zbMATH — the first resource for mathematics

Statistics, causality and Bell’s theorem. (English) Zbl 1331.81048
Summary: J. L. Bell’s theorem [“On the Einstein Podolsky Rosen paradox”, Physics 1, 195–200 (1964)] is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell’s inequality in experiments such as that of A. Aspect et al. [“Experimental test of Bell’s inequalitiies using time-varying analyzers”, Phys. Rev. Lett 49, 1804–1807 (1982)] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell’s theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell’s inequality and thereby also of Bell’s theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom.
Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed.
The paper argues that Bell’s theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.

81P15 Quantum measurement theory, state operations, state preparations
62P35 Applications of statistics to physics
62A01 Foundations and philosophical topics in statistics
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
Full Text: DOI Euclid
[1] Almeida, M. L., Bancal, J.-D., Brunner, N., Acin, A., Gisin, N. and Pironio, S. (2010). Guess your neighbour’s input: A multipartite non-local game with no quantum advantage. Phys. Rev. Lett. 104 230404, 4 pp.
[2] Aspect, A., Dalibard, J. and Roger, G. (1982). Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49 1804-1807.
[3] Bedingham, D. J. (2011). Relativistic state reduction dynamics. Found. Phys. 41 686-704. · Zbl 1214.81010
[4] Belavkin, V. P. (2000). Dynamical solution to the quantum measurement problem, causality, and paradoxes of the quantum century. Open Syst. Inf. Dyn. 7 101-129. · Zbl 1063.81510
[5] Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics 1 195-200.
[6] Bohm, D. (1951). Quantum Theory . Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0048.21802
[7] Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. and Wehner, S. (2014). Bell nonlocality. Rev. Modern Phys. 86 419; Erratum Rev. Modern Phys. 86 839.
[8] Christensen, B. G., McCusker, K. T., Altepeter, J., Calkins, B., Gerrits, T., Lita, A., Miller, A., Shalm, L. K., Zhang, Y., Nam, S. W., Brunner, N., Lim, C. C. W., Gisin, N. and Kwiat, P. G. (2013). Detection-loophole-free test of quantum nonlocality, and applications. Phys. Rev. Lett. 111 130406.
[9] Cirel’son, B. S. (1980). Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4 93-100.
[10] Clauser, J. F., Horne, M. A., Shimony, A. and Holt, R. A. (1969). Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 49 1804-1806. · Zbl 1371.81014
[11] Einstein, A., Podolsky, B. and Rosen, N. (1935). Can quantum-mechanical description of reality be considered complete? Phys. Rev. 47 777-780. · Zbl 0012.04201
[12] Gallicchio, J., Friedman, A. S. and Kaiser, D. I. (2014). Testing Bell’s inequality with cosmic photons: Closing the setting-independence loophole. Phys. Rev. Lett. 112 110405.
[13] Gill, R. D. (2009). Schrödinger’s cat meets Occam’s razor. Preprint, available at . arXiv:0905.2723
[14] Gill, R. D. (2003). Time, finite statistics, and Bell’s fifth position. In Foundations of Probability and Physics , 2 ( Växjö , 2002). Math. Model. Phys. Eng. Cogn. Sci. 5 179-206. Växjö Univ. Press, Växjö.
[15] Gill, R. D. (2007). Better Bell inequalities (passion at a distance). In Asymptotics : Particles , Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes-Monograph Series 55 135-148. IMS, Beachwood, OH. · Zbl 1180.81017
[16] Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S. W., Ursin, R. and Zeilinger, A. (2013). Bell violation using entangled photons without the fair-sampling assumption. Nature 497 227-230.
[17] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. · Zbl 0127.10602
[18] Larsson, J.- Å. (1998). Bell’s inequality and detector inefficiency. Phys. Rev. A (3) 57 3304-3308.
[19] Larsson, J.- Å. (1999). Modeling the singlet state with local variables. Phys. Lett. A 256 245-252.
[20] Larsson, J.- Å. and Gill, R. D. (2004). Bell’s inequality and the coincidence-time loophole. Europhysics Letters 67 707-713.
[21] Larsson, J.- Å., Giustina, M., Kofler, J., Wittman, B., Ursin, R. and Ramelow, S. (2013:) Bell violation with entangled photons, free of the coincidence-time loophole. Available at . arXiv:1309.0712
[22] Masanes, Ll., Acin, A. and Gisin, N. (2006). General properties of nonsignaling theories. Phys. Rev. A (3) 73 012112, 9 pp.
[23] Merali, Z. (2011). Quantum mechanics braces for the ultimate test. Science 331 1380-1382. · Zbl 1226.81057
[24] Pearle, P. (1997). True collapse and false collapse. In Quantum Classical Correspondence ( Philadelphia , PA , 1994) 69-86. Int. Press, Cambridge, MA. · Zbl 0916.03006
[25] Pearle, P. (2012). Collapse miscellany. Available at . arXiv:1209.5082 · Zbl 1243.83084
[26] Robins, J. M., VanderWeele, T. J. and Gill, R. D. (2015). A proof of Bell’s inequality in quantum mechanics using causal interactions. Scand. J. Stat. To appear, available at . · Zbl 1368.81045
[27] Santos, E. (2005). Bell’s theorem and the experiments: Increasing empirical support for local realism? Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 36 544-565. · Zbl 1222.81117
[28] Spelke, E. S. and Kinzler, K. D. (2007). Core knowledge. Developmental Science 10 89-96.
[29] van Dam, W., Gill, R. D. and Grünwald, P. D. (2005). The statistical strength of nonlocality proofs. IEEE Trans. Inform. Theory 51 2812-2835. · Zbl 1286.81017
[30] Ver Steeg, G. and Galstyan, A. (2011). A sequence of relaxations constraining hidden variable models. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence ( UAI 2011). Available at [cs.AI]. arXiv:1106.1636
[31] Vongehr, S. (2012). Quantum Randi challenge. Available at . arXiv:1207.5294
[32] Vongehr, S. (2013). Exploring inequality violations by classical hidden variables numerically. Ann. Physics 339 81-88. · Zbl 1343.81010
[33] Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. and Zeilinger, A. (1998). Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81 5039-5043. · Zbl 0947.81013
[34] Zohren, S. and Gill, R. D. (2008). On the maximal violation of the CGLMP inequality for infinite dimensional states. Phys. Rev. Lett. 100 120406, 4 pp.
[35] Zohren, S., Reska, P., Gill, R. D. and Westra, W. (2010). A tight Tsirelson inequality for infinitely many outcomes. Europhysics Letters 90 10002, 4 pp.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.