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Journal of Mathematical Physics

Non-standard thermal statistics with q-entropies

Gustavo Guerberoff, Pedro A. Pury, G. A. Raggio

Published Apr 1, 1996• 21 citations
Publisher

Abstract

We consider the quantum thermal statistics à la Gibbs–Shannon–Szilard–Jaynes based on q-entropies Sq[ρ]=(q−1)−1(1−tr(ρq)) (0<q≠1) and the non-standard ‘‘internal energy’’ functionals Uq[ρ]=tr(ρqH) proposed by C. Tsallis [J. Stat. Phys. 52, 479–487 (1988)].
DOI
10.1063/1.531480

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In this paper we investigate the inequality Sq(ρ123) + Sq(ρ2) ≤ Sq(ρ12) + Sq(ρ23) (∗) where ρ123 is a state on a finite dimensional Hilbert space H1 ⊗ H2 ⊗ H3, and Sq is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (*), which is an inequality of relative quasi-entropies. We derive an inequality of the form Sq(ρ123) + Sq(ρ2) ≤ Sq(ρ12) + Sq(ρ23) + fq(ρ123), where f1(ρ123) = 0. Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that (*) does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.