# Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario

Victor Gitton and Mischa P. Woods

Institute for Theoretical Physics, ETH Zürich, Switzerland

### Abstract

Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called $\textit{unit separability}$, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called unit separability, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

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### Cited by

[1] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens, "A structure theorem for generalized-noncontextual ontological models", arXiv:2005.07161.

[2] John H. Selby, Elie Wolfe, David Schmid, and Ana Belén Sainz, "An open-source linear program for testing nonclassicality", arXiv:2204.11905.

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

[4] David Schmid, John Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens, "The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories", arXiv:1911.10386.

[5] Roberto D. Baldijão, Rafael Wagner, Cristhiano Duarte, Bárbara Amaral, and Marcelo Terra Cunha, "Emergence of Noncontextuality under Quantum Darwinism", PRX Quantum 2 3, 030351 (2021).

[6] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", arXiv:2112.04521.

[7] Sergii Strelchuk and Mischa P. Woods, "Measuring time with stationary quantum clocks", arXiv:2106.07684.

[8] Carlos de Gois, George Moreno, Ranieri Nery, Samuraí Brito, Rafael Chaves, and Rafael Rabelo, "General Method for Classicality Certification in the Prepare and Measure Scenario", PRX Quantum 2 3, 030311 (2021).

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