Zhaohui Liu

We present a unified analytical framework for mode‑pairing quantum key distribution (MP QKD) that incorporates both discrete phase randomisation and finite‑size statistical fluctuations. By mapping MP QKD to an equivalent two‑mode measurement-device-independent QKD, we derive closed‑form bounds for key‑rate estimation without numerical optimisation. Using Chernoff bounds, we evaluate realistic system performance and show how the interplay between phase‑slice number and block size impacts whether the protocol can surpass the PLOB bound. The results highlight the practicality of our approach for real‑time evaluation and implementation of DPR MP QKD in a finite-key setting.

We consider discrete phase randomisation (DPR) for several quantum key distribution (QKD) protocols. Full continuous phase randomisation of weak laser pulses (WCPs) would create an output state that is diagonal in Fock basis. This will simplify the security proof of QKD protocols that rely on WCPs or decoy states. In practice, however, such an ideal phase randomisation may not be achievable. Instead, we may actively choose a discrete number of global phase values for our WCPs. The security proof with DPR has been reported for several QKD protocols, which often requires numerical optimisation. In this work, we develop analytical bounds on the secret key generation rate for BB84 and measurement-device-independent (MDI) QKD protocols with DPR. These analytical bounds closely match the numerical results. We then extend our results to the newly proposed mode-pairing (MP) QKD protocols, which offer favourable rate-versus-distance scaling, with DPR. We show that the number of phase slices needed for MP-QKD to approach the ideal case is larger than that of MDI-QKD.