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

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