Proposed Test of Quantum Nonlocality for Continuous Variables

We propose a test of nonlocality for continuous variables using a two-mode squeezed state as the source of nonlocal correlations and a measurement scheme based on conditional homodyne detection. Both the Clauser-Horne-Shimony-Holt and the Clauser-Horne inequality are constructed from the conditional homodyne data and found to be violated for a squeezing parameter larger than $r\approx 0.48$.

Figure 1

Schematic diagram of the conditional homodyne detection of an EPR-correlated state. Two beams of light in the correlated state ${\rho }_{12}$ are shared at beam splitters $\mathrm{B}\mathrm{S}\mathrm{\left(}\mathrm{\theta }\mathrm{\right)}$ between photodetectors ${\mathrm{P}\mathrm{D}}_3$ and ${\mathrm{P}\mathrm{D}}_4$, and homodyne detectors ${\mathrm{H}\mathrm{D}}_1\left({\phi }_1\right)$ and ${\mathrm{H}\mathrm{D}}_2\left({\phi }_2\right)$. The homodyne data are accepted when both photodetectors fire (each at least once).

Figure 2

$B_{\mathrm{C}\mathrm{H}\mathrm{S}\mathrm{H}}$ as a function of the squeezing parameter $r$ and the reflectance $R={sin}^2\theta$, for quantum efficiency $\eta =1$. The CHSH inequality is violated for $B_{\mathrm{C}\mathrm{H}\mathrm{S}\mathrm{H}}>2$.

Figure 3

(a) $B_{\mathrm{C}\mathrm{H}\mathrm{S}\mathrm{H}}$ as a function of the squeezing parameter $r$ and the reflectance $R={sin}^2\theta$, for quantum efficiency $\eta =0.3$. (b) $B_{\mathrm{C}\mathrm{H}\mathrm{S}\mathrm{H}}$ as a function of $\eta$, for $r=0.6$ and $R\approx 0.9891$.