Skip to main content
SpringerLink
Log in

Bell Non-locality and Kochen–Specker Contextuality: How are They Connected?

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Bell non-locality and Kochen–Specker (KS) contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality and KS contextuality for ideal measurements, we show that, in quantum theory, due to Neumark’s dilation theorem, every quantum Bell non-local behavior can be mapped to a formally identical KS contextual behavior produced in a scenario with identical relations of compatibility but where measurements are ideal and no space-like separation is required. A more difficult problem is identifying connections in the opposite direction. We show that there are “one-to-one” and partial connections between KS contextual behaviors and Bell non-local behaviors for some KS scenarios, but not for all of them. However, there is also a method that transforms any KS contextual behavior for quantum systems of dimension d into a Bell non-local behavior between two quantum subsystems each of them of dimension d. We collect all these connections in map and list some problems which can benefit from this map.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)

    Article  MathSciNet  Google Scholar 

  2. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  ADS  MATH  Google Scholar 

  3. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014)

    Article  ADS  Google Scholar 

  4. Specker, E. P.: Die Logik nicht gleichzeitig entscheidbarer Aussagen, Dialectica 14, 239 (1960) [English version: the logic of non-simultaneously decidable propositions, arXiv:1103.4537.]

  5. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967)

    MathSciNet  MATH  Google Scholar 

  7. Klyachko, A.A., Can, M.A., Binicioğlu, S., Shumovsky, A.S.: Simple test for hidden variables in spin-1 systems. Phys. Rev. Lett. 101, 020403 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Cabello, A.: Experimentally testable state-independent quantum contextuality. Phys. Rev. Lett. 101, 210401 (2008)

    Article  ADS  Google Scholar 

  9. Cabello, A.: Quantum correlations from simple assumptions. Phys. Rev. A 100, 032120 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  10. Barrett, J., Kent, A.: Non-contextuality, finite precision measurement and the Kochen–Specker theorem. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 35, 151 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kleinmann, M.: Sequences of projective measurements in generalized probabilistic models. J. Phys. A Math. Theor. 47, 455304 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Chiribella, G., Yuan, X.: Measurement sharpness cuts nonlocality and contextuality in every physical theory. arXiv:1404.3348

  13. Chiribella, G., Yuan, X.: Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality. Inf. Comput. 250, 15 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Barrett, J., Hardy, L., Kent, A.: No signaling and quantum key distribution. Phys. Rev. Lett. 95, 010503 (2005)

    Article  ADS  Google Scholar 

  16. Acín, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)

    Article  ADS  Google Scholar 

  17. Brukner, Č., Żukowski, M., Pan, J.-W., Zeilinger, A.: Bell’s inequalities and quantum communication complexity. Phys. Rev. Lett. 92, 127901 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Colbeck, R.: Quantum and Relativistic Protocols for Secure Multi-Party Computation, Ph.D. thesis, University of Cambridge (2006). arXiv:0911.3814

  19. Pironio, S., Acín, A., Massar, S., de la Giroday, A.B., Matsukevich, D. N., Maunz, P., Olmschenk, S., Hayes, D., Luo, L., Manning, T. A., Monroe, C., : Random numbers certified by Bell’s theorem. Nature (London) 464, 1021 (2010)

  20. Summers, S.J., Werner, R.: Maximal violation of Bell’s inequalities is generic in quantum field theory. Commun. Math. Phys. 110, 247 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Popescu, S., Rohrlich, D.: Which states violate Bell’s inequality maximally? Phys. Lett. A 169, 411 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  22. Mayers, D., Yao, A.: Quantum cryptography with imperfect apparatus. In: Proceedings 39th Annual Symposium on Foundations of Computer Science, p. 503. Los Alamitos, CA, IEEE (1998)

  23. Howard, M., Wallman, J., Veitch, V., Emerson, J.: Contextuality supplies the ‘magic’ for quantum computation. Nature (London) 510, 351 (2014)

    Article  ADS  Google Scholar 

  24. Delfosse, N., Guerin, P.A., Bian, J., Raussendorf, R.: Wigner function negativity and contextuality in quantum computation on rebits. Phys. Rev. X 5, 021003 (2015)

    Google Scholar 

  25. Raussendorf, R., Browne, D.E., Delfosse, N., Okay, C., Bermejo-Vega, J.: Contextuality as a resource for models of quantum computation with qubits. Phys. Rev. Lett. 119, 120505 (2017)

