Strong superadditivity and monogamy of the Renyi measure of entanglement

Employing the quantum R\'enyi $\alpha$-entropies as a measure of entanglement, we numerically find the violation of the strong superadditivity inequality for a system composed of four qubits and $\alpha>1$. This violation gets smaller as $\alpha\rightarrow 1$ and vanishes for $\alpha=1$ when the measure corresponds to the Entanglement of Formation (EoF). We show that the R\'enyi measure aways satisfies the standard monogamy of entanglement for $\alpha = 2$, and only violates a high order monogamy inequality, in the rare cases in which the strong superadditivity is also violated. The sates numerically found where the violation occurs have special symmetries where both inequalities are equivalent. We also show that every measure satisfing monogamy for high dimensional systems also satisfies the strong superadditivity inequality. For the case of R\'enyi measure, we provide strong numerical evidences that these two properties are equivalent.