Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles

TL;DR

This paper demonstrates that using multiple copies of maximally entangled two-qubit states can lower the detector efficiency threshold needed to observe Bell nonlocality, surpassing the traditional CHSH limit.

Contribution

The authors introduce a method to reduce the detection efficiency threshold in Bell tests by entangling multiple copies of two-qubit states, using convex optimization techniques.

Findings

Thresholds decrease exponentially with the number of copies

Achieved thresholds of 80.86%, 73.99%, and 69.29% for 2, 3, and 4 copies

Utilized large-scale convex optimization and linear programming

Abstract

In this paper, we investigate the critical efficiency of detectors to observe Bell nonlocality using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles, such as hyperentangled states, and the product of Pauli measurements. It is known that in a Clauser-Horne-Shimony-Holt (CHSH) Bell test the symmetric detection efficiency of $82.84\%$ can be tolerated for the two-qubit maximally entangled state. We beat this enigmatic threshold by entangling two particles with multiple degrees of freedom. The obtained upper bounds of the symmetric detection efficiency thresholds are $80.86\%$, $73.99\%$ and $69.29\%$ for two, three and four copies of the two-qubit maximally entangled state, respectively. The number of measurements and outcomes in the respective cases are 4, 8 and 16. To find the improved thresholds, we use large-scale convex optimization tools, which allows us to significantly go beyond state-of-the-art results. The proof is exact up to three copies, while for four copies it is due to reliable numerical computations. Specifically, we used linear programming to obtain the two-copy threshold and the corresponding Bell inequality, and convex optimization based on Gilbert's algorithm for three and four copies of the two-qubit state. We show analytically that the symmetric detection efficiency threshold decays exponentially with the number of copies of the two-qubit state. Our techniques can also be applied to more general Bell nonlocality scenarios with more than two parties.