Abstract
We derive a sufficient condition for advantage distillation to be secure against collective attacks in device-independent quantum key distribution (DIQKD), focusing on the repetition-code protocol. In addition, we describe a semidefinite programming method to check whether this condition holds for any probability distribution obtained in a DIQKD protocol. Applying our method to various probability distributions, we find that advantage distillation is possible up to depolarising-noise values of q ≈ 9.1% or limited detector efficiencies of η ≈ 89.1% in a 2-input 2-output scenario. This exceeds the noise thresholds of q ≈ 7.1% and η ≈ 90.7% respectively for DIQKD with one-way error correction using the CHSH inequality, thereby showing that it is possible to distill secret key beyond those thresholds.