Abstract
A general algorithm is given for determining whether or not a given set of pair distributions allows for the construction of all the members of a specified set of higher-order distributions which return the given pair distributions as marginals. This mathematical question underlies studies of quantum correlation experiments such as those of Bell or of Clauser and Horne, or their higher-spin generalizations. The algorithm permits the analysis of rather intricate versions of such problems, in a form readily adaptable to the computer. The general procedure is illustrated by simple derivations of the results of Mermin and Schwarz for the symmetric spin-1 and spin-3/2 Einstein-Podolsky-Rosen problems. It is also used to extend those results to the spin-2 and spin-5/2 cases, providing further evidence that the range of strange quantum theoretic correlations does not diminish with increasing s. The algorithm is also illustrated by giving an alternative derivation of some recent results on the necessity and sufficiency of the Clauser-Horne conditions. The mathematical formulation of the algorithm is given in general terms without specific reference to the quantum theoretic applications.
Similar content being viewed by others
References
J. S. Bell,Physics (N.Y.) 1, 195 (1964).
J. F. Clauser and M. A. Horne,Phys. Rev. D 10, 526 (1974).
J. F. Clauser and A. Shimony,Rep. Prog. Phys. 41, 1881 (1978).
N. D. Mermin,Phys. Rev. D 22, 356 (1980).
N. D. Mermin and G. M. Schwarz,Found. Phys. 12, 101 (1982).
A. Garg and N. D. Mermin,Phys. Rev. Lett. 49, 901 (1982);49, 1220 (1982).
A. Garg and N. D. Mermin,Phys. Rev. D 27, 339 (1983).
R. T. Rockafellar,Convex Analsysis (Princeton U.P., Princeton, New Jersey, 1970), Sections 18, 19, and 22.
D. Gale,The Theory of Linear Economic Models (McGraw-Hill, New York, 1970), Chapter 2.
A. Einstein, B. Podolsky, and N. Rosen,Phys. Rev. 47, 777 (1935).
D. Bohm,Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951), pp. 614–619.
N. Bohr,Phys. Rev. 48, 696 (1935).
J. S. Bell,J. Phys. (Paris), Colloq. 42, C2–41 (1981).
A. Fine,Phys. Rev. Lett. 48, 291 (1982).
A. Garg and N. D. Mermin,Phys. Rev. Lett. 49, 242 (1982); A. Fine,Phys. Rev. Lett. 49, 243 (1982).
N. V. Chernikova,USSR Comp. Math. Math. Phys. 5, 228 (Engl. Transl.) (1965).
H. Greenberg,Numer. Math. 24, 19 (1975).
M. E. Dyer and L. G. Proll,Math. Prog. 12, 81 (1977).
P. Suppes and M. Zanotti, inLogic and Probability in Quantum Mechanics, P. Suppes, ed. (Reidel, Dordrecht, 1976), pp. 445–455.
L. D. Landau and E. M. Lifshitz,Quantum Mechanics (Pergamon Press, Elmsford, New York, 1977), 3rd edn.
G. Szego,Orthogonal Polynomials (American Mathematical Society Colloquium Publications XXIII, 1939).
G. Dall'Aglio,Symp. Math. IX, 131 (1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Garg, A., Mermin, N.D. Farkas's Lemma and the nature of reality: Statistical implications of quantum correlations. Found Phys 14, 1–39 (1984). https://doi.org/10.1007/BF00741645
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00741645