The inequality with Rényi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a d -level system and for the angle and angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.