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In this paper we investigate the inequality $S_q(\rho_{123})+S_q(\rho_2)\leqS_q(\rho_{12})+S_q(\rho_{23}) \, (*)$ where $\rho_{123}$ is a state on a finitedimensional 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 thestrong subadditivity of the von Neumnann entropy can be derived from themonotonicity of the Umegaki relative entropy. Now, we present an equivalentform of $(*)$, which is an inequality of relative quasi-entropies. We derive aninequality of the form $S_q(\rho_{123})+S_q(\rho_2)\leqS_q(\rho_{12})+S_q(\rho_{23})+f_q(\rho_{123})$, where $f_1(\rho_{123})=0$. Sucha result can be considered as a generalization of the strong subadditivity ofthe von Neumnann entropy. One can see that $(*)$ does not hold in general (apicturesque example is included in this paper), but we give a sufficientcondition for this inequality, as well.

Some inequalities for quantum Tsallis entropy related to the strong subadditivity