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 state. We exhibit a tripartite $2\times 2\times 2$ UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e., all three corresponding $2\times 4$ bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local positive operator valued measurements and classical communication.