- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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

Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories

David Schmid1,*, John H. Selby1,2,†, Elie Wolfe1, Ravi Kunjwal1,3, and Robert W. Spekkens1

  • 1Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 Canada
  • 2ICTQT, University of Gdańsk, Wita Stwosza 63, Gdańsk 80-308, Poland
  • 3Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, Brussels 1050, Belgium

  • *dschmid@perimeterinstitute.ca
  • †john.h.selby@gmail.com
PDF

PRX Quantum 2, 010331 – Published 25 February, 2021

DOI: https://doi.org/10.1103/PRXQuantum.2.010331

Abstract

To make precise the sense in which the operational predictions of quantum theory conflict with a classical worldview, it is necessary to articulate a notion of classicality within an operational framework. A widely applicable notion of classicality of this sort is whether or not the predictions of a given operational theory can be explained by a generalized-noncontextual ontological model. We here explore what notion of classicality this implies for the generalized probabilistic theory (GPT) that arises from a given operational theory, focusing on prepare-measure scenarios. We first show that, when mapping an operational theory to a GPT by quotienting relative to operational equivalences, the constraint of explainability by a generalized-noncontextual ontological model is mapped to the constraint of explainability by an ontological model. We then show that, under the additional assumption that the ontic state space is of finite cardinality, this constraint on the GPT can be expressed as a geometric condition which we term simplex embeddability. Whereas the traditional notion of classicality for a GPT is that its state space be a simplex and its effect space be the dual of this simplex, simplex embeddability merely requires that its state space be embeddable in a simplex and its effect space in the dual of that simplex. We argue that simplex embeddability constitutes an intuitive and freestanding notion of classicality for GPTs. Our result also has applications to witnessing nonclassicality in prepare-measure experiments.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum theory is famously counterintuitive, admitting of many phenomena that do not obviously have counterparts in classical physics. Nonetheless, some of these initially surprising features can in fact be given a very natural account in terms of an underlying classical theory, called a noncontextual model. Proving that a given quantum experiment does not admit of a noncontextual model formally establishes that it is strongly nonclassical, a feature that is increasingly being found to be associated with quantum advantages for information processing.

The framework of generalized probabilistic theories (GPTs) provides a means of summarizing the scope of possible statistics that can be achieved in any physical theory, including alternatives to quantum theory. It also provides the means of doing so for any subtheory of quantum theory, that is, any closed subset of quantum states and measurements, and therefore for any experiment. These summaries are termed GPTs and for prepare-measure experiments are represented by a pair of geometric objects: a state space and an effect space.

The question we address here is: which GPTs may be said to exhibit strong nonclassicality? More precisely, we ask: which GPTs correspond to operational theories that do not admit of a noncontextual model? It turns out that, for prepare-measure experiments, the answer has an elegant geometric characterization in terms of whether or not it is possible to find a simplex and a hypercube dual to this simplex that respectively contain the state and effect spaces of the GPT. This result opens up new avenues for experimentally witnessing strong nonclassicality, including the possibility of doing so on data from experiments that were not specifically designed to test noncontextuality.

See Also

Contextuality of General Probabilistic Theories

Farid Shahandeh
PRX Quantum 2, 010330 (2021)

Article Text

References (66)

