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arXiv:quant-ph/0411066 · Physical Review Letters

Tight Multipartite Bell's Inequalities Involving Many Measurement Settings

Wiesław Laskowski, Tomasz Paterek, Marek Żukowski, Časlav Brukner

Published Nov 8, 2004• 74 citations
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Abstract

We derive tight Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give a necessary and sufficient condition for a general quantum state to violate the new inequalities. The inequalities are violated by some classes of states, for which all standard Bell's inequalities with two measurement settings per observer are satisfied.

DOI
10.1103/physrevlett.93.200401

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We derive tight Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give a necessary and sufficient condition for a general quantum state to violate the new inequalities. The inequalities are violated by some classes of states, for which all standard Bell's inequalities with two measurement settings per observer are satisfied.
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).
We present a family of Bell inequalities involving only two measurement settings of each party for N>2N>2N>2 qubits. Our inequalities include all the standard ones with fewer than NNN qubits and thus give a natural generalization. It is shown that all the Greenberger-Horne-Zeilinger states violate the inequalities maximally, with an amount that grows exponentially as ${2}^{(N\ensuremath{-}2)∕2}$. The inequalities are also violated by some states that do satisfy all the standard Bell inequalities. Remarkably, our results yield in an efficient and simple way an implementation of nonlocality tests of many qubits favorably within reach of the well-established technology of linear optics.
Based on a geometrical argument introduced by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski, a new multisetting Bell inequality is derived, for the scenario in which many parties make measurements on two-level systems. This generalizes and unifies some previous results. Moreover, a necessary and sufficient condition for the violation of this inequality is presented. It turns out that the class of nonseparable states which do not admit local realistic description is extended when compared to the two-setting inequalities. However, supporting the conjecture of Peres, quantum states with positive partial transposes with respect to all subsystems do not violate the inequality. Additionally, we follow a general link between Bell inequalities and communication complexity problems, and present a quantum protocol linked with the inequality, which outperforms the best classical protocol.
It has been believed that statistical inequalities such as the Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that the Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive the Bell inequality when both determinism and setting independence, usually called local realism, are satisfied. We thus propose a condition necessary and sufficient for deriving the Bell inequality, called concealed measurement dependence.