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Quantum key distribution (QKD) stands as a pioneering method for establishinginformation-theoretically secure communication channels by utilizing theprinciples of quantum mechanics. In the security proof of QKD, the phase errorrate serves as a critical indicator of information leakage and directlyinfluences the security of the shared key bits between communicating parties,Alice and Bob. In estimating the upper bound of the phase error rate, phaserandomization and subsequent postselection mechanisms serve pivotal rolesacross numerous QKD protocols. Here we make a precise phase error rate analysisfor QKD protocols with phase postselection, which helps us to accurately boundthe amount of information an eavesdropper may obtain. We further apply ouranalysis in sending-or-not-sending twin-field quantum key distribution(SNS-TFQKD) and mode-pairing quantum key distribution (MP-QKD). The simulationresults confirm that our precise phase error analysis can noticeably improvethe key rate performance especially over long distances in practice. Note thatour method does not require alterations to the existing experimental hardwareor protocol steps. It can be readily applied within current SNS-TF-QKD andMP-QKD for higher key rate generation.

Precise Phase Error Rate Analysis for Quantum Key Distribution with Phase Postselection