Blind post-processing for the unbalanced BB84

For the realization of quantum key distribution, it is important to investigate its security based on a mathematical model that captures properties of the actual devices used by the legitimate users. Recently, Ferenczi, et. al. (Phys. Rev. A 86 042327 (2012)) pointed out potential influences that the losses in phase modulators and/or the unbalance in the transmission rate of beam splitters may have on the security of the phase-encoded BB84 and analyzed the security of this scheme, which is called the unbalanced BB84. In this paper, we ask whether blindly applying the post-processing of the balanced BB84 to the unbalanced BB84 would lead to an insecure key or not, and we conclude that we can safely distill a secure key even with this post-processing. It follows from our proof that as long as the unbalances are basis-independent, our conclusion holds even if the unbalances are unknown and fluctuate in time.