  1. J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195 (1964).
  2. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  3. R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements, Phys. Rev. A 71, 052108 (2005).
  4. Refs. [48, 49, 50, 51, 52, 53] provided alternative proposals for how to operationalize the Kochen-Specker notion of noncontextuality [54]. See Refs. [21, 55, 56] for a critique of these.
  5. R. W. Spekkens, Noncontextuality: how we should define it, why it is natural, and what to do about its failure, PIRSA:17070035 (2017).
  6. R. W. Spekkens, Negativity and Contextuality are Equivalent Notions of Nonclassicality, Phys. Rev. Lett. 101, 020401 (2008).
  7. C. Ferrie and J. Emerson, Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations, J. Phys. A 41, 352001 (2008).
  8. M. F. Pusey, Anomalous Weak Values Are Proofs of Contextuality, Phys. Rev. Lett. 113, 200401 (2014).
  9. R. Kunjwal, M. Lostaglio, and M. F. Pusey, Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts, Phys. Rev. A 100, 042116 (2019).
  10. R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Preparation Contextuality Powers Parity-Oblivious Multiplexing, Phys. Rev. Lett. 102, 010401 (2009).
  11. A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, Optimal bounds for parity-oblivious random access codes, New J. Phys. 18, 045003 (2016).
  12. A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai, Parity Oblivious d-Level Random Access Codes and Class of Noncontextuality Inequalities, arXiv:1607.05490 (2016).
  13. D. Saha, P. Horodecki, and M. Pawłowski, State independent contextuality advances one-way communication, New J. Phys. 21, 093057 (2019).
  14. D. Schmid and R. W. Spekkens, Contextual Advantage for State Discrimination, Phys. Rev. X 8, 011015 (2018).
  15. M. Lostaglio and G. Senno, Contextual advantage for state-dependent cloning, arXiv:1905.08291 (2019).
  16. M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’ for quantum computation, Nature 510, 351 (2014).
  17. R. Raussendorf, Contextuality in measurement-based quantum computation, Phys. Rev. A 88, 022322 (2013).
  18. M. J. Hoban, E. T. Campbell, K. Loukopoulos, and D. E. Browne, Non-adaptive measurement-based quantum computation and multi-party Bell inequalities, New J. Phys. 13, 023014 (2011).
  19. S. A. Yadavalli and R. Kunjwal, Contextuality in entanglement-assisted one-shot classical communication, arXiv:2006.00469 (2020).
  20. R. W. Spekkens, The ontological identity of empirical indiscernibles: Leibniz’s methodological principle and its significance in the work of Einstein, arXiv:1909.04628 (2019).
  21. M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, An experimental test of noncontextuality without unphysical idealizations, Nat. Comm. 7, 1 (2016).
  22. L. Hardy, Quantum Theory From Five Reasonable Axioms, quant-ph/0101012 (2001).
  23. J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007).
  24. In previous work (and here), these relations are only stipulated in the “ordinary language” description of the procedures. In forthcoming work [57], we show how they can be incorporated into the formal structure of an operational theory.
  25. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Probabilistic theories with purification, Phys. Rev. A 81, 062348 (2010).
  26. This uniqueness is referred to as the “causality axiom” in Ref. [25].
  27. O. Magidor, in The Stanford Encyclopedia of Philosophy (Metaphysics Research Lab, Stanford University, 2019).
  28. E. G. Beltrametti and S. Bugajski, A classical extension of quantum mechanics, J. Phys. A 28, 3329 (1995).
  29. R. W. Spekkens, in Quantum Theory: Informational Foundations and Foils, edited by G. Chiribella and R. W. Spekkens (Springer Netherlands, Dordrecht, 2016), p. 83.
  30. Note, however, that there it was the nonclassicality of operational phenomena rather than of operational theories that was at issue.
  31. In our formalization of GPTs, a measurement is any collection of effects that sum to the unit effect.
  32. M. Plávala, All measurements in a probabilistic theory are compatible if and only if the state space is a simplex, Phys. Rev. A 94, 042108 (2016).
  33. F. Shahandeh, Contextuality of general probabilistic theories, arXiv:1911.11059 (2020).
  34. For many years, the no-restriction hypothesis was taken to be a fundamental part of the definition of a GPT (indeed, it was only in Ref. [25] in 2009 that the term “no-restriction hypothesis” was coined), but it was later acknowledged to be a substantial restriction on the sorts of GPTs that could be considered. Examples of GPTs that violate the no-restriction hypothesis include: the GPT associated to Spekkens’ toy model [47], the stabilizer subtheories of quantum theory as well as the Gaussian and quadrature subtheories of quantum mechanics [29], and the GPTs considered in, for example, Refs. [36, 58, 59, 60, 61, 62].
