# Continuous-variable nonlocality and contextuality

## TL;DR

This paper develops a framework for analyzing quantum contextuality in continuous-variable systems, extending key theorems and introducing measures to quantify nonlocality and contextuality.

## Contribution

It extends the Fine--Abramsky--Brandenburger theorem to continuous variables and introduces the contextual fraction as a quantifiable measure in this setting.

## Key findings

- The theorem extension links Bell nonlocality to contextuality in continuous variables.
- The contextual fraction is shown to be a non-increasing monotone under classical operations.
- A hierarchy of semi-definite programs is developed to compute the contextual fraction accurately.

## Abstract

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine--Abramsky--Brandenburger theorem extends to this setting, an important consequence of which is that Bell nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, is also defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program. Through Lasserre relaxations, we are able to express this infinite linear program as a hierarchy of semi-definite programs that allow to calculate the contextual fraction with increasing accuracy.