Some inequalities for quantum Tsallis entropy related to the strong subadditivity

In this paper we investigate the inequality $S_q(ρ_{123})+S_q(ρ_2)\leq S_q(ρ_{12})+S_q(ρ_{23}) \, (*)$ where $ρ_{123}$ 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 $(*)$, which is an inequality of relative quasi-entropies. We derive an inequality of the form $S_q(ρ_{123})+S_q(ρ_2)\leq S_q(ρ_{12})+S_q(ρ_{23})+f_q(ρ_{123})$, 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.

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