# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 3 Simulation and Discussion
[17; 18] assumes that the decoy-state parameter estimation either meets idealized conditions or is performed with perfect accuracy, which is not representative of practical scenarios. For the gaps in their work, we have performed a thorough analysis addressing the effects of asymmetric channels on decoy-state parameters under practical conditions. Here, we can consider the SKR as a function of the source parameters  
\[R(\vec{g})=R\big{(}\mu_{a},\
u_{a},o_{a},p_{\mu_{a}},p_{\
u_{a}},p_{\alpha_{a} },p_{b},\
u_{b},o_{b},p_{\mu_{b}},p_{\
u_{b}},p_{\alpha_{b}}\big{)}. \tag{14}\]  
Given the uncertainty of the convex form of function \(R\), we employ the modified PSO algorithm (detailed in Appendix A) for global optimization of the 10 parameters rather than the local search algorithm (LSA) [33; 34; 35]. The LSA is highly sensitive to the selection of the initial point, as a randomly chosen initial point can often result in an invalid or infeasible outcome. The modified PSO algorithm can optimize non-smooth and non-convex functions to search for the optimal \(\vec{g}\) that maximizes \(R\). It is especially effective for asymmetric MP-QKD because it does not depend on the specific form or gradient information of the SKR function, and it does not require an initial starting point, showing robust global search abilities. In comparison with the original PSO algorithm [36], the modified version demonstrates an improved balance between global exploration and local exploitation, strengthened particle coordination, and enhanced optimization efficiency and convergence speed. We note that Ref. [10] employs a genetic algorithm for parameter optimization, whereas our use of a modified PSO algorithm offers a simpler implementation with faster convergence and lower computational complexity, making it more suitable for practical deployment scenarios. In the simulation, we employed an asymmetric intensity strategy to address the asymmetric channel: by deploying the PSO algorithm to optimize \(\vec{g}\) for compensating the channel asymmetry and achieving the optimal SKR. The simulation formulas for the observed values in asymmetric MP-QKD are provided in Appendix B. Furthermore, we also compare this approach with the strategy of adding extra attenuation. It is important to note that in the simulation, we consider [PERSON]'s and [PERSON]'s source configurations as independent, satisfying  
\[0=o_{a(b)}<\
u_{a(b)}<\mu_{a(b)}<1,\] \[p_{\alpha_{a(b)}}+p_{\
u_{a(b)}}+p_{\mu_{a(b)}}=1, \tag{15}\] \[0\leq p_{\mu_{a(b)}},p_{\alpha_{a(b)}},p_{\
u_{a(b)}}\leq 1\]  
without imposing constraints such as \(\eta_{a}\eta_{a}\approx\eta_{b}\eta_{b}\) (or \(\eta_{a}v_{a}\approx\eta_{b}v_{b}\)), where \(\eta_{a}\) and \(\eta_{b}\) are the channel transmittance of [PERSON] and [PERSON]. The formula \(\eta_{a}\mu_{a}\approx\eta_{b}\mu_{b}\) (or \(\eta_{a}v_{a}\approx\eta_{b}v_{b}\)) is a rule of thumb on the ratio of intensities between [PERSON]'s and [PERSON]'s light [37]. Equation (15)imposes three constraints: the first line ensures that the intensity of the signal state is greater than that of the decoy state, which in turn is greater than that of the vacuum state; the second line enforces that the sum of the probabilities for the three states equals 1; and the third line guarantees that the individual probabilities for the signal, decoy, and vacuum states are all strictly between 0 and 1. The parameters employed for the numerical simulations are detailed in Table 3. Additionally, we set \(\epsilon_{\text{cor}}=\epsilon=\epsilon_{\text{PA}}=\xi_{ee}=\epsilon_{e}/12= \epsilon_{1}/7=3.28\times 10^{-23}\), where \(\epsilon_{e}=11\epsilon_{\text{CB}}+\xi_{ee}\), \(\epsilon_{1}=7\epsilon_{\text{CB}}\). We plot the optimized SKR for the asymmetric scenario in Figure 1 and the detailed optimization results at a distance of 200 km in Table 4. From Figure 1, it is evident that the SKR of the asymmetric intensities strategy consistently outperforms that of the adding extra attenuation, for both \(\Delta L=50\) and \(\Delta L=100\).