Exact and asymptotic measures of multipartite pure-state entanglement - art. no. 012307

Hoping to simplify the classification of pure entangled states of multi (mj -partite quantum systems, we study exactly and asymptotically (in n) revers ible transformations among nth tensor powers of such states (i.e., n copies of the state shared among the same In parties) under local quantum operati ons and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenhereer-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classif ication than exact transformations; for example, they allow all pure bipart ite states to be characterized by a single parameter-their partial entropy- which may be interpreted as the number of EPR pairs asymptotically intercon vertible to the state in question by LOCC transformations. We show that m-p artite pure states having an m-way Schmidt decomposition are similarly para metrizable! with the partial entropy across any nontrivial partition repres enting the number of standard quantum superposition or "cat" states \0(circ le timesm)>+\1(circle timesm)) asymptotically interconvertible to the state in question. For general m-partite states, partial entropies across differ ent partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglemen t measure is needed, with each scalar component representing a different ki nd of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the m=4 cat state is not isentropic to, and the refore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m= 4 cat state can be prepared from bipartite EPR states, the preparation proc ess is necessarily irreversible, and remains so even asymptotically. For ea ch number of parties,n we define a minimal reversible entanglement generati ng set (MREGS) as a set of states of minimal cardinality sufficient to gene rate all m-partite pure states by asymptotically reversible LOCC transforma tions. Partial entropy arguments provide lower bounds on the size of the MR EGS, but for m>2 we know no upper bounds. We briefly consider several gener alizations of LOCC transformations, including transformations with some pro bability of failure, transformations with the catalytic assistance of state s other than the states we are trying to transform, and asymptotic LOCC tra nsformations supplemented by a negligible [o(n)] amount of quantum communic ation.