    Article  MathSciNet  Google Scholar 

  26. Bravyi, S., Gosset, D., König, R.: Quantum advantage with shallow circuits. Science 362, 308 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Maudlin, T.: Bell’s inequality, information transmission, and prism models. In: Hull, D., Forbes, M., and Okruhlik, K. (eds) PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, vol. 1992, p. 404. Philosophy of Science Association, East Lansing, MI (1992)

  28. Brassard, G., Cleve, R., Tapp, A.: Cost of exactly simulating quantum entanglement with classical communication. Phys. Rev. Lett. 83, 1874 (1999)

    Article  ADS  Google Scholar 

  29. Toner, B.F., Bacon, D.: Communication cost of simulating Bell correlations. Phys. Rev. Lett. 91, 187904 (2003)

    Article  ADS  Google Scholar 

  30. Kleinmann, M., Gühne, O., Portillo, J.R., Larsson, J.-Å., Cabello, A.: Memory cost of quantum contextuality. New J. Phys. 13, 113011 (2011)

    Article  ADS  MATH  Google Scholar 

  31. Cabello, A., Gu, M., Gühne, O., Xu, Z.-P.: Optimal classical simulation of state-independent quantum contextuality. Phys. Rev. Lett. 120, 130401 (2018)

    Article  ADS  Google Scholar 

  32. Budroni, C.: Contextuality, memory cost and non-classicality for sequential measurements. Philos. Trans. A 377, 20190141 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Cabello, A., Gu, M., Gühne, O., Larsson, J.-Å., Wiesner, K.: Thermodynamical cost of some interpretations of quantum theory. Phys. Rev. A 94, 052127 (2016)

    Article  ADS  Google Scholar 

  34. Cabello, A.: "All versus nothing" inseparability for two observers. Phys. Rev. Lett. 87, 010403 (2001)

    Article  ADS  Google Scholar 

  35. Cabello, A.: Proposal for revealing quantum nonlocality via local contextuality. Phys. Rev. Lett. 104, 220401 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  36. Aolita, L., Gallego, R., Acín, A., Chiuri, A., Vallone, G., Mataloni, P., Cabello, A.: Fully nonlocal quantum correlations. Phys. Rev. A 85, 032107 (2012)

    Article  ADS  Google Scholar 

  37. Cabello, A., Amselem, E., Blanchfield, K., Bourennane, M., Bengtsson, I.: Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality. Phys. Rev. A 85, 032108 (2012)

    Article  ADS  Google Scholar 

  38. Liu, B.-H., Hu, X.-M., Chen, J.-S., Huang, Y.-F., Han, Y.-J., Li, C.-F., Guo, G.-C., Cabello, A.: Nonlocality from local contextuality. Phys. Rev. Lett. 117, 220402 (2016)

    Article  ADS  Google Scholar 

  39. Suarez, A.: All-possible-worlds: Unifying many-worlds and Copenhagen, in the light of quantum contextuality. arXiv:1712.06448

  40. Peres, A.: Incompatible results of quantum measurements. Phys. Lett. A 151, 107 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  41. Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Neumark, M. A.: On the self-adjoint extensions of the second kind of a symmetric operator, Izv. Akad. Nauk S.S.S.R. [Bull. Acad. Sci. U.S.S.R.] Sér. Mat. 4, 53 (1940) (Russian with English summary); Spectral functions of a symmetric operator, Izv. Akad. Nauk S.S.S.R. [Bull. Acad. Sci. U.S.S.R.] Sér. Mat. 4, 277 (1940) (Russian with English summary); On a representation of additive operator set functions, C.R. (Dokl.) Acad. Sci. U.R.S.S. (N.S.) 41, 359 (1943)

  43. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory, p. 55. Scuola Normale Superiore Pisa, Pisa (2011)

    Book  Google Scholar 

  44. Peres, A.: Quantum Theory: Concepts and Methods, p. 285. Kluwer, New York (1995)

    MATH  Google Scholar 

  45. Froissart, M.: Constructive generalization of Bell’s inequalities. Nuov. Cim. B 64, 241 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  46. Suppes, P., Zanotti, M.: When are probabilistic explanations possible? Synthese 48, 191 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  47. Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23, 1306 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  48. Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  49. Pitowsky, I.: Quantum Probability-Quantum Logic. Lecture Notes in Physics. Springer, Berlin (1989)

