Closing the Detection Loophole in Bell Experiments Using Qudits

We show that the detection efficiencies required for closing the detection loophole in Bell tests can be significantly lowered using quantum systems of dimension larger than two. We introduce a series of asymmetric Bell tests for which an efficiency arbitrarily close to $1/N$ can be tolerated using $N$-dimensional systems, and a symmetric Bell test for which the efficiency can be lowered down to 61.8% using four-dimensional systems. Experimental perspectives for our schemes look promising considering recent progress in atom-photon entanglement and in photon hyperentanglement.

Figure 1

Asymmetric Bell test with qutrits. Threshold efficiency ${\eta }_B$ as a function of the degree of entanglement $ϵ$ for different values of background noise $p$. For $p=0$, the efficiency tends to 33.3% when $ϵ\to 0$, and is equal to 66.7% at the value $ϵ=0$ (i.e., for maximally entangled qubits). For $p>0$, the curve exhibits a plateau when the efficiency becomes equal to the one given by maximally entangled qubits.

Figure 2

Symmetric Bell test with four-dimensional systems. Threshold efficiency $\eta$ as a function of the degree of entanglement $ϵ$ for different values of background noise $p$. For maximally entangled states ($ϵ=\frac{1}{2}$) an efficiency of 76.98% can be tolerated; for very partially entangled states ($ϵ\to 0$), the efficiency drops down to 61.8%.

Figure 3

Minimal detection efficiency $\eta$ for a given amount of noise $p$. The curve for the CHSH inequality is Eberhard’s result [12]. For a realistic amount of background noise ($p<2\%$), our models can tolerate efficiencies 5%–8% lower than the CHSH inequality.