Noncontextuality inequalities for prepare-transform-measure scenarios

David Schmid; Roberto D. Baldijão; John H. Selby; Ana Belén Sainz; Robert W. Spekkens

doi:10.22331/q-2026-10-27-2250

Noncontextuality inequalities for prepare-transform-measure scenarios

David Schmid^1,2, Roberto D. Baldijão^1,2, John H. Selby¹, Ana Belén Sainz^1,3, and Robert W. Spekkens²

¹International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-309 Gdańsk, Poland
²Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5
³Basic Research Community for Physics e.V., Germany

Published:	2026-10-27, volume 10, page 2250
Editor:	Leon Loveridge
Eprint:	arXiv:2407.09624v3
Doi:	https://doi.org/10.22331/q-2026-10-27-2250
Citation:	Quantum 10, 2250 (2026).

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Abstract

We provide the first systematic technique for deriving witnesses of contextuality in prepare-transform-measure scenarios. More specifically, we show how linear quantifier elimination can be used to compute a polytope of correlations consistent with generalized noncontextuality in such scenarios. This polytope is specified as a set of noncontextuality inequalities that are necessary and sufficient conditions for observed data in the scenario to admit of a classical explanation relative to any linear operational identities, if one ignores some constraints from diagram preservation. While including these latter constraints generally leads to tighter inequalities, it seems that nonlinear quantifier elimination would be required to systematically include them. We also provide a linear program which can certify the nonclassicality of a set of numerical data arising in a prepare-transform-measure experiment. We apply our results to get a robust noncontextuality inequality for transformations that can be violated within the stabilizer subtheory. Finally, we give a simple algorithm for computing all the linear operational identities holding among a given set of states, of transformations, or of measurements.

Featured image: An ontological model for a scenario with transformations. We derive constraints on the data that can be generated in any such model that is consistent with the principle of noncontextuality, which captures the class of scenarios that are classically explainable.

Popular summary

For any experiment on a quantum system, a natural question is whether the observed data can be reproduced by a classical model. By this, we mean that each system has definite properties at all times, that these change probabilistically each time a laboratory procedure is applied to the system, and that the manner in which they change is consistent with a principle known as generalized noncontextuality. This principle demands that procedures that cannot be distinguished by any experiment should be represented identically in the underlying model. Until now, practical tests of this kind of nonclassicality have been developed mainly for the simplest class of experiments, in which a single system is prepared and then immediately measured. In this work, we extend develop systematic methods for studying experiments in which the system also undergoes a transformation between preparation and measurement. We give methods for deriving noncontextuality inequalities, namely constraints on the observed statistics that any classical model must satisfy, and which remain applicable to the noisy and imperfect data of real experiments. We also provide an efficient computational test that, given a table of experimental data, either constructs a classical model for it, or identifies an inequality demonstrating that no such model exists. As an application, we show that transformations within the single-qubit stabilizer subtheory can violate one of our inequalities in a noise-robust manner. Our tests are not yet complete, as they do not incorporate certain constraints arising from the sequential composition of operations; nonetheless, they constitute the first general method for certifying nonclassicality in experiments involving transformations, and hence a first step toward understanding nonclassicality in general circuits and computations.

► BibTeX data

@article{Schmid2026noncontextuality,
  doi = {10.22331/q-2026-10-27-2250},
  url = {https://doi.org/10.22331/q-2026-10-27-2250},
  title = {Noncontextuality inequalities for prepare-transform-measure scenarios},
  author = {Schmid, David and Baldij{\~{a}}o, Roberto D. and Selby, John H. and Sainz, Ana Bel{\'{e}}n and Spekkens, Robert W.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2250},
  month = oct,
  year = {2026}
}

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[3] 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).

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