Detection-efficiency mismatch in a satellite-to-ground quantum communication

The detection efficiency mismatch is one of the issues with practical quantum key distribution. The challenge is also inherent in satellite quantum communication due to the detectors and optical elements imperfections in the different quantum channels. Here, we generalize the theory developed for optical fiber QKD to satellite-to-ground QKD with four unbalanced polarization channels to estimate the secret key length. We simulate satellite quantum communication for the measured parameters of the realistic receiving ground station and calculate bounds for the secret key rate and length using two approaches: the first one when you separate the data on bases and the second when calculate the key from the full amount of data. We discuss the advantages and disadvantages of each approach and describe ways to operate with them. For our ground station, the secret key rate turned out to be 20% less when the detection-efficiency mismatch was at 1:2 and we used the optimal way to calculate the key.