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arXiv:quant-ph/0611086

All tight multipartite Bell correlation inequalities for three dichotomic observables per observer

Marek Żukowski

Published Nov 8, 2006• 4 citations
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Abstract

A derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and three possible local settings. The procedure can be extended straightforwardly to multiparty correlations.
DOI
10.48550/arxiv.quant-ph/0611086

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Derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and N possible local settings is not as easy as it seemed. The proof of v1 is wrong. Additionaly one can find a counterexample, which will be presented soon. Thus our thesis is dead. Still the series of Bell inequalities discussed in the manuscript (v1) form a necessary condition for local realism, and are tight. They are tight and complete (sufficient) only for N=3 settings per observer (as shown in quant-ph/0611086, fortunately using an entirely different approach).
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.
In this work we develop two methods to construct Bell inequalities for multipartite systems. By considering non-Hermitian operators we study Bell inequalities for the cases of three settings, three outcomes, and three to six parties. The maximal value achieved in the framework of quantum theory is computed for subsystems with three levels each. The other technique, based on a mapping from pure entangled states to Bell operators, allows us to construct further multipartite Bell inequalities. As a consequence, we reproduce some known results in a different way and find some multipartite Bell inequalities for systems having three settings and three outcomes per party.
Non-trivial facet inequalities play important role in detecting and quantifying the nonolocality of a state -- specially a pure state. Such inequalities are expected to be tight. Number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, dimensionality of the Hilbert space, or/and the number of measurements, there are more nontrivial facet inequalities. By considering a specific measurement scenario, we find that for any multipartite qubit state, local polytope can have only one nontrivial facet. Therefore there exist a possibility that only one Bell inequality, and its permutations, would be able to detect the nonlocality of a pure state. The scenario involves two dichotomic measurement settings for two parties and one dichotomic measurement by other parties. This measurement scenario for a multipartite state may be considered as minimal scenario involving multipartite correlations that can detect nonlocality. We present detailed results for three-qubit states.
Current understanding of correlations and quantum phase transitions in many-body systems has significantly improved thanks to the recent intensive studies of their entanglement properties. In contrast, much less is known about the role of quantum non-locality in these systems. On the one hand, standard, "theorist- and experimentalist-friendly" many-body observables involve correlations among only few (one, two, rarely three...) particles. On the other hand, most of the available multipartite Bell inequalities involve correlations among many particles. Such correlations are notoriously hard to access theoretically, and even harder experimentally. Typically, there is no Bell inequality for many-body systems built only from low-order correlation functions. Recently, however, it has been shown in [J. Tura et al., Science 344, 1256 (2014)] that multipartite Bell inequalities constructed only from two-body correlation functions are strong enough to reveal non-locality in some many-body states, in particular those relevant for nuclear and atomic physics. The purpose of this lecture is to provide an overview of the problem of quantum correlations in many-body systems - from entanglement to nonlocality - and the methods for their characterization.