General theory for decoy-state quantum key distribution with arbitrary number of intensities

We develop a general theory for quantum key distribution (QKD) in both the forward error correction and the reverse error correction cases when the QKD system is equipped with phase-randomized coherent light with arbitrary number of decoy intensities. For this purpose, generalizing Wang's expansion, we derive a convex expansion of the phase-randomized coherent state. We also numerically check that the asymptotic key generation rates are almost saturated when the number of decoy intensities is three.