Tight Bell inequalities with no quantum violation from qubit unextendible product bases

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.

Figure 1

Schematic depiction of the sets of classical CC, quantum QC, and nonsignalling correlations NC. Tight Bell inequalities correspond to facets of the classical set. B denotes a Bell inequality with no quantum violation which is (a) not tight and (b) tight. Note that only tightness guarantees the existence of a nontrivial region in which quantum and classical correlations coincide.