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Polyhedral and semidefinite approaches to classical and quantum Bell inequalities
 

Summary: Polyhedral and semidefinite approaches
to classical and quantum Bell inequalities
David Avis1
Tsuyoshi Ito2
1
School of Computer Science, McGill University,
3480 University, Montreal, Quebec, Canada H3A 2A7.
2
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.

  

Source: Avis, David - School of Computer Science, McGill University

 

Collections: Computer Technologies and Information Sciences