Testing for Multipartite Quantum Nonlocality Using Functional Bell Inequalities

We show that arbitrary functions of continuous variables, e.g., position and momentum, can be used to generate tests that distinguish quantum theory from local hidden variable theories. By optimizing these functions, we obtain more robust violations of local causality than obtained previously. We analytically calculate the optimal function and include the effect of nonideal detectors and noise, revealing that optimized functional inequalities are resistant to standard forms of decoherence. These inequalities could allow a loophole-free Bell test with efficient homodyne detection.

Figure 1

Maximum violations of functional cv inequality with GHZ states as a function of the number of modes. The violations using the optimal function (solid) are much stronger than the CFRD result (dashed).

Figure 2

(a) The critical minimum detection efficiency ${\eta }_{\mathrm{crit}}$ for pure state, and (b) the critical purity $p_{\mathrm{crit}}$ for ideal detectors required for violation of functional moment, CFRD and MK inequalities with optimal choice of parameters.