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.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.

arXiv:quant-ph/0508210Physical Review AApr 21, 200620 citations

On Bell’s inequality

Ivanović, I. D.

Jan 1, 19800 citations

Variants of Bell inequalities

Zeqian Chen

A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.

arXiv:quant-ph/0611126Nov 13, 20060 citations

Stronger Hardy-type paradox based on the Bell inequality and its experimental test

Mu Yang, Hui-Xian Meng, Jie Zhou +7 more

Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.

arXiv:1809.04289Physical Review AMar 5, 201926 citations

A limitation on Bell's inequality

V. Buonomano

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 19789 citations

{\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.