Deriving Tight Bell Inequalities for 2 Parties with Many 2-valued Observables from Facets of Cut Polytopes

Abstract

Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, I_{mm22}, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2-valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove a theorem allowing us to derive new facets of cut polytopes from facets of smaller polytopes by a process derived from Fourier-Motzkin elimination, which we call triangular elimination. These new facets in turn give new tight Bell inequalities. We give additional results for projections, liftings, and the complexity of membership testing for the associated Bell polytope.

24 pages, LaTeX. Asymmetric setting of observables is added in the introduction, section 5.4 and section 7.2. Some references and typos are corrected. A description on the I_{3422}^2 inequality is corrected in section 5.4 and added to section 5.1