Testing locality and noncontextuality with the lowest moments

TL;DR

This paper investigates the minimal statistical moments needed to demonstrate quantum nonlocality and noncontextuality, revealing that fourth-order correlations are sufficient for such proofs and proposing a new Bell inequality for continuous variables.

Contribution

It identifies that fourth-order correlations can violate classical bounds, providing a new Bell inequality for position and momentum in continuous-variable systems.

Findings

Second-order correlations are classically explainable.

Third-order correlations generally do not violate classical inequalities.

Fourth-order correlations can demonstrate quantum nonlocality and noncontextuality.

Abstract

The quest for fundamental test of quantum mechanics is an ongoing effort. We here address the question of what are the lowest possible moments needed to prove quantum nonlocality and noncontextuality without any further assumption -- in particular without the often assumed dichotomy. We first show that second order correlations can always be explained by a classical noncontextual local-hidden-variable theory. Similar third-order correlations also cannot violate classical inequalities in general, except for a special state-dependent noncontextuality. However, we show that fourth-order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fourth-order continuous-variable Bell inequality for position and momentum, which can be violated and might be useful in Bell tests closing all loopholes simultaneously.