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arXiv:quant-ph/0506230 · Physical Review A

Bell inequalities for three particles

Jing‐Ling Chen, Chunfeng Wu, L. C. Kwek, C. H. Oh

Published Sep 8, 2008• 19 citations
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Abstract

We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.

DOI
10.1103/physreva.78.032107

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We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three ddd-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.
We propose a generalized structure of Bell inequalities for arbitrary ddd-dimensional bipartite systems, which includes the existing two types of Bell inequalities introduced by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)] and Son et al. [Phys. Rev. Lett. 96, 060406 (2006)]. We analyze Bell inequalities in terms of correlation functions and joint probabilities, and show that the coefficients of correlation functions and those of joint probabilities are in Fourier transform relations. We finally show that the coefficients in the generalized structure determine the characteristics of quantum violation and tightness.
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.
We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real-component Hilbert spaces using numerical optimization. Out of these inequalities 129 have been introduced here. In 43 cases higher-dimensional component spaces gave larger violation than qubits, and in three occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.