  The BB84 quantum key distribution protocol is semi device independent in the
sense that it can be shown to be secure if just one of the users' devices is
restricted to a qubit Hilbert space. Here, we derive an analytic lower bound on
the asymptotic secret key rate for the entanglement-based version of BB84
assuming only that one of the users performs unknown qubit POVMs. The result
holds against the class of collective attacks and reduces to the well known
Shor-Preskill key rate for correlations corresponding to the ideal BB84
correlations mixed with any amount of random noise.
