Finite-Key Analysis for Quantum Key Distribution with Discrete-Phase Randomization

Quantum key distribution (QKD) allows two remote parties to share information-theoretic secret keys. Many QKD protocols assume the phase of encoding state can be continuous randomized from 0 to 2 pi, which, however, may be questionable in the experiment. This is particularly the case in the recently proposed twin-field (TF) QKD, which has received a lot of attention since it can increase the key rate significantly and even beat some theoretical rate-loss limits. As an intuitive solution, one may introduce discrete-phase randomization instead of continuous randomization. However, a security proof for a QKD protocol with discrete-phase randomization in the finite-key region is still missing. Here, we develop a technique based on conjugate measurement and quantum state distinguishment to analyze the security in this case. Our results show that TF-QKD with a reasonable number of discrete random phases, e.g., 8 phases from {0,pi/4,pi/2, horizontal ellipsis ,7 pi/4}, can achieve satisfactory performance. On the other hand, we find the finite-size effects become more notable than before, which implies that more pulses should be emit in this case. More importantly, as a the first proof for TF-QKD with discrete-phase randomization in the finite-key region, our method is also applicable in other QKD protocols.