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.
Similar content being viewed by others
Notes
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.
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].
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.
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.
See Feintzeig [20] for an extensive discussion of generalized probability spaces and a slight variation on the main theorem applied to them.
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.
Srinivasan [69] also considers quaternionic probability spaces, which he sometimes calls extended measure spaces. The same results below apply to these spaces as well.
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.
One can also generalize to even weaker additivity constraints, to which our results apply equally.
See also [20] for a discussion of Dutch Books in the context of generalized probability spaces.
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
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)
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)
Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982)
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)
Bassi, A., Ghirardi, G.: Can the decoherent histories description of reality be considered satisfactory? Phys. Lett. A 257, 247–263 (1999)
Bassi, A., Ghirardi, G.: Decoherent histories and realism. J. Stat. Phys. 98, 457–494 (2000)
Bell, J.: On the Einstein Podolsky Rosen paradox. Physics 1(3), 195–200 (1964)
Bell, J.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447–451 (1966)
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)
Bradley, S.: Imprecise probabilities. In: Zalta, E. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Stanford (2015)
Clauser, J., Horne, M.: Experimental consequences of objective local theories. Phys. Rev. D 10, 526–535 (1974)
Clauser, J., Horne, M., Shimony, A., Holt, R.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969)
Cox, R.: Probability, frequency and reasonable expectation. Am. J. Phys. 14, 1–10 (1946)
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)
de Barros, J.A., Suppes, P.: Some conceptual issues involving probability in quantum mechanics (2000). arXiv:quant-ph/0001017
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)
Dirac, P.: Bakerian lecture: the physical interpretation of quantum mechanics. Proc. R. Soc. Lond. A 180, 1–40 (1942)
Dowker, F., Ghazi-Tabatabai, Y.: The Kochen-Specker theorem revisited in quantum measure theory. J. Phys. A 41, 105301 (2008)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)
Feintzeig, B.: Hidden variables and incompatible observables in quantum mechanics. Br. J. Philos. Sci. 66(4), 905–927 (2015)
Feynman, R.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367–387 (1948)
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)
Fine, A.: Antinomies of entanglement: the puzzling case of the tangled statistics. J. Philos. 79(12), 733–747 (1982)
Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48(5), 291–294 (1982)
Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23(7), 1306–1310 (1982)
Gell-Mann, M., Hartle, J.: Decoherent histories quantum mechanics with one real fine-grained history. Phys. Rev. A 85, 062120 (2012)
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)
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)
Greenberger, D., Horne, M., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990)
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)
Griffiths, R.: Consistent histories, quantum truth functionals, and hidden variables. Phys. Lett. A 265, 12–19 (2000)
Griffiths, R.: Consistent quantum realism: A reply to Bassi and Ghirardi. J. Stat. Phys. 99, 1409–1425 (2000)
Gudder, S.: Quantum Probability. Academic Press, San Diego, CA (1988)
Gudder, S.: Finite quantum measure spaces. Am. Math. Mon. 117(6), 512–527 (2010)
Gudder, S.: Quantum measure theory. Math. Slovaca 60(5), 681–700 (2010)
Halliwell, J., Yearsley, J.: Negative probabilities, Fine’s theorem, and linear positivity. Phys. Rev. A 87, 022114 (2013)
Han, Y.D., Hwang, W., Koh, I.: Explicit solutions for negative probability measures for all entangled states. Phys. Lett. A 221, 283–286 (1996)
Hartle, J.: Linear positivity and virtual probability. Phys. Rev. A 70, 022104 (2004)
Hartle, J.: Quantum mechanics with extended probabilities. Phys. Rev. A 78, 012108 (2008)
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)
Home, D., Lepore, V., Selleri, F.: Local realistic models and non-physical probabilities. Phys. Lett. A 158, 357–360 (1991)
Home, D., Selleri, F.: Bell’s theorem and the EPR paradox. Revista del Nuovo Cimento 14(9), 1–95 (1991)
