The verification of a requirement of entanglement measures

The quantification of quantum entanglement is a difficult and fundamental question in quantum information theory. We devote to the refinement of axiomatic approach of quantifying entanglement. Recently, Gao et al. (Phys Rev Lett 112:180501, 2014) pointed out that the maximum of entanglement measure of the permutational invariant part \(\rho ^{\mathrm {PI}}\) of a state \(\rho \) ought to be a lower bound on entanglement measure of the original state \(\rho \). They further argued to add this result as requirement on any (multipartite) entanglement measure. Whether any individual proposed entanglement measure satisfies the new requirement still has to be proved. In this paper, we show that most existing entanglement measures of bipartite quantum systems satisfy the new criterion, including all convex-roof entanglement measures, the relative entropy of entanglement, the negativity, the logarithmic negativity and the logarithmic convex-roof extended negativity. Our approach gives a refinement in quantifying entanglement and provides new insights on better understanding of entanglement properties of composite quantum systems.