A Structure Theorem For Generalized-noncontextual Ontological Models

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David Schmid^1,2,3, John H. Selby¹, Matthew F. Pusey⁴ja Robert W. Spekkens²

¹Rahvusvaheline kvanttehnoloogiate teooria keskus, Gdański ülikool, 80-308 Gdańsk, Poola
²Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Kanada N2L 2Y5
³Kvantarvutite instituut ning füüsika ja astronoomia osakond, Waterloo ülikool, Waterloo, Ontario N2L 3G1, Kanada
⁴Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

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Abstraktne

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.

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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-i andmed

@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} }

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