Secure Quantum Key Distribution with an Uncharacterized Source

We prove the security of the Bennett-Brassard (BB84) quantum key distribution protocol for an arbitrary source whose averaged states are basis independent, a condition that is automatically satisfied if the source is suitably designed. The proof is based on the observation that, to an adversary, the key extraction process is equivalent to a measurement in the ${\widehat{\sigma }}_x$ basis performed on a pure ${\widehat{\sigma }}_z$-basis eigenstate. The dependence of the achievable key length on the bit error rate is the same as that established by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)] for a perfect source, indicating that the defects in the source are efficiently detected by the protocol.