# Phase-Matching Quantum Key Distribution with Discrete Phase Randomization
## 1 Introduction  
Quantum key distribution (QKD) can offer information theoretically secure means to distribute secret keys between two remote parties [1], but the performance is restricted by the fundamental rate-loss limit [2, 3]. Recently, a novel twin-field QKD (TF-QKD) protocol [4] is proposed to surpass the linear Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound [2], which shows the superiority relation between key rate and channel transmittance, \(R\sim O(\sqrt{\eta})\). However, the security proof is not completed in the original TF-QKD protocol [4]. In order to present a more rigorous security proof, various variations [5, 6, 7, 8, 9, 10] of the original TF-QKD protocol have been proposed. The related experimental works have also been extensively studied [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. All of these variant TF-QKD protocols have their own advantages. The no-phase-post-selection TF-QKD (NPP-TF-QKD) protocol [5, 6] provides better key rate performance in closer-to-mid distance, but it needs phase locking and pre-phase feedback in the experiment, so it is hard to implement [5, 6, 21]. The sending-or-not-sending TF-QKD (SNS-TF-QKD) protocol [10] can tolerate large misalignment errors and provide better performance in long distance [10, 21]. The phase-matching QKD (PM-QKD) protocol [8] has no phase locking with phase slices and employs a phase post-compensation technique, so it can be easily experimentally implemented without pre-phase feedback [13, 21]. In reality, the decoy state method is adopted to ensure the security of imperfect single photon source [22; 23; 24; 25] in the actual QKD system. An important theoretical premise and assumption of the method is that the global phase of coherent sources should be continuously randomized [26; 27; 28]. However, perfect phase randomization is very difficult to achieve. In an actual experiment, there are two means to randomize the global phase. One means is to turn the laser on and off by controlling the current, but it is not suitable for PM-QKD with the phase post-compensation technique--the reason for this is that we do not know the precise phase slices. Moreover, experiments show that residual phase correlations may exist between adjacent pulses [29]. The other one is to actively modulate the phase of coherent sources controlled by a phase modulator with a true random number generator; this method is suitable for PM-QKD, but the phase randomization is not continuous. Thus, neither of these two means satisfy the assumption of the decoy state method, which may introduce a potential loophole that threatens the security of the actual protocol [30]. Then, the unambiguous state discrimination (USD) measurement [31] and the photon-number-splitting (PNS) attack [32] can be used against the imperfect phase randomization. An earlier security analysis of discrete phase randomization appears in the decoy state Bennet-Brassard-1984 (BB84) in Reference [33], which points out, when the number of discrete phase values is larger, that the performance of discrete phase randomization is close to that of continuous phase randomization, and the number is said to be ten [33]. Similar security analysis methods are used for several other protocols, the measurement-device-independent (MDI) QKD in Reference [34], the NPP-TF-QKD in References [35; 36], the SNS-TF-QKD in Reference [37], the PM-QKD in Reference [38]. Therein, Reference [38] uses a different security proof method with Reference [8], and there is no in-depth formula derivation in the decoy state PM-QKD with discrete phase randomization. In this paper, we focus on these discrete global phase randomization issues in the PM-QKD protocol [39], study a concrete attack against PM-QKD with imperfect phase randomization, apply the decoy-state method to derive the single photon yield formula to exhibit performance of the key rate and compare the yield difference of continuous phase randomization with discrete phase randomization.