The Significance of Measurement Independence for Bell Inequalities and Locality

  • Michael J. W. HallEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 183)


A local and deterministic model of quantum correlations is always possible, as shown explicitly by Brans in 1988: one simply needs the physical systems being measured to have a suitable statistical correlation with the physical systems performing the measurement, via some common cause. Hence, to derive no-go results such as Bell inequalities, an assumption of measurement independence is crucial. It is a surprisingly strong assumption—less than 1 / 15 bits of prior correlation suffice for a local model of the singlet state of two qubits—with ramifications for the security of quantum communication protocols. Indeed, without this assumption, any statistical correlations whatsoever—even those which appear to allow explicit superluminal signalling—have a corresponding local deterministic model. It is argued that ‘quantum nonlocality’ is bad terminology, and that measurement independence does not equate to ‘experimental free will’. Brans’ 1988 model is extended to show that no more than \(2\log d\) bits of prior correlation are required for a local deterministic model of the correlations between any two d-dimensional quantum systems.


Measurement Dependence Singlet State Quantum Correlation Bell Inequality Statistical Completeness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I thank Cyril Branciard, Antonio Di Lorenzo, Jason Gallicchio, Nicolas Gisin, Valerio Scarani, Joan Vaccaro, Mark Wilde and Howard Wiseman for various stimulating discussions on this topic over the past few years. This work was supported by the ARC Centre of Excellence CE110001027.


  1. 1.
    W. Heisenberg, Physics and Beyond (Harper and Row, New York, 1971), p. 206Google Scholar
  2. 2.
    J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)Google Scholar
  3. 3.
    M.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969)ADSCrossRefGoogle Scholar
  4. 4.
    J.S. Bell, The theory of local beables. Preprint CERN-TH 2053/75, reprinted in. Dialectica 39, 86–96 (1985)Google Scholar
  5. 5.
    J.S. Bell, A. Shimony, M.A. Horne, J.F. Clauser, An exchange on local beables. Dialectica 39, 85–110 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    C. Brans, Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theoret. Phys. 27, 219–226 (1988)CrossRefGoogle Scholar
  8. 8.
    J. Degorre, S. Laplante, J. Roland, Simulating quantum correlations as a distributed sampling problem. Phys. Rev. A 72, 062314 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    M.J.W. Hall, Local deterministic model of singlet state correlations based on relaxing measurement independence. Phys. Rev. Lett. 105, 250404 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    J. Barrett, N. Gisin, How much measurement independence is needed in order to demonstrate nonlocality? Phys. Rev. Lett. 106, 100406 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    M.J.W. Hall, Relaxed Bell inequalities and Kochen-Specker theorems. Phys. Rev. A 84, 022102 (2011)ADSCrossRefGoogle Scholar
  12. 12.
    M. Banik, M.D.R. Gazi, S. Das, A. Rai, S. Kunkri, Optimal free will on one side in reproducing the singlet correlation. J. Phys. A 45, 205301 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    D.E. Koh, M.J.W. Hall, J.E. Pope, C. Marletto, A. Kay, V. Scarani, A.E. Ekert, Effects of reduced measurement independence on Bell-based randomness expansion. Phys. Rev. Lett. 109, 160404 (2012)Google Scholar
  14. 14.
    L.P. Thinh, L. Sheridan, V. Scarani, Bell tests with min-entropy sources. Phys. Rev. A 87, 062121 (2013)ADSCrossRefGoogle Scholar
  15. 15.
    J.E. Pope, A. Kay, Limited measurement dependence in multiple runs of a Bell test. Phys. Rev. A 88, 032110 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    G. Pütz, D. Rosset, T.J. Barnea, Y.-C. Liang, N. Gisin, Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. Phys. Rev. Lett. 113, 190404 (2014)CrossRefGoogle Scholar
  17. 17.
    R. Chaves, R. Kueng, J.B. Brask, D. Gross, Unifying framework for relaxations of the causal assumptions in Bell’s theorem. Phys. Rev. Lett. 114, 140403 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    J. Gallicchio, A.S. Friedman, D.I. Kaiser, Testing Bell’s inequality with cosmic photons: closing the setting-independence loophole. Phys. Rev. Lett. 112, 110405 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    D. Aktas, S. Tanzilli, A. Martin, G. Pütz, R. Thew, N. Gisin, Demonstration of quantum nonlocality in the presence of measurement dependence. Phys. Rev. Lett. 114, 220404 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    J.P. Jarrett, On the physical significance of the locality conditions in the Bell arguments. Noûs 18, 569–589 (1984)MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. Fine, Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23, 1306–1310 (1982)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    H.M. Wiseman, E.C.G. Cavalcanti, Causarum Investigatio and the two Bell’s theorems of John Bell (2015). Eprint: arXiv:1503.06413 [quant-ph]
  23. 23.
    A. Aspect, G. Dalibard, G. Roger, Experimental test of Bell inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    B. Hensen, H. Bernien, A.E. Dréau, N. Reiserer et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526, 682–686 (2015)ADSCrossRefGoogle Scholar
  25. 25.
    A. Fine, Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291–295 (1982)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Garg, N.D. Mermin, Farkas’s lemma and the nature of reality: statistical implications of quantum correlations. Found. Phys. 14, 1–39 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    I. Pitowsky, George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci. 45, 95–125 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    M.J.W. Hall, Imprecise measurements and non-locality in quantum mechanics. Phys. Lett. A 125, 89–91 (1987)ADSCrossRefGoogle Scholar
  29. 29.
    N.D. Mermin, What do these correlations know about reality? Nonlocality and the absurd. Found. Phys. 29, 571–587 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Kent, Locality and reality revisited, in Quantum Locality and Modality, ed. by T. Placek, J. Butterfield (Kluwer, Dordrecht, 2002), pp. 163–171CrossRefGoogle Scholar
  31. 31.
    M. Żukowski, C. Brukner, Quantum nonlocality–it ain’t necessarily so. J. Phys. A 47, 424009 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    M.J.W. Hall, Comment on ‘Non-realism: deep thought or soft option?’, by N. Gisin. Eprint: arXiv:0909.0015 [quant-ph] (2009)
  33. 33.
    R.E. Blahut, Principles and Practice of Information Theory (Addison Wesley, New York, 1987)zbMATHGoogle Scholar
  34. 34.
    G. ’t Hooft, Models on the boundary between classical and quantum mechanics. Phil. Trans. R. Soc. A 373, 20140236 (2015)Google Scholar
  35. 35.
    A. Di Lorenzo, Private communication (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith UniversityBrisbaneAustralia

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