Accessibility of physical states and non-uniqueness of entanglement measure

Ordering physical states is the key to quantifying some physical property of the states uniquely. Bipartite pure entangled states are totally ordered under local operations and classical communication (LOCC) in the asymptotic limit and uniquely quantified by the well-known entropy of entanglement. However, we show that mixed entangled states are partially ordered under LOCC even in the asymptotic limit. Therefore, non-uniqueness of entanglement measure is understood on the basis of an operational notion of asymptotic convertibility.