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Constraints on Determinism: Bell Versus Conway–Kochen


Bell’s Theorem from Physics 36:1–28 (1964) and the (Strong) Free Will Theorem of Conway and Kochen from Notices AMS 56:226–232 (2009) both exclude deterministic hidden variable theories (or, in modern parlance, ‘ontological models’) that are compatible with some small fragment of quantum mechanics, admit ‘free’ settings of the archetypal Alice and Bob experiment, and satisfy a locality condition akin to parameter independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the ‘ontological state’ initially determines both the settings and the outcome of the experiment. The usual status of the settings as ‘free’ parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell’s later generalization of his theorem to stochastic hidden variable theories.

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  1. Analogous earlier results were obtained, in chronological order, by Heywood and Redhead [20], Stairs [34], Brown and Svetlichny [7], and Clifton [11] (of which only [20] was cited by Conway and Kochen).

  2. Bell [3] even attributes it to Einstein. See [37] for a detailed analysis of the way this condition is actually used by Bell in [3, 5], and of the way it has been (mis)perceived by others. In particular, one should distinguish it from the locality condition usually named after Bell [6]. The latter, also called local causality, is a conjunction of two (probabilistic) notions that are now generally called Parameter Independence (pi) and Outcome Independence (oi); see [8, 22, 23, 27, 31, 33]. The latter is automatically satisfied in the type of deterministic theories studied in [3, 13, 14], upon which the former reduces to the condition stated in the main text above, but now conditioned on certain values of the hidden variables. Note that our definition of the term pi will be different from the literature so far, though in the same spirit.

  3. The only significant exception we could find is the small and otherwise interesting book by Hemmick and Shakur [17], whose scathing treatment of the Free Will Theorem is somewhat undermined by their claim (p. 90) that the assumption of determinism follows from the other assumptions in the Strong Free Will Theorem (notably pi and perfect correlation). This seems questionable [37]: either Bell’s (later) locality condition (i.e., pi plus oi) in conjunction with perfect correlation implies determinism, or pi plus determinism implies oi (and hence Bell Locality). Perhaps our view (which is certainly shared by Conway and Kochen!) that the assumptions of the Strong Free Will Theorem have been chosen quite carefully is clearer from our reformulation below than from even their second paper [14] (not to speak of their first [13]). Indeed, if valid, the objection of Hemmick and Shakur could just as well be raised against Bell’s Theorem [3], where it would be equally misguided if both results are construed as attempts to put constraints on determinism in the first place. Our treatment of parameter settings will also be different from [17].

  4. This even led them to their curious way of paraphrasing their theorem as showing that ‘If we humans have free will, then elementary particles already have their own small share of this valuable commodity’.

  5. See also [6, 10] and most recently [26] for the interpretation of hidden variables as ontological states.

  6. See also Colbeck and Renner [12] for at least the first step of this strategy in the context of stochastic hidden variable theories. Using settings as labels, on the other hand, is defended in e.g. [8, 10, 32].

  7. Equivalently, \(\alpha \) and \(\beta \) could stand for the corresponding unit vectors \(\vec {a}\) and \(\vec {b}\), defined up to a sign.

  8. Here \(P_E(F\ne G|A=\alpha ,B=\beta )\equiv P_E(F=0,G=1|A=\alpha ,B=\beta )+P_E(F=1,G=0|A=\alpha ,B=\beta )\). The complete statistics are: \(P_E(F=1,G=1|A=\alpha ,B=\beta )=P_E(F=0,G=0|A=\alpha ,B=\beta )={\frac{1}{2}} \cos ^2(\alpha -\beta )\) and \(P_E(F=0,G=1|A=\alpha ,B=\beta )=P_E(F=1,G=0|A=\alpha ,B=\beta )={\frac{1}{2}} \sin ^2(\alpha -\beta )\).

  9. This formulation incorporates the assumption that \(P\) is independent of \(A,B,F,G\), and vice versa.

  10. Here \(P(F=\lambda ,G=\mu |A=\alpha ,B=\beta )\equiv P(F=\lambda ,G=\mu ,A=\alpha ,B=\beta )/P(A=\alpha ,B=\beta )\) and \(P(F=\lambda ,G=\mu ,A=\alpha ,B=\beta )\equiv P(\{x\in X\mid F(x)=\lambda ,G(x)=\mu ,A(x)=\alpha ,B(x)=\beta \})\), etc.

