Multipartite entanglement gambling: The power of asymptotic state transformations assisted by a sublinear amount of quantum communication

Reversible state transformations under entanglement nonincreasing operations give rise to entanglement measures. It is well known that asymptotic local operations and classical communication (LOCC) are required to get a simple operational measure of bipartite pure state entanglement. For bipartite mixed states and multipartite pure states it is likely that a more powerful class of operations will be needed. To this end more powerful versions of state transformations (or reducibilities), namely, LOCCq (asymptotic LOCC with a sublinear amount of quantum communication) and CLOCC (asymptotic LOCC with catalysis) have been considered in the literature. In this paper we show that LOCCq state transformations are only as powerful as asymptotic LOCC state transformations for multipartite pure states. The basic tool we use is multipartite entanglement gambling: Any pure multipartite entangled state can be transformed to an Einstein-Podolsky-Rosen pair shared by some pair of parties and any irreducible m-party pure state $(m>~2)$ can be used to create any other state (pure or mixed) using LOCC. We consider applications of multipartite entanglement gambling to multipartite distillability and to characterizations of multipartite minimal entanglement generating sets. We briefly consider generalizations of this result to mixed states by defining the class of cat-distillable states, i.e., states from which cat states $(|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉)$ may be distilled.