  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.
