Some inequalities for quantum tsallis entropy related to the strong subadditivity

Abstract

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.