# Phase-Matching Quantum Key Distribution with Discrete Phase Randomization
## 5 Numerical Results  
Let's suppose the transmittances between [PERSON]/[PERSON] and [PERSON] are \(\eta_{\rm{A}}=\eta_{\rm{B}}=\eta_{f}\), the detection efficiency of detectors is \(\eta_{d}\), after the channel and detection losses, \(\eta=\eta_{f}\eta_{d}\), the detection click probabilities are given by  
\[ \begin{split} P_{\rm{A}}(L)&=(1-p_{d})e^{-\eta_{\rm{ A}}\cos^{2}\frac{\Phi_{\rm{AB}}}{2}}\\ P_{\rm{A}}(L)&=1-P_{\rm{A}}(L)\\ P_{\rm{A}}(\bar{R})&=(1-p_{d})e^{-\eta_{\rm{AB}} \sin^{2}\frac{\Phi_{\rm{AB}}}{2}}\\ P_{\rm{A}}(R)&=1-P_{\rm{A}}(\bar{R})\end{split}  \tag{25}\]  
where \(P_{\rm{A}}(L)/P_{\rm{A}}(R)\) and \(P_{\rm{A}}(L)/P_{\rm{A}}(\bar{R})\) are the detection click probabilities of the L/R click and no L/R click, \(\phi_{\rm{AB}}\) is the phase mismatch between [PERSON] and [PERSON]. Due to the discrete phase randomization, we can obtain \(D\) phase slices. Although we keep the phase match pairs and discard all of the others, there is still an intrinsic bit error rate [4], \(E_{D}=\frac{D}{2\pi}\int_{0}^{2\pi/D}\sin^{2}\frac{\Phi_{\rm{AB}}}{2}d\phi_{ \rm{AB}}\). Significantly, this is very different from BB84 protocol with the global phase mismatch value \(\phi_{\text{AB}}=0\). When we use discrete phase randomization, we must consider the intrinsic bit error rate, which will deeply affect the bit error rate and phase error rate. The error gain can be given by  
\[ \begin{split} Q_{\alpha}^{E}&\!=\!\frac{D}{2\pi}\int _{0}^{\frac{2\pi}{D}}P_{\alpha}(R)P_{\alpha}(\bar{L})d\phi_{\text{AB}}\\ &=\frac{D}{2\pi}\int_{0}^{\frac{2\pi}{D}}(1-p_{d})e^{-\eta a\cos^ {2}\frac{2\Delta\text{B}}{2}}d\phi_{\text{AB}}-(1-p_{d})^{2}e^{-\eta a}\end{split}  \tag{26}\]  
We can derive the total gain \(Q_{\alpha}\) as  
\[ \begin{split} Q_{\alpha}&\!=\!\frac{D}{2\pi}\int _{0}^{\frac{2\pi}{D}}[P_{\alpha}(L)P_{\alpha}(\bar{R})\!+\!P_{\alpha}(R)P_{ \alpha}(\bar{L})]d\phi_{\text{AB}}\\ &=\frac{D}{2\pi}\int_{0}^{\frac{2\pi}{D}}(1-p_{d})e^{-\eta a\sin^ {2}\frac{2\Delta\text{B}}{2}}d\phi_{\text{AB}}-(1-p_{d})^{2}e^{-\eta a}\!+\!Q_ {\alpha}^{E}\end{split}  \tag{27}\]  
The bit error rate of signal states is given by  
\[E_{\mu}\!=\!\frac{Q_{\mu}^{E}(1-2e_{opt})+e_{opt}Q_{\mu}}{Q_{\mu}} \tag{28}\]  
The simulate parameters are listed in Table 1.  
In the key rate versus the transmission distance of the finite decoy states PM protocol with a different number of phase values, as shown in Figure 1, the PLOB bound is plotted for comparison. The smaller \(D\), the lower the key rate; the reason is that the smaller the \(D\), the larger the intrinsic bit error rate. \(D=8\) can break the PLOB bound, and meanwhile, we can find that there is an optimal \(D=10\), which can guarantee better performance. With the increase of \(D\), the key rate will become lower due to the sifting factor \(2/D\). Hence, in an actual experiment of PM-QKD, we must find the suitable discrete phases value to guarantee security and performance. When \(D\rightarrow\infty\), the key rate will tend to 0; we do not present it here. Moreover, we compare the performance of PM-QKD with discrete phase randomization between infinite decoy states and vacuum and one decoy state. As depicted in Figure 2, when we adopt vacuum and one decoy state and small \(D\), the key rate exhibits poor performance. As \(D\) increases, the key rate of adopting vacuum and one decoy state approaches infinite decoy states. Combining the conclusion of Figure 1, we find that the discrete phase \(D=10\) still maintains good security and performance when the finite decoy states are implemented. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{p_{d}}\) & \(\mathbf{e_{opt}}\) & \(\mathbf{\eta_{d}}\) & \(\mathbf{f}\) & \(\mathbf{\eta_{f}}\) \\ \hline \(1\times 10^{-8}\) & 1.5\% & 0.2 & 1.1 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of parameters used in numerical simulations.