  We Haar uniformly generate random states of various ranks and study their
performance in an entanglement-based quantum key distribution (QKD) task. In
particular, we analyze the efficacy of random two-qubit states in realizing
device-independent (DI) QKD. We first find the normalized distribution of
entanglement and Bell-nonlocality which are the key resource for DI-QKD for
random states ranging from rank-1 to rank-4. The number of entangled as well as
Bell-nonlocal states decreases as rank increases. We observe that decrease of
the secure key rate is more pronounced in comparison to that of the quantum
resource with increase in rank. We find that the pure state and Werner state
provide the upper and lower bound, respectively, on the minimum secure key rate
of all mixed two-qubit states possessing the same magnitude of entanglement
under general as well as optimal collective attack strategies.
