Exponentially Decreasing Critical Detection Efficiency for Any Bell Inequality

We address the problem of closing the detection efficiency loophole in Bell experiments, which is crucial for real-world applications. Every Bell inequality has a critical detection efficiency $\eta$ that must be surpassed to avoid the detection loophole. Here, we propose a general method for reducing the critical detection efficiency of any Bell inequality to arbitrary low values. This is accomplished by entangling two particles in $N$ orthogonal subspaces (e.g., $N$ degrees of freedom) and conducting $N$ Bell tests in parallel. Furthermore, the proposed method is based on the introduction of penalized $N$-product (PNP) Bell inequalities, for which the so-called simultaneous measurement loophole is closed, and the maximum value for local hidden-variable theories is simply the $N$th power of the one of the Bell inequality initially considered. We show that, for the PNP Bell inequalities, the critical detection efficiency decays exponentially with $N$. The strength of our method is illustrated with a detailed study of the PNP Bell inequalities resulting from the Clauser-Horne-Shimony-Holt inequality.

Figure 1

(Solid line) Critical detection efficiency ${\eta }_{\mathrm{crit}}$ for the PNP Bell inequality as a function of $N$. (Dashed line) Visibility (per qubit pair) required for a loophole-free Bell test when $\eta =0.75$ as a function of $N$. Perfect statistics is assumed, i.e., the penalty term $\mathrm{A}+\mathrm{B}=0$.

Figure 2

Required detection efficiency for different values of the visibility $v$, as a function of $N$. Lines in the plot are arranged from bottom to top as $v$ changes from 1 to 0.85. The value of the penalty term $\kappa (\mathrm{A}+\mathrm{B})$ in Eq. (4) is taken to be ${10}^{-5}\kappa$.