Some inequalities for quantum Tsallis entropy related to the strong subadditivity

Abstract

In this paper we investigate the inequality $S_q({\rho }_{123})+S_q({\rho }_2)\le S_q({\rho }_{12})+S_q({\rho }_{23})(\ast )$ where ${\rho }_{123}$ is a state on a finite dimensional Hilbert space ${\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes {\mathcal{H}}_3,$ 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 $(\ast )$, which is an inequality of relative quasi-entropies. We derive an inequality of the form $S_q({\rho }_{123})+S_q({\rho }_2)\le S_q({\rho }_{12})+S_q({\rho }_{23})+f_q({\rho }_{123})$, where $f_1({\rho }_{123})=0$. Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that $(\ast )$ does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.

arXiv commentv2: the introductory part reorganized v3: the published version

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