# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 3 Simulation and Discussion
The optimal selection of signal states \(\mu_{a}\) (\(\mu_{b}\)) is a balance among quantum bit error rate, the probability of single-photon transmission, and error correction. Figure 3 illustrates the variations in the probabilities of the signal, decoy, and vacuum states with asymmetric intensity strategy as the distance increases, when \(\Delta L=100\) km. It can be observed that as the distance increases, the probability of the signal states remains relatively stable in most cases, while the probability of the decoy states notably increases. As the distance increases and losses grow, an inevitable increase in the probability of the decoy states is necessary to achieve better HOM interference effects and lower error rates. We \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Point** & \(L_{A}+L_{B}\) & \(\Delta L\) & **Strategy** & \(\mu_{a}\) & \(\
u_{a}\) & \(\mu_{b}\) & \(\
u_{b}\) & \(P_{pa}\) & \(P_{pa}\) & \(P_{pa}\) & \(P_{pa}\) & \(P_{p_{b}}\) & \(P_{p_{b}}\) & \(R\) **(bit/pulse)** \\ \hline \(\blacklozenge\blacklozenge\blacklozenge\blacklozenge\blacklozenge\) & 200 km & 0 km & Sym & 0.424 & 0.0213 & 0.424 & 0.0213 & 0.254 & 0.180 & 0.566 & 0.254 & 0.180 & 0.566 & \(2.95\times 10^{-5}\) \\ \(\blacklozenge\blacklozenge\blacklozenge\blacklozenge\) & 200 km & 50 km & This work & 0.216 & 0.00449 & 0.621 & 0.0376 & 0.170 & 0.229 & 0.601 & 0.305 & 0.192 & 0.503 & \(1.84\times 10^{-5}\) \\ \(\blacklozenge\blacklozenge\blacklozenge\blacklozenge\) & 200 km & 50 km & Add att & 0.492 & 0.0258 & 0.492 & 0.0258 & 0.271 & 0.220 & 0.509 & 0.271 & 0.220 & 0.509 & \(5.71\times 10^{-6}\) \\ \(\blacklozenge\blacklozenge\blacklozenge\) & 200 km & 100 km & This work & 0.107 & 0.000624 & 0.718 & 0.0549 & 0.0902 & 0.309 & 0.6008 & 0.327 & 0.230 & 0.443 & \(5.89\times 10^{-6}\) \\ \(\blacklozenge\blacklozenge\) & 200 km & 100 km & Add att & 0.560 & 0.0321 & 0.560 & 0.0321 & 0.278 & 0.281 & 0.441 & 0.278 & 0.281 & 0.441 & \(6.37\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Examples of optimal parameters in Figure 1. The numerical values in the table presented here are rounded to three significant figures. also notice a relatively high proportion of vacuum states, attributable to their contribution in the pairing of the \(Z\)-basis, estimation of single-photon counts, and calculation of phase error rates. It is important to emphasize that for each optimization point in Figures 2 and 3, the intensities and probabilities of the signal, decoy, and vacuum states strictly adhere to the constraints specified in Equation (15), while the counts and count rates are required to satisfy the physical validity conditions outlined in Equation (A1). Figure 4 demonstrates the influence of different maximum pairing intervals \(l\) on the SKR with asymmetric intensity strategy. With the increase in \(l\), the SKR in the asymmetric scenario shows a significant improvement but remains below that of the symmetric case, unable to surpass the PLOB bound. Additionally, as shown in Figure 5, at long distances, the infinite-key scenario slightly surpasses the PLOB bound, representing the theoretical performance limit of MP-QKD in this regime. In contrast, under the finite-key condition, increasing the total number of pulses \(N\) makes it challenging to surpass the PLOB bound due to statistical fluctuations and finite-size constraints. Figure 4: Optimized SKR with asymmetric intensity strategy versus the total distance (\(L_{A}+L_{B}\)) between Alice and Bob, at different maximal pairing intervals \(l\). Figure 3: The optimized probabilities of the signal, decoy, and vacuum states with an asymmetric intensity strategy are plotted versus the total distance (\(L_{A}+L_{B}\)) between Alice and Bob, where \(\Delta L=100\) km and the maximum pairing interval \(l=2000\).