  One of the challenges in practical quantum key distribution is dealing with
efficiency mismatch between different threshold single-photon detectors. There
are known bounds for the secret key rate for the BB84 protocol with
detection-efficiency mismatch provided that the eavesdropper sends no more than
one photon to the legitimate receiver. Here we improve these bounds and give
tight bounds for the secret key rate with a constant detection-efficiency
mismatch under the same single-photon assumption. We propose a method based on
the analytical minimization of the relative entropy of coherence, which can be
used in other problems in quantum key distribution. Also we propose an
adaptation of the decoy state method to proof the security in the case of weak
coherent pulses on the source side.
