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arXiv:quant-ph/0601107 · Physical Review A

Bell inequality with an arbitrary number of settings and its applications

Koji Nagata, Wiesław Laskowski, Tomasz Paterek

Published Dec 18, 2006• 25 citations
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Abstract

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.

DOI
10.1103/physreva.74.062109

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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.
We investigate the nonlocal properties of graph states. To this aim, we derive a family of Bell inequalities which require three measurement settings for each party and are maximally violated by graph states. In turn, for each graph state there is an inequality maximally violated only by that state. We show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits. We also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement.
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.
In the first part of this thesis Bell's theorem is revisited. It points at a difference between the quantum and the classical world. This difference is often behind the advantages of solutions using quantum mechanics. New and more general versions of Bell inequalities are presented. These inequalities involve multiple settings per observer. Compared with the two-setting inequalities, the new ones reveal the non-classical character of a broader class of states. Some of them are also proven to be optimal (tight). Next, we go beyond Bell's theorem. It is shown, both in theory and in experiment, that incompatibility between quantum mechanics and realistic theories can be extended into an important class of nonlocal models. We also show that the violation of Bell inequalities disqualifies local realistic models with a limited lack of the experimenter's freedom. This, at first glance quite philosophical result, has its down-to-earth implications for quantum communication. In the second part of the thesis well-known examples of quantum communication are reviewed. Next, new results concerning quantum cryptography and quantum communication complexity are given.
We derive a single general Bell inequality which is a sufficient and necessary condition for the correlation function for N particles to be describable in a local and realistic picture, for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables. We also derive a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this inequality. This condition is a generalization and reformulation of the Horodecki family condition for two qubits.