Skip to main content
Log in

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (Br J Philos Sci 66(4): 905–927, 2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Kolmogorovian probability theory typically assumes a stronger \(\sigma \)-algebra, which requires the probability measure to be \(\sigma \)-additive, that is, additive over countable collections of (disjoint) sets, rather than just finite ones. (Cf. Definition 3). Our results, however, nowhere appeal to this stronger condition, so we have dropped it everywhere for clarity of expression.

  2. See Malament [53] and Feintzeig [20, Sect. 2.2] for more on the relationship of this presentation with more standard presentations of Bell’s theorem.

  3. It is “restricted” since one might demand the second condition hold for all collections of compatible observables, not just pairs. Cf. Definitions 5 and 6 of [20].

  4. Notice that if \((\mathcal {H},\psi ,\mathcal {O}_n)\) is a KS witness, then so is \((\mathcal {H},\psi ',\mathcal {O}_n)\) for any distinct \(\psi '\in \mathcal {H}\) because the KS theorem does not depend on any choice of state vector. We nevertheless include an arbitrary state vector \(\psi \) in all of our results to highlight the structural similarity with Bell-type no-go theorems. In addition, it will be apparent in the proof of Theorem 3 below that nothing depends on whether the state chosen is pure or mixed. The result applies to density operator states just as well.

  5. Note that this is not the solution Fine favors; rather, he suggests the use of what he calls prism models [23], which we will not analyze in this paper.

  6. See Feintzeig [20] for an extensive discussion of generalized probability spaces and a slight variation on the main theorem applied to them.

  7. See, for example, [1, 37, 41, 57, 58, 61, 63, 64, 71, 78], and [49,50,51,52].

  8. Kronz [49,50,51,52] requires that the range of \(\mu \) is contained in \([-1,\infty )\), calling the resulting spaces non-monotonic probability spaces. This restriction makes no difference for our remarks.

  9. In this context especially, Halliwell and Yearsley [36] suggest restricting attention to negative probability spaces, which they call quasiprobability distributions, whose marginal distributions match those of some Kolmogorovian probability space. This restriction makes no difference for our remarks.

  10. Srinivasan [69] also considers quaternionic probability spaces, which he sometimes calls extended measure spaces. The same results below apply to these spaces as well.

  11. Hartle [38] calls probability assignments lying outside the unit interval virtual probabilities, and Hartle [39] and Gell-Mann and Hartle [26] call them extended probabilities, but the formal definition is the same.

  12. Gudder [35] also includes an extra condition called regularity. For our purposes it suffices to consider the simpler and more general spaces defined here, of which Gudder’s are a special case.

  13. One can also generalize to even weaker additivity constraints, to which our results apply equally.

  14. See also [20] for a discussion of Dutch Books in the context of generalized probability spaces.

  15. Such an additional assignment to realize a finite null cover does not appear to be unique in general, although it may be so in specific cases.

References

  1. Agarwal, G., Home, D., Schleich, W.: Einstein-Podolsky-Rosen correlation-parallelism between the Wigner function and the local hidden variable approaches. Phys. Lett. A 170, 359–362 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aspect, A.: Trois tests expérimentaux des inégalités de Bell par corrélation de polarisation de photons. PhD thesis, Universite de Paris-Sud, Cenre d’Orsay, Orsay (1983)

  3. Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  4. Aspect, A., Grangier, P., Roger, G.: Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49, 91–94 (1982)

    Article  ADS  Google Scholar 

  5. Bassi, A., Ghirardi, G.: Can the decoherent histories description of reality be considered satisfactory? Phys. Lett. A 257, 247–263 (1999)

    Article  ADS  Google Scholar 

  6. Bassi, A., Ghirardi, G.: Decoherent histories and realism. J. Stat. Phys. 98, 457–494 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1(3), 195–200 (1964)

    Google Scholar 

  8. Bell, J.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447–451 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bell, J.: Introduction to the hidden-variable question. In: d’Espagnat, B. (ed.) Foundations of Quantum Mechanics (Proceedings of the International School of Physics ’Enrico Fermi’. course IL), pp. 171–181. Academic Press, New York (1971)

