Graph-theoretic characterization of unextendible product bases

Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. Here we provide an equivalent characterization UPBs graph-theoretic terms. Different from previous investigations UPBs, which focused mostly on orthogonality relations between different states, our includes reformulation unextendibility condition. Building this characterization, develop constructive method for building low dimensions shed light open question whether there exist genuinely unextendible (GUPBs), that is, multipartite are with respect to every possible bipartition. We derive lower bound size any candidate GUPB, significantly improving over state art. Moreover, show minimal GUPB saturating must be associated regular graphs discuss path towards construction tripartite system local dimension. Finally, apply problem distinguishing UPB states under operations classical communication, deriving necessary condition reliable discrimination asymptotic limit.