QUROPE Quantum Information Processing and Communication in Europe

We investigate the relation between unextendible product bases (UPB) and Bell inequalities found recently in R. Augusiak et al. [ Phys. Rev. Lett. 107 070401 (2011)]. We, first, review the procedure introduced there that associates to any set of mutually orthogonal product vectors in a many-qubit Hilbert space a Bell inequality. We then show that if a set of mutually orthogonal product vectors can be completed to a full basis, then the associated Bell inequality is trivial, in the sense of not being violated by any nonsignalling correlations. This implies that the relevant Bell inequalities that arise from the construction all come from UPBs, which adds additional weight to the significance of UPBs for Bell inequalities. Then, we provide additional examples of tight Bell inequalities with no quantum violation constructed from UPBs in this way. Finally, it is proven that the Bell inequalities with no quantum violation introduced recently in M. Almeida et al. [ Phys. Rev. Lett. 104 230404 (2010)] are tight for any odd number of parties.