  10. Bradley, S.: Imprecise probabilities. In: Zalta, E. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Stanford (2015)

    Google Scholar 

  11. Clauser, J., Horne, M.: Experimental consequences of objective local theories. Phys. Rev. D 10, 526–535 (1974)

    Article  ADS  Google Scholar 

  12. Clauser, J., Horne, M., Shimony, A., Holt, R.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969)

    Article  ADS  Google Scholar 

  13. Cox, R.: Probability, frequency and reasonable expectation. Am. J. Phys. 14, 1–10 (1946)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Craig, D., Dowker, F., Henson, J., Major, S., Rideout, D., Sorkin, R.: A Bell inequality analog in quantum measure theory. J. Phys. A 40, 501–523 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. de Barros, J.A., Suppes, P.: Some conceptual issues involving probability in quantum mechanics (2000). arXiv:quant-ph/0001017

  16. de Barros, J.A., Suppes, P.: Probabilistic inequalities and upper probabilities in quantum mechanical entanglement. Manuscrito — Revista Internacional de Filosofia 33(1), 55–71 (2010)

    Google Scholar 

  17. Dirac, P.: Bakerian lecture: the physical interpretation of quantum mechanics. Proc. R. Soc. Lond. A 180, 1–40 (1942)

    Article  ADS  MATH  Google Scholar 

  18. Dowker, F., Ghazi-Tabatabai, Y.: The Kochen-Specker theorem revisited in quantum measure theory. J. Phys. A 41, 105301 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)

    Article  ADS  MATH  Google Scholar 

  20. Feintzeig, B.: Hidden variables and incompatible observables in quantum mechanics. Br. J. Philos. Sci. 66(4), 905–927 (2015)

    Article  MathSciNet  Google Scholar 

  21. Feynman, R.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367–387 (1948)

    Article  ADS  MathSciNet  Google Scholar 

  22. Feynman, R.: Negative probability. In: Hiley, B., Peat, F.D. (eds.) Quantum Implications: Essays in Honour of David Bohm, pp. 235–248. Routledge, New York (1987)

    Google Scholar 

  23. Fine, A.: Antinomies of entanglement: the puzzling case of the tangled statistics. J. Philos. 79(12), 733–747 (1982)

    MathSciNet  Google Scholar 

  24. Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48(5), 291–294 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  25. Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23(7), 1306–1310 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  26. Gell-Mann, M., Hartle, J.: Decoherent histories quantum mechanics with one real fine-grained history. Phys. Rev. A 85, 062120 (2012)

    Article  ADS  Google Scholar 

  27. Giustina, M., Mech, A., Ramelow, S., Wittman, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S., Ursin, R., Zeilinger, A.: Bell violation using entangled photons without the fair-sampling assumption. Nature 497, 227–230 (2013)

    Article  ADS  Google Scholar 

  28. Giustina, M., Versteegh, M., Wengerowsky, S., Handsteiner, J., Hochrainer, A., Phelan, K., Steinlechner, F., Kofler, J., Larsson, J., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M., Beyer, J., Gerrits, T., Lita, A., Shalm, L., Nam, S., Scheidl, T., Ursin, R., Wittmann, B., Zeilinger, A.: Significant-loophole-free test of Bell’s theorem with entangled photons. Phys. Rev. Lett. 115, 250401 (2015)

    Article  ADS  Google Scholar 

  29. Greenberger, D., Horne, M., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Greenberger, D., Horne, M., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.) Bell’s Theorem. Quantum Theory and Conceptions of the Universe, pp. 69–72. Kluwer, Dordrecht (1989)

    Google Scholar 

  31. Griffiths, R.: Consistent histories, quantum truth functionals, and hidden variables. Phys. Lett. A 265, 12–19 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Griffiths, R.: Consistent quantum realism: A reply to Bassi and Ghirardi. J. Stat. Phys. 99, 1409–1425 (2000)

    Article  MATH  Google Scholar 

  33. Gudder, S.: Quantum Probability. Academic Press, San Diego, CA (1988)

    MATH  Google Scholar 

  34. Gudder, S.: Finite quantum measure spaces. Am. Math. Mon. 117(6), 512–527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gudder, S.: Quantum measure theory. Math. Slovaca 60(5), 681–700 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Halliwell, J., Yearsley, J.: Negative probabilities, Fine’s theorem, and linear positivity. Phys. Rev. A 87, 022114 (2013)

