# Asymmetric Measurement-Device-Independent Quantum Key Distribution through Advantage Distillation
## 3 Results  
In this work, we explore the combination of a QKD and a post-processing method. We adopt the asymmetric seven-intensity MDI-QKD protocol and the AD method, which can improve the performance of asymmetric MDI-QKD protocol greatly. In this section, numerical simulations of the asymmetric seven-intensity MDI-QKD protocol with AD method are given and the simulation parameters are shown in Table 1. After analyzing the simulation results, we obtained the following significant research results. We analyze the secret key rate of the asymmetric MDI-QKD protocol with and without the AD method, and the corresponding comparison results are shown in Figure 1 under different conditions \(L_{asy}\) = 0 dB, 12 dB, 24 dB. The figure shows that the key rate with and without AD are consistent within a short distance. However, for example, the red line with \(L_{asy}\) = 12 dB, the AD method has a clear advantage at a transmission loss of about 33 dB, and a final transmission loss reaching 39 dB as well as the secret key rate showing a clear improvement. For a more obvious exploration of the reason, we present Figure 2 with respect to \(b\). We can observe that, in the above example, the value of \(b\) at about 33 dB has changed from 1 to 2, indicating that the AD method begins to work. With the increase of transmission loss, the AD method requires a larger \(b\) value to obtain a tight correlation from weak correlation. Furthermore, the results of the above case are similar to the other two cases (\(L_{asy}\) = 0 dB, \(L_{asy}\) = 24 dB). Therefore, the AD method can improve the key generation rate of asymmetric MDI-QKD over a long distance. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{e_{d}}\) & \(\mathbf{\eta_{D}}\) & \(\mathbf{Y_{0}}\) & \(\mathbf{f_{e}}\) & \(\mathbf{N}\) \\ \hline
0.5\% & 65\% & \(8\times 10^{-7}\) & 1.16 & \(10^{11}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The basic system parameters used in our numerical simulations. \(\eta_{D}\) and \(Y_{0}\) are the efficiency and dark count rate of detectors at [PERSON]’s side; \(e_{d}\): the misalignment error of the QKD system; \(f_{e}\): the error correction efficiency; \(N\): the number of pulse pairs [PERSON] and [PERSON] send. Figure 1: Comparison of the secret key generation rate versus the transmission loss. The value \(L_{asy}\) is the loss difference of [PERSON] to [PERSON] and [PERSON] to [PERSON]. The different colors represent loss difference, which is \(L_{asy}\) = 0 dB, \(L_{asy}\) = 12 dB, and \(L_{asy}\) = 24 dB, respectively. The solid line represents the secret key without the AD method, and the dotted line represents the secret key with the AD method. Figure 2: Results of the optimal \(b\) versus the transmission loss. The black, red, and blue represent the values \(L_{asy}\) = 0 dB, \(L_{asy}\) = 12 dB, and \(L_{asy}\) = 24 dB, respectively. When value \(b\) is not equal to 1, the AD method can further improve the secret key rate and transmission distance of the asymmetric MDI-QKD protocol. Additionally, we also further investigate the specific effects of the AD method on an asymmetric MDI-QKD under various values \(L_{asy}\), and the results are shown in Figure 3. We describe the meaning of Figure 3 and give a detailed definition of the improved percentage. Generally, when the degree of asymmetry is large, the deterioration of the key rate becomes more obvious. However, after the AD method is used, it can be clearly seen in Figure 3 that the improvement effect of the AD method becomes more obvious with the increase of the degree of asymmetry. For example, the improved percentage can reach about 35% when the value \(L_{asy}=35\) dB, which means that AD method can better solve some transmission performance bottlenecks of the entire network. By the above analysis, the AD method indeed can increase the propagation distance when the number of pulse pairs \(N=10^{11}\). In order to further analyze the finite size effects, we give the simulation results in Figure 4 under different values of \(N\) when the value \(L_{asy}=12\) dB.