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Do the Bell Inequalities Require the Existence of Joint Probability Distributions?

Published online by Cambridge University Press:  01 April 2022

George Svetlichny ,
Michael Redhead ,
Harvey Brown  and
Jeremy Butterfield
George Svetlichny
Affiliation:
Departamento de Matematica Pontificia Universidade Católica do Rio de Janeiro
Michael Redhead
Affiliation:
History and Philosophy of Science Department Cambridge University
Harvey Brown
Affiliation:
Philosophy Sub-Faculty Oxford University
Jeremy Butterfield
Affiliation:
Philosophy Faculty Cambridge University
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Abstract

Fine has recently proved the surprising result that satisfaction of the Bell inequality in a Clauser-Horne experiment implies the existence of joint probabilities for pairs of noncommuting observables in the experiment. In this paper we show that if probabilities are interpreted in the von Mises-Church sense of relative frequencies on random sequences, a proof of the Bell inequality is nonetheless possible in which such joint probabilities are assumed not to exist. We also argue that Fine's theorem and related results do not impugn the common view that local realists are committed to the Bell inequality.

Type
Research Article
Information
Philosophy of Science , Volume 55 , Issue 3 , September 1988 , pp. 387 - 401
Copyright
Copyright © 1988 by the Philosophy of Science Association

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Footnotes

†

Ken Regan made extremely useful suggestions related to the mathematical issues involved in the paper, and David Wood helped us see further implications of the inequality-conforming cases discussed in Section 2. We are very grateful also to Arthur Fine, Robert Weingard, Thomas Brody and Nancy Cartwright for comments on an earlier version. One of us (G. S.) thanks CNPq and FINEP, agencies of the Brazilian government, for financial support.

References

REFERENCES

Aspect, A., Grangier, P., and Roger, G. (1981), “Experimental Tests of Realistic Local Theories via Bell's Theorem”, Physical Review Letters 47: 460–467.CrossRefGoogle Scholar
Aspect, A., Grangier, P., and Roger, G. (1982), “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities”, Physical Review Letters 49: 91–94.CrossRefGoogle Scholar
Aspect, A., Dalibard, J., and Roger, G. (1982a), “Experimental Test of Bell's Inequalities Using Variable Analyzers”, Physical Review Letters 49: 1804–1807.CrossRefGoogle Scholar
Bell, J. (1964), “On the Einstein-Podolsky-Rosen Paradox”, Physics 1: 195–200.CrossRefGoogle Scholar
Clauser, J., and Shimony, A. (1978), “Bell's Theorem: Experimental Tests and Implications”, Reports on Progress in Physics 41: 1881–1297.CrossRefGoogle Scholar
Cohen, L. (1966), “Generalized Phase Space Distribution Functions”, Journal of Mathematical Physics 7: 781–786.CrossRefGoogle Scholar
Cohen, L. (1966a), “Can Quantum Mechanics Be Formulated As a Classical Probability Theory?”, Philosophy of Science 33: 317–322.CrossRefGoogle Scholar
De Muynck, M. (1986), “The Bell Inequalities and Their Irrelevance to the Problem of Locality in Quantum Mechanics”, Physics Letters 114A: 65–67.CrossRefGoogle Scholar
Eberhard, P. (1977), “Bell's Theorem Without Hidden Variables”, Il Nuovo Cimento 38B: 75–80.
Eberhard, P. (1982), “Constraints of Determinism and of Bell's Inequalities are not Equivalent”, Physical Review Letters 49: 1474–1477.CrossRefGoogle Scholar
Fine, A. (1973), “Probability and the Interpretation of Quantum Mechanics”, British Journal for the Philosophy of Science 24: 1–37.CrossRefGoogle Scholar
Fine, A. (1976), “On the Completeness of Quantum Theory”, Synthese 29: 257–289; Reprinted in P. Suppes (ed.), Logic and Probability in Quantum Mechanics. Dordrecht: Reidel, pp. 249–281. Page references in text to reprint.CrossRefGoogle Scholar
Fine, A. (1982a), “Hidden Variables, Joint Probability and the Bell Inequalities”, Physical Review Letters 48: 291–295.CrossRefGoogle Scholar
Fine, A. (1982b), “Joint Distributions, Quantum Correlations, and Commuting Observables”, Journal of Mathematical Physics 23: 1306–1310.CrossRefGoogle Scholar
Fine, A. (1982c), “Fine Responds; Comments on Mermin and Garg 1982, Stapp 1982, Eberhard 1982”, Physical Review Letters 49: 243, 1536.Google Scholar
Fine, A. (1982d), “Some Local Models of Correlation Experiments”, Synthese 50: 279–294.CrossRefGoogle Scholar
Fry, E., and Thompson, R. (1976), “Experimental Test of Local Hidden Variable Theories”, Physical Review Letters 37: 465–468.CrossRefGoogle Scholar
Knuth, D. (1981), The Art of Computer Programming, Vol. 2, 2nd ed. New York: Addison-Wesley.Google Scholar
Landau, L. (1987), “On the Violation of Bell's Inequality in Quantum Theory”, Physics Letters A120: 54–56.CrossRefGoogle Scholar
Mehlberg, J. (1968), “On the Set Theoretical Approach to Probability”, Methodology and Science 1: 179–189.Google Scholar
Mermin, N. (1983), “Pair Distributions and Conditional Independence: Some Hints about the Structure of Strange Quantum Correlations”, Philosophy of Science 50: 359–373.CrossRefGoogle Scholar
Mermin, N., and Garg, A. (1982), “Comment on Fine 1982a”, Physical Review Letters 49: 242.Google Scholar
Nelson, E. (1967), Dynamical Theories of Brownian Motion. Princeton: Princeton University Press.CrossRefGoogle Scholar
Redhead, M. (1983), “Relativity, Causality and the Einstein-Podolsky-Rosen Paradox: Nonlocality and Peaceful Coexistence”, in R. Swinburne (ed.), Space, Time and Causality. Dordrecht: Reidel, pp. 151–189.CrossRefGoogle Scholar
Redhead, M. (1984), “Undressing Baby Bell”, unpublished manuscript.Google Scholar
Redhead, M. (1987), Incompleteness, Nonlocality and Realism: a Prolegomenon to the Philosophy of Quantum Mechanics. Oxford: Oxford University Press.Google Scholar
Schnorr, C. (1971), Zufalligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, vol. 218. Berlin: Springer Verlag.CrossRefGoogle Scholar
Sharp, W., and Shanks, N. (1985), “Fine's Prism Models for Quantum Correlation Statistics”, Philosophy of Science 52: 538–564.CrossRefGoogle Scholar
Shimony, A. (1984), “Contextual Hidden Variable Theories and Bell's Inequalities”, British Journal for the Philosophy of Science 35: 25–45.CrossRefGoogle Scholar
Stapp, H. (1982), “Bell's Theorem as a Nonlocality Property of Quantum Theory”, Physical Review Letters 49: 1470–1474.CrossRefGoogle Scholar
Wald, A. (1937), “Die Widerspruchsfreiheit des Kollektivbegriffes”, Ergebnisse eines mathematische Kolloquiums 8: 38–72, Vienna.Google Scholar
Wigner, E. (1970), “On Hidden Variables and Quantum Mechanical Probabilities”, American Journal of Physics 38: 1005–1009.CrossRefGoogle Scholar