    Article  ADS  Google Scholar 

  37. Han, Y.D., Hwang, W., Koh, I.: Explicit solutions for negative probability measures for all entangled states. Phys. Lett. A 221, 283–286 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Hartle, J.: Linear positivity and virtual probability. Phys. Rev. A 70, 022104 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. Hartle, J.: Quantum mechanics with extended probabilities. Phys. Rev. A 78, 012108 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Hartmann, S., Suppes, P.: Entanglement, upper probabilities and decoherence in quantum mechanics. In: Suárez, M., Dorato, M., Rédei, M. (eds.) EPSA Philosophical Issues in the Sciences: Launch of the European Philosophy of Science Association, pp. 93–103. Springer, Berlin (2010)

    Chapter  Google Scholar 

  41. Home, D., Lepore, V., Selleri, F.: Local realistic models and non-physical probabilities. Phys. Lett. A 158, 357–360 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  42. Home, D., Selleri, F.: Bell’s theorem and the EPR paradox. Revista del Nuovo Cimento 14(9), 1–95 (1991)

    Article  MathSciNet  Google Scholar 

  43. Ivanović, I.: On complex Bell’s inequality. Lettere al Nuovo Cimento 22(1), 14–16 (1978)

    Article  Google Scholar 

  44. Khrennikov, A.: p-adic probability distributions of hidden variables. Physica A 215(4), 577–587 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  45. Khrennikov, A.: p-Adic probability interpretation of Bell’s inequality. Phys. Lett. A 200, 219–223 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Khrennikov, A.: Interpretations of Probability, 2nd edn. de Gruyter, Berlin (2009)

    Book  MATH  Google Scholar 

  47. Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)

    MathSciNet  MATH  Google Scholar 

  48. Krantz, D., Luce, D., Suppes, P., Tversky, A.: Foundations of Measurement, vol. I. Dover, Mineola, NY (1971)

    MATH  Google Scholar 

  49. Kronz, F.: A nonmonotonic theory of probability for spin-1/2 systems. Int. J. Theor. Phys. 44(11), 1963–1976 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kronz, F.: Non-monotonic probability theory and photon polarization. J. Philos. Logic 36, 446–472 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kronz, F.: Non-monotonic probability theory for n-state quantum systems. Stud. Hist. Philos. Mod. Phys. 39, 259–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kronz, F.: Actual and virtual events in the quantum domain. Ontol. Stud. 9, 209–220 (2009)

    Google Scholar 

  53. Malament, D.: Notes on Bell’s theorem (2012). http://www.socsci.uci.edu/~dmalamen/courses/prob-determ/PDnotesBell.pdf

  54. Mermin, N.: Generalizations of Bell’s theorem to higher spins and higher correlations. In: Roth, L., Inomato, A. (eds.) Fundamental Questions in Quantum Mechanics, pp. 7–20. Gordon and Breach, New York (1986)

    Google Scholar 

  55. Mermin, N.: Quantum mysteries revisited. Am. J. Phys. 58, 731–733 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  56. Miller, D.: Realism and time symmetry in quantum mechanics. Phys. Lett. A 222, 31–36 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Mückenheim, W.: A resolution of the Einstein-Podolsky-Rosen paradox. Lettere al Nuovo Cimento 35(9), 300–304 (1982)

    Article  Google Scholar 

  58. Mückenheim, W.: A review of extended probabilities. Phys. Rep. 133(6), 337–401 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  59. Pitowsky, I.: Deterministic model of spin and statistics. Phys. Rev. D 27(10), 2316–2326 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  60. Pitowsky, I.: Quantum Probability-Quantum Logic. Springer, New York (1989)

    MATH  Google Scholar 

  61. Polubarinov, I.: Continuous Representation for Spin 1/2, Quantum Probability and Bell Paradox, p. E2-88-80. Communication of the Joint Institute for Nuclear Research, Dubna (1988)

