New Classes of Facets of Cut Polytope and Tightness of I_{mm22} Bell Inequalities

Abstract

The Grishukhin inequality Gr_7 is a facet of CutP_7, the cut polytope on seven points, which is ``sporadic'' in the sense that its proper generalization has not been known. In this paper, we extend Gr_7 to an inequality I(G,H) valid for CutP_{n+1} where G and H are graphs with n nodes satisfying certain conditions, and prove a necessary and sufficient condition for I(G,H) to be a facet. This result combined with the triangular elimination theorem of Avis, Imai, Ito and Sasaki settles Collins and Gisin's conjecture in quantum theory affirmatively: the I_{mm22} Bell inequality is a facet of the correlation polytope CorP(K_{m,m}) of the complete bipartite graph K_{m,m} for all m>=1. We also extend the Gr_8 facet inequality of CutP_8 to an inequality I'(G,H,C) valid for CutP_{n+2}, and provide a sufficient condition for I'(G,H,C) to be a facet.