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arXiv:quant-ph/0210073 · Quantum Information and Computation

Tight Bell inequality for d-outcome measurements correlations

Lluís Masanes

Published Jul 1, 2003• 92 citations
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Abstract

In this paper we prove that the inequality introduced by Collins, Gisin, Linden, Massar and Popescu is tight, or in other words, it is a facet of the convex polytope generated by all local-realistic joint probabilities of d-outcomes. This means that this inequality is optimal. We also show that, for correlation functions generalized to deal with three-outcome measurements, the satisfyability of this inequality is a necessary and sufficient condition for the existence of a local-realistic model accounting for them.

DOI
10.26421/qic3.4-4

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In this paper we prove that the inequality introduced by Collins, Gisin, Linden, Massar and Popescu is tight, or in other words, it is a facet of the convex polytope generated by all local-realistic joint probabilities of d-outcomes. This means that this inequality is optimal. We also show that, for correlation functions generalized to deal with three-outcome measurements, the satisfyability of this inequality is a necessary and sufficient condition for the existence of a local-realistic model accounting for them.
Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins–Gisin–Linden–Massar–Popescu type.
We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}d$ Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of ddd the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ scenario.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite ddd-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.
We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin-Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can be reached only using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that our inequalities can be used as multipartite dimension witnesses.