Bell Inequalities for Continuous-Variable Correlations

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 noncommutativity for violations of local realism.

Figure 1

Minimum state preparation fidelity ${ϵ}_{\min }$ for ideal detectors (solid line), and minimum detection efficiency ${\eta }_{\min }$ for ideal state preparation (dashed line) required for violation of (8) as a function of the number of modes. The asymptotic value of ${\eta }_{\min }$ is indicated by the dash-dotted line.