Decoy-state methods [1, 2] are essential to perform quantum key distribution
(QKD) at large distances in the absence of single photon sources. However, the
standard techniques apply only if laser pulses are used that are independent
and identically distributed (iid). Moreover, they require that the laser pulses
are fully phase-randomised. However, realistic high-speed QKD setups do not
meet these stringent requirements [3]. In this work, we generalise decoy-state
analysis to accommodate laser sources that emit imperfectly phase-randomised
states. We also develop theoretical tools to prove the security of protocols
with lasers that emit pulses that are independent, but not identically
distributed. These tools can be used with recent work [4] to prove the security
of laser sources with correlated phase distributions as well. We quantitatively
demonstrate the effect of imperfect phase-randomisation on key rates by
computing the key rates for a simple implementation of the three-state
protocol