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Joint distributions, quantum correlations, and commuting observables
1.See K. Urbanik, Studia Math. 21, 117 (1961);
1.L. Cohen, J. Math. Phys. 7, 781 (1966);
1.E. Nelson, Dynamical Theories of Brownian Motion (Princeton, U.P., Princeton, 1967), pp. 117–119;
1.S. Gudder, J. Math. Mech. 18, 325 (1968);
1.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;
1.S. Bugajski, Z. Naturforsch. Teil A 33, 1383 (1978).
2.See the survey of this literature by J. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978).
3.S. Kochen and E. Specker, J. Math. Mech. 17, 59 (1967).
4.For the special case where the range of the observables is and where 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.
5.We can add a fifth equivalent statement to this list; namely, (5) there exists a factorizable (so‐called “local”) stochastic hidden variables theory for 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).
5.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).
6.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.
7.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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