Improved security analysis for phase-encoding twin-field quantum key distribution

Twin-field (TF) quantum key distribution (QKD) has been extensively researched because of its long transmission distance. The phase-encoding TF-QKD can realize both a high key rate and a long transmission distance, so it has attracted a lot of attention. In this article, we are inspired by the sending-or-not-sending TF-QKD, in which the information leakages of correct rounds and erroneous rounds are calculated separately. And, we notice that this kind of method has been used in many QKD protocols---for example, the six-state QKD. We apply a similar method to the phase-matching TF-QKD and find some improvement in it. With the new analysis, the transmission distance can be improved without any changes in experiments. We also apply the new analysis in the four-phase partial-phase-postselection protocol, and a similar improvement can be achieved.