A structure theorem for generalized-noncontextual ontological models

David Schmid; John H. Selby; Matthew F. Pusey; Robert W. Spekkens

doi:10.22331/q-2024-03-14-1283

A structure theorem for generalized-noncontextual ontological models

David Schmid^1,2,3, John H. Selby¹, Matthew F. Pusey⁴, and Robert W. Spekkens²

¹International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
²Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5
³Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
⁴Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

Published:	2024-03-14, volume 8, page 1283
Eprint:	arXiv:2005.07161v3
Doi:	https://doi.org/10.22331/q-2024-03-14-1283
Citation:	Quantum 8, 1283 (2024).

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Abstract

It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model. Existing works on generalized noncontextuality have focused on experimental scenarios having a simple structure: typically, prepare-measure scenarios. Here, we formally extend the framework of ontological models as well as the principle of generalized noncontextuality to arbitrary compositional scenarios. We leverage a process-theoretic framework to prove that, under some reasonable assumptions, every generalized-noncontextual ontological model of a tomographically local operational theory has a surprisingly rigid and simple mathematical structure — in short, it corresponds to a frame representation which is not overcomplete. One consequence of this theorem is that the largest number of ontic states possible in any such model is given by the dimension of the associated generalized probabilistic theory. This constraint is useful for generating noncontextuality no-go theorems as well as techniques for experimentally certifying contextuality. Along the way, we extend known results concerning the equivalence of different notions of classicality from prepare-measure scenarios to arbitrary compositional scenarios. Specifically, we prove a correspondence between the following three notions of classical explainability of an operational theory: (i) existence of a noncontextual ontological model for it, (ii) existence of a positive quasiprobability representation for the generalized probabilistic theory it defines, and (iii) existence of an ontological model for the generalized probabilistic theory it defines.

Featured image: In this work, we formally extend the definition of ontological models as well as the principle of generalized noncontextuality to arbitrary compositional scenarios. An ontological model of a generalized probabilistic theory (GPT) is a linear and diagram-preserving map taking GPT systems to classical random variables and taking GPT processes to stochastic matrices, such that the GPT probabilities are reproduced by the composition of the stochastic maps. Such a model corresponds to a noncontextual ontological model for the unquotiented operational theory that gave rise to the GPT.

► BibTeX data

@article{Schmid2024structuretheorem,
  doi = {10.22331/q-2024-03-14-1283},
  url = {https://doi.org/10.22331/q-2024-03-14-1283},
  title = {A structure theorem for generalized-noncontextual ontological models},
  author = {Schmid, David and Selby, John H. and Pusey, Matthew F. and Spekkens, Robert W.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1283},
  month = mar,
  year = {2024}
}

► References

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

[2] R. W. Spekkens, Phys. Rev. Lett. 101, 020401 (2008).
https://doi.org/10.1103/PhysRevLett.101.020401

[3] C. Ferrie and J. Emerson, J. Phys. A: Math. Theor. 41, 352001 (2008).
https://doi.org/10.1088/1751-8113/41/35/352001

[4] D. Schmid, J. H. Selby, E. Wolfe, R. Kunjwal, and R. W. Spekkens, PRX Quantum 2, 010331 (2021a).
https://doi.org/10.1103/PRXQuantum.2.010331

[5] F. Shahandeh, PRX Quantum 2, 010330 (2021).
https://doi.org/10.1103/PRXQuantum.2.010330

[6] J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Phys. Rev. Lett. 130, 230201 (2023a).
https://doi.org/10.1103/PhysRevLett.130.230201

[7] J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Phys. Rev. A 107, 062203 (2023b).
https://doi.org/10.1103/PhysRevA.107.062203

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

[9] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014).
https://doi.org/10.1103/RevModPhys.86.419

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

[11] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, Nat. Commun. 7, 11780 (2016).
https://doi.org/10.1038/ncomms11780

[12] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).
https://doi.org/10.1103/PhysRevLett.102.010401

[13] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, New J. Phys. 18, 045003 (2016).
https://doi.org/10.1088/1367-2630/18/4/045003

[14] A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai, Quant. Inf. Process. 18, 111 (2019).
https://doi.org/10.1007/s11128-019-2228-3

[15] D. Saha, P. Horodecki, and M. Pawłowski, New J. Phys. 21, 093057 (2019).
https://doi.org/10.1088/1367-2630/ab4149

[16] D. Saha and A. Chaturvedi, Phys. Rev. A 100, 022108 (2019).
https://doi.org/10.1103/PhysRevA.100.022108

