  Device-independent quantum key distribution is a secure quantum cryptographic
paradigm that allows two honest users to establish a secret key, while putting
minimal trust in their devices. Most of the existing protocols have the
following structure: first, a bipartite nonlocal quantum state is distributed
between the honest users, who perform local projective measurements to
establish nonlocal correlations. Then, they announce the implemented
measurements and extract a secure key by post-processing their measurement
outcomes. We show that no protocol of this form allows for establishing a
secret key when implemented on certain entangled nonlocal states, namely on a
range of entangled two-qubit Werner states. To prove this result, we introduce
a technique for upper-bounding the asymptotic key rate of device-independent
quantum key distribution protocols, based on a simple eavesdropping attack. Our
results imply that either different tools -- such as different reconciliation
techniques or non-projective measurements -- are needed for device-independent
quantum key distribution in the large-noise regime, or Bell nonlocality is not
sufficient for this task.
