Tight scaling of key rate for differential-phase-shift quantum key distribution

The performance of quantum key distribution (QKD) protocols is evaluated based on the ease of implementation and key generation rate. Among major protocols, the differential-phase-shift (DPS) protocol has the advantage of simple implementation using a train of coherent pulses and a passive detection unit. Unfortunately, however, its key rate is known to be at least proportional to η² with respect to channel transmission η→0. If one can only prove the rate proportional to η² and cannot improve the analysis beyond that, then the DPS protocol will be deemed inferior to other major protocols, such as the decoy BB84 protocol. In this paper, we consider a type of DPS protocol in which the phase of each emitted block comprising n pulses is randomized and significantly improve the analysis of its key rate. Specifically, we reveal that the key rate is proportional to η1+1/n-2 and this rate is tight. This implies that the DPS protocol can achieve a key rate proportional to η for a large number of n, which is the same scaling as the decoy BB84 protocol. Our result suggests that the DPS protocol can achieve a combination of both advantages of ease of implementation and a high key generation rate.