Foundations of Physics

, Volume 44, Issue 7, pp 736–761

A Rigorous Analysis of the Clauser–Horne–Shimony–Holt Inequality Experiment When Trials Need Not be Independent


DOI: 10.1007/s10701-014-9811-3

Cite this article as:
Bierhorst, P. Found Phys (2014) 44: 736. doi:10.1007/s10701-014-9811-3


The Clauser–Horne–Shimony–Holt (CHSH) inequality is a constraint that local hidden variable theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show that the CHSH inequality holds even for completely general state variables in the measure-theoretic setting, and second, demonstrate how to drop the assumption of independence of subsequent trials while still being able to perform a hypothesis test that will distinguish Quantum Mechanics from local theories. The statistical strength of such a test is computed.


Quantum theory Bell’s theorem Measure-theoretic probability  Bell inequalities Hypothesis test Hidden-variable theories 

Mathematics Subject Classification

81P15 81Qxx 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematics DepartmentTulane UniversityNew OrleansUSA

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