Detection Loophole in Asymmetric Bell Experiments

The problem of closing the detection loophole with asymmetric systems, such as entangled atom-photon pairs, is addressed. We show that, for the Bell inequality $I_{3322}$, a minimal detection efficiency of 43% can be tolerated for one of the particles, if the other one is always detected. We also study the influence of noise and discuss the prospects of experimental implementation.

Figure 1

Numerical optimization of the threshold efficiency ${\eta }_B^{\mathrm{th}}$ as a function of $\theta$. Thick lines: results for pure states, for CHSH (dashed-dotted line) and $I_{3322}$ (full line). For $I_{3322}$ the threshold efficiency ${\eta }_B^{\mathrm{th}}$ goes down to $\sim 43\%$ in the limit of weakly entangled states. Thin full lines: $I_{3322}$ with background noise, i.e., states (6); thin dashed lines: $I_{3322}$ with detection errors, i.e., states (7); with error value for both.

Figure 2

Minimal detection efficiency ${\eta }_B$ required for a given noise power. The curves for the symmetric case, ${\eta }_A={\eta }_B$, are also plotted, the curve for CHSH being Eberhard’s result. Though CHSH provides a smaller threshold efficiency for any noise power than $I_{3322}$ in the symmetric case, $I_{3322}$ can tolerate smaller efficiencies than CHSH when $p<6\%$ and ${\eta }_A=1$. This is probably due to the fact that inequality $I_{3322}$ is asymmetric, contrary to CHSH.