Unextendible Product Bases and Bound Entanglement

Abstract

We obtain easy examples of ``bound'' entanglement---entangled mixed states from which no pure entanglement is distillable---using the notion of an unextendible product basis, of which we give many examples. We exhibit a bound-entangled state of a tripartite $2\times2\times2$ system that has no bipartite entanglement, in the sense that all three corresponding $2\times4$ bipartite states are unentangled. If a set of orthogonal product states is distinguishable by local von Neumann measurements and classical communication, then this set can be completed to a full basis. We give a set of states on $3\times4$ that is distinguishable by generalized measurements and can be completed in an extended Hilbert space.