Bell inequalities and the cut polytope : bridging quantum information science and combinatorial optimization

Yunpyo Hong, Jaejoon Gill, Daeyeun Kim +3 more

David Avis, Hiroshi Imai, Tsuyoshi Ito

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 et al ( 2005 J. Phys. A: Math. Gen. 38 10971–87 ) 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 (1993 Hadronic J. Suppl. 8 329–45) to represent hidden deterministic behaviours, quantum behaviours and no-signalling behaviours. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviours. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with ( m , n ) dichotomic observables when m = 4, n = 4 and when min{ m , n } ⩽ 3, and give a new general family of correlation inequalities.

David Avis, Hiroshi Imai, Tsuyoshi Ito +1 more

We establish a relation between the two-party Bell inequalities for two-valued measurements and a high-dimensional convex polytope called the cut polytope in polyhedral combinatorics. Using this relation, we propose a method, triangular elimination , to derive tight Bell inequalities from facets of the cut polytope. This method gives two hundred million inequivalent tight Bell inequalities from currently known results on the cut polytope. In addition, this method gives general formulae which represent families of infinitely many Bell inequalities. These results can be used to examine general properties of Bell inequalities.

Emilio Santos

David Avis, Hiroshi Imai, Tsuyoshi Ito +1 more

Bell inequalities, originally introduced as a method to prove that some quantum states show nonlocal behavior, are now studied as a method to capture the extent of the nonlocality of quantum states. Tight Bell inequalities are considered to be more important than redundant ones. Despite the increasing importance of the study of Bell inequalities, few kinds of tight Bell inequalities have been found. Examples include the Clauser-Horne-Shimony-Holt inequality, the I_{mmvv} inequalities, the CGLMP inequalities, and the Bell inequalities in systems small enough to generate all the Bell inequalities by exhaustive search. In this paper, we establish a relation between the two-party Bell inequalities for two-valued measurements and a high-dimensional convex polytope called the cut polytope in polyhedral combinatorics. Using this relation, we propose a method, triangular elimination, to derive tight Bell inequalities from facets of the cut polytope. This method gives two hundred million inequivalent tight Bell inequalities from currently known results on the cut polytope. In addition, this method gives general formulas which represent families of infinitely many Bell inequalities. These results can be used to examine general properties of Bell inequalities.

Alexandre Gondran, Michel Gondran