{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T01:00:11Z","timestamp":1774573211067,"version":"3.50.1"},"reference-count":73,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T00:00:00Z","timestamp":1710374400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004442","name":"National Science Centre","doi-asserted-by":"crossref","award":["project no. 2021\/41\/B\/ST2\/03149"],"award-info":[{"award-number":["project no. 2021\/41\/B\/ST2\/03149"]}],"id":[{"id":"10.13039\/501100004442","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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 \u2014 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.<\/jats:p>","DOI":"10.22331\/q-2024-03-14-1283","type":"journal-article","created":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T12:58:37Z","timestamp":1710421117000},"page":"1283","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":["A structure theorem for generalized-noncontextual ontological models"],"prefix":"10.22331","volume":"8","author":[{"given":"David","family":"Schmid","sequence":"first","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, 80-308 Gda\u0144sk, Poland"},{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5"},{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada"}]},{"given":"John H.","family":"Selby","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gda\u0144sk, 80-308 Gda\u0144sk, Poland"}]},{"given":"Matthew F.","family":"Pusey","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom"}]},{"given":"Robert W.","family":"Spekkens","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5"}]}],"member":"9598","published-online":{"date-parts":[[2024,3,14]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).","DOI":"10.1103\/PhysRevA.71.052108"},{"key":"1","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, Phys. Rev. Lett. 101, 020401 (2008).","DOI":"10.1103\/PhysRevLett.101.020401"},{"key":"2","doi-asserted-by":"publisher","unstructured":"C. Ferrie and J. Emerson, J. Phys. A: Math. Theor. 41, 352001 (2008).","DOI":"10.1088\/1751-8113\/41\/35\/352001"},{"key":"3","doi-asserted-by":"publisher","unstructured":"D. Schmid, J. H. Selby, E. Wolfe, R. Kunjwal, and R. W. Spekkens, PRX Quantum 2, 010331 (2021a).","DOI":"10.1103\/PRXQuantum.2.010331"},{"key":"4","doi-asserted-by":"publisher","unstructured":"F. Shahandeh, PRX Quantum 2, 010330 (2021).","DOI":"10.1103\/PRXQuantum.2.010330"},{"key":"5","doi-asserted-by":"publisher","unstructured":"J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Phys. Rev. Lett. 130, 230201 (2023a).","DOI":"10.1103\/PhysRevLett.130.230201"},{"key":"6","doi-asserted-by":"publisher","unstructured":"J. H. Selby, D. Schmid, E. Wolfe, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Phys. Rev. A 107, 062203 (2023b).","DOI":"10.1103\/PhysRevA.107.062203"},{"key":"7","doi-asserted-by":"publisher","unstructured":"J. S. Bell, Physics 1, 195 (1964).","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195"},{"key":"8","doi-asserted-by":"publisher","unstructured":"N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014).","DOI":"10.1103\/RevModPhys.86.419"},{"key":"9","unstructured":"R. W. Spekkens, arXiv:1909.04628 [physics.hist-ph] (2019)."},{"key":"10","doi-asserted-by":"publisher","unstructured":"M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, Nat. Commun. 