  In recent years, the gap between theory and practice in quantum key
distribution (QKD) has been significantly narrowed, particularly for QKD
systems with arbitrarily awed optical receivers. The status for QKD systems
with imperfect light sources is however less satisfactory, in the sense that
the resulting secure key rates are often overly-dependent on the quality of
state preparation. This is especially the case when the channel loss is high.
Very recently, to overcome this limitation, Tamaki et al proposed a QKD
protocol based on the so-called rejected data analysis, and showed that its
security|in the limit of infinitely long keys|is almost independent of any
encoding flaw in the qubit space, being this protocol compatible with the decoy
state method. Here, as a step towards practical QKD, we show that a similar
conclusion is reached in the finite-key regime, even when the intensity of the
light source is unstable. More concretely, we derive security bounds for a wide
class of realistic light sources and show that the bounds are also efficient in
the presence of high channel loss. Our results strongly suggest the feasibility
of long distance provably-secure communication with imperfect light sources.