[17] D. Schmid and R. W. Spekkens, Phys. Rev. X 8, 011015 (2018).
https://doi.org/10.1103/PhysRevX.8.011015

[18] M. Lostaglio and G. Senno, Quantum 4, 258 (2020).
https://doi.org/10.22331/q-2020-04-27-258

[19] D. Schmid, H. Du, J. H. Selby, and M. F. Pusey, arXiv:2101.06263 (2021b).
https://doi.org/10.1103/PhysRevLett.129.120403
arXiv:2101.06263

[20] P. Lillystone, J. J. Wallman, and J. Emerson, Phys. Rev. Lett. 122, 140405 (2019).
https://doi.org/10.1103/PhysRevLett.122.140405

[21] M. S. Leifer and R. W. Spekkens, Phys. Rev. Lett. 95, 200405 (2005), arXiv:quant-ph/0412178.
https://doi.org/10.1103/PhysRevLett.95.200405
arXiv:quant-ph/0412178

[22] M. F. Pusey and M. S. Leifer, in Proceedings of the 12th International Workshop on Quantum Physics and Logic, Electron. Proc. Theor. Comput. Sci., Vol. 195 (2015) pp. 295–306.
https://doi.org/10.4204/EPTCS.195.22

[23] M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014).
https://doi.org/10.1103/PhysRevLett.113.200401

[24] R. Kunjwal, M. Lostaglio, and M. F. Pusey, Phys. Rev. A 100, 042116 (2019).
https://doi.org/10.1103/PhysRevA.100.042116

[25] B. Coecke and A. Kissinger, in Categories for the Working Philosopher, edited by E. Landry (Oxford University Press, 2017) pp. 286–328.
https://doi.org/10.1093/oso/9780198748991.003.0012

[26] B. Coecke and A. Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning (Cambridge University Press, 2017).
https://doi.org/10.1017/9781316219317

[27] J. H. Selby, C. M. Scandolo, and B. Coecke, Quantum 5, 445 (2021).
https://doi.org/10.22331/q-2021-04-28-445

[28] S. Gogioso and C. M. Scandolo, in Proceedings of the 14th International Workshop on Quantum Physics and Logic, Electron. Proc. Theor. Comput. Sci., Vol. 266 (2018) pp. 367–385.
https://doi.org/10.4204/EPTCS.266.23

[29] L. Hardy, arXiv:quant-ph/0101012 (2001).
arXiv:quant-ph/0101012

[30] J. Barrett, Phys. Rev. A 75, 032304 (2007).
https://doi.org/10.1103/PhysRevA.75.032304

[31] L. Hardy, arXiv:1104.2066 [quant-ph] (2011).
arXiv:1104.2066

[32] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348

[33] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Physical Review A 84, 012311 (2011).
https://doi.org/10.1103/PhysRevA.84.012311

[34] G. Chiribella, G. M. DAriano, and P. Perinotti, in Quantum theory: informational foundations and foils (Springer, 2016) pp. 171–221.
https://doi.org/10.48550/arXiv.1506.00398

[35] D. Schmid, J. H. Selby, and R. W. Spekkens, arXiv:2009.03297 (2020).
arXiv:2009.03297

[36] A. Gheorghiu and C. Heunen, in Proceedings of the 16th International Workshop on Quantum Physics and Logic, Electron. Proc. Theor. Comput. Sci., Vol. 318 (2020) pp. 196–212.
https://doi.org/10.4204/EPTCS.318.12

[37] J. van de Wetering, in Proceedings of the 14th International Workshop on Quantum Physics and Logic, Electron. Proc. Theor. Comput. Sci., Vol. 266 (2018) pp. 179–196.
https://doi.org/10.4204/EPTCS.266.12

[38] C. Ferrie and J. Emerson, New J. Phys. 11, 063040 (2009).
https://doi.org/10.1088/1367-2630/11/6/063040

[39] L. Hardy, Stud. Hist. Phil. Mod. Phys. 35, 267 (2004).
https://doi.org/10.1016/j.shpsb.2003.12.001

[40] P.-A. Mellies, in International Workshop on Computer Science Logic (Springer, 2006) pp. 1–30.
https://doi.org/10.1007/11874683_1

[41] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Physical review letters 101, 060401 (2008a).
https://doi.org/10.1103/PhysRevLett.101.060401

[42] G. Chiribella, G. M. D'Ariano, and P. Perinotti, EPL (Europhysics Letters) 83, 30004 (2008b).
https://doi.org/10.1209/0295-5075/83/30004

[43] M. Wilson and G. Chiribella, in \rm Proceedings 18th International Conference on Quantum Physics and Logic, \rm Gdansk, Poland, and online, 7-11 June 2021, Electronic Proceedings in Theoretical Computer Science, Vol. 343, edited by C. Heunen and M. Backens (Open Publishing Association, 2021) pp. 265–300.
https://doi.org/10.4204/EPTCS.343.12

[44] T. Fritz and P. Perrone, in Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXIV), Electron. Notes Theor. Comput. Sci., Vol. 341 (2018) pp. 121 – 149.
https://doi.org/10.1016/j.entcs.2018.11.007

[45] S. Mac Lane, Categories for the working mathematician, Vol. 5 (Springer Science & Business Media, 2013).

