# Classical causal models cannot faithfully explain Bell nonlocality or Kochen-Specker contextuality in arbitrary scenarios

J. C. Pearl and E. G. Cavalcanti

Centre for Quantum Dynamics, Griffith University, Gold Coast, QLD 4222, Australia

### Abstract

In a recent work, it was shown by one of us (EGC) that Bell-Kochen-Specker inequality violations in phenomena satisfying the no-disturbance condition (a generalisation of the no-signalling condition) cannot in general be explained with a faithful classical causal model---that is, a classical causal model that satisfies the assumption of $\textit{no fine-tuning}$. The proof of that claim however was restricted to Bell scenarios involving 2 parties or Kochen-Specker-contextuality scenarios involving 2 measurements per context. Here we show that the result holds in the general case of arbitrary numbers of parties or measurements per context; it is not an artefact of the simplest scenarios. This result unifies, in full generality, Bell nonlocality and Kochen-Specker contextuality as violations of a fundamental principle of classical causality. We identify, however, an implicit assumption in the former proof, making it explicit here: that certain operational symmetries of the phenomenon are reflected in the model, rather than requiring fine-tuned choices of model parameters. This clarifies a subtle but important distinction between Bell nonlocality and Kochen-Specker contextuality.

Bell nonlocality and Kochen-Specker contextuality are two classes of quantum phenomena where the departure from classicality becomes most striking. They have also been identified as key resources in powering quantum advantage over classical information processing. Nevertheless, many questions remain about their meaning and the precise relationship between these phenomena. On one hand, their formal similarity has led many to think of Bell nonlocality as nothing but a special case of Kochen-Specker contextuality. On the other hand, their conceptual basis is very distinct: whereas Bell nonlocality is now well understood as the quantum violation of classical causal constraints applied to a relativistic causal structure, K-S contextuality has not been traditionally understood in causal terms at all.

In this paper, based on a causal framework for contextuality introduced by EGC in Phys. Rev. X 8, 021018 (2018), we prove in full generality that both Bell and KS contextuality can be understood as the impossibility to simulate quantum correlations with any classical causal model satisfying a principle of no-fine-tuning – regardless of any assumptions about the causal structure. We conjecture that this opens a path to better understand the resource overheads required for the classical simulations of these key quantum phenomena. On the other hand, we identify the need for a slightly stronger notion of no-fine-tuning for KS contextuality than for Bell scenarios, requiring that certain operational symmetries in the phenomenon be reflected in the model. This clarifies a subtle but important distinction between the two types of scenarios.

### ► References

[1] J. S. Bell, Physics Physique Fizika 1, 195 (1964).
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[2] S. Kochen and E. P. Specker, Journal of Mathematics and Mechanics 17, 59 (1967).
http:/​/​www.jstor.org/​stable/​24902153

[3] H. M. Wiseman and E. G. Cavalcanti, Causarum Investigatio and the Two Bell's Theorems of John Bell,'' in Quantum [Un]Speakables II: Half a Century of Bell's Theorem, edited by R. Bertlmann and A. Zeilinger (Springer International Publishing, Cham, 2017) pp. 119–142.
https:/​/​doi.org/​10.1007/​978-3-319-38987-5_6

[4] R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.052108

[5] Č. Brukner, M. Żukowski, J.-W. Pan, and A. Zeilinger, Phys. Rev. Lett. 92, 127901 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.92.127901

[6] A. K. Ekert, Quantum Cryptography and Bell's Theorem,'' in Quantum Measurements in Optics, edited by P. Tombesi and D. F. Walls (Springer US, Boston, MA, 1992) pp. 413–418.
https:/​/​doi.org/​10.1007/​978-1-4615-3386-3_34

[7] G. Brassard, R. Cleve, and A. Tapp, Phys. Rev. Lett. 83, 1874 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.1874

[8] B. F. Toner and D. Bacon, Phys. Rev. Lett. 91, 187904 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.187904

[9] J. Anders and D. E. Browne, Phys. Rev. Lett. 102, 050502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.050502

[10] R. Raussendorf, Phys. Rev. A 88, 022322 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.022322

[11] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Nature 510, 351 (2014).
https:/​/​doi.org/​10.1038/​nature13460

[12] S. Mansfield and E. Kashefi, Phys. Rev. Lett. 121, 230401 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.230401

[13] J. Pearl, Causality: Models, Reasoning and Inference (Cambridge University Press, 2000).

[14] C. J. Wood and R. W. Spekkens, New Journal of Physics 17, 033002 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​3/​033002

[15] E. G. Cavalcanti, Physical Review X 8, 021018 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.021018

[16] H. Reichenbach, The Direction of Time (University of Los Angeles Press, Berkeley, 1956).

