  Mode-pairing (MP) quantum key distribution (QKD) eliminates the requirements
of phase locking and phase tracking compared with twin-field (TF) QKD while
still surpassing the fundamental rate-distance limit of QKD. The complexity of
the experimental implementation is reduced while the efficiency is also
guaranteed. The security of MP-QKD is proved rigorously by examining the
consistency of the states detailly between MP-QKD and the fixed-pairing scheme
under all of Eve's possible interference, where the latter is equivalent to
measurement-device-independent (MDI) QKD. Here we propose a simple and
straightforward method to prove the information-theoretic security of MP-QKD.
Specifically, an entanglement scheme for MP-QKD is proposed and its security is
proved using entanglement purification. Then the security of MP-QKD can be
guaranteed with the equivalence of the entanglement scheme and
prepare-and-measure scheme for MP-QKD. With this approach, it is beneficial to
analyze and understand the performance and security of MP-QKD. We explain why
the pairing rounds in MP-QKD can be decoupled and determined by the measurement
results announced by a third party, which is the key difference between MP-QKD
and MDI-QKD. Moreover, we analyze the security of MP-QKD with the allowed
optimal pairing strategy, which is significant for the secret key rate, under
collective and coherent attacks.
