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Two-Party Bell Inequalities Derived from Combinatorics via Triangular Elimination

David Avis, Hiroshi Imai, Tsuyoshi Ito, Yuuya Sasaki

Published Jan 1, 2005• 39 citations
OpenAlex

Abstract

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.

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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.

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Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, Imm22, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2-valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove a theorem allowing us to derive new facets of cut polytopes from facets of smaller polytopes by a process derived from FourierMotzkin elimination, which we call triangular elimination. These new facets in turn give new tight Bell inequalities. We give additional results for projections, liftings, and the complexity of membership testing for the associated Bell polytope.

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Deriving Tight Bell Inequalities for 2 Parties with Many 2-valued Observables from Facets of Cut Polytopes

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We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for $3\ensuremath{\bigotimes}3$ isotropic states in the sense that they are violated by some isotropic states in the $3\ensuremath{\bigotimes}3$ system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the ${\mathrm{I}}_{3322}$ Bell inequality for $3\ensuremath{\bigotimes}3$ isotropic states. On the other hand, we found no Bell inequalities stronger than the CHSH inequality for $2\ensuremath{\bigotimes}2$ isotropic states. In addition, we illustrate an inclusion relation among some Bell inequalities derived by triangular elimination.

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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.
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.
Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, Imm22, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2-valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove a theorem allowing us to derive new facets of cut polytopes from facets of smaller polytopes by a process derived from FourierMotzkin elimination, which we call triangular elimination. These new facets in turn give new tight Bell inequalities. We give additional results for projections, liftings, and the complexity of membership testing for the associated Bell polytope.
Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, I_{mm22}, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2-valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove a theorem allowing us to derive new facets of cut polytopes from facets of smaller polytopes by a process derived from Fourier-Motzkin elimination, which we call triangular elimination. These new facets in turn give new tight Bell inequalities. We give additional results for projections, liftings, and the complexity of membership testing for the associated Bell polytope.
We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for $3\ensuremath{\bigotimes}3$ isotropic states in the sense that they are violated by some isotropic states in the $3\ensuremath{\bigotimes}3$ system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the I3322{\mathrm{I}}_{3322}I3322​ Bell inequality for $3\ensuremath{\bigotimes}3$ isotropic states. On the other hand, we found no Bell inequalities stronger than the CHSH inequality for $2\ensuremath{\bigotimes}2$ isotropic states. In addition, we illustrate an inclusion relation among some Bell inequalities derived by triangular elimination.
We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario (X,Y,A,B)(X,Y,A,B)(X,Y,A,B) is defined by the numbers of settings and outcomes per party, X,AX,AX,A, and Y,BY,BY,B for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of (6,3,2,2), (3,3,3,2), (3,2,3,3), and (2,2,3,5). We provide extensive lists of facets for (2,2,4,4), (3,3,4,2), and (4,3,3,2). For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits, and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to the Clauser-Horne-Shimony-Holt inequality. Such scenarios could find important applications in quantum communication.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where GGG denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of GGG. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.