Unextendible product bases and bound entanglement

An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled stat e. We exhibit a tripartite 2 x 2 x 2 UPB whose complementary mixed state ha s tripartite entanglement but no bipartite entanglement, i.e., all three co rresponding 2 x 4 bipartite mixed states are unentangled. We show that memb ers of a UPB are not perfectly distinguishable by local positive operator v alued measurements and classical communication.