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arXiv:quant-ph/0512176 · Physical Review A

Generalized structure of Bell inequalities for bipartite arbitrary-dimensional systems

Seungwoo Lee, Yong Wook Cheong, Jinhyoung Lee

Published Sep 7, 2007• 18 citations
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Abstract

We propose a generalized structure of Bell inequalities for arbitrary $d$-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.

DOI
10.1103/physreva.76.032108

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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 analyze the correlation structure of bipartite arbitrary-dimensional Bell inequalities via novel conditions of correlations in terms of differences of joint probabilities called correlators. The conditions of correlations are shown to be necessary for the multi-level Bell state. In particular, we find that the bipartite arbitrary-dimensional Bell-type inequalities introduced by Collins-Gisin-Linden-Massar-Popescu [Phys. Rev. Lett. 88, 040404 (2002)] and Son-Lee-Kim [Phys. Rev. Lett. 96, 060406 (2006)] are composed of correlators, and we reveal that the maximal violations by the Bell state just fit the conditions of quantum correlations. Correlators can be considered as essential elements of Bell inequalities.
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 give a multidimensional generalization of the complete set of Bell-correlation inequalities given by Werner and Wolf (2001 Phys. Rev. A 64 032112) and by Zˆukowski and Brukner (2002 Phys. Rev. Lett. 88 210401), for the two-dimensional case. Our construction applies to the n -party, two-observable case, where each observable is d -valued. The inequalities obtained involve homogeneous polynomials. They define the facets of a polytope in a complex vector space of dimension d n . We detail the inequalities obtained in the case d = 3 and, from them, we recover known inequalities. We finally explain how the violations of our inequalities by quantum mechanics can be computed and could be observed, when using unitary observables.
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 Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.