    Google Scholar 

  62. Pusey, M., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nat. Phys. 8, 475–478 (2012)

    Article  Google Scholar 

  63. Rothman, T., Sudarshan, E.: Hidden variables or positive probabilities? Int. J. Theor. Phys. 40(8), 1525–1543 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. Scully, M., Walther, H., Schleich, W.: Feynman’s approach to negative probability in quantum mechanics. Phys. Rev. A 49(3), 1562–1566 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  65. Sorkin, R.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(33), 3119–3127 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Sorkin, R.: Quantum measure theory and its interpretation. In: Feng, D., Hu, B. (eds.) Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, pp. 229–251, International Press, Cambridge, MA (1997)

  67. Srinivasan, S.: Complex measureable processes and path integrals. In: Sridhar, R., Srinivasa Rao, K., Lakshminarayanan, V. (eds.) Selected Topics in Mathematical Physics: Professor R. Vasudevan Memorial Volume, pp. 13–27. Allied Publishers, New Delhi (1995)

    Google Scholar 

  68. Srinivasan, S.: Complex measure, coherent state and squeezed state representation. J. Phys. A 31, 4541–4553 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Srinivasan, S.: Quantum phenomena via complex measure: holomorphic extension. Fortschritte der Physik 54(7), 580–601 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Srinivasan, S., Sudarshan, E.: Complex measures and amplitudes, generalized stochastic processes and their application to quantum mechanics. J. Phys. A 27, 517–534 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  71. Sudarshan, E., Rothman, T.: A new interpretation of Bell’s inequalities. Int. J. Theor. Phys. 32(7), 1077–1086 (1993)

    Article  MathSciNet  Google Scholar 

  72. Suppes, P.: Logics appropriate to empirical theories. In Symposium on the Theory of Models. North Holland, Amsterdam (1965)

  73. Suppes, P.: The probablistic argument for a nonclassical logic of quantum mechanics. Philos. Sci. 33, 14–21 (1966)

    Article  MathSciNet  Google Scholar 

  74. Suppes, P., Zanotti, M.: Existence of hidden variables having only upper probabilities. Found. Phys. 21(2), 1479–1499 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  75. Surya, S., Wallden, P.: Quantum covers in quantum measure theory. Found. Phys. 40, 585–606 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. Van Wesep, R.: Hidden variables in quantum mechanics: generic models, set-theoretic forcing, and the appearance of probability. Ann. Phys. 321, 2453–2475 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. Van Wesep, R.: Hidden variables in quantum mechanics: noncontextual generic models. Ann. Phys. 321, 2476–2490 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. Wódkiewicz, K.: On the equivalence of nonlocality and nonpositivity of quasi-distributions in EPR correlations. Phys. Lett. A 121(1), 1–3 (1988)

    Article  Google Scholar 

  79. Youssef, S.: A reformulation of quantum mechanics. Mod. Phys. Lett. A 6(3), 225–235 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. Youssef, S.: Quantum mechanics as Bayesian complex probability theory. Mod. Phys. Lett. A 9(28), 2571–2586 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Youssef, S.: Is complex probability theory consistent with Bell’s theorem? Phys. Lett. A 204, 181–187 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. Youssef, S.: Quantum mechanics as an exotic probability theory. In: Hanson, K., Silver, R. (eds.) Maximum Entropy and Bayesian Methods, pp. 237–244. Kluwer, Dordrecht (1996)

    Chapter  Google Scholar 

Download references

Acknowledgements

SCF would like to thank audiences at Budapest, Tübingen, and Saig, Germany, especially Guido Bacciagalupi and Fay Dowker, for their comments. BHF would like to thank audiences at the Perimeter Institute and Chapman University. Both authors acknowledge the support of National Science Foundation Graduate Research Fellowships. In addition, the authors would like to thank an anonymous referee for helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Benjamin H. Feintzeig.