7, 11780 (2016).","DOI":"10.1038\/ncomms11780"},{"key":"11","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).","DOI":"10.1103\/PhysRevLett.102.010401"},{"key":"12","doi-asserted-by":"publisher","unstructured":"A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, New J. Phys. 18, 045003 (2016).","DOI":"10.1088\/1367-2630\/18\/4\/045003"},{"key":"13","doi-asserted-by":"publisher","unstructured":"A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai, Quant. Inf. Process. 18, 111 (2019).","DOI":"10.1007\/s11128-019-2228-3"},{"key":"14","doi-asserted-by":"publisher","unstructured":"D. Saha, P. Horodecki, and M. Paw\u0142owski, New J. Phys. 21, 093057 (2019).","DOI":"10.1088\/1367-2630\/ab4149"},{"key":"15","doi-asserted-by":"publisher","unstructured":"D. Saha and A. Chaturvedi, Phys. Rev. A 100, 022108 (2019).","DOI":"10.1103\/PhysRevA.100.022108"},{"key":"16","doi-asserted-by":"publisher","unstructured":"D. Schmid and R. W. Spekkens, Phys. Rev. X 8, 011015 (2018).","DOI":"10.1103\/PhysRevX.8.011015"},{"key":"17","doi-asserted-by":"publisher","unstructured":"M. Lostaglio and G. Senno, Quantum 4, 258 (2020).","DOI":"10.22331\/q-2020-04-27-258"},{"key":"18","doi-asserted-by":"publisher","unstructured":"D. Schmid, H. Du, J. H. Selby, and M. F. Pusey, arXiv:2101.06263 (2021b).","DOI":"10.1103\/PhysRevLett.129.120403"},{"key":"19","doi-asserted-by":"publisher","unstructured":"P. Lillystone, J. J. Wallman, and J. Emerson, Phys. Rev. Lett. 122, 140405 (2019).","DOI":"10.1103\/PhysRevLett.122.140405"},{"key":"20","doi-asserted-by":"publisher","unstructured":"M. S. Leifer and R. W. Spekkens, Phys. Rev. Lett. 95, 200405 (2005), arXiv:quant-ph\/0412178.","DOI":"10.1103\/PhysRevLett.95.200405"},{"key":"21","doi-asserted-by":"publisher","unstructured":"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\u2013306.","DOI":"10.4204\/EPTCS.195.22"},{"key":"22","doi-asserted-by":"publisher","unstructured":"M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014).","DOI":"10.1103\/PhysRevLett.113.200401"},{"key":"23","doi-asserted-by":"publisher","unstructured":"R. Kunjwal, M. Lostaglio, and M. F. Pusey, Phys. Rev. A 100, 042116 (2019).","DOI":"10.1103\/PhysRevA.100.042116"},{"key":"24","doi-asserted-by":"publisher","unstructured":"B. Coecke and A. Kissinger, in Categories for the Working Philosopher, edited by E. Landry (Oxford University Press, 2017) pp. 286\u2013328.","DOI":"10.1093\/oso\/9780198748991.003.0012"},{"key":"25","doi-asserted-by":"publisher","unstructured":"B. Coecke and A. Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning (Cambridge University Press, 2017).","DOI":"10.1017\/9781316219317"},{"key":"26","doi-asserted-by":"publisher","unstructured":"J. H. Selby, C. M. Scandolo, and B. Coecke, Quantum 5, 445 (2021).","DOI":"10.22331\/q-2021-04-28-445"},{"key":"27","doi-asserted-by":"publisher","unstructured":"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\u2013385.","DOI":"10.4204\/EPTCS.266.23"},{"key":"28","unstructured":"L. Hardy, arXiv:quant-ph\/0101012 (2001)."},{"key":"29","doi-asserted-by":"publisher","unstructured":"J. Barrett, Phys. Rev. A 75, 032304 (2007).","DOI":"10.1103\/PhysRevA.75.032304"},{"key":"30","unstructured":"L. Hardy, arXiv:1104.2066 [quant-ph] (2011)."},{"key":"31","doi-asserted-by":"publisher","unstructured":"G. Chiribella, G. M. D&apos;Ariano, and P. Perinotti, Phys. Rev. A 81, 062348 (2010).","DOI":"10.1103\/PhysRevA.81.062348"},{"key":"32","doi-asserted-by":"publisher","unstructured":"G. Chiribella, G. M. D&apos;Ariano, and P. Perinotti, Physical Review A 84, 012311 (2011).","DOI":"10.1103\/PhysRevA.84.012311"},{"key":"33","doi-asserted-by":"publisher","unstructured":"G. Chiribella, G. M. DAriano, and P. Perinotti, in Quantum theory: informational foundations and foils (Springer, 2016) pp. 171\u2013221.","DOI":"10.48550\/arXiv.1506.00398"},{"key":"34","unstructured":"D. Schmid, J. H. Selby, and R. W. Spekkens, arXiv:2009.03297 (2020)."