## 3 Description of the Discrete-Modulated Underwater CV-QKD  
There are three main types of communication links in the end-to-end underwater continuous variable quantum key distribution system, namely the direct link, the feedback link and the reflection link, which are depicted in Figure 3. Direct link is the most common link structure in which signal light is transmitted from one side and received by the other. In this link structure, it is necessary to ensure accurate orientation between the sender and the receiver, so that the receiver can detect the light beam propagating along the direction of the transmitter. Feedback link can be viewed as a plug-and-play configuration [27; 28], in which the signal light emitted by the sender is incident on a mirror, such as a Faraday mirror, and the receiver modulates the received signal light. The quantum information is then loaded onto the reflected light and fed back to the sender via the reverse channel. It should be noted here that the backscattering of the signal light on the forward channel may interfere with the reflected optical signal on the reverse channel, which may result in an increase in the bit error rate (BER) and a decrease in the SNR [29]. In addition, since the optical signal will pass through the seawater channel twice, the received signal will undergo extra attenuation. Reflective link utilizes the reflection of the sea surface to transmit information in case the signal light cannot travel in a straight line [30]. In this configuration, the transmitter projects the beam at an incident angle greater than the critical angle to the sea surface, causing it to experience total reflection. The receiver should keep the direction facing the sea surface roughly parallel to the reflected light to ensure proper reception of the signal. This structure is susceptible to the fluctuation of the sea surface. Figure 3: Three link models of end-to-end underwater CV-QKD system. Figure 2: (Color online.) Light intensity distribution at different transmission distances of the four-state protocol over a pure sea water channel: (a–c) the three-dimensional views of the intensity distribution of the signal light at the position of initial, 8 m, and 20 m of the seawater channel, respectively; and (d–f) are their corresponding plane views. In this paper, we only consider the direct link structure of the end-to-end underwater continuous variable quantum key distribution for simplicity. In the previous investigation, the Gaussian modulated CV-QKD over the seawater channel has been investigated [20; 21]. Although the transmission distance of the secret key is much shorter than that in optical fiber, the research still provides a new idea for us to study the underwater quantum communication and opens a new way for establishing a global quantum communication network. In what follows, the researchers should find ways to further promote the system so as to improve the performance of the system and finally realize practical application. Studies have shown that discrete modulation is more suitable for long-distance communication than Gaussian modulation because of the relatively large key generation rate even at a lower SNR [12; 31]. Up to now, different types of discretely modulated CV-QKD schemes have been investigated, including the four-state protocol, eight-state protocol, sixteen-state protocol, and so on. Here, we consider introducing this modulation scheme into the underwater CV-QKD system. The PM scheme can be described as follows:  
**Step 1**: [PERSON] firstly choose a random number \(k\) from the set \(\{0,1,\ldots,N\}\) with equal probability to modulate the weak coherent light to prepare coherent states \(|\alpha_{k}\rangle=|\alpha\ e^{i(2k+1)\pi/N}\rangle\), where \(\alpha\) is a positive number related to the modulation variance \(V_{A}\) and satisfies \(V_{A}=2\alpha^{2}\), and then sends them to the seawater channel. **Step 2**: Due to the absorption and scattering of seawater, the quantum signal transmitted in the seawater channel will be attenuated.