## 4 Simulations and Performance Estimation
### The Influence of the Pulse Width and Research on Repetition Frequency
The corresponding time-domain attenuation factor \(q_{1}=9.1832\times 10^{-8}\), or \(-\)70.4 dB, and \(q_{2}=7.1116\times 10^{-29}\), or \(-\)281.5 dB. Furthermore, as Figure 7 shows, when the residual ratio of the frequency  
Figure 6: Secure key rates versus pulse repetition frequency curves under s = 0.01 corresponding to different polarization extinction ratios. (The transmission distance is 25 km, the pulse width is 1 ns, s = 0.01, the finite-size block is 10\({}^{11}\), and VA = 1.6). modulation s is set to 0.01, the maximum SKR can achieve 12.801 Mbps@25 km, and the optimal repetition frequency increases to 334 MHz correspondingly. By comparing Figures 6 and 7, we can conclude that the optimal repetition frequencies and SKRs increase due to the application of the frequency division multiplexing technique. By optimizing the frequency-domain residual ratio from s = 0.01 to s = 0.001, the scale of optimal repetition frequency increases to 319.0-334.0 MHz, and the SKRs increase correspondingly. The reason for this is that the use of frequency division multiplexing reduces the high requirement for time-domain isolation degree. As is shown in Table 1, we set the residual ratio of the frequency modulation as \(-20\) dB and \(-30\) dB to evaluate the overall isolation degree. By comparing the cases of s = \(-20\) dB and s = \(-30\) dB, we can conclude that the isolation degrees of frequency and time division multiplexing are correlated with each other. When the frequency-domain residual ratio s reduces, the time-domain attenuation factors \(q_{1}\) and \(q_{2}\) increase accordingly, and the result is that the repetition frequencies increase to the optimal values, and the SKRs increase to the maximum. Furthermore, from Equations (15)-(20), we can conclude that \(<n_{e2}>\) is the main influencing factor that increases the total excess noise. Thus, the total degree of isolation is mainly determined by s and \(q_{1}\). Based on the dual-polarization LLO CV-QKD model we proposed in Section 3, we achieved an overall isolation degree higher than 84.0 dB-90.4 dB, and the crosstalk between pilot tones and quantum signals can be limited to a very small range. Our work lays an important theoretical foundation for the practical LLO CV-QKD system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{s}\) & **PER** & **Optimal Repetition Frequency** & \(\boldsymbol{q_{1}}\) & \(\boldsymbol{q_{2}}\) & **The Total Degree of Isolation** & **SKR** \\ \hline \(-20\) dB & inf & 302.5 MHz & \(-65.8\) dB & \(-263.1\) dB & \(\approx\)\(-85.8\) dB & 11.740 Mbit/s \\ \(-20\) dB & 20 dB & 297.5 MHz & \(-68.0\) dB & \(-272.1\) dB & \(\approx\)\(-88.0\) dB & 5.863 Mbit/s \\ \(-20\) dB & 18 dB & 292.5 MHz & \(-70.4\) dB & \(-281.5\) dB & \(\approx\)\(-90.4\) dB & 2.933 Mbit/s \\ \hline \(-30\) dB & inf & 334.0 MHz & \(-54.0\) dB & \(-215.9\) dB & \(\approx\)\(-84.0\) dB & 12.801 Mbit/s \\ \(-30\) dB & 20 dB & 325.0 MHz & \(-57.0\) dB & \(-228.0\) dB & \(\approx\)\(-87.0\) dB & 6.373 Mbit/s \\ \(-30\) dB & 18 dB & 319.0 MHz & \(-59.2\) dB & \(-236.7\) dB & \(\approx\)\(-89.2\) dB & 3.175 Mbit/s \\ \hline \hline \end{tabular}  
* The factor s denotes the residual ratio of the frequency modulation, inf means positive infinity. PER denotes the polarization perturbation in the quantum channel. \(q_{1}\) and \(q_{2}\) are the time-domain attenuation factor due to the finite time-domain extinction ratio of PTI and PTZ, respectively.  
\end{table} 
Table 1: The total degree of isolation between the pilot tones and quantum signals. Figure 7: Secure key rates versus pulse repetition frequency curves under s = 0.001 corresponding to different polarization extinction ratios. (The transmission distance is 25 km, the pulse width is 1 ns, s = 0.01, the finite-size block is \(10^{11}\), and VA = 1.6). By the above research and analysis, the orthogonal dual-polarization multiplexing LLO CV-QKD system under relatively low PER conditions has been achieved, which is of great significance for the practicality of the dual-polarization CV-QKD system.