Exponentially decreasing critical detection efficiency for any Bell inequality

TL;DR

The paper introduces a general method to exponentially lower the critical detection efficiency in Bell tests by entangling particles across multiple subspaces and conducting parallel Bell tests, thus closing loopholes.

Contribution

It proposes a novel approach using penalized N-product Bell inequalities to reduce detection efficiency thresholds exponentially, enhancing Bell test robustness.

Findings

Critical detection efficiency decays exponentially with the number of subspaces N.

The method closes the simultaneous measurement loophole in Bell inequalities.

Detailed analysis applied to CHSH-based PNP Bell inequalities.

Abstract

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.