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Violation of Bell’s Inequality in the Clauser-Horne-Shimony-Holt Form with Entangled Quantum States Revisited

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Abstract

Scientific imagination and experimental ingenuity are at the heart of physics. One of the most known instances where this interplay between theory (i.e., foundations) and experiments (i.e., technology) occurs is in the discussion of Bell’s inequalities. In this paper, we present a revisitation of the violation of Bell’s inequality in the Clauser-Horne-Shimony-Holt (CHSH) form with entangled quantum states. First, we begin with a discussion of the 1935 Einstein-Podolski-Rosen (EPR) paradox (i.e., incompleteness of quantum mechanics) that emerges from putting the emphasis on Einstein’s locality and the absolute character of physical phenomena. Second, we discuss Bell’s 1971 derivation of the 1969 CHSH form of the original 1964 Bell inequality in the context of a realistic local hidden-variable theory (RLHVT). Third, identifying the quantum-mechanical spin correlation coefficient with the RLHVT one, we follow Gisin’s 1991 analysis to show that quantum mechanics violates Bell’s inequality when systems are in entangled quantum states. For pedagogical purposes, we show how the extent of this violation depends both on the orientation of the polarizers and the degree of entanglement of the quantum states. Fourth, we discuss the basics of the experimental verification of Bell’s inequality in an actual laboratory as presented in the original 1982 Aspect-Grangier-Roger (AGR) experiment. Finally, we provide an outline of some essential take home messages from this wonderful example of physics at its best.

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Acknowledgements

C. Cafaro thanks Ariel Caticha, Domenico Felice, Nicholas Gisin, Yanjun He, and Elena R. Loubenets for valuable comments and helpful electronic mail exchanges throughout the last years that helped shaping this paper in its current form.

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Authors and Affiliations

  1. University at Albany-SUNY, Albany, NY, 12222, USA

    Carlo Cafaro & Ayhan Bingolbali

  2. SUNY Polytechnic Institute, Utica, NY, 13502, USA

    Carlo Cafaro & Christian Corda

  3. Istituto Livi, 59100, Prato, Italy

    Christian Corda

  4. International Institute for Applicable Mathematics and Information Sciences, 500063, Adarsh Nagar, Hyderabad, India

    Christian Corda

  5. Physics Program, Graduate Center of the City University of New York, New York, NY, 10016, USA

    Philip Cairns

  6. Yildiz Technical University, 34220, Istanbul, Turkey

    Ayhan Bingolbali

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  1. Carlo Cafaro
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  4. Ayhan Bingolbali
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Contributions

Conceptualization, C.C.; Writing-original draft, C.C., Ch.C., P.C., and A.B..; Writing—review and editing, C.C., Ch.C., P.C., and A.B.. All authors have read and agreed to the submitted version of the manuscript.

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Correspondence to Carlo Cafaro.

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C.C. is an editorial board member of the Int. J. Theor. Phys.

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Cafaro, C., Corda, C., Cairns, P. et al. Violation of Bell’s Inequality in the Clauser-Horne-Shimony-Holt Form with Entangled Quantum States Revisited. Int J Theor Phys 63, 112 (2024). https://doi.org/10.1007/s10773-024-05627-6

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