  11. On the usual definition, this also implies that the pairs \((A,B)\), \((A,Z)\), and \((B,Z)\) are independent.

  12. This is true even if \(F=F(A,B,Z)\) and \(G=G(A,B,Z)\) rather than \(F=F(A,Z)\) and \(G=G(B,Z)\).

  13. What is being measured here by say Alice with setting \(\mathbf {a}\) is the triple \((\langle \vec {a}_1,\vec {J}\rangle ^2,\langle \vec {a}_2,\vec {J}\rangle ^2,\langle \vec {a}_3,\vec {J}\rangle ^2)\), where \(\vec {J}\) is the angular momentum operator for spin one. Each operator \(\langle \vec {a}_i,\vec {J}\rangle \) has spectrum \(\{-1, 0,1\}\), so each square \(\langle \vec {a}_i,\vec {J}\rangle ^2\) can be 0 or 1. Since \(\vec {J}^2=2\), one has \(\langle \vec {a}_1,\vec {J}\rangle ^2+\langle \vec {a}_2,\vec {J}\rangle ^2+\langle \vec {a}_3,\vec {J}\rangle ^2=2\), which gives (3.1).

  14. The complete (theoretical) statistics are: \(P_{QM}(F_i=1,G_j=1|A=\mathbf {a},B=\mathbf {b}) ={\frac{1}{3}}(1+\langle \vec {a}_i,\vec {b}_j\rangle ^2)\),

    \(P_{QM}(F_i=0,G_j=0|A=\mathbf {a},B=\mathbf {b})= {\frac{1}{3}}\langle \vec {a}_i,\vec {b}_j\rangle ^2\), \(P_{QM}(F_i=1,G_j=0|A=\mathbf {a},B=\mathbf {b})= {\frac{1}{3}}(1-\langle \vec {a}_i,\vec {b}_j\rangle ^2)\), and \(P_{QM}(F_i=0,G_j=1|A=\mathbf {a},B=\mathbf {b})= {\frac{1}{3}}(1-\langle \vec {a}_i,\vec {b}_j\rangle ^2)\). See footnote 8 for notation like \(P(F_i\ne G_j|\cdot )\).

  15. See Definition 4.1 below for Determinism, and Definition 2.1 for the others, mutatis mutandis.

  16. Here \(P_{QM}(F_i=G_j|A_i=B_j)\) denotes \(P_{QM}(F_i=0,G_j=0|A_i=B_j)+P_{QM}(F_i=1,G_j=1|A_i=B_j)\), where the setting \(A_i=B_j\) stands for \((A=\mathbf {a},B=\mathbf {b})\) subject to \(\vec {a}_i=\pm \vec {b}_j\). It follows from (3.3) or the previous footnote that \(P_{QM}(F_i=G_j|A_i=B_j)={\frac{1}{3}}(1+2\cos ^2 \theta _{\vec {a}_i,\vec {b}_j})\), which for \(\vec {a}_i=\pm \vec {b}_j\) equals unity.

  17. To keep matters simple, we will not be bothered with the notational difference between frames \([\vec {a}_1,\vec {a}_2,\vec {a}_3]\) and orthonormal bases \((\vec {a}_1,\vec {a}_2,\vec {a}_3)\), and similarly for \(\mathbf {b}\), until the end of the proof.


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The authors are indebted to Jeff Bub, Jeremy Butterfield, Dennis Dieks, Richard Gill, Hans Maassen, and Matt Leifer for various comments on this work (including predecessors).

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Correspondence to Klaas Landsman.

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Dedicated to Professor Hans Maassen, on the occasion of his inaugural lecture (15-01-2014).

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Cator, E., Landsman, K. Constraints on Determinism: Bell Versus Conway–Kochen. Found Phys 44, 781–791 (2014).

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  • Free will theorem
  • Bell’s theorem
  • Hidden variable theories
  • Determinism