    MATH  Google Scholar 

  50. Kleinmann, M., Budroni, C., Larsson, J.-Å., Gühne, O., Cabello, A.: Optimal inequalities for state-independent contextuality. Phys. Rev. Lett. 109, 250402 (2012)

    Article  ADS  Google Scholar 

  51. Araújo, M., Quintino, M.T., Budroni, C., Cunha, M.Terra, Cabello, A.: All noncontextuality inequalities for the n-cycle scenario. Phys. Rev. A 88, 022118 (2013)

    Article  ADS  Google Scholar 

  52. Michler, M., Weinfurter, H., Żukowski, M.: Experiments towards falsification of noncontextual hidden variable theories. Phys. Rev. Lett. 84, 5457 (2000)

    Article  ADS  Google Scholar 

  53. Hasegawa, Y., Loidl, R., Badurek, G., Baron, M., Rauch, H.: Violation of a Bell-like inequality in single-neutron interferometry. Nature (London) 425, 45 (2003)

    Article  ADS  MATH  Google Scholar 

  54. Pearle, P.M.: Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418 (1970)

    Article  ADS  Google Scholar 

  55. Braunstein, S.L., Caves, C.M.: Wringing out better Bell inequalities. Ann. Phys. 202, 22 (1990)

    Article  ADS  MATH  Google Scholar 

  56. Boschi, D., Branca, S., De Martini, F., Hardy, L.: Ladder proof of nonlocality without inequalities: theoretical and experimental results. Phys. Rev. Lett. 79, 2755 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Xu, Z.-P., Cabello, A.: Quantum correlations with a gap between the sequential and spatial cases. Phys. Rev. A 96, 012127 (2017)

    Article  ADS  Google Scholar 

  58. Cabello, A.: Converting contextuality into nonlocality. arXiv:2011.13790

  59. Badzia̧g, P., Bengtsson, I., Cabello, A., Pitowsky, I.: Universality of State-independent violation of correlation inequalities for noncontextual theories. Phys. Rev. Lett. 103, 050401 (2009)

  60. Yu, S., Oh, C.H.: State-independent proof of Kochen–Specker Theorem with 13 rays. Phys. Rev. Lett. 108, 030402 (2012)

    Article  ADS  Google Scholar 

  61. Cabello, A., Kleinmann, M., Budroni, C.: Necessary and sufficient condition for quantum state-independent contextuality. Phys. Rev. Lett. 114, 250402 (2015)

    Article  ADS  Google Scholar 

  62. Amaral, B., Cunha, M.Terra, Cabello, A.: Quantum theory allows for absolute maximal contextuality. Phys. Rev. A 92, 062125 (2015)

    Article  ADS  Google Scholar 

  63. Cabello, A., Severini, S., Winter, A.: Graph-theoretic approach to quantum correlations. Phys. Rev. Lett. 112, 040401 (2014)

    Article  ADS  Google Scholar 

  64. Grudka, A., Horodecki, K., Horodecki, M., Horodecki, P., Horodecki, R., Joshi, P., Kłobus, W., Wójcik, A.: Quantifying contextuality. Phys. Rev. Lett. 112, 120401 (2014)

    Article  ADS  Google Scholar 

  65. Bharti, K., Ray, M., Varvitsiotis, A., Warsi, N.A., Cabello, A., Kwek, L.-C.: Robust self-testing of quantum systems via noncontextuality inequalities. Phys. Rev. Lett. 122, 250403 (2019)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author thanks Matthew Pusey and Zhen-Peng Xu for comments on an earlier version of this work. This work was supported by Project Qdisc (Project No. US-15097), with FEDER funds, MINECO Project No. FIS2017-89609-P, with FEDER funds, and QuantERA Grant SECRET, by MINECO (Project No. PCI2019-111885-2).

Author information

Authors and Affiliations

  1. Departamento de Física Aplicada II and Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla, 41012, Sevilla, Spain

    Adán Cabello

Authors
  1. Adán Cabello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adán Cabello.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cabello, A. Bell Non-locality and Kochen–Specker Contextuality: How are They Connected?. Found Phys 51, 61 (2021). https://doi.org/10.1007/s10701-021-00466-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-021-00466-5

Keywords

Search

Navigation