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

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## Abstract

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.

## Keywords

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

81P15 81Qxx## Notes

### Acknowledgments

The author would like to thank Michael Mislove and Keye Martin for their support and guidance, as well as Gustavo Didier and Lev Kaplan for their helpful comments and suggestions. This work was partially supported by grant FA9550-13-1-0135 from the US Air Force Office of Scientific Research and Grant N00014-10-1-0329 P00004 from the US Office of Naval Research.

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