## 3 Performance Analysis for Underwater CVQKD  
In this section, we analyze the performance of the underwater CVQKD system based on the parameters discussed above. The theoretical asymptotic secret key rate under collective attacks is obtained by [42; 43]  
\[K=\beta I_{AB}-\chi_{BE}, \tag{4}\]  
where \(\beta\) is the reverse reconciliation efficiency, \(I_{AB}\) is the [PERSON] mutual information between [PERSON] and [PERSON], and \(\chi_{BE}\) is the Holevo quantity which bounds the maximum information available to [PERSON]. The calculations about \(I_{AB}\) and \(\chi_{BE}\) are the same as fiber-based CVQKD system because in this paper we only consider a deep sea environment without turbulent effects. Therefore, the mutual information \(I_{AB}\) is calculated by  
\[I_{AB}=\frac{1}{2}\log_{2}\frac{V_{B}}{V_{B|A}}=\frac{1}{2}\log_{2}\frac{V+ \chi_{tot}}{1+\chi_{tot}}, \tag{5}\]  
with \(V=V_{A}+1\) and \(\chi_{tot}=\chi_{line}+\chi_{hom}^{\prime}/T\), where \(\chi_{hom}^{\prime}=(1+v_{el})/\eta_{t}-1\). \(V_{B}=\eta_{t}T(V+\chi_{tot})\) is the variance of the quadratures measured by [PERSON], and \(V_{B|A}\) is the conditional variance. The Holevo quantity \(\chi_{BE}\) is given by  
\[\chi_{BE}=S(\rho_{E})-\int\!dm_{B}p(m_{B})S(\rho_{E}^{m_{B}}), \tag{6}\]  
where \(p(m_{B})\) is the probability distribution of [PERSON]'s measurement, \(\rho_{E}^{m_{B}}\) is [PERSON]'s state conditional on [PERSON]'s measurement \(m_{B}\), and \(S(\rho_{E})\) is the Von Neumann entropy of the quantum state \(\rho_{E}\). In the case of Gaussian attack, the above equation can be further simplified as follows:  
\[\chi_{BE}=\sum_{i=1}^{2}G\left(\frac{\lambda_{i}-1}{2}\right)-\sum_{i=3}^{5}G \left(\frac{\lambda_{i}-1}{2}\right), \tag{7}\]  
where \(G(x)=(x+1)\log_{2}(x+1)-x\log_{2}(x)\). \(\lambda_{1,2}\) are the symplectic eigenvalues given by  
\[\lambda_{1,2}^{2}=\frac{1}{2}\left(A\pm\sqrt{A^{2}-4B}\right), \tag{8}\]  
with  
\[A =V^{2}+T^{2}(V+\chi_{line})^{2}+2T(1-V^{2}), \tag{9}\] \[B =T^{2}(1+V\chi_{line})^{2}.\]  
\(\lambda_{3,4}\) are the symplectic eigenvalues given by  
\[\lambda_{3,4}^{2}=\frac{1}{2}\left(C\pm\sqrt{C^{2}-4D}\right), \tag{10}\]  
with  
\[C =\frac{AX_{hom}+V\sqrt{B}+T(V+\chi_{line})}{T(V+\chi_{tot})}, \tag{11}\] \[D =\frac{\sqrt{B}V+B\chi_{hom}}{T(V+\chi_{tot})}.\]  
The last symplectic eigenvalue \(\lambda_{5}=1\). In our simulation, the fixed parameters mentioned above are set to \(V_{A}=20\), \(\beta=0.95\), \(\epsilon=0.01\), \(\eta=0.6\), and \(v_{el}=0.01\), corresponding to the standard realistic assumption for CVQKD implementations [44]. Figure 6a compares the secret key rate for different types of water with parameters calculated by [PERSON]'s law and Monte Carlo. Because [PERSON]'s law case only considers the influence of transmittance and does not consider other parameters. To better compare the two methods, in Monte Carlo case we analyze two situations. One considers the impact of the receiver aperture and FOV, and the other does not consider the impact of them with \(\eta_{r}=1\). We find that when \(\eta_{r}=1\)the key rate and transmission distance of the Monte Carlo method are higher than that of the [PERSON]'s law method, while when considering the influence of receiver aperture and FOV, the key rate and transmission distance are significant reduced. In particular, in pure sea water, clear ocean water and coastal ocean water, the secret key rate calculated by Monte Carlo method with considering receiver parameters is slightly smaller than [PERSON]'s law method, but the transmission distance is still longer than that of the [PERSON]'s law method. However, the results are very different in turbid harbor water. We can see that the key rate of harbor water calculated by our method is apparently lower than that obtained by [PERSON]'s law. This is because the value of \(\eta_{r}\) in turbid coastal water is so small that almost all of the photons arriving the reception plane can not be successfully detected.