Bell inequalities for Continuous-Variable Measurements

TL;DR

This paper develops Bell inequalities for continuous-variable quantum measurements, comparing methods with and without binning, and demonstrates that optimized moment-based inequalities offer more robust violations, enabling potential loophole-free tests.

Contribution

It introduces extended functional Bell inequalities for continuous variables and compares their robustness to noise and loss against existing methods.

Findings

Optimized moment-based inequalities outperform binning methods.

Unbinned inequalities show greater robustness to noise and detection loss.

Potential for loophole-free Bell tests with continuous-variable systems.

Abstract

Tests of local hidden variable theories using measurements with continuous variable (CV) outcomes are developed, and a comparison of different methods is presented. As examples, we focus on multipartite entangled GHZ and cluster states. We suggest a physical process that produces the states proposed here, and investigate experiments both with and without binning of the continuous variable. In the former case, the Mermin-Klyshko inequalities can be used directly. For unbinned outcomes, the moment-based CFRD inequalities are extended to functional inequalities by considering arbitrary functions of the measurements at each site. By optimising these functions, we obtain more robust violations of local hidden variable theories than with either binning or moments. Recent inequalities based on the algebra of quaternions and octonions are compared with these methods. Since the prime advantage of CV experiments is to provide a route to highly efficient detection via homodyne measurements, we analyse the effect of noise and detection losses in both binned and unbinned cases. The CV moment inequalities with an optimal function have greater robustness to both loss and noise. This could permit a loophole-free test of Bell inequalities.