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

The performance of quantum key distribution 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 improves the analysis of its key rate. Specifically, we reveal that the key rate is proportional to η1+1n-2 and this scaling 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.