Schrödinger’s Paradox and Proofs of Nonlocality Using Only Perfect Correlations

  • Jean BricmontEmail author
  • Sheldon Goldstein
  • Douglas Hemmick


We discuss proofs of nonlocality based on a generalization by Erwin Schrödinger of the argument of Einstein, Podolsky and Rosen. These proofs do not appeal in any way to Bell’s inequalities. Indeed, one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations. First we explain that Schrödinger’s argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems imply the existence of a non-contextual value-map for quantum observables; non-contextual means that the observable has a particular value before its measurement, for any given quantum system, and that any experiment “measuring this observable” will reveal that value. Then, we establish the impossibility of a non-contextual value-map for quantum observables without invoking any further quantum predictions. Combining this with Schrödinger’s argument implies nonlocality. Finally, we illustrate how Bohmian mechanics is compatible with the impossibility of a non-contextual value-map.


Nonlocality No hidden variables theorems Bohmian mechanics Perfect correlations 



  1. 1.
    Albert, D.: Quantum Mechanics and Experience. Harvard University Press, Cambridge (1992)Google Scholar
  2. 2.
    Aravind, P.K.: Bell’s theorem without inequalities and only two distant observers. Found. Phys. Lett. 15, 399–405 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bassi, A., Ghirardi, G.C.: The Conway-Kochen argument and relativistic GRW models. Found. Phys. 37, 169–185 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bell, J.S.: On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964). Reprinted as Chap. 2 in [8]MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966). Reprinted as Chap. 1 in [8]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bell, J.S.: Bertlmann’s socks and the nature of reality. J. Phys. 42(C2), 41–61 (1981). Reprinted as Chap. 16 in [8]Google Scholar
  7. 7.
    Bell, J.S.: On the impossible pilot wave. Found. Phys. 12, 989–999 (1982). Reprinted as Chap. 17 in [8]ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Collected Papers on Quantum Philosophy, 2nd edn, with an introduction by Alain Aspect. Cambridge University Press, Cambridge, 2004; 1st edn (1987)Google Scholar
  9. 9.
    Bohm, D.: Quantum Theory, New edition. Dover, New York (1989). First edition: Prentice Hall, Englewood Cliffs (NJ), 1951Google Scholar
  10. 10.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden variables”, Parts 1 and 2. Phys. Rev. 89, 166–193 (1952). Reprinted in [49] pp. 369–390Google Scholar
  11. 11.
    Bohm, D., Hiley, B.J.: The Undivided Universe. Routledge, London (1993)zbMATHGoogle Scholar
  12. 12.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Bohr, N.: Discussion with Einstein on epistemological problems in atomic physics. In: Schilpp, P.A. (ed.) Albert Einstein, Philosopher-Scientist, pp. 201–241. The Library of Living Philosophers, Evanston (1949)Google Scholar
  14. 14.
    Bricmont, J.: Making Sense of Quantum Mechanics. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  15. 15.
    Bricmont, J.: Quantum Sense and Nonsense. Springer, Basel (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Brown, H.R., Svetlichny, G.: Nonlocality and Gleason’s lemma. Part I: Deterministic theories. Found. Phys. 20, 1379–1386 (1990)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Cabello, A.: Bell’s theorem without inequalities and without probabilities for two observers. Phys. Rev. Lett. 86, 1911–1914 (2001)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Conway, J. H., Kochen, S.: The free will theorem. Found. Phys. 36, 1441–1473 (2006); Reply to comments of Bassi, Ghirardi, and Tumulka on the free will theorem. Found. Phys. 37, 1643–1647 (2007); The strong free will theorem. Not. Am. Math. Soc. 56, 226–232 (2009); The free will theorem. Series of 6 public lectures delivered by J. Conway at Princeton University, March 23–April 27, 2009.
  19. 19.
    Daumer, M., Dürr, D., Goldstein, S., Zanghì, N.: Naive realism about operators. Erkenntnis 45, 379–397 (1996)MathSciNetzbMATHGoogle Scholar
  20. 20.
    DeWitt, B., Graham, R.N. (eds.): The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, Princeton (1973)Google Scholar
  21. 21.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843–907 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: John Bell and Bell’s theorem. In: Borchert, D.M. (ed.) Encyclopedia of Philosophy. Macmillan, New York (2005)Google Scholar
  23. 23.
    Dürr, D., Teufel, S.: Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, Berlin (2009)zbMATHGoogle Scholar
  24. 24.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum Physics Without Quantum Philosophy. Springer, Berlin (2012)zbMATHGoogle Scholar
  25. 25.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Elby, A.: Nonlocality and Gleason’s Lemma. Part 2. Found. Phys. 20, 1389–1397 (1990)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Everett, H.: ‘Relative state’ formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957). Reprinted in [20, pp. 141–149]ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Goldstein, S., Tausk, D.V., Tumulka, R., Zanghì, N.: What does the free will theorem actually prove? Not. Am. Math. Soc. 57, 1451–1453 (2010)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Goldstein, S.: Bohmian mechanics and quantum information. Found. Phys. 40, 335–355 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goldstein, S., Norsen, T., Tausk, D.V., Zanghì, N.: Bell’s theorem. Scholarpedia 6(10), 8378 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Goldstein, S.: Bohmian mechanics. In: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy. Spring 2013 Edition. Google Scholar
  32. 32.
    Hemmick, D.L.: Hidden variables and nonlocality in quantum mechanics. Doctoral thesis, Rutgers University. and arXiv:quant-ph/0412011v1 (1996)
  33. 33.
    Hemmick, D.L., Shakur, A.M.: Bell’s Theorem and Quantum Realism. Reassessment in Light of the Schrödinger Paradox. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  34. 34.
    Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker paradox. Found. Phys. 13, 48–499 (1983)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kochen, S.: Private communication to Abner Shimony, see [34, footnote 2]Google Scholar
  37. 37.
    Maudlin, T.: Quantum Nonlocality and Relativity. Blackwell, Cambridge, 1st edn, 1994, 3rd edn, 2011Google Scholar
  38. 38.
    Mermin, D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  40. 40.
    Norsen, T.: Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory. Springer, Basel (2017)CrossRefzbMATHGoogle Scholar
  41. 41.
    Peres, A.: Incompatible results of quantum measurements. Phys. Lett. A 151, 107–108 (1990)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Peres, A.: Two simple proofs of the Kochen-Specker theorem. J. Phys. A 24, L175–L178 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Schrödinger, E.: Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften 23, 807–812; 823–828; 844–849 (1935). English translation: The present situation in quantum mechanics, translated by J.D. Trimmer, Proceedings of the American Philosophical Society 124, 323–338 (1980). Reprinted. In: Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement. Princeton University Press, Princeton, 152–167 (1983)Google Scholar
  44. 44.
    Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Cambrid. Philos. Soc. 31, 555–563 (1935)ADSCrossRefzbMATHGoogle Scholar
  45. 45.
    Schrödinger, E.: Probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 32, 446–452 (1936)ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Stairs, A.: Quantum logic, realism and value-definiteness. Philos. Sci. 50, 578–602 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Tumulka, R.: Understanding Bohmian mechanics—a dialogue. Am. J. Phys. 72, 1220–1226 (2004)ADSCrossRefGoogle Scholar
  48. 48.
    Tumulka, R.: Comment on “The Free Will Theorem”. Found. Phys. 37, 186–197 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wheeler, J.A., Zurek, W.H. (eds.): Quantum Theory and Measurement. Princeton University Press, Princeton (1983)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IRMPUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA
  3. 3.BerlinUSA

Personalised recommendations