Strong superadditivity and monogamy of the Renyi measure of entanglement

TL;DR

This paper investigates the properties of the quantum Rényi α-entropies as entanglement measures, revealing conditions under which they violate strong superadditivity and satisfy monogamy, with implications for high-dimensional quantum systems.

Contribution

It demonstrates the violation of strong superadditivity for Rényi entropies with α>1 and establishes the numerical equivalence of monogamy and strong superadditivity for this measure.

Findings

Violation of strong superadditivity for α>1 diminishes as α approaches 1.

Rényi measure satisfies monogamy at α=2.

Violations occur in states with special symmetries.

Abstract

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.