  Parametric down-conversion (PDC) sources can be used for quantum key
distribution (QKD). One can use a PDC source as a triggered single photon
source. Recently, there are various practical proposals of the decoy state QKD
with triggering PDC sources. In this paper, we generalize the passive decoy
state idea, originally proposed by Mauerer and Silberhorn. The generalized
passive decoy state idea can be applied to cases where either threshold
detectors or photon number resolving detectors are used. The decoy state
protocol proposed by Adachi, Yamamoto, Koashi and Imoto (AYKI) can be treated
as a special case of the generalized passive decoy state method. By simulating
a recent PDC experiment, we compare various practical decoy state protocols
with the infinite decoy protocol and also compare the cases using threshold
detectors and photon-number resolving detectors. Our simulation result shows
that with the AYKI protocol, one can achieve a key generation rate that is
close to the theoretical limit of infinite decoy protocol. Furthermore, our
simulation result shows that a photon-number resolving detector appears to be
not very useful for improving QKD performance in this case. Although our
analysis is focused on the QKD with PDC sources, we emphasize that it can also
be applied to other QKD setups with triggered single photon sources.
