Entangled multipartite systems

Abstract

One unsatisfied desideratum for entanglement measures is that of full generality. There is no known single good entanglement measure applicable to all mixed states of systems with arbitrary numbers of subsystems. At present, the bipartite case is the only one in which definitive results may be said to have been obtained, by reference to the number of Bell states asymptotically interconvertible by local operations and classical communication to other states. The von Neumann entropy used in the previous chapter is a reliable measure only of bipartite entanglement. The partial entropies, defined as the number of Bell-state pairs convertible to subsystem states, can be unequal for distinct portions of a multipartite quantum system of more than two components. Because partial entropies are conserved by asymptotically reversible local operations and classical communication (LOCC) involved in the pertinent interconversions, they can therefore no longer be viewed as absolute entanglement measures beyond the bipartite case, in which there is only one way of partitioning the composite system [48]. This prevents the straightforward extension of the standard entanglement measure, the entanglement of formation, to the general multipartite case, as would be natural given its utility in characterizing bipartite entanglement. Schmidt number, a coarse measure, has been generalized to n-parties and then applied independently to various entanglement classes but, although it satisfies most of the conditions on entanglement monotones, it fails to satisfy condition (v) of Section 6.7 [148], [149].