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