Strong superadditivity and monogamy of the Rényi measure of entanglement

Employing the quantum Rényi $\alpha$ 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 $\alpha \to 1$ and vanishes for $\alpha =1$ when the measure corresponds to the entanglement of formation. We show that the Rényi 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 Rényi measure, we provide strong numerical evidences that these two properties are equivalent.

Figure 1

Evolution of the minimization. Each point is a new state found with a smaller residual entanglement than the previous one. The total number of states generated in this example was about 10 000. One can see that the residues of monogamy and strong additivity start to coincide when they are close to zero.

Figure 2

Violation of SS for some states numerically found. These states are found successively minimizing the residual entanglement for SS for $\alpha =2$, 1.5, 1.2, 1.1, and 1.05.