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

TL;DR

This paper explores the connection between unextendible product bases (UPB) and Bell inequalities, demonstrating that certain Bell inequalities derived from UPBs are tight and exhibit no quantum violation, highlighting UPBs' significance in quantum nonlocality.

Contribution

The paper shows that Bell inequalities from UPBs are tight and provides new examples of such inequalities with no quantum violation, extending previous results.

Findings

Bell inequalities from UPBs are tight and non-violable by quantum mechanics.

Constructed new examples of tight Bell inequalities with no quantum violation from UPBs.

Proved that certain Bell inequalities are tight for any odd number of parties.

Abstract

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 new 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.