  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 Alice's measurements is used to generate the key, and the CHSH
expression, which is symmetric with respect to Alice's two measurements. We
therefore investigate the impact of using an extended family of Bell
expressions where we give different weights to Alice's 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 Alice's 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 Alice's 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.
