  Differential-phase-shift (DPS) quantum key distribution (QKD) is one of the
major QKD protocols that can be implemented with a simple setup using a laser
source and a passive detection unit. Recently, an information-theoretic
security proof of this protocol has been established in [npj Quant. Inf. 5, 87
(2019)] assuming the infinitely large number of emitted pulses. To implement
the DPS protocol in a real-life world, it is indispensable to analyze the
security with the finite number of emitted pulses. The extension of the
security proof to the finite-size regime requires the accommodation of the
statistical fluctuations to determine the amount of privacy amplification. In
doing so, Azuma's inequality is often employed, but unfortunately we show that
in the case of the DPS protocol, this results in a substantially low key rate.
This low key rate is due to a loose estimation of the sum of probabilities
regarding three-photon emission whose probability of occurrence is very small.
The main contribution of our work is to show that this obstacle can be overcome
by exploiting the recently found novel concentration inequality, Kato's
inequality. As a result, the key rate of the DPS protocol is drastically
improved. For instance, assuming typical experimental parameters, a 3 Mbit
secret key can be generated over 77 km for 8.3 hours, which shows the
feasibility of DPS QKD under a realistic setup.

Quantum Physics