- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Clauser-Horne-Shimony-Holt game, Tsirelson's bound, and causal locality

Jacob A. Barandes*

Mahmudul Hasan† and David Kagan‡

  • Departments of Philosophy and Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA

  • *Contact author: jacob_barandes@harvard.edu
  • †Contact author: mahmudul.hasan@umassd.edu
  • ‡Contact author: dkagan@umassd.edu
PDF

Phys. Rev. A 114, 022208 – Published 17 August, 2026

DOI: https://doi.org/10.1103/xfpf-734d

Abstract

Starting with heuristic assumptions about locality and causality, Bell established a numerical upper bound on statistical correlations between spatially separated systems and showed that quantum systems could violate it. This upper bound, also known as the Bell or Clauser-Horne-Shimony-Holt (CHSH) inequality, can be derived from an experimental setup known as the CHSH game. Tsirelson used the CHSH formulation to derive a higher, maximal bound on quantum correlations. Later, Popescu and Rohrlich (PR) studied correlations beyond quantum theory that were constrained only by the no-signaling principle, a condition that disallows superluminal communication between observers. They showed that such correlations could theoretically violate the Tsirelson bound. Is there some conceptually simple or transparent principle that explains the large gap between the Tsirelson bound and the PR bound? Or do we have to appeal to the technical axiomatic structure of quantum theory? In this paper, we reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a proof of the Tsirelson bound. In particular, we show that, unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (29)

  1. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  2. B. Tsirelson, Quantum information processing lecture notes, 2. Quanta in networks (unpublished), https://www.webcitation.org/5fl2WZOMI.
  3. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  4. J. S. Bell, The theory of local beables, CERN Technical Report No. 2053-CERN, 1975, https://cds.cern.ch/record/980036.
  5. J. S. Bell, La Nouvelle Cuisine (Cambridge University Press, Cambridge, 2004).
  6. B. S. Cirel'son, Quantum generalizations of Bell's inequality, Lett. Math. Phys. 4, 93 (1980).
  7. G. C. Ghirardi, A. Rimini, and T. Weber, A general argument against superluminal transmission through the quantum mechanical measurement process, Lett. Nuovo Cimento 27, 293 (1980).
  8. T. F. Jordan, Quantum correlations do not transmit signals, Phys. Lett. A 94, 264 (1983).
  9. S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Found. Phys. 24, 379 (1994).
  10. P. Rastall, Locality, Bell's theorem, and quantum mechanics, Found. Phys. 15, 963 (1985).
  11. A. Cabello, Simple explanation of the quantum violation of a fundamental inequality, Phys. Rev. Lett. 110, 060402 (2013).
  12. M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, Information causality as a physical principle, Nature (London) 461, 1101 (2009).
  13. N. Miklin and M. Pawłowski, Information causality without concatenation, Phys. Rev. Lett. 126, 220403 (2021).
  14. L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012.
  15. J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007).
  16. G. Chiribella, G. M. D'Ariano, and P. Perinotti, Probabilistic theories with purification, Phys. Rev. A 81, 062348 (2010).
  17. G. Chiribella, G. M. D'Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011).
  18. J. A. Barandes, The stochastic-quantum correspondence, Philos. Phys. 3, 6 (2025).
  19. J. A. Barandes, The stochastic-quantum theorem, arXiv:2309.03085.
  20. J. A. Barandes, Quantum systems as indivisible stochastic processes, arXiv:2507.21192.
  21. J. A. Barandes, New prospects for a causally local formulation of quantum theory, arXiv:2402.16935.
  22. Ll. Masanes, A. Acin, and N. Gisin, General properties of nonsignaling theories, Phys. Rev. A 73, 012112 (2006).
  23. J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, Nonlocal correlations as an information-theoretic resource, Phys. Rev. A 71, 022101 (2005).
  24. A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Am. J. Math. 76, 620 (1954).
  25. J. Pearl, Causality (Cambridge University Press, Cambridge, 2009).
  26. H. Buhrman and S. Massar, Causality and Tsirelson's bounds, Phys. Rev. A, 72, 052103 (2005).
  27. G. Chiribella, G. M. D' Ariano, and P. Perinotti, Quantum circuit architecture, Phys. Rev. Lett. 101, 060401 (2008).
  28. S. Milz, D. Egloff, P. Taranto, T. Theurer, M. B. Plenio, A. Smirne, and S. F. Huelga, When is a non-Markovian quantum process classical? Phys. Rev. X 10, 041049 (2020).
  29. S. Milz and K. Modi, Quantum stochastic processes and quantum non-Markovian phenomena, PRX Quantum 2, 030201 (2021).

Outline

Information

Phys. Rev. A 114, 022208– Published 17 August, 2026
  • Received 2 January 2026
  • Revised 29 April 2026
  • Accepted 9 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation