Sending-or-not-sending twin-field quantum key distribution with discrete-phase-randomized weak coherent states

Abstract

In many security proofs of quantum key distribution, the random phases of coherent states are assumed to be continuously modulated. However, in practice, we can only take discrete phase randomization to coherent-state sources. In this paper, we study the sending-or-not-sending (SNS) protocol with discrete-phase-randomized coherent states. We present the security proof of the SNS protocol with discrete phase modulation. We then present analytic formulas for key rate calculation. With the decoy-state method and the properties of trace distance, we get the analytical formula of the upper bound of the phase-flip error rate. We also get the lower bound of the yield of untagged bits, which can be calculated by either analytical formula or linear programming. Our numerical simulation results show that with only six phase values, the key rates of the SNS protocol can exceed the linear bound and, with 12 phase values, the key rates are very close to the results of the SNS protocol with continuously modulated phase randomization.