Ivanović, I.: On complex Bell’s inequality. Lettere al Nuovo Cimento 22(1), 14–16 (1978)
Khrennikov, A.: p-adic probability distributions of hidden variables. Physica A 215(4), 577–587 (1995)
Khrennikov, A.: p-Adic probability interpretation of Bell’s inequality. Phys. Lett. A 200, 219–223 (1995)
Khrennikov, A.: Interpretations of Probability, 2nd edn. de Gruyter, Berlin (2009)
Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
Krantz, D., Luce, D., Suppes, P., Tversky, A.: Foundations of Measurement, vol. I. Dover, Mineola, NY (1971)
Kronz, F.: A nonmonotonic theory of probability for spin-1/2 systems. Int. J. Theor. Phys. 44(11), 1963–1976 (2005)
Kronz, F.: Non-monotonic probability theory and photon polarization. J. Philos. Logic 36, 446–472 (2007)
Kronz, F.: Non-monotonic probability theory for n-state quantum systems. Stud. Hist. Philos. Mod. Phys. 39, 259–272 (2008)
Kronz, F.: Actual and virtual events in the quantum domain. Ontol. Stud. 9, 209–220 (2009)
Malament, D.: Notes on Bell’s theorem (2012). http://www.socsci.uci.edu/~dmalamen/courses/prob-determ/PDnotesBell.pdf
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)
Mermin, N.: Quantum mysteries revisited. Am. J. Phys. 58, 731–733 (1990)
Miller, D.: Realism and time symmetry in quantum mechanics. Phys. Lett. A 222, 31–36 (1996)
Mückenheim, W.: A resolution of the Einstein-Podolsky-Rosen paradox. Lettere al Nuovo Cimento 35(9), 300–304 (1982)
Mückenheim, W.: A review of extended probabilities. Phys. Rep. 133(6), 337–401 (1986)
Pitowsky, I.: Deterministic model of spin and statistics. Phys. Rev. D 27(10), 2316–2326 (1983)
Pitowsky, I.: Quantum Probability-Quantum Logic. Springer, New York (1989)
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)
Pusey, M., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nat. Phys. 8, 475–478 (2012)
Rothman, T., Sudarshan, E.: Hidden variables or positive probabilities? Int. J. Theor. Phys. 40(8), 1525–1543 (2001)
Scully, M., Walther, H., Schleich, W.: Feynman’s approach to negative probability in quantum mechanics. Phys. Rev. A 49(3), 1562–1566 (1994)
Sorkin, R.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(33), 3119–3127 (1994)
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)
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)
Srinivasan, S.: Complex measure, coherent state and squeezed state representation. J. Phys. A 31, 4541–4553 (1998)
Srinivasan, S.: Quantum phenomena via complex measure: holomorphic extension. Fortschritte der Physik 54(7), 580–601 (2006)
Srinivasan, S., Sudarshan, E.: Complex measures and amplitudes, generalized stochastic processes and their application to quantum mechanics. J. Phys. A 27, 517–534 (1994)
Sudarshan, E., Rothman, T.: A new interpretation of Bell’s inequalities. Int. J. Theor. Phys. 32(7), 1077–1086 (1993)
Suppes, P.: Logics appropriate to empirical theories. In Symposium on the Theory of Models. North Holland, Amsterdam (1965)
Suppes, P.: The probablistic argument for a nonclassical logic of quantum mechanics. Philos. Sci. 33, 14–21 (1966)
Suppes, P., Zanotti, M.: Existence of hidden variables having only upper probabilities. Found. Phys. 21(2), 1479–1499 (1991)
Surya, S., Wallden, P.: Quantum covers in quantum measure theory. Found. Phys. 40, 585–606 (2010)
Van Wesep, R.: Hidden variables in quantum mechanics: generic models, set-theoretic forcing, and the appearance of probability. Ann. Phys. 321, 2453–2475 (2006)
Van Wesep, R.: Hidden variables in quantum mechanics: noncontextual generic models. Ann. Phys. 321, 2476–2490 (2006)
Wódkiewicz, K.: On the equivalence of nonlocality and nonpositivity of quasi-distributions in EPR correlations. Phys. Lett. A 121(1), 1–3 (1988)
Youssef, S.: A reformulation of quantum mechanics. Mod. Phys. Lett. A 6(3), 225–235 (1991)
Youssef, S.: Quantum mechanics as Bayesian complex probability theory. Mod. Phys. Lett. A 9(28), 2571–2586 (1994)
Youssef, S.: Is complex probability theory consistent with Bell’s theorem? Phys. Lett. A 204, 181–187 (1995)
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)
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
Corresponding author
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
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
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:
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 A, B, C 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 \(S \in \Sigma \):
They also make the following (partial) joint assignments:
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:
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:
Hence
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-017-0061-z