New classes of facets of the cut polytope and tightness of [formula omitted] Bell inequalities

Abstract

The Grishukhin inequality is a facet of the cut polytope CUT 7 □ of the complete graph K 7 , for which no natural generalization to a family of inequalities has previously been found. On the other hand, the I mm 22 Bell inequalities of quantum information theory, found by Collins and Gisin, can be seen as valid inequalities of the cut polytope CUT □ ( ∇ K m , m ) of the complete tripartite graph ∇ K m , m = K 1 , m , m . They conjectured that they are facet inducing. We prove their conjecture by relating the I mm 22 inequalities to a new class of facets of CUT N □ that are a natural generalization of the Grishukhin inequality. An important component of the proof is the use of a method called triangular elimination, introduced by Avis, Imai, Ito and Sasaki, for producing facets of CUT □ ( ∇ K m , m ) from facets of CUT N □ .