  Quantum key distribution (QKD) is a method that distributes a secret key to a
sender and a receiver by the transmission of quantum particles (e.g. photons).
Device-independent quantum key distribution (DIQKD) is a version of QKD with a
stronger notion of security, in that the sender and receiver base their
protocol only on the statistics of input and outputs of their devices as
inspired by Bell's theorem. We study the rate at which DIQKD can be carried out
for a given bipartite quantum state distributed between the sender and receiver
or a quantum channel connecting them. We provide upper bounds on the achievable
rate going beyond upper bounds possible for QKD. In particular, we construct
states and channels where the QKD rate is significant while the DIQKD rate is
negligible. This gap is illustrated for a practical case arising when using
standard post-processing techniques for entangled two-qubit states.
