Generalised contextuality of continuous variable quantum theory can be revealed with a single projective measurement

TL;DR

This paper extends the concept of generalized contextuality to continuous variable quantum systems, showing that classical measurements can reveal quantum contextuality through a modified definition based on finite measurement effects.

Contribution

It introduces a new definition of generalized contextuality for continuous variables, addressing limitations of the standard approach and linking it to classical measurement schemes.

Findings

Classical, commuting measurements can demonstrate quantum contextuality in continuous systems.

A physically-motivated approximation procedure aligns the new definition with noncontextual models.

Structural results on entanglement breaking channels in infinite dimensions are established.

Abstract

Generalized contextuality is a possible indicator of non-classical behaviour in quantum information theory. In finite-dimensional systems, this is justified by the fact that noncontextual theories can be embedded into some simplex, i.e. into a classical theory. We show that a direct application of the standard definition of generalized contextuality to continuous variable systems does not envelope the statistics of some basic measurements, such as the position observable. In other words, we construct families of fully classical, i.e. commuting, measurements that nevertheless can be used to show contextuality of quantum theory. To overcome the apparent disagreement between the two notions of classicality, that is commutativity and noncontextuality, we propose a modified definition of generalised contextuality for continuous-variable systems. The modified definition is based on a physically-motivated approximation procedure, that uses only finite sets of measurement effects. We prove that in the limiting case this definition corresponds exactly to an extension of noncontextual models that benefits from non-constructive response functions. In the process, we discuss the extension of a known connection between contextuality and no-broadcasting to the continuous-variable scenario, and prove structural results regarding fixed points of infinite-dimensional entanglement breaking channels.