  We present improved results and more general results for the decoy-state
measurement device independent quantum key distribution (MDI-QKD) where the
intensities of all 3 pulses can be non-zero. We present a more tightened
explicit formula for the lower bound of the yield of two single-photon pulses
for this generalized 3-intensity protocol, which can be applied to the recently
proposed MDI-QKD with imperfect single-photon sources such as the coherent
states or the heralded states from the parametric down-conversion. We strictly
prove that our result is better than the prior art result. With this result,
the final key rate rises drastically.
