A Characterization of Partial Entanglement Recovery

Nielsen’s theorem indicates that a bipartite entangled pure state can be exactly converted into another pure one by local quantum operations and classical communication if and only if the Schmidt coefficient vector of the former is majorized by that of the latter [Phys. Rev. Lett. 83, 436 (1999)]. As a consequence of Nielsen’s majorization criterion, the quantity of entanglement is decreased during the conversion process. On the other hand, it was observed that the entanglement lost in this process may be partially recovered by using an auxiliary entangled state [Phys. Rev. Lett. 84, 3189 (2000)]. However, it is usually not easy to find an auxiliary state that can be used to do partial entanglement recovery for a given transformation. The major difficulty seems due to the lack of suitable mathematical tools. In this paper, we introduce three indices of uniformity for bipartite entangled pure states. These indices enable us to find some convenient conditions for partial entanglement recovery. In the case that the Schmidt coefficient vector of the source state is strictly majorized by that of the target state, a complete characterization of auxiliary states for partial entanglement recovery is obtained. For the general case, two sufficient conditions for such auxiliary states are given. A combination of these results implies that the choice of the auxiliary state can only depend on the target state and the presence of the equalities in the majorization relation, and the source state is in fact irrelevant. As applications, we establish some interesting connections of partial entanglement recovery to the generation of maximally entangled states, entanglement catalysis, and multiple-copy entanglement transformation. Index Terms — Quantum information processing, Quantum entanglement, Entanglement transformation, Partial entanglement recovery, Majorization relation.