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Arijit Dutta, Jinhyoung Lee, and Marek Zukowski

Geometric extension of Clauser-Horne inequality to tripartite systems

We propose extension of Clauser Horne (CH) inequality [1] to a tri-partite system. Our extension is based on the geometric interpretation of Bell inequalities in Kolmogorov theory of probability (see Refs.[2], [3] and references therein). Adopting the geometric interpretation [3], we introduce statistical separations between probabilistic events such that the separations obey triangle inequalities. We show that consecutive applications of such triangle inequalities result in Bell inequalities. In particular we find an extension of CH inequality for a tripartite system. Our tripartite CH inequality is reduced to Mermin inequality of three qubits [4], as to CHSH inequality of two qubits in Ref. [1]. We show also that it is reduced to a bipartite CH inequality if one party is eliminated. In the senses we call our tripartite inequality by CH-Mermin. We analyze the quantum violation by GHZ-type and W-type states with respect to two parameters, detection efficiency and the visibility associated with white noise. We find minimal detection efficiencies, below which there is no conclusive evidence of quantum violation regardless of settings in local measurements. We extend our method to present inequalities for four and five qubits.

[1] J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526-535 (1974).
[2] A. Khrennikov, Entropy 10, 19-32 (2008).
[3] M. Zukowski, A. Dutta. Phys. Rev. A 90, 012106 (2014). 
[4] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).

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