## 2 Monte Carlo-Based Channel Parameter Characteristics Analysis
### Transmittance Analysis  
As defined in Figure 1, the coherent states are sent from [PERSON] to [PERSON] through a quantum channel, which is characterized by its transmission efficiency \(T\) and its excess noise \(\varepsilon\). Therefore, the total channel added noise referred to the channel input can be expressed as a combination of the noise caused by loss and excess noise, given by \(\chi_{line}=(1-T)/T+\varepsilon\), where \(\varepsilon\) is related to [PERSON]'s attack and \(T\) is related to the extinction coefficient \(c\). As introduced in Section 2.1, the value of \(c\) is determined by the chlorophyll concentration \(C\), which is a parameter that can describe the type of the water medium. In this section, we investigate the transmittance \(T\) of the underwater links in four typical types of water, including pure sea water, clear ocean water, coastal ocean water, and turbid harbor water. The coefficient values of \(a,b\) and \(c\) of these four water types are listed in Table 1[29; 41]. For each water types, the transmittance is collected in two ways: Monte Carlo simulation and [PERSON]'s law calculation, and the results are shown in Figure 2. We can find that for the case of pure sea water, the difference between the transmittance obtained by Monte Carlo simulation and [PERSON]'s law calculation is negligible,because there are few particles in the pure sea water and the probability of photons being scattered is very low. The trajectory of photons in pure sea water is approximately a straight line. However, for the cases of clear ocean water, coastal ocean water, and turbid harbor water, the results of [PERSON]'s law are much smaller than the results of Monte Carlo simulation, which is due to the scattering coefficient of these water types is higher than pure sea water, and a significant part of scattered photons may still be received by the receiver, while [PERSON]'s law considers all the scattered photons to be lost. Therefore, the more turbid seawater photons are scattered more times, and accordingly, the greater the difference between the transmittance estimated by [PERSON]'s law and Monte Carlo.  
Figure 1: Schematic diagram of an underwater CVQKD system. The sender Alice separates coherent light pulses generated by a laser diode into weak signal pulses and strong LO pulses. Then the signal pulses are loaded with key information by using an amplitude modulator and a phase modulator, and the LO pulses are delayed by a delay line so as to be transmitted together with the signal pulses through time multiplexing. After being transmitted through the seawater, the pulses reach [PERSON]’s telescope. [PERSON] first separates the LO and signal pulses by using a beam splitter, and then randomly measures one of the quadratures of the signal states by homodyne detection, with the help of the LO as a phase reference. The phase modulator in Bob is used to randomly select the quadrature by imposing a \(\pi/2\) phase shift to the LO. The dark spots between Alice and Bob indicate the water components such as phytoplankton, suspended sediments and detritus, and colored dissolved organic matter. AM: amplitude modulator. PM: phase modulator. Tele.: telescope. BS: beam splitter. PD: photodiode. LO: local oscillator.  
Figure 2: Transmittance of four types of water obtained by Beer’s law and Monte Carlo simulation. The solid lines correspond to the transmittance obtained by Monte Carlo simulation, and the dashed lines correspond to the transmittance obtained by Beer’s law. From left to right, the lines correspond, respectively, to turbid harbor water, coastal ocean water, clear ocean water, and pure sea water case. MC: results obtained by Monte Carlo simulation. BL: results calculated by Beer’s law.