Polyhedral and semidefinite approaches to classical and quantum Bell inequalities

摘要

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 10971-10987, 2005) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those de ned by Tsirelson (Hadronic J. S. 8 329-345, 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 abody that contains the set of quantum behaviors. Optimization over this latter body may be performed e ciently by semide nite programming.