[17] E. G. Cavalcanti and R. Lal, Journal of Physics A: Mathematical and Theoretical 47, 424018 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424018

[18] J. Henson, R. Lal, and M. F. Pusey, New Journal of Physics 16, 113043 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​11/​113043

[19] J. Pienaar and Č. Brukner, New Journal of Physics 17, 073020 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​7/​073020

[20] R. Chaves, C. Majenz, and D. Gross, Nature Communications 6, 5766 (2015).
https:/​/​doi.org/​10.1038/​ncomms6766

[21] F. Costa and S. Shrapnel, New Journal of Physics 18, 063032 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​6/​063032

[22] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, Physical Review X 7, 031021 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.031021

[23] J. Barrett, R. Lorenz, and O. Oreshkov, (2019), arXiv:1906.10726 [quant-ph].
arXiv:1906.10726

[24] L. Catani and M. Leifer, (2020), arXiv:2003.10050 [quant-ph].
arXiv:2003.10050

[25] S. Abramsky and A. Brandenburger, New Journal of Physics 13, 113036 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​11/​113036

[26] A. Fine, Phys. Rev. Lett. 48, 291 (1982).
https:/​/​doi.org/​10.1103/​PhysRevLett.48.291

[27] A. A. Klyachko, M. A. Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett. 101, 020403 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.020403

[28] A. Cabello, Phys. Rev. Lett. 101, 210401 (2008a).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.210401

[29] S. Abramsky and L. Hardy, Physical Review A 85, 062114 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.062114

[30] I. Pitowsky, Quantum Probability - Quantum Logic, Lecture Notes in Physics (Springer-Verlag, 1989).
https:/​/​doi.org/​10.1007/​BFb0021186

[31] A. Peres, Foundations of Physics 29, 589 (1999).
https:/​/​doi.org/​10.1023/​A:1018816310000

[32] M. Jones, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, 20190133 (2019).
https:/​/​doi.org/​10.1098/​rsta.2019.0133

[33] R. W. Spekkens, Foundations of Physics 44, 1125 (2014).
https:/​/​doi.org/​10.1007/​s10701-014-9833-x

[34] R. W. Spekkens, (2019), arXiv:1909.04628 [physics.hist-ph].
arXiv:1909.04628

[35] D. Bohm, Physical Review 85, 166 (1952).
https:/​/​doi.org/​10.1103/​PhysRev.85.166

[36] H. Price, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 39, 752 (2008).
https:/​/​doi.org/​10.1016/​j.shpsb.2008.05.006

[37] G. 't Hooft, (2007), arXiv:quant-ph/​0701097 [quant-ph].
arXiv:quant-ph/0701097

[38] Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506, 1 (2011).
https:/​/​doi.org/​10.1016/​j.physrep.2011.05.001

[39] A. Peres, Journal of Physics A: Mathematical and General 24, L175 (1991).
https:/​/​doi.org/​10.1088/​0305-4470/​24/​4/​003

[40] N. D. Mermin, Reviews of Modern Physics 65, 803 (1993).
https:/​/​doi.org/​10.1103/​RevModPhys.65.803

[41] A. Krishna, R. W. Spekkens, and E. Wolfe, New Journal of Physics 19, 1 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa9168

[42] A. Cabello, Phys. Rev. Lett. 101, 210401 (2008b).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.210401

[43] E. N. Dzhafarov and J. V. Kujala, Fortschritte der Physik 65, 1600040 (2017).
https:/​/​doi.org/​10.1002/​prop.201600040

[44] Physical Review Letters 123, 140401 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.140401

[45] T. Fritz, New Journal of Physics 14, 103001 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​10/​103001

### Cited by

[1] Zheng-Hao Liu, Qiang Li, Bi-Heng Liu, Yun-Feng Huang, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, "Twenty years of quantum contextuality at USTC", JUSTC 52 10, 1 (2022).

[2] Ehtibar N. Dzhafarov, "Assumption-Free Derivation of the Bell-Type Criteria of Contextuality/Nonlocality", Entropy 23 11, 1543 (2021).

[3] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Quantum Causal Models", arXiv:1906.10726, (2019).

[4] Eric G. Cavalcanti and Howard M. Wiseman, "Implications of Local Friendliness Violation for Quantum Causality", Entropy 23 8, 925 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2023-03-31 03:19:49) and SAO/NASA ADS (last updated successfully 2023-03-31 03:19:50). The list may be incomplete as not all publishers provide suitable and complete citation data.