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arXiv:quant-ph/0303187

Multiparticle Bell's inequalities involving many measurement settings

Časlav Brukner, Wiesław Laskowski, Tomasz Paterek, Marek Żukowski

Published Mar 31, 2003• 0 citations
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Abstract

We present a prescription for obtaining Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give examples of some families of such inequalities. The inequalities are violated by certain classes of states for which all standard Bell's inequalities with two measurement settings per observer are satisfied.

DOI
10.48550/arxiv.quant-ph/0303187

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We present a prescription for obtaining Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give examples of some families of such inequalities. The inequalities are violated by certain classes of states for which all standard Bell's inequalities with two measurement settings per observer are satisfied.
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We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.
Bell's theorem for systems more complicated than two qubits faces a hidden, as yet undiscussed, problem. One of the methods to derive Bell inequalities is to assume the existence of a joint probability distribution for measurement results for all settings in the given experiment. However, for spin-1 systems, one faces the problem that the eigenvalues of observables do not allow a consistent algebra if one allows all possible settings on each side (Bell's 1966 contradiction), or some specific sets (leading to a Kochen-Specker 1967 contradiction). We show here that by choosing a special set of settings which never lead to inconsistent algebra of eigenvalues, one can still derive multisetting Bell inequalities, and that they are robustly violated. Violation factors increase with the number of subsystems. The inequalities involve only spin observables, we do not allow all possible qutrit observables, still the violations are strong.
It was shown by Dur [Phys. Rev. Lett. 87,87,87, 230402 (2001)] that N (N> 4)(N>~4)(N> 4) qubits described by a certain one-parameter family F\mathcal{F}F of bound entangled states violate the Mermin-Klyshko inequality for N> 8.N>~8.N> 8. In this paper we prove that the states from the family F\mathcal{F}F violate Bell inequalities derived by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski and Kaszlikowski [Phys. Rev. A 56,56,56, R1682 (1997)], in which each observer measures three noncommuting sets of orthogonal projectors, for N> 7.N>~7.N> 7. We also derive a simple one-parameter family of entanglement witnesses that detect entanglement for all the states belonging to F.\mathcal{F}.F. It is possible that these entanglement witnesses could be generated by some Bell inequalities.