Abstract
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.
Similar content being viewed by others
Notes
For the reader who might worry that, according to special relativity, the temporal order, “before” and “after,” depends on the reference frame for spatially separated events, let us say that everything here refers to the reference frame where the laboratory in which the measurements are performed is at rest. See [37] for a discussion of the tensions that exist, due to such experiments, between quantum mechanics and relativity.
This argument is discussed in detail by Maudlin [37, 3rd edn, pp. 128–132].
For the proofs of nonlocality, it will be sufficient to consider finite dimensional spaces, which allows us to avoid all mathematical subtleties.
The correlations mentioned here are often called anti-correlations, for example when \({\tilde{O}}=-O\), as in the example of the spin in Sect. 2.
This is one way to understand Bohr’s response to EPR [12], although that response was not very clear.
The correct answer here being the value v(O) or \(v({\tilde{O}}')\) which is determined by the measurement of \({\tilde{O}}\) or of \( {\tilde{O}}'\).
This is obvious for (4.2), a special case of (3.2.1). For (4.1) we observe that, since O is self adjoint, we can write \(O = \sum _i \lambda _i P_{\lambda _i}\) where \(P_{\lambda _i}\) is the projector on the subspace of eigenvectors of eigenvalue \(\lambda _i\) of O and thus we have that \(f(O) = \sum _i f(\lambda _i) P_{\lambda _i}\). If we choose any f whose range is the set of eigenvalues of O and is such that \(f(\lambda _i)= \lambda _i\)\(\forall i\), we have that \(O=f(O)\) and, by (3.2.1), we obtain that \(v(O)=v(f(O))=f(v(O))\) and thus v(O) is an eigenvalue of O.
This is not quite true. For the choice \(f(x, y)=xy\) we must have \(\dim \mathcal{H} \ge 4\).
Here, “determinism” refers to the idea of there being pre-existing values v(A) or of the existence of a non-contextual value-map.
In fact, this is the property that we discussed in remark 4 of Sect. 3.2, and, as Schrödinger’s example of the schoolchildren shows, it does not depend on humans having “genuine free will” (whatever that means).
As we mentioned in Sect. 3.2, if there exists a pre-existing value v(A), then the result of any experiment \(\mathcal{E}_A\) measuring A must be v(A).
References
Albert, D.: Quantum Mechanics and Experience. Harvard University Press, Cambridge (1992)
Aravind, P.K.: Bell’s theorem without inequalities and only two distant observers. Found. Phys. Lett. 15, 399–405 (2002)
Bassi, A., Ghirardi, G.C.: The Conway-Kochen argument and relativistic GRW models. Found. Phys. 37, 169–185 (2007)
Bell, J.S.: On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964). Reprinted as Chap. 2 in [8]
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]
Bell, J.S.: Bertlmann’s socks and the nature of reality. J. Phys. 42(C2), 41–61 (1981). Reprinted as Chap. 16 in [8]
Bell, J.S.: On the impossible pilot wave. Found. Phys. 12, 989–999 (1982). Reprinted as Chap. 17 in [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)
Bohm, D.: Quantum Theory, New edition. Dover, New York (1989). First edition: Prentice Hall, Englewood Cliffs (NJ), 1951
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. 369–390
Bohm, D., Hiley, B.J.: The Undivided Universe. Routledge, London (1993)
Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935)
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)
Bricmont, J.: Making Sense of Quantum Mechanics. Springer, Berlin (2016)
Bricmont, J.: Quantum Sense and Nonsense. Springer, Basel (2017)
Brown, H.R., Svetlichny, G.: Nonlocality and Gleason’s lemma. Part I: Deterministic theories. Found. Phys. 20, 1379–1386 (1990)
Cabello, A.: Bell’s theorem without inequalities and without probabilities for two observers. Phys. Rev. Lett. 86, 1911–1914 (2001)
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. http://www.math.princeton.edu/facultypapers/Conway/
Daumer, M., Dürr, D., Goldstein, S., Zanghì, N.: Naive realism about operators. Erkenntnis 45, 379–397 (1996)
DeWitt, B., Graham, R.N. (eds.): The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, Princeton (1973)
Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843–907 (1992)
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)
Dürr, D., Teufel, S.: Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, Berlin (2009)
Dürr, D., Goldstein, S., Zanghì, N.: Quantum Physics Without Quantum Philosophy. Springer, Berlin (2012)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)
Elby, A.: Nonlocality and Gleason’s Lemma. Part 2. Found. Phys. 20, 1389–1397 (1990)
Everett, H.: ‘Relative state’ formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957). Reprinted in [20, pp. 141–149]
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)
Goldstein, S.: Bohmian mechanics and quantum information. Found. Phys. 40, 335–355 (2010)
Goldstein, S., Norsen, T., Tausk, D.V., Zanghì, N.: Bell’s theorem. Scholarpedia 6(10), 8378 (2011)
Goldstein, S.: Bohmian mechanics. In: E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy. Spring 2013 Edition. plato.stanford.edu/archives/spr2013/entries/qm-bohm/
Hemmick, D.L.: Hidden variables and nonlocality in quantum mechanics. Doctoral thesis, Rutgers University. https://sites.google.com/site/dlhquantum/doctoral-thesis and arXiv:quant-ph/0412011v1 (1996)
Hemmick, D.L., Shakur, A.M.: Bell’s Theorem and Quantum Realism. Reassessment in Light of the Schrödinger Paradox. Springer, Heidelberg (2012)
Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker paradox. Found. Phys. 13, 48–499 (1983)
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
Kochen, S.: Private communication to Abner Shimony, see [34, footnote 2]
Maudlin, T.: Quantum Nonlocality and Relativity. Blackwell, Cambridge, 1st edn, 1994, 3rd edn, 2011
Mermin, D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Norsen, T.: Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory. Springer, Basel (2017)
Peres, A.: Incompatible results of quantum measurements. Phys. Lett. A 151, 107–108 (1990)
Peres, A.: Two simple proofs of the Kochen-Specker theorem. J. Phys. A 24, L175–L178 (1991)
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)
Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Cambrid. Philos. Soc. 31, 555–563 (1935)
Schrödinger, E.: Probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 32, 446–452 (1936)
Stairs, A.: Quantum logic, realism and value-definiteness. Philos. Sci. 50, 578–602 (1983)
Tumulka, R.: Understanding Bohmian mechanics—a dialogue. Am. J. Phys. 72, 1220–1226 (2004)
Tumulka, R.: Comment on “The Free Will Theorem”. Found. Phys. 37, 186–197 (2007)
Wheeler, J.A., Zurek, W.H. (eds.): Quantum Theory and Measurement. Princeton University Press, Princeton (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Giovanni Gallavotti.
Dedicated to our mentor, Joel L. Lebowitz, a master of statistical physics who, when it came to foundational issues of quantum mechanics, was always willing to listen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bricmont, J., Goldstein, S. & Hemmick, D. Schrödinger’s Paradox and Proofs of Nonlocality Using Only Perfect Correlations. J Stat Phys 180, 74–91 (2020). https://doi.org/10.1007/s10955-019-02361-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02361-w