[46] G. Chiribella, in Proceedings of the 11th workshop on Quantum Physics and Logic, Electron. Notes Theor. Comput. Sci., Vol. 172 (2014) pp. 1 – 14.
https://doi.org/10.4204/EPTCS.172.1

[47] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).
https://doi.org/10.1017/CBO9780511976667

[48] D. Schmid, K. Ried, and R. W. Spekkens, Phys. Rev. A 100, 022112 (2019).
https://doi.org/10.1103/PhysRevA.100.022112

[49] M. Appleby, C. A. Fuchs, B. C. Stacey, and H. Zhu, Eur. Phys. J. D 71, 197 (2017).
https://doi.org/10.1140/epjd/e2017-80024-y

[50] R. W. Spekkens, Phys. Rev. A 75, 032110 (2007).
https://doi.org/10.1103/PhysRevA.75.032110

[51] D. Gottesman, in 22nd International Colloquium on Group Theoretical Methods in Physics (1999) pp. 32–43, arXiv:quant-ph/9807006.
arXiv:quant-ph/9807006

[52] L. Hardy and W. K. Wootters, Found. Phys. 42, 454 (2012).
https://doi.org/10.1007/s10701-011-9616-6

[53] N. Harrigan, T. Rudolph, and S. Aaronson, arXiv:0709.1149 (2007).
arXiv:0709.1149

[54] R. W. Spekkens, Noncontextuality: how we should define it, why it is natural, and what to do about its failure (2017), PIRSA:17070035.
http://pirsa.org/17070035

[55] E. G. Beltrametti and S. Bugajski, J. Phys. A 28, 3329 (1995).
https://doi.org/10.1088/0305-4470/28/12/007

[56] J. J. Wallman and S. D. Bartlett, Phys. Rev. A 85, 062121 (2012).
https://doi.org/10.1103/PhysRevA.85.062121

[57] F. Riesz, in Annales scientifiques de l'École Normale Supérieure, Vol. 31 (1914) pp. 9–14.

[58] V. Gitton and M. P. Woods, Quantum 6, 732 (2022).
https://doi.org/10.22331/q-2022-06-07-732

[59] A. Karanjai, J. J. Wallman, and S. D. Bartlett, arXiv:1802.07744 (2018).
arXiv:1802.07744

[60] R. W. Spekkens, in Quantum Theory: Informational Foundations and Foils, edited by G. Chiribella and R. W. Spekkens (Springer Netherlands, Dordrecht, 2016) pp. 83–135.
https://doi.org/10.1007/978-94-017-7303-4_4

[61] R. W. Spekkens, The paradigm of kinematics and dynamics must yield to causal structure, in Questioning the Foundations of Physics: Which of Our Fundamental Assumptions Are Wrong?, edited by A. Aguirre, B. Foster, and Z. Merali (Springer International Publishing, Cham, 2015) pp. 5–16.
https://doi.org/10.1007/978-3-319-13045-3_2

[62] N. Harrigan and R. W. Spekkens, Found. Phys. 40, 125 (2010).
https://doi.org/10.1007/s10701-009-9347-0

[63] R. W. Spekkens, Found. Phys. 44, 1125 (2014).
https://doi.org/10.1007/s10701-014-9833-x

[64] M. F. Pusey, J. Barrett, and T. Rudolph, Nat. Phys. 8, 475 (2012).
https://doi.org/10.1038/nphys2309

[65] K. Husimi, Proc. Physico-Mathematical Soc. Jpn. 3rd Series 22, 264 (1940).
https://doi.org/10.11429/ppmsj1919.22.4_264

[66] R. J. Glauber, Phys. Rev. 131, 2766 (1963).
https://doi.org/10.1103/PhysRev.131.2766

[67] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
https://doi.org/10.1103/PhysRevLett.10.277

[68] K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004).
https://doi.org/10.1103/PhysRevA.70.062101

[69] D. Gross, J. Math. Phys. 47, 122107 (2006).
https://doi.org/10.1063/1.2393152

