Phase-error-rate analysis for quantum key distribution with phase postselection

Quantum key distribution (QKD) stands as a pioneering method for establishing information-theoretically secure communication channels by utilizing the principles of quantum mechanics. In the security proof of QKD, the phase error rate serves as a critical indicator of information leakage and directly influences the security of the shared key bits between communicating parties, Alice and Bob. In estimating the upper bound of the phase error rate, phase randomization and subsequent postselection mechanisms serve pivotal roles across numerous QKD protocols. However, the nonzero interval of phase postselection will introduce intrinsic errors, leading to an overestimation of phase error rate. Here we make a precise phase-error-rate analysis for QKD protocols with phase postselection, which eliminates error rate associated with nonzero interval and helps us to accurately bound the amount of information an eavesdropper may obtain. We further apply our analysis in sending-or-not-sending twin-field quantum key distribution (SNS-TFQKD) and mode-pairing quantum key distribution (MP-QKD). The simulation results confirm that our precise phase error analysis can noticeably improve the key rate performance especially over long distances in practice. Note that our method does not require alterations to the existing experimental hardware or protocol steps. It can be readily applied within current SNS-TF-QKD and MP-QKD for higher key rate generation.