Some inequalities for quantum Tsallis entropy related to the strong subadditivity

Petz, Dénes and Virosztek, Dániel (2015) Some inequalities for quantum Tsallis entropy related to the strong subadditivity. MATHEMATICAL INEQUALITIES & APPLICATIONS, 18 (2). pp. 555-568. ISSN 1331-4343

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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.

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