[70] A. Krishna, R. W. Spekkens, and E. Wolfe, New J, Phys. 19, 123031 (2017).
https://doi.org/10.1088/1367-2630/aa9168

[71] D. Schmid, R. W. Spekkens, and E. Wolfe, Phys. Rev. A 97, 062103 (2018).
https://doi.org/10.1103/PhysRevA.97.062103

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

[73] M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, PRX Quantum 2, 020302 (2021).
https://doi.org/10.1103/PRXQuantum.2.020302

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[10] Albert Aloy, Matteo Fadel, Thomas D. Galley, Caroline L. Jones, and Markus P. Müller, "Theory-independent monitoring of the decoherence of a superconducting qubit with generalized contextuality", Nature Communications 17 1, 2474 (2026).

[11] Jonathan J. Thio, Wilfred Salmon, Crispin H. W. Barnes, Stephan De Bièvre, and David R. M. Arvidsson-Shukur, "Verification of contextuality by noncontextual experiments", Physical Review A 112 6, 062230 (2025).

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[13] Giulio Chiribella, Lorenzo Giannelli, and Carlo Maria Scandolo, "Bell Nonlocality in Classical Systems Coexisting with Other System Types", Physical Review Letters 132 19, 190201 (2024).

[14] Holger F. Hofmann, "Statistical Signatures of Quantum Contextuality", Entropy 26 9, 725 (2024).

[15] Sagar Silva Pratapsi, Sebastian Deffner, and Stefano Gherardini, "Quantum speed limit for Kirkwood–Dirac quasiprobabilities", Quantum Science and Technology 10 3, 035019 (2025).

[16] Laurens Walleghem, Rui Soares Barbosa, Matthew F. Pusey, and Stefan Weigert, "A refined Frauchiger–Renner paradox based on strong contextuality", Quantum 10, 2116 (2026).

[17] David Schmid, John H. Selby, Vinicius P. Rossi, Roberto D. Baldijão, and Ana Belén Sainz, "Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs", Quantum 9, 1880 (2025).

[18] Alisson Tezzin, Bárbara Amaral, and Jonte R. Hance, "Sufficiency of the counterfactual account of Lüders' rule to rule out ontological models of quantum mechanics", Physical Review A 112 5, 052208 (2025).

[19] Caroline Lima, María Rosa Preciado-Rivas, and Sanchit Srivastava, "Contextuality from quantum field vacua using qutrit Unruh-DeWitt detectors", Physical Review D 113 6, 065025 (2026).

[20] Pauli Jokinen, Mirjam Weilenmann, Martin Plávala, Juha-Pekka Pellonpää, Jukka Kiukas, and Roope Uola, "No-Broadcasting Characterizes Operational Contextuality", Physical Review Letters 133 24, 240201 (2024).

[21] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, "Kochen-Specker contextuality", Reviews of Modern Physics 94 4, 045007 (2022).

[22] Martin Plávala, "General probabilistic theories: An introduction", Physics Reports 1033, 1 (2023).

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[52] Russell P Rundle and Mark J Everitt, "Overview of the phase space formulation of quantum mechanics with application to quantum technologies", arXiv:2102.11095, (2021).

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[54] Marco Erba, Paolo Perinotti, Davide Rolino, and Alessandro Tosini, "Measurement incompatibility is strictly stronger than disturbance", Physical Review A 109 2, 022239 (2024).

[55] John H. Selby, Maria E. Stasinou, Matt Wilson, and Bob Coecke, "Generalised Process Theories", arXiv:2502.10368, (2025).

[56] Martin Plávala, "Incompatibility in restricted operational theories: connecting contextuality and steering", Journal of Physics A Mathematical General 55 17, 174001 (2022).

[57] Rafael Wagner, "Coherence and contextuality as quantum resources", arXiv:2511.16785, (2025).

[58] Laurens Walleghem, Shashaank Khanna, and Rutvij Bhavsar, "Comment on a no-go theorem for $\psi$-ontic models", arXiv:2402.13140, (2024).

[59] Lorenzo Catani, "Relationship between covariance of Wigner functions and transformation noncontextuality", arXiv:2004.06318, (2020).

[60] Soumyabrata Hazra, Debashis Saha, Anubhav Chaturvedi, Subhankar Bera, and A. S. Majumdar, "Efficient Computation of Generalized Noncontextual Polytopes and Quantum violation of their Facet Inequalities", Quantum 10, 2015 (2026).

[61] Farid Shahandeh, Theodoros Yianni, and Mina Doosti, "A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality", arXiv:2512.10000, (2025).

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