Constructive generalization of Bell’s inequalities

Froissart, M.

doi:10.1007/bf02903286

Cited by 150 publications

(161 citation statements)

References 8 publications

(9 reference statements)

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“…III, that for n A = n B we have max I n,n = max{n 2 − 2(k − l) 2 }, by adding the marginals max I ′ n,n = max{n 2 − 2(k − l) 2 + (2k − n) − (2l − n)} = max{n 2 − 2(k − l)(k − l + 1)} = n 2 , thus the local bound does not change. Let us notice, that I ′ 2,2 specified by n = 2 in (14) is just the I 3322 Bell inequality [40,41], which is known to be tight [42]. In both cases, I n,n−1 and I ′ n,n , we suspect that these Bell inequalities are tight for any higher values of n > 4, as well.…”

Section: Tightness and Relevance Of Bell Inequalitiesmentioning

confidence: 94%

More efficient Bell inequalities for Werner states

Vértesi

2008

Phys. Rev. A

107

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In this paper we study the nonlocal properties of two-qubit Werner states parameterized by the visibility parameter 0 ≤ p ≤ 1. New family of Bell inequalities are constructed which prove the twoqubit Werner states to be nonlocal for the parameter range 0.7056 < p ≤ 1. This is slightly wider than the range 0.7071 < p ≤ 1, corresponding to the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. This answers a question posed by Gisin in the positive, i.e., there exist Bell inequalities which are more efficient than the CHSH inequality in the sense that they are violated by a wider range of two-qubit Werner states.

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Section: Tightness and Relevance Of Bell Inequalitiesmentioning

confidence: 94%

More efficient Bell inequalities for Werner states

Vértesi

2008

Phys. Rev. A

107

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“…These formulae turn out to have exactly the same form as the ones derived for even i values, that is Eqs. (19)(20)(21)(22) to introduce the notation c 0 ≡ 1. The value B 1 nn = 0 is the optimum value.…”

Section: Family Of Constructionsmentioning

confidence: 99%

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

2010

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The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the ClauserHorne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.

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“…In fact, recent extensive numerical investigations [22] suggest that the simplest extremal inequality after CHSH, the I 3322 inequality (first introduced in [23], its name refers to the fact it has three settings and two outcomes per site), allows for a surprisingly complex structure of the maximally violating states.…”

Section: A the I 3322 Inequalitymentioning

confidence: 99%

More nonlocality with less entanglement

Vidick

Wehner

2011

Phys. Rev. A

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Recent numerical investigations [K. Pál and T. Vértesi, Phys. Rev. A 82, 022116 (2010)] suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.

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Constructive generalization of Bell’s inequalities

Abstract: The amount of contextuality is quantified in terms of the probability of the necessary violations of noncontextual assignments to counterfactual elements of physical reality.

Cited by 150 publications

References 8 publications

More efficient Bell inequalities for Werner states

More efficient Bell inequalities for Werner states

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

More nonlocality with less entanglement

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