},{"key":"35","doi-asserted-by":"publisher","unstructured":"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\u2013212.","DOI":"10.4204\/EPTCS.318.12"},{"key":"36","doi-asserted-by":"publisher","unstructured":"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\u2013196.","DOI":"10.4204\/EPTCS.266.12"},{"key":"37","doi-asserted-by":"publisher","unstructured":"C. Ferrie and J. Emerson, New J. Phys. 11, 063040 (2009).","DOI":"10.1088\/1367-2630\/11\/6\/063040"},{"key":"38","doi-asserted-by":"publisher","unstructured":"L. Hardy, Stud. Hist. Phil. Mod. Phys. 35, 267 (2004).","DOI":"10.1016\/j.shpsb.2003.12.001"},{"key":"39","doi-asserted-by":"publisher","unstructured":"P.-A. Mellies, in International Workshop on Computer Science Logic (Springer, 2006) pp. 1\u201330.","DOI":"10.1007\/11874683_1"},{"key":"40","doi-asserted-by":"publisher","unstructured":"G. Chiribella, G. M. D&apos;Ariano, and P. Perinotti, Physical review letters 101, 060401 (2008a).","DOI":"10.1103\/PhysRevLett.101.060401"},{"key":"41","doi-asserted-by":"publisher","unstructured":"G. Chiribella, G. M. D&apos;Ariano, and P. Perinotti, EPL (Europhysics Letters) 83, 30004 (2008b).","DOI":"10.1209\/0295-5075\/83\/30004"},{"key":"42","doi-asserted-by":"publisher","unstructured":"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\u2013300.","DOI":"10.4204\/EPTCS.343.12"},{"key":"43","doi-asserted-by":"publisher","unstructured":"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 \u2013 149.","DOI":"10.1016\/j.entcs.2018.11.007"},{"key":"44","unstructured":"S. Mac Lane, Categories for the working mathematician, Vol. 5 (Springer Science & Business Media, 2013)."},{"key":"45","doi-asserted-by":"publisher","unstructured":"G. Chiribella, in Proceedings of the 11th workshop on Quantum Physics and Logic, Electron. Notes Theor. Comput. Sci., Vol. 172 (2014) pp. 1 \u2013 14.","DOI":"10.4204\/EPTCS.172.1"},{"key":"46","doi-asserted-by":"publisher","unstructured":"M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).","DOI":"10.1017\/CBO9780511976667"},{"key":"47","doi-asserted-by":"publisher","unstructured":"D. Schmid, K. Ried, and R. W. Spekkens, Phys. Rev. A 100, 022112 (2019).","DOI":"10.1103\/PhysRevA.100.022112"},{"key":"48","doi-asserted-by":"publisher","unstructured":"M. Appleby, C. A. Fuchs, B. C. Stacey, and H. Zhu, Eur. Phys. J. D 71, 197 (2017).","DOI":"10.1140\/epjd\/e2017-80024-y"},{"key":"49","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, Phys. Rev. A 75, 032110 (2007).","DOI":"10.1103\/PhysRevA.75.032110"},{"key":"50","unstructured":"D. Gottesman, in 22nd International Colloquium on Group Theoretical Methods in Physics (1999) pp. 32\u201343, arXiv:quant-ph\/9807006."},{"key":"51","doi-asserted-by":"publisher","unstructured":"L. Hardy and W. K. Wootters, Found. Phys. 42, 454 (2012).","DOI":"10.1007\/s10701-011-9616-6"},{"key":"52","unstructured":"N. Harrigan, T. Rudolph, and S. Aaronson, arXiv:0709.1149 (2007)."},{"key":"53","unstructured":"R. W. Spekkens, Noncontextuality: how we should define it, why it is natural, and what to do about its failure (2017), PIRSA:17070035."