The simplest device-independent quantum key distribution protocol is based on the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and allows two users, Alice and Bob, to generate a secret key if they observe sufficiently strong correlations. There is, however, a mismatch between the protocol, in which only one of Alices measurements is used to generate the key, and the CHSH expression, which is symmetric with respect to Alices two measurements. We therefore investigate the impact of using an extended family of Bell expressions where we give different weights to Alices measurements. Using this family of asymmetric Bell expressions improves the robustness of the key distribution protocol for certain experimentally-relevant correlations. As an example, the tolerable error rate improves from 7.15% to about 7.42% for the depolarising channel. Adding random noise to Alices key before the postprocessing pushes the threshold further to more than 8.34%. The main technical result of our work is a tight bound on the von Neumann entropy of one of Alices measurement outcomes conditioned on a quantum eavesdropper for the family of asymmetric CHSH expressions we consider and allowing for an arbitrary amount of noise preprocessing.
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