Summary: Polyhedral and semidefinite approaches
to classical and quantum Bell inequalities
School of Computer Science, McGill University,
3480 University, Montreal, Quebec, Canada H3A 2A7.
Department of Computer Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
Abstract. In this paper we explore further the connections between convex bodies related to quantum
correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization,
especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (J. Phys. A:
Math. Gen. 38 1097110987, 2005) with respect to Bell inequalities. We show that several well known
bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (Hadronic
J. S. 8 329345, 1993) to represent hidden deterministic behaviors, quantum behaviors, and no-signaling
behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary
condition for vertices of the no-signaling polytope, and give a method for bounding the quantum violation
of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this
latter body may be performed efficiently by semidefinite programming.