Appendix: Null Covers for Some Upper Probability Space Hidden Variable Models

Appendix: Null Covers for Some Upper Probability Space Hidden Variable Models

1.1 EPR Setup

In the Bell-EPR setup [12], Alice and Bob have two possible “yes/no” measurements they can make on the Bell state. Let \(A,A'\) and \(B,B'\), respectively, denote the events that these measurements return the value “yes”. The projection operators corresponding with each element of \(\{A,A'\}\) commutes with that of each element of \(\{B,B'\}\), but that of A (B) does not commute with that of \(A'\) (\(B'\)). Despite this non-commutativity, [40] propose to assign an upper probability to the algebra of events generated from these four that also reproduces the predictions of quantum mechanics. That is, they consider an upper probability space \((X,\Sigma ,\mu )\) and assign the following upper probabilities \(\mu (C)\) to the atomic events C of X, where for any \(S \in \Sigma \), \(S^c = X \backslash S\):

\(\cap \)

\(A \cap A'\)

\(A^c \cap A'\)

\(A \cap A'^c\)

\(A^c \cap A'^c\)

\(B \cap B'\)

0

\(\frac{1}{16}\)

\(\frac{1}{8}\)

\(\frac{1}{8} + \frac{\sqrt{3}}{8}\)

\(B^c \cap B'\)

\(\frac{1}{8}\)

\(\frac{1}{8} - \frac{\sqrt{3}}{8}\)

0

\(\frac{1}{16}\)

\(B \cap B'^c\)

\(\frac{1}{16}\)

0

\(\frac{1}{8} - \frac{\sqrt{3}}{8}\)

\(\frac{1}{8}\)

\(B^c \cap B'^c\)

\(\frac{1}{8} + \frac{\sqrt{3}}{8}\)

\(\frac{1}{8}\)

\(\frac{1}{16}\)

0

They also assign the following (partial) joint probabilities:

\(\cap \)

A

\(A^c\)

\(A'\)

\(A'^c\)

B

\(\frac{1}{4} - \frac{\sqrt{3}}{8}\)

\(\frac{1}{4} + \frac{\sqrt{3}}{8}\)

0

\(\frac{1}{2}\)

\(B^c\)

\(\frac{1}{4} + \frac{\sqrt{3}}{8}\)

\(\frac{1}{4} - \frac{\sqrt{3}}{8}\)

\(\frac{1}{2}\)

0

\(B'\)

\(\frac{1}{8}\)

\(\frac{3}{8}\)

\(\frac{1}{4} - \frac{\sqrt{3}}{8}\)

\(\frac{1}{4} + \frac{\sqrt{3}}{8}\)

\(B'^c\)

\(\frac{3}{8}\)

\(\frac{1}{8}\)

\(\frac{1}{4} + \frac{\sqrt{3}}{8}\)

\(\frac{1}{4} - \frac{\sqrt{3}}{8}\)

In addition, they set

$$\begin{aligned} \mu (A) = \mu (A^c) = \mu (B) = \mu (B^c) = \mu (A') = \mu (A'^c) = \mu (B') = \mu (B'^c) = 1/2. \end{aligned}$$

They verify that these assignments are consistent and reproduce the predictions of quantum mechanics for a certain Bell-EPR experiment, even though they do not uniquely specify an upper probability space, which requires \(2^{16} - 2 = 65,534\) assignments to make. Note as well that the symmetry of the EPR state requires that these assignments are invariant under the atomic complementation map.

Theorem 3 implies that the upper probability space hidden variable theory described above must contain a finite null cover if it is not inconsistent, since it satisfies WC. The following additional assignments witness that possibility:Footnote 15

$$\begin{aligned} \mu (A^c \cap ((A'^c \cap B) \cup (A' \cap B^c \cap B') )) = \mu (A \cap ((A' \cap B^c) \cup (A'^c \cap B \cap B'^c) )) = 0, \\ \mu (A^c \cap ((A' \cap B^c) \cup (A'^c \cap B \cap B'^c))) = \mu (A \cap ((A'^c \cap B) \cup (A' \cap B^c \cap B'))) = 0. \end{aligned}$$

