The verification of a requirement of entanglement measures

TL;DR

This paper investigates a new criterion for entanglement measures based on the permutational invariant part of quantum states, showing most known measures satisfy this criterion, thus refining entanglement quantification.

Contribution

It proves that many established bipartite entanglement measures meet the new lower bound criterion, enhancing understanding of entanglement properties.

Findings

Most known bipartite entanglement measures satisfy the new criterion.

Convex-roof entanglement measures, relative entropy, negativity, and logarithmic negativity meet the criterion.

The results refine the quantification and understanding of quantum entanglement.

Abstract

The quantification of quantum entanglement is a central issue in quantum information theory. Recently, Gao \emph{et al}. ( \href{http://dx.doi.org/10.1103/PhysRevLett.112.180501}{Phys. Rev. Lett. \textbf{112}, 180501 (2014)}) pointed out that the maximum of entanglement measure of the permutational invariant part of $\rho$ ought to be a lower bound on entanglement measure of the original state $\rho$, and proposed that this argument can be used as an additional requirement for (multipartite) entanglement measures. Whether any individual proposed entanglement measure satisfies the requirement still has to prove. In this work, we show that most known entanglement measures of bipartite quantum systems satisfy the new criterion, include all convex-roof entanglement measures, the relative entropy of entanglement, the negativity, the logarithmic negativity and the logarithmic convex-roof extended negativity. Our results give a refinement in quantifying entanglement and provide new insights into a better understanding of entanglement properties of quantum systems.