## 4 Security Analysis and Numerical Simulation
### Asymptotic Security Analysis
The numerical simulation results indicate that the tolerable excess noise and the maximum transmission distance decrease with the increase of the complexity of the seawater. The sensitivity of the pure sea water, clear ocean water, coastal ocean water, and turbid harbor water channels to the excess noise increases gradually, which is mainly reflected in the steepness of the curve in the figures. Besides, the tolerable excess noise in the case of homodyne detection is more than that in the case of heterodyne. By comparing Figures 7 and 8, it can be found that the tolerance to the noise of the eight-state protocol surpasses that of the four-state protocol, and the maximum transmission distance of the former is much larger than that of the latter. In addition, we can know that the four different seawater channels are all sensitive to excess noise; especially the turbid harbor water, coastal ocean water, and clear ocean water have very low tolerance to the additional noise, resulting in a short secure transmission distance for the quantum key. Figure 5: (Color online.) Secret key rate of the four-state protocol for realistic reconciliation efficiency of 90% and a quantum efficiency of Bob’s detection equal to 0.6 with thermal noise \(v_{el}~{}=~{}0.01\). \(V_{A}~{}=~{}0.6,\varepsilon=0.01\) (in shot-noise unit). Figure 6: (Color online.) Secret key rate of the eight-state protocol for realistic reconciliation efficiency of 90% and a quantum efficiency of Bob’s detection equal to 0.6 with thermal noise \(v_{el}=0.01\). \(V_{A}~{}=~{}0.6,\varepsilon=0.01\) (in shot-noise unit). To facilitate the complete analysis of the influence of modulation variance on the performance of the protocols with four different channels, we take the extra noise value of 0.005 to provide a relatively wide range of transmission distance values for the analysis under the conditions of four different channels. Since the four kinds of seawater channels have different transmittances, the corresponding optimal modulation variances are naturally different. Hence, we need to study the effect of modulation variance on the performance of the four systems with different channels, so that we can find the optimal variance of each system to improve its performance. Figures 9 and 10 separately show the relationship between modulation variance and secret key rate of the four-state and the eight-state protocols when the four kinds of typical seawater are used as transmission channels. Combined with the transmission characteristics of the four types of seawater, we selected different transmission distances to study the influence of modulation variance on secret key rate with homodyne detection and heterodyne detection. It should be noted that when the variance \(V_{A}\) of the discrete modulation is small, the lower bound of the secret key rate is close to the secret key rate of the case in which [PERSON] uses Gaussian modulation with variance \(V_{A}\)[35; 36]. The results of the simulation show that the maximum modulation variance and the optimal modulation variance decrease with the increase of the complexity of seawater and the transmission distance of the quantum signals. The maximum modulation variance in the case of homodyne detection is slightly larger than  
Figure 4:0 (Color online.) The maximum tolerable excess noise at each distance for eight-state protocol. \(\beta=0.9\), \(V_{A}=0.6\), \(\eta=0.6\), \(v_{d}=0.01\). Figure 4:1 (Color online.) The maximum tolerable excess noise at each distance for four-state protocol. \(\beta=0.9\), \(V_{A}=0.6\), \(\eta=0.6\), \(v_{d}=0.01\). that of heterodyne detection, and the difference between them decreases with the increase of the complexity of seawater and the transmission distance.