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Bell Bi-Inequalities for Bell Local Correlation Tensors

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Abstract

When an n-partite physical system is measured by n observers, the joint probabilities of outcomes conditioned on the observables chosen by the n- parties form a nonnegative tensor P, called an n-partite correlation tensor (CT). According to the special relativity, CTs can be classified as signaling and nonsignaling ones; and from the point of view of hidden variable theory, CTs can be divided into Bell local and Bell nonlocal ones. In this paper, we aim to establish some Bell bi-inequalities for n-partite Bell local correlation tensors. A Bell bi-inequality consists of two Bell inequalities that hold only for Bell local CTs. First, we obtain an inequality for n-partite nonsignaling CTs, which can be used to check the nonsignaling property of a CT. Second, we recall the mathematical definition of Bell locality of CTs and prove global properties of the set of all Bell local CTs over an index set Δn. Then we establish a series tight Bell bi-inequalities and prove that a CT P is Bell local if and only if it satisfies all tight Bell bi-inequalities. Lastly, we list some examples to illustrate how to establish a Bell bi-inequality.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11871318, 11771009, 12001480), the Fundamental Research Funds for the Central Universities (GK202103003, GK202107014) and and the Special Plan for Young Top-notch Talent of Shaanxi Province (1503070117).

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  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China

    Wen-Qian Zhu, Di Hu, Zhi-Hua Guo & Huai-Xin Cao

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  1. Wen-Qian Zhu
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  2. Di Hu
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Zhu, WQ., Hu, D., Guo, ZH. et al. Bell Bi-Inequalities for Bell Local Correlation Tensors. Int J Theor Phys 62, 68 (2023). https://doi.org/10.1007/s10773-023-05320-0

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