Bell inequalities for continuous-variable correlations

TL;DR

This paper introduces a new class of Bell inequalities applicable to continuous-variable systems, demonstrating that violations can persist at high detector efficiencies and large numbers of parties, highlighting the role of non-commutativity.

Contribution

It derives a novel class of correlation Bell inequalities for continuous and unbounded observables, emphasizing second-moment correlations and the significance of non-commutativity.

Findings

Bell violations persist at high detector efficiencies

Violations can occur even in the macroscopic limit of large n

The method links Bell inequalities to variance inequalities

Abstract

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.