Some inequalities for quantum Tsallis entropy related to the strong subadditivity

Abstract

Abstract. In this paper we investigate the inequality S q (ρ 123 ) + S q (ρ 2 ) S q (ρ 12 ) + S q (ρ 23 )( * ) where ρ 123 is a state on a finite dimensional Hilbert space H 1 ⊗ H 2 ⊗ H 3 , and S q 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 S q (ρ 123 ) + S q (ρ 2 ) S q (ρ 12 ) + S q (ρ 23 ) + f q (ρ 123 ) , where f 1 (ρ 123 ) = 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.Mathematics subject classification (2010): 46N50, 94A15, 46L30, 47L90.

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