## 4 Security Analysis and Numerical Simulation
### Asymptotic Security Analysis  
With a reverse reconciliation efficiency \(\beta\), the secret key rate of the protocol is [12; 32; 33]  
\[K\geq\beta\ I(A:B)-S_{N}(E:B), \tag{8}\]  
Figure 4: The entanglement-based scheme of the discrete-modulated underwater CV-QKD. [PERSON] randomly prepares one of the \(N\) states and sends it to [PERSON] through the untrusted seawater channel. [PERSON] detects the received model to derive a sequence of bits shared with [PERSON] by using a homodyne detector or a heterodyne detector. The seawater channel is assumed to be a linear channel. where \(I(A:B)\) is the Shannon mutual information between [PERSON] and [PERSON], and it can be calculated as  
\[I(A:B)=\frac{1}{2}log_{2}\frac{V+\chi_{tot}}{1+\chi_{tot}}, \tag{9}\]  
for the homodyne detection case, and  
\[I(A:B)=log2\frac{V+\chi_{tot}}{1+\chi_{tot}}, \tag{10}\]  
for the heterodyne detection case. Besides, \(S_{N}(E:B)\) is the Holevo bound of the mutual information between [PERSON] and [PERSON], and there is  
\[S_{N}(E:B)=S(E)-S(E|B). \tag{11}\]  
Therefore, the key to calculate the secret key rate is how to calculate \(S(E)\) and \(S(E|B)\). After [PERSON] applies homodyne or heterodyne measurement, [PERSON] purifies the whole system so that \(S(E)\) can be replaced by \(S(AB_{1})\), namely  
\[S(E)=S(AB_{1})=\sum_{i=1}^{2}G(\lambda_{i}), \tag{12}\]  
where  
\[G(\lambda_{i})=\frac{\lambda_{i}+1}{2}log\frac{\lambda_{i}+1}{2}-\frac{ \lambda_{i}-1}{2}log\frac{\lambda_{i}-1}{2}, \tag{13}\]  
and \(\lambda_{i}(i=1,2)\) is the symplectic eigenvalues of the covariance matrix \(\gamma_{AB_{1}}\). The conditional entropy \(S(E|B)\) satisfies  
\[S(E|B)=S(\rho_{AFG}^{B})=\sum_{i=3}^{5}G(\lambda_{i}), \tag{14}\]  
where \(\lambda_{i}(i=3,4,5)\) is the symplectic eigenvalues of the conditional covariance matrix \(\gamma_{AFG}^{B}\). The specific calculation process of the symplectic eigenvalues is shown in Appendix A. Based on the above discussion, we can analyze the performance of the four-state and eight-state underwater CV-QKD protocols with homodyne detection and heterodyne detection for the four typical seawater types. For the underwater CV-QKD protocol, its performance is mainly affected by three parameters, namely channel transmissivity \(T_{sea}\), excess noise \(\epsilon\), and Alice's modulated variance \(V_{A}\)[34]. Therefore, we can analyze the performance of the protocol from these three aspects. Figures 5 and 6, respectively, show the relationship between secret key rate and distance for the underwater four-state and eight-state protocols in different types of seawater. Here, homodyne detection and heterodyne detection are considered. It is obvious that the secret key rate and the transmission distance decrease sharply with the increase of complexity of seawater; especially when the turbid harbor water is used as the quantum channel, the secure transmission distance of the key is less than 3 m. This result is what we expect. The main reason is that the complexity of seawater increases the attenuation of light, thus reducing the transmittance of signal light in the seawater channel. This indicates that channel transmittance has a great influence on the performance of the system. In addition, by comparing the performance of the protocol in the homodyne and heterodyne detection conditions, we can find that there is little difference between them, which is associated with the relatively small modulation variance. Compared with Figures 5 and 6, we can also find that the four-state protocol is obviously outperformed by the eight-state protocol. Under the eight-state protocol, the secure transmission distance of the key in the pure seawater increases most obviously, and its added value is about 30 m. At the same time, the secret key rate of the eight-state protocol also increases at the same transmission distance. Figures 7 and 8 show the maximum tolerable excess noise as a function of the transmission distance for four-state and eight-state protocols, respectively.