Asymptotic Entanglement Preservability of LOCC Conversions

Recently, entanglement concentration was shown to be irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. Moreover, conversion can be asymptotically achieved perfectly and reversibly with only local unitary operation under such condition. Astonishingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting.