All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences

TL;DR

This paper develops a comprehensive algorithmic framework to derive all noncontextuality inequalities for finite prepare-and-measure experiments, enabling complete characterization of noncontextual models and their quantum representations.

Contribution

It introduces a systematic method to generate necessary and sufficient noncontextuality inequalities based on operational equivalences, applicable to any finite experiment scenario.

Findings

The space of noncontextual data forms a polytope.

The method provides necessary and sufficient conditions for noncontextual models.

Efficient algorithms for testing noncontextuality in experimental data.

Abstract

Within the framework of generalized noncontextuality, we introduce a general technique for systematically deriving noncontextuality inequalities for any experiment involving finitely many preparations and finitely many measurements, each of which has a finite number of outcomes. Given any fixed sets of operational equivalences among the preparations and among the measurements as input, the algorithm returns a set of noncontextuality inequalities whose satisfaction is necessary and sufficient for a set of operational data to admit of a noncontextual model. Additionally, we show that the space of noncontextual data tables always defines a polytope. Finally, we provide a computationally efficient means for testing whether any set of numerical data admits of a noncontextual model, with respect to any fixed operational equivalences. Together, these techniques provide complete methods for characterizing arbitrary noncontextuality scenarios, both in theory and in practice. Because a quantum prepare-and-measure experiment admits of a noncontextual model if and only if it admits of a positive quasiprobability representation, our techniques also determine the necessary and sufficient conditions for the existence of such a representation.