Note that these assignments are also preserved under the atomic complementation map. To show this assignment is consistent with the previous assignments, we must check that \(\mu \) is still subadditive on disjoint sets. This is automatic for the decomposition of these newly assigned null sets. The only superset of any of these null sets that is assigned a measure so far is the whole space. So, for example:

$$\begin{aligned} 1&= \mu (X) \\&\le \mu (A^c \cap ((A'^c \cap B) \cup (A' \cap B^c \cap B') )) + \mu ((A^c \cap ((A'^c \cap B) \cup (A' \cap B^c \cap B') ))^c) \\&= \mu ((A^c \cap ((A'^c \cap B) \cup (A' \cap B^c \cap B') ))^c) \\&\le \sum _{\text {atomic } C \in X} \mu (C) - \mu (A^c \cap A'^c \cap B \cap B') - \mu (A^c \cap A'^c \cap B \cap B'^c)\\&\qquad - \mu (A^c \cap A' \cap B^c \cap B') \\&= 1 + \frac{3}{8} - \left( \frac{1}{8} + \frac{\sqrt{3}}{8}\right) - \frac{1}{8} - \left( \frac{1}{8} - \frac{\sqrt{3}}{8}\right) = 1, \end{aligned}$$

where in the last line we have used the calculation at the bottom of p. 97 of [40]. The remaining verifications are similar. Since subadditivity holds for this decomposition into atoms, it holds for decomposition into larger subsets as well.

1.2 GHZ Setup

For the GHZ setup [29, 30, 55], there are three “yes/no” observables, none of which pairwise commute. Let ABC denote the events that these observables return the value “yes.” In order to reproduce the predictions of quantum mechanics, [16] make the following assignments to the atoms of an upper probability space hidden variable model \((X,\Sigma ,\mu )\) for this setup, where again \(S^c = X \backslash S\) for any \(S \in \Sigma \):

$$\begin{aligned} \mu (A \cap B \cap C) = \mu (A^c \cap B^c \cap C^c) = 1, \\ \mu (A^c \cap B \cap C) = \mu (A \cap B^c \cap C) = \mu (A \cap B \cap C^c) = 0, \\ \mu (A \cap B^c \cap C^c) = \mu (A^c \cap B \cap C^c) = \mu (A^c \cap B^c \cap C) = 0. \end{aligned}$$

They also make the following (partial) joint assignments:

$$\begin{aligned} \mu (A) = \mu (B) = \mu (C) = 1, \\ \mu (A^c) = \mu (B^c) = \mu (C^c) = 0. \end{aligned}$$

Since their assignments are designed to satisfy WC and reproduce the quantum mechanical expectations, on pain of contradiction they must be compatible with the existence of a finite null cover by Theorem 3. Their assignments do not completely determine the values the upper probability measure takes on all \(2^8-2=254\) nontrivial measurable sets, but by fixing the following assignment the finite null cover is achieved:

$$\begin{aligned} \mu ((A \cap B \cap C) \cup (A^c \cap B^c \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c) ) = 0. \end{aligned}$$

To show this assignment is consistent with the previous assignments, we must check that the measure is still subadditive on disjoint sets. This is automatic for its decomposition into disjoint sets because it is null. The only superset of this null set assigned a measure so far is the whole space:

$$\begin{aligned} 1&= \mu (X) \\&\le \mu ((A \cap B \cap C) \cup (A^c \cap B^c \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c)) \\&\quad + \mu (((A \cap B \cap C) \cup (A^c \cap B^c \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c))^c) \\&= \mu (((A \cap B \cap C) \cup (A^c \cap B^c \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c))^c). \end{aligned}$$

Hence

$$\begin{aligned} 1&= \mu (((A \cap B \cap C) \cup (A^c \cap B^c \cap C) \cup (A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c))^c) \\&= \mu ((A^c \cap B^c \cap C^c) \cup (A \cap B \cap C^c) \cup (A^c \cap B \cap C) \cup (A \cap B^c \cap C)) \\&\le \mu (A^c \cap B^c \cap C^c) + \mu (A \cap B \cap C^c) + \mu (A^c \cap B \cap C) + \mu (A \cap B^c \cap C) \\&= 1 + 0 + 0+ 0 =1. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feintzeig, B.H., Fletcher, S.C. On Noncontextual, Non-Kolmogorovian Hidden Variable Theories. Found Phys 47, 294–315 (2017). https://doi.org/10.1007/s10701-017-0061-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-017-0061-z

Keywords

Navigation