Time-synchronization error in twin-field quantum key distribution

For twin-field quantum key distribution (TF-QKD), it is commonly known that erroneous time synchronization leads to decoherence of the twin fields, compromising the performance of the TF-QKD. However, more quantitative information about such an effect still needs further revelation in order to understand the detailed requirement of time synchronization in TF-QKD and to further optimize the TF-QKD system for better performance. Here, we provide an analytical model describing TF-QKD with time-synchronization errors, choosing phase-matching quantum key distribution for our simulation. The model is constructed by a relation between the fringe visibility of the interferometry and the time-synchronization error described by the time difference between the arrival of twin-field signals from both users. By a numerical simulation, we demonstrate that our model can dramatically improve the tolerance of the system against the time-synchronization error, achieving a much higher secure key rate and communication distance with inaccurate time synchronization. Through this work, we intend to start a further discussion on the synchronization demand of TF-QKD.