  35. H. Barnum and L. Lami, Classical embeddability of general probabilistic theories, Forthcoming.
  36. M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories, arXiv:1710.05948 (2017).
  37. M. F. Pusey, L. del Rio, and B. Meyer, Contextuality without access to a tomographically complete set, arXiv:1904.08699 (2019).
  38. D. Schmid, R. W. Spekkens, W. Robert, and E. Wolfe, All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences, Phys. Rev. A 97, 062103 (2018).
  39. This set is usually inferred from a preexisting no-go theorem for generalized noncontextuality in quantum theory, for instance, those inspired by Kochen-Specker contradictions [55, 63, 64, 65].
  40. In fact, because the procedures that are experimentally realized always deviate from the targets, to satisfy the operational equivalences, one must shift attention to “secondary” procedures that lie within the convex hull of the realized ones, a step that compromises some purity [66].
  41. Nor postprocess the data to satisfy them.
  42. V. Gitton and M. P. Woods, Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario, arXiv:2003.06426 (2020).
  43. D. Schmid, J. H. Selby, M. F. Pusey, and R. W. Spekkens, A structure theorem for generalized-noncontextual ontological models, arXiv:2005.07161 (2020).
  44. L. Hardy, Reformulating and reconstructing quantum theory, arXiv:1104.2066 (2011).
  45. D. Schmid, H. Du, J. H. Selby, and M. F. Pusey, The only noncontextual model of the stabilizer subtheory is Gross's, arXiv:2101.06263 (2021).
  46. For a GPT G that is simplex-embeddable but not itself simplicial, the embedding relation cannot be interpreted as merely due to technological limitations. To posit that a world is governed by the GPT G is to posit that the states and effects that are physically realizable are all and only the states and effects included in G. Consequently, any states and effects that are included in the simplicial GPT into which G embeds, but are outside of G, are by assumption not physically realizable in a world governed by the GPT G. The embedding relation must therefore be understood as a fundamental (rather than practical) limitation on what is possible in such a world.
  47. R. W. Spekkens, Evidence for the epistemic view of quantum states: A toy theory, Phys. Rev. A 75, 032110 (2007).
  48. A. Cabello and G. García-Alcaine, Proposed Experimental Tests of the Bell-Kochen-Specker Theorem, Phys. Rev. Lett. 80, 1797 (1998).
  49. A. Cabello, Kochen–Specker theorem and experimental test on hidden variables, Int. J. Mod. Phys. A 15, 2813 (2000).
  50. C. Simon, M. Żukowski, H. Weinfurter, and A. Zeilinger, Feasible “Kochen-Specker” Experiment with Single Particles, Phys. Rev. Lett. 85, 1783 (2000).
  51. C. Simon, i. c. v. Brukner, and A. Zeilinger, Hidden-Variable Theorems for Real Experiments, Phys. Rev. Lett. 86, 4427 (2001).
  52. J.-Å.. Larsson, A Kochen-Specker inequality, Europhys. Lett. 58, 799 (2002).
  53. A. Cabello, S. Filipp, H. Rauch, and Y. Hasegawa, Proposed Experiment for Testing Quantum Contextuality with Neutrons, Phys. Rev. Lett. 100, 130404 (2008).
  54. S. Kochen and E. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17, 59 (1967), also available from the Indiana Univ. Math. J.
  55. R. Kunjwal and R. W. Spekkens, From the Kochen-Specker Theorem to Noncontextuality Inequalities Without Assuming Determinism, Phys. Rev. Lett. 115, 110403 (2015).
  56. R. Kunjwal, Contextuality beyond the Kochen-Specker theorem, arXiv:1612.07250 (2016).
  57. D. Schmid, J. Selby, and R. W. Spekkens, Unscrambling the omelette of causation and inference: The framework of causal-inferential theories, arXiv:2009.03297 (2020).
  58. P. Janotta and R. Lal, Generalized probabilistic theories without the no-restriction hypothesis, Phys. Rev. A 87, 052131 (2013).
  59. P. Janotta and R. Lal, Non-locality in theories without the no-restriction hypothesis, arXiv:1412.8524 (2014).
  60. A. B. Sainz, Y. Guryanova, A. Acín, and M. Navascués, Almost-Quantum Correlations Violate the No-Restriction Hypothesis, Phys. Rev. Lett. 120, 200402 (2018).
  61. S. N. Filippov, S. Gudder, T. Heinosaari, and L. Leppäjärvi, Operational restrictions in general probabilistic theories, Found. Phys. 50, 850 (2020).
  62. V. J. Wright and S. Weigert, General Probabilistic Theories with a Gleason-type Theorem, arXiv:2005.14166 (2020).
  63. R. Kunjwal and R. W. Spekkens, From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities, arXiv:1708.04793 (2017).
  64. R. Kunjwal, Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements, Quantum 3, 184 (2019).
  65. R. Kunjwal, Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem, Quantum 4, 219 (2020).
  66. M. F. Pusey, The robust noncontextuality inequalities in the simplest scenario, arXiv:1506.04178 (2015).

Outline

Information

PRX Quantum 2, 010331– Published 25 February, 2021
  • Received 27 January 2020
  • Revised 28 July 2020
  • Accepted 11 January 2021
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