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Joint distributions, quantum correlations, and commuting observables

J. Math. Phys. 23, 1306 (1982); doi:10.1063/1.525514

Issue Date: July 1982

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Arthur Fine
University of Illinois at Chicago Circle, Chicago, Illinois 60680
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/23/1306/1
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KEYWORDS and PACS

Keywords
PACS
  • 02.50.-r
    Mathematical methods in physics Probability theory, stochastic processes, and statistics
  • 03.65.Bz
    Classical and quantum physics; mechanics and fields Quantum theory; quantum mechanics Foundations, theory of measurement, miscellaneous theories
  • YEAR: 1982

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (7)
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  1. See K. Urbanik, Studia Math. 21, 117 (1961);
  2. L. Cohen, J. Math. Phys. 7, 781 (1966);
    3. E. Nelson, Dynamical Theories of Brownian Motion (Princeton, U.P., Princeton, 1967), pp. 117–119;
    S. Gudder, J. Math. Mech. 18, 325 (1968);
    V. V. Kuryshkin, in The Uncertainly Principle and Foundations of Quantum Mechanics, edited by W. Price and S. Chissick (Wiley, New York, 1977), pp. 61–83;
    S. Bugajski, Z. Naturforsch. Teil A 33, 1383 (1978).
  3. See the survey of this literature by J. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978).
  4. S. Kochen and E. Specker, J. Math. Mech. 17, 59 (1967).
  5. For the special case where the range of the observables is ±1 and where (1/2) = P(A) = P(B) = P(B[prime]), a system of inequalities equivalent to (2) was first discovered by P. Suppes and M. Zanotti, Synthese (to be published). I want to thank Suppes and Zanotti for sharing their work with me, and for some very stimulating exchanges of ideas.
  6. We can add a fifth equivalent statement to this list; namely, (5) there exists a factorizable (so-called “local”) stochastic hidden variables theory for A1,A2,B1,B2 returning the observed single and double distributions. See J. Clauser and M. Horne, Phys. Rev. D 10, 526 (1974), for the definitions here. It is well known that (3) implies (5), and Clauser and Horne show that (5) implies (1). Thus the equivalence follows from our proof that (1) implies (3). It is easier, however, indeed trivial, to show that (5) implies (2), and to get the equivalence from that of (2) to (3).
    7. See my “Hidden variables, joint probability and the Bell inequalities,” Phys. Rev. Lett. 48, 291 (1982), which also contains another derivation of (3) from (1).
  7. A. Fine, Synthese 29, 257 (1974). This is reprinted, with a relevant correction to the proof, in Logic and Probability in Quantum Mechanics, edited by P. Suppes (Reidel, Dordrecht, 1976), pp. 249–281.
  8. For pairs of discrete observables a computational proof is contained in A. Fine, Brit. J. Philos. Sci. 24, 1 (1973). want to thank Robert Latzer for correspondence that helped me find the simple, general derivation below.

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