# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 3 Simulation and Discussion
However, regardless of the strategy employed, the SKR does not achieve a level comparable to that of the symmetric channel case. The SKR for \(\Delta L=100\) experiences a significant decrease compared to \(\Delta L=50\), indicating that an increase in channel asymmetry leads to a reduction in the SKR. As evident from Table 4, the asymmetric intensities strategy exhibits a substantial increase of approximately an order of magnitude compared to the adding extra attenuation strategy when \(L_{a}+L_{b}=200\) km. All data in Table 4 are optimized using the modified PSO algorithm. Furthermore, from points A, B, and D, it can be observed that Alice and Bob adjusted their intensities (probabilities) to compensate for the channel asymmetry. As \(\Delta L\) increases, to achieve the optimal Hong-Ou-Mandel (HOM) interference effect, \(\mu_{a}\) and \(\
u_{a}\) continuously decrease, while \(\mu_{b}\) and \(\
u_{b}\) increase. Figure 1: The optimized SKR versus the total distance (\(L_{A}+L_{B}\)) between [PERSON] and [PERSON]. The maximum pairing interval is fixed to 2000. The SKR is calculated for two cases, i.e., (**a**) a 50 km length difference between [PERSON] and [PERSON], and (**b**) a 100 km length difference between [PERSON] and [PERSON]. Label ‘Sym’ represents the symmetric case. Label ‘This work’ refers to the asymmetric intensity method in this paper. Label ‘Add att’ refers to the processing method, with added extra attenuation directly. The details of points A, B, C, D, and E are given in Table 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{p_{d}}\) & \(\mathbf{\eta_{d}}\) & \(\mathbf{\alpha}\) & \(\mathbf{f}\) & \(\mathbf{e_{d}^{X}}\) & \(\mathbf{e_{d}^{Z}}\) & \(\mathbf{N}\) \\ \hline \(10^{-8}\) & 75\% & 0.2 dB/km & 1.1 & 0.1 & \(10^{-6}\) & \(10^{13}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters used in numerical simulation. \(p_{d}\) and \(\eta_{d}\) represent the dark counting rate per pulse and detection efficiency of SPD, respectively. \(\alpha\) is the transmission fiber loss. \(f\) is the error correction efficiency. \(e_{d}^{X}\) and \(e_{d}^{Z}\) are the misalignment-error of the \(X\)-pair and \(Z\)-pair, respectively. \(N\) is the number of total pulses. In the asymmetric intensity strategy for MP-QKD, a primary concern is the optimal selection of weak coherent pulse intensities by [PERSON] and [PERSON]. While \(\eta_{a}\mu_{a}\approx\eta_{b}\mu_{b}\) (or \(\eta_{a}\upsilon_{a}\approx\eta_{b}\upsilon_{b}\)) may seem natural, we do not impose this constraint in the modified PSO algorithm. Figure 2 depicts the variation of \(\frac{\eta_{a}\
u_{a}}{\eta_{b}\
u_{b}}\) and \(\frac{\eta_{a}\
u_{b}}{\eta_{b}\
u_{b}}\) for \(\Delta L=100\) km. It is evident that the magnitude of \(\frac{\eta_{a}\
u_{a}}{\eta_{b}\
u_{b}}\) remains predominantly near 1, although some deviations occur as the distance increases. The decoy state \(\
u_{a}\) (\(\
u_{b}\)) are primarily employed for estimating the phase error rate. The relationship \(\eta_{a}\upsilon_{a}\approx\eta_{b}\upsilon_{b}\) helps maintain a balance in the photon intensities reaching Charlie, ensuring good HOM visibility and low error rates. However, this relationship is derived under the assumptions of the infinite-key size, an infinite number of decoy states, and the neglect of dark counts. Therefore, discarding the ideal assumption mentioned above, slight deviations in \(\eta_{a}\upsilon_{a}\) and \(\eta_{b}\upsilon_{b}\) are reasonable for the scenario depicted in Figure 2. Additionally, from Figure 2, it can be observed that the value of \(\frac{\eta_{a}\mu_{a}}{\eta_{b}\
u_{b}}\) is significantly distant from 1, largely deviating from the condition \(\eta_{a}\mu_{a}\approx\eta_{b}\mu_{b}\). The reason for this phenomenon is that the signal state is primarily used for key generation, where the intensity \(\mu_{a}\) (\(\mu_{b}\)) not only impacts the quantum bit error rate but also influences the probability of sending single photons and error correction.