},{"key":"54","doi-asserted-by":"publisher","unstructured":"E. G. Beltrametti and S. Bugajski, J. Phys. A 28, 3329 (1995).","DOI":"10.1088\/0305-4470\/28\/12\/007"},{"key":"55","doi-asserted-by":"publisher","unstructured":"J. J. Wallman and S. D. Bartlett, Phys. Rev. A 85, 062121 (2012).","DOI":"10.1103\/PhysRevA.85.062121"},{"key":"56","doi-asserted-by":"crossref","unstructured":"F. Riesz, in Annales scientifiques de l&apos;\u00c9cole Normale Sup\u00e9rieure, Vol. 31 (1914) pp. 9\u201314.","DOI":"10.24033\/asens.669"},{"key":"57","doi-asserted-by":"publisher","unstructured":"V. Gitton and M. P. Woods, Quantum 6, 732 (2022).","DOI":"10.22331\/q-2022-06-07-732"},{"key":"58","unstructured":"A. Karanjai, J. J. Wallman, and S. D. Bartlett, arXiv:1802.07744 (2018)."},{"key":"59","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, in Quantum Theory: Informational Foundations and Foils, edited by G. Chiribella and R. W. Spekkens (Springer Netherlands, Dordrecht, 2016) pp. 83\u2013135.","DOI":"10.1007\/978-94-017-7303-4_4"},{"key":"60","doi-asserted-by":"publisher","unstructured":"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\u201316.","DOI":"10.1007\/978-3-319-13045-3_2"},{"key":"61","doi-asserted-by":"publisher","unstructured":"N. Harrigan and R. W. Spekkens, Found. Phys. 40, 125 (2010).","DOI":"10.1007\/s10701-009-9347-0"},{"key":"62","doi-asserted-by":"publisher","unstructured":"R. W. Spekkens, Found. Phys. 44, 1125 (2014).","DOI":"10.1007\/s10701-014-9833-x"},{"key":"63","doi-asserted-by":"publisher","unstructured":"M. F. Pusey, J. Barrett, and T. Rudolph, Nat. Phys. 8, 475 (2012).","DOI":"10.1038\/nphys2309"},{"key":"64","doi-asserted-by":"publisher","unstructured":"K. Husimi, Proc. Physico-Mathematical Soc. Jpn. 3rd Series 22, 264 (1940).","DOI":"10.11429\/ppmsj1919.22.4_264"},{"key":"65","doi-asserted-by":"publisher","unstructured":"R. J. Glauber, Phys. Rev. 131, 2766 (1963).","DOI":"10.1103\/PhysRev.131.2766"},{"key":"66","doi-asserted-by":"publisher","unstructured":"E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).","DOI":"10.1103\/PhysRevLett.10.277"},{"key":"67","doi-asserted-by":"publisher","unstructured":"K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004).","DOI":"10.1103\/PhysRevA.70.062101"},{"key":"68","doi-asserted-by":"publisher","unstructured":"D. Gross, J. Math. Phys. 47, 122107 (2006).","DOI":"10.1063\/1.2393152"},{"key":"69","doi-asserted-by":"publisher","unstructured":"A. Krishna, R. W. Spekkens, and E. Wolfe, New J, Phys. 19, 123031 (2017).","DOI":"10.1088\/1367-2630\/aa9168"},{"key":"70","doi-asserted-by":"publisher","unstructured":"D. Schmid, R. W. Spekkens, and E. Wolfe, Phys. Rev. A 97, 062103 (2018).","DOI":"10.1103\/PhysRevA.97.062103"},{"key":"71","doi-asserted-by":"publisher","unstructured":"M. Howard, J. Wallman, V. Veitch, and J. Emerson, Nature 510, 351 (2014).","DOI":"10.1038\/nature13460"},{"key":"72","doi-asserted-by":"publisher","unstructured":"M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, PRX Quantum 2, 020302 (2021).","DOI":"10.1103\/PRXQuantum.2.020302"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-03-14-1283\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T09:22:48Z","timestamp":1710753768000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-03-14-1283\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,14]]},"references-count":73,"URL":"https:\/\/doi.org\/10.22331\/q-2024-03-14-1283","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,14]]},"article-number":"1283"}}