  Decoy-state quantum key distribution (QKD) is undoubtedly the most efficient
solution to handle multi-photon signals emitted by laser sources, and provides
the same secret key rate scaling as ideal single-photon sources. It requires,
however, that the phase of each emitted pulse is uniformly random. This might
be difficult to guarantee in practice, due to inevitable device imperfections
and/or the use of an external phase modulator for phase randomization, which
limits the possible selected phases to a finite set. Here, we investigate the
security of decoy-state QKD with arbitrary, continuous or discrete, non-uniform
phase randomization, and show that this technique is quite robust to deviations
from the ideal uniformly random scenario. For this, we combine a novel
parameter estimation technique based on semi-definite programming, with the use
of basis mismatched events, to tightly estimate the parameters that determine
the achievable secret key rate. In doing so, we demonstrate that our analysis
can significantly outperform previous results that address more restricted
scenarios.
