Statistical Science

Statistics, Causality and Bell’s Theorem

Richard D. Gill

Full-text: Open access


Bell’s [Physics 1 (1964) 195–200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell’s inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804–1807] 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.

Article information

Statist. Sci., Volume 29, Number 4 (2014), 512-528.

First available in Project Euclid: 15 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Counterfactuals Bell inequality CHSH inequality Tsirelson inequality Bell’s theorem Bell experiment Bell test loophole nonlocality local hidden variables quantum Randi challenge


Gill, Richard D. Statistics, Causality and Bell’s Theorem. Statist. Sci. 29 (2014), no. 4, 512--528. doi:10.1214/14-STS490.

Export citation


  • 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.
  • Aspect, A., Dalibard, J. and Roger, G. (1982). Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49 1804–1807.
  • Bedingham, D. J. (2011). Relativistic state reduction dynamics. Found. Phys. 41 686–704.
  • 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.
  • Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics 1 195–200.
  • Bohm, D. (1951). Quantum Theory. Prentice-Hall, Englewood Cliffs, NJ.
  • 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.
  • 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.
  • Cirel’son, B. S. (1980). Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4 93–100.
  • 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.
  • Einstein, A., Podolsky, B. and Rosen, N. (1935). Can quantum-mechanical description of reality be considered complete? Phys. Rev. 47 777–780.
  • 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.
  • Gill, R. D. (2009). Schrödinger’s cat meets Occam’s razor. Preprint, available at arXiv:0905.2723.
  • 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.
  • 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.
  • 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.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • Larsson, J.- Å. (1998). Bell’s inequality and detector inefficiency. Phys. Rev. A (3) 57 3304–3308.
  • Larsson, J.- Å. (1999). Modeling the singlet state with local variables. Phys. Lett. A 256 245–252.
  • Larsson, J.- Å. and Gill, R. D. (2004). Bell’s inequality and the coincidence-time loophole. Europhysics Letters 67 707–713.
  • 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.
  • Masanes, Ll., Acin, A. and Gisin, N. (2006). General properties of nonsignaling theories. Phys. Rev. A (3) 73 012112, 9 pp.
  • Merali, Z. (2011). Quantum mechanics braces for the ultimate test. Science 331 1380–1382.
  • Pearle, P. (1997). True collapse and false collapse. In Quantum Classical Correspondence (Philadelphia, PA, 1994) 69–86. Int. Press, Cambridge, MA.
  • Pearle, P. (2012). Collapse miscellany. Available at arXiv:1209.5082.
  • 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 biostats.bepress:cobra/83.
  • 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.
  • Spelke, E. S. and Kinzler, K. D. (2007). Core knowledge. Developmental Science 10 89–96.
  • van Dam, W., Gill, R. D. and Grünwald, P. D. (2005). The statistical strength of nonlocality proofs. IEEE Trans. Inform. Theory 51 2812–2835.
  • 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 arXiv:1106.1636 [cs.AI].
  • Vongehr, S. (2012). Quantum Randi challenge. Available at arXiv:1207.5294.
  • Vongehr, S. (2013). Exploring inequality violations by classical hidden variables numerically. Ann. Physics 339 81–88.
  • 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.
  • 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.
  • 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.