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arXiv:0809.1713 · Physical Review A

Tight correlation-function Bell inequality for multipartite d -dimensional systems

Jing‐Ling Chen, Dong-Ling Deng

Published Jan 23, 2009• 11 citations
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Abstract

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite $d$-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.

DOI
10.1103/physreva.79.012111

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We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite ddd-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two dimensions. This inequality is optimal in the same sense as the CHSH inequality for two-dimensional systems, namely, the maximal amount by which the inequality is violated consists of the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]].
Abstract No matter for a scientific or technological reason, constructing Bell inequalities for multi-partite and high-dimensional systems is a significant task. The Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) inequality is expressed in the form of iteration formula and correlation functions, and is the generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multi-partite cases. In the sense of detecting the nonlocality of the noisy maximally entangled states with the most white noises for the corresponding quantum system, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality, expressed in the joint probabilities form, is the high-dimensional analogue of the Clauser-Horne (CH) inequalities on joint probabilities. In the sense of detecting the nonlocality of the noisy Greenberger-Horne-Zeilinger(GHZ) states with the most white noises for the n-qudit system, it is a challenging task to construct the multi-partite analogue of the CGLMP inequality with iteration formula and correlation functions. In this paper, we generalize the multi-component correlation functions [Phys. Rev. A 71, 032 107 (2005)] for bipartite d-dimensional systems to n-partite d-dimensional ones, introduce the general full-correlated multi-component Bell function I n , d , and construct the corresponding Bell inequality. By this way, we can reproduce both the CGLMP inequality and the Bell inequality in [Phys. Rev. A 71, 032 107 (2005)] for the case n = 2, and the MABK inequality for the case d = 2. Inspired by the iteration formula form of the MABK inequality, we prove that for prime d the general Bell function I n , d can be reformulated by iterating two Bell functions I n − 1 , d . As applications, for prime d, confined to the unbiased symmetric (d × 2)-port beam splitters and the noisy n-qudit GHZ states, we recover the most robust Bell inequalities for small n and d, such as for the (3, 2, 3), (4, 2, 3), (5, 2, 3), and (3, 2, 5) Bell scenarios, with the iteration formula and the most robust Bell inequalities for the (2, 2, d) scenario. This implies that the iteration formula is an efficient way of constructing the multi-partite analogues of the CGLMP inequality with correlation functions. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
It is very difficult and important to construct Bell inequalities for n-partite, k-settings of measurement, and d-dimensional (n,k,d) systems. Inspired by the iteration formula form of the Mermin-Ardehali-Belinski{\uı}-Klyshko (MABK) inequality, we generalize the multi-component correlation functions for bipartite d-dimensional systems to n-partite ones, and construct the corresponding Bell inequality. The Collins-Gisin-Linden-Massar-Popescu inequality can be reproduced by this way. The most important result is that for prime d the general Bell function in full-correlated multi-component correlation function form for (n,2,d) systems can be reformulated in iteration formula by two full-correlated multi-component Bell functions for (n-1,2,d) systems. As applications, we recover the MABK inequality and the most robust coincidence Bell inequalities for (3,2,3),(4,2,3),(5,2,3), and (3,2,5) Bell scenarios with this iteration formula. This implies that the iteration formula is an efficient way of constructing multi-partite Bell inequalities. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.
A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving (N+1)(N+1)(N+1)-partite CHSH-type Bell inequalities from NNN-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.