  Measurement-device-independent quantum key distribution (MDI-QKD) is immune
to all the detection attacks; thus when it is combined with the decoy-state
method, the final key is unconditionally secure, even if a practical weak
coherent source is used by Alice and Bob. However, until now, the analysis of
decoy-state MDI-QKD with a weak coherent source is incomplete. In this paper,
we derive, with only vacuum+weak decoy state, some tight formulas to estimate
the lower bound of yield and the upper bound of error rate for the fraction of
signals in which both Alice and Bob send a single-photon pulse to the untrusted
third party Charlie. The numerical simulations show that our method with only
vacuum+weak decoy state can asymptotically approach the theoretical limit of
the infinite number of decoy states. Furthermore, the statistical fluctuation
due to the finite length of date is also considered based on the standard
statistical analysis.

Quantum Physics