Bell inequalities stronger than the Clauser-Horne-Shimony-Holt inequality for three-level isotropic states

We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for $3\otimes 3$ isotropic states in the sense that they are violated by some isotropic states in the $3\otimes 3$ system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the ${\mathrm{I}}_{3322}$ Bell inequality for $3\otimes 3$ isotropic states. On the other hand, we found no Bell inequalities stronger than the CHSH inequality for $2\otimes 2$ isotropic states. In addition, we illustrate an inclusion relation among some Bell inequalities derived by triangular elimination.

Figure 1

Inclusion relation among 89 Bell inequalities, with at most 5 measurements per party, obtained by tringular elimination from facets of ${\mathrm{CUT}}_9^{◻}$. A asterisk $\left({}^{*}\right)$ on the right of the serial number indicates that the inequality is relevant to the CHSH inequality for $3\otimes 3$ isotropic states and that it is a candidate for being relevant to ${\mathrm{I}}_{3322}$.