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

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.

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Fig. 1
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Notes

  1. 1.

    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.

  2. 2.

    This argument is discussed in detail by Maudlin [37, 3rd edn, pp. 128–132].

  3. 3.

    For the proofs of nonlocality, it will be sufficient to consider finite dimensional spaces, which allows us to avoid all mathematical subtleties.

  4. 4.

    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.

  5. 5.

    This is one way to understand Bohr’s response to EPR [12], although that response was not very clear.

  6. 6.

    The correct answer here being the value v(O) or \(v({\tilde{O}}')\) which is determined by the measurement of \({\tilde{O}}\) or of \( {\tilde{O}}'\).

  7. 7.

    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.

  8. 8.

    This is not quite true. For the choice \(f(x, y)=xy\) we must have \(\dim \mathcal{H} \ge 4\).

  9. 9.

    In their original argument, Kochen and Specker used 117 such sets [35], but that number was reduced to 33 by Peres [41, 42].

  10. 10.

    See [4] for the original inequalities and [30] for a detailed discussion of them. For a simple version of those inequalities see [22] or [14, Chap. 4].

  11. 11.

    For elementary introductions to that theory, see [15, 47] and for more advanced ones, see [8, 10, 11, 14, 21, 23, 24, 31, 40]. There are also pedagogical videos made by students in Munich, available at: https://cast.itunes.uni-muenchen.de/vod/playlists/URqb5J7RBr.html.

  12. 12.

    One might object that adherents of the “Many-Worlds Interpretation” of quantum mechanics [20, 27] do not share that view, since, for them, experiments do not have a single outcome. But they nevertheless agree that measuring devices indicate definite results in each World.

  13. 13.

    Here, “determinism” refers to the idea of there being pre-existing values v(A) or of the existence of a non-contextual value-map.

  14. 14.

    Reference [4], reprinted as Chap. 2 in [8].

  15. 15.

    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).

  16. 16.

    They also claim that their results rule out certain “spontaneous collapse theories,” a statement criticized in [3, 28, 48].

  17. 17.

    Since, combined with locality, they imply the existence of a non-contextual value-map, satisfying (3.2.1) and thus also (4.14.2), whose existence is impossible because of theorem 4.1.

  18. 18.

    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).

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Correspondence to Jean Bricmont.

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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.

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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

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Keywords

  • Nonlocality
  • No hidden variables theorems
  • Bohmian mechanics
  • Perfect correlations