Two-Party Bell Inequalities Derived from Combinatorics via Triangular Elimination

Abstract

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.

Part of results in Section 2 appeared in quant-ph/0404014, but this paper gives new proofs using only elementary mathematics. The results in Section 3 was presented in EQIS'04, Kyoto, Sept. 2004. 20 pages with 2 figures. Submitted to Journal of Physics A. v2: introduction rewritten, typos corrected, some detail of proof of Th 2.1 added, relation to open problems mentioned in concluding remarks. v3: URL of the page containing list of Bell inequalities and program to produce it added, typos corrected

References in corpus (1)

• Lifting Bell inequalities

Cited by in corpus (33)

• Bell nonlocality
• Specker's Parable of the Over-protective Seer: A Road to Contextuality, Nonlocality and Complementarity (PLUS AN ERRATUM)
• Maximal violation of the I3322 inequality using infinite dimensional quantum systems
• More efficient Bell inequalities for Werner states
• Quantum Bounds on Bell inequalities
• Classifying 50 years of Bell inequalities
• Looking for symmetric Bell inequalities
• Causal structures from entropic information: Geometry and novel scenarios
• On the Relationship between Convex Bodies Related to Correlation Experiments with Dichotomic Observables
• Reexamination of a multisetting Bell inequality for qudits
• Efficiency of higher dimensional Hilbert spaces for the violation of Bell inequalities
• Leggett-Garg inequalities and the geometry of the cut polytope
• Generalized Clauser-Horne-Shimony-Holt inequalities maximally violated by higher dimensional systems
• Locality, detection efficiencies, and probability polytopes
• Bell inequalities from variable elimination methods
• Bell inequalities stronger than the CHSH inequality for 3-level isotropic states
• A geometric comparison of entanglement and quantum nonlocality in discrete systems
• Compact Bell inequalities for multipartite experiments
• On the inequivalence of the CH and CHSH inequalities due to finite statistics
• Bipartite nonlocality with a many-body system
• Extending Bell inequalities to more parties
• Optimal and tight Bell inequalities for state-independent contextuality sets
• CHSH type Bell inequalities involving a party with two or three local binary settings
• Upper bounds on key rates in device-independent quantum key distribution based on convex-combination attacks
• Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles
• On the tightness of correlation inequalities with no quantum violation
• Varieties of contextuality based on probability and structural nonembeddability
• All two-party facet Bell inequalities are violated by Almost Quantum correlations
• Class of Bell-Clauser-Horne inequalities for testing quantum nonlocality
• Lifting noncontextuality inequalities
• Homogenization of Bell inequalities
• Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables
• Linear Program Witness for Network Nonlocality in Arbitrary Networks