## 3 Security Analysis  
In this section, we demonstrate the effect of the DM source preparation on CVQKD in both asymptotic [32; 33] and finite-size cases [33] with numeric simulation results. For the detailed derivation of the secret key rate of the CVQKD system in asymptotic and finite-size cases, please see Appendices A and B, respectively. For the ZPC-based DM source preparation, we show the success probability of ZPC operation in Figure 4. We find that the success probability increases with the increased modulation variance, and simultaneously increases with the increased transmittance. In the security analysis, the transmittance \(\tau\) is tunable to satisfy the optimality of the ZPC operation. Taking \(V_{A}\leq 0.5\) into account, no matter what the value of \(V_{A}\) is, the success probability can remain more than 0.75. This is a feasible solution to improve the CVQKD system. As usual, it is necessary to select an optimal modulation variance \(V_{A}\) for the performance improvement of CVQKD involving the ZPC operation. As shown in Figure 5, solid lines denote the secret key rate of the eight state CVQKD system, while dashed lines represent that of the four-state CVQKD system. In addition, we let \(\beta=90\%\), \(\eta=0.6\) and \(v_{el}=0.05\). We find that for \(\varepsilon=0.01\), the secret key rate decreases as the transmission distance, while the inserted subgraph shows that the secret key rate decreases with the excess noise at the distance of 80 km. Moreover, for \(V_{A}=0.5\) the secret key rate can reach the optimal value, and hence we let \(V_{A}=0.5\) in the subsequent simulations. In Figure 6a, we show the asymptotic secret key rate of the ZPC-based CVQKD system, where solid lines denote homodyne detection and dashed lines represent heterodyne detection. It seems that performances of homodyne and heterodyne detectors are almost similar, and thus we employ the homodyne detector in security analysis. The performance of the ZPC-based CVQKD is better than that of the traditional CVQKD whether it is four-state modulation or not. The eight-state modulation scheme demonstrates better performance than the four-state modulation scheme in the traditional CVQKD system. However, after performing the ZPC operation, the performance of the ZPC-based four-state modulation scheme exceeds the eight-state modulation scheme. Therefore, the ZPC-based source preparation is an effective way to improve the DM-CVQKD system. We note  
Figure 5: Asymptotic secret key rate of the CVQKD system with the ZPC-based DM source preparation for \(\varepsilon=0.01\), where solid lines denote the eight-state DM source preparation, while dashed lines represent the four-state DM source preparation. The inserted subgraph shows that asymptotic secret key rate of the CVQKD system for \(L=80\) km. Figure 6: (**a**) Asymptotic secret key rate of DM–CVQKD involving the ZPC operation for \(\varepsilon=0.01\), where solid lines denote homodyne detection, while dashed lines heterodyne detection. (**b**) Corresponding to (**a**), the transmittance \(\tau\) varies with the transmission distance. that the effect of the ZPC operation does not work over the whole transmission distance. In the short distance range, there are few catalytic effects. As shown in Figure 6b, the performance of the ZPC-based system is almost similar to that of the traditional system for \(\tau=1\). For the four-state DM source preparation, the ZPC effect occurs at the distance of \(L=60\) km, whereas for the eight-state DM source preparation, it occurs at \(L=30\) km. Usually, one assumes that the quantum channel is given in an asymptotic regime. However, one does not know the quantum channel in advance in the finite-size scenario. Therefore, we take into account effect of the finite-size on the ZPC-based CVQKD. As shown in Figure 7, the proposed CVQKD system in finite-size scenario performs worse than that in the asymptotic limit. As the number \(N\) of the exchanged signals is increased, the corresponding curves approach to the asymptotic scenario.