Advantage distillation for device-independent quantum key distribution

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 $q \approx 9.1\%$ or limited detector efficiencies of $η\approx 89.1\%$ in a 2-input 2-output scenario. This exceeds the noise thresholds of $q \approx 7.1\%$ and $η\approx 90.7\%$ 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.

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