Strong superadditivity and monogamy of the Renyi measure of entanglement

Employing the quantum Renyi 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 -> 1 and vanishes for alpha = 1 when the measure corresponds to the entanglement of formation. We show that the Renyi 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 satisfying monogamy for high-dimensional systems also satisfies the strong superadditivity inequality. For the case of Renyi measure, we provide strong numerical evidences that these two properties are equivalent.