# Performance Analysis of the Shore-to-Reef Atmospheric Continuous-Variable Quantum Key Distribution
## 2 Shore-to-Reef Atmospheric CVQKD  
In this section, we analyze the secret key rate of atmospheric CVQKD. A shore-to-reef atmospheric CVQKD protocol is shown in Figure 1. This model is based on the model in ref [8]. In this paper, we use a different channel. It is composed of shore-based atmospheric channel and oceanic atmospheric channel. Einstein-Podolsky-Rosen (EPR) source generates an entangled state with variance V. The \(B_{0}\) is one mode of the entangled state that is transmitted to [PERSON] through a fading channel, which is depicted by a distribution of transmittance \(T\), and [PERSON] performs the heterodyne detection to measure the quadrature. The detection efficiency \(\eta\) is used to depict the imperfection of the detector and the electronic noise \(v_{el}\) contained in variance \(\
u\)[8]. The secret key rate \(K\) is given as [21]  
\[K=\beta_{1}\,I_{AB}-\chi_{BE}, \tag{1}\]  
Figure 1: Continuous-variable quantum key distribution (CVQKD) through shore-to-reef atmospheric channel. where \(\beta_{1}\) is the reconciliation efficiency, \(I_{AB}\) is the [PERSON] mutual information of [PERSON] and [PERSON], and \(\chi_{BE}\) is the Holevo quantity, which can be expressed as [22]  
\[\chi_{BE}=S(\rho_{E})-\int dm_{B}p(m_{B})S(\rho_{E}^{m_{B}}), \tag{2}\]  
where \(m_{B}\) represents measurement of [PERSON], \(p(m_{B})\) represents the probability density of the measurement, \(\rho_{E}^{m_{B}}\) represents the state of eavesdropper conditional on the measurement result of [PERSON], and \(S(\cdot)\) represents the Von Neumann entropy. Then, we should calculate \(I_{AB}\) and \(\chi_{BE}\) in order to obtain the secret key rate. The covariance matrix of a two-mode squeezed vacuum state with variance [22]:  
\[\gamma_{AB}=\left( \begin{array}{cc}VI&\sqrt{V^{2}-1}\sigma_{z}\\ \sqrt{V^{2}-1}\sigma_{z}&VI\end{array} \right), \tag{3}\]  
where \(V\) represents [PERSON]'s modulation variance, \(\sigma_{z}=diag(1,-1)\) and \(I=diag(1,1)\). On the assumption that the state is described by probability distribution \(\{p_{i}\}\) of transmittance \(\{T_{i}\}\) through a fluctuating channel, and we can use the mean value of transmittance \(<T>\) and the mean of the square root of transmittance \(<\sqrt{T}>\) to replace the fluctuating transmittance [7], the covariance matrix can be expressed as:  
\[\gamma^{{}^{\prime}}_{AB}=\left( \begin{array}{cc}VI&<\sqrt{T}>\sqrt{V^{2}-1 }\sigma_{z}\\ <\sqrt{T}>\sqrt{V^{2}-1}\sigma_{z}&<T>(V+1/<T>-1+\varepsilon)I\end{array} \right). \tag{4}\]  
Now, we can read the mutual information as [8]:  
\[I_{AB}=log_{2}\frac{1}{1-\frac{<\sqrt{T}>^{2}(V-1)}{<T>(V+\chi)}}, \tag{5}\]  
where \(\chi=2(1+\
u_{el})/\eta<T>-1+\varepsilon\). Furthermore, \(\chi_{BE}\) can also be obtained [21]:  
\[\chi_{BE}=\sum_{i=1}^{2}G(\frac{\lambda_{i}-1}{2})-\sum_{i=3}^{5}G(\frac{ \lambda_{i}-1}{2}), \tag{6}\]  
where \(G(x)=(x+1)log_{2}(x+1)-xlog_{2}(x)\). \(\lambda_{1,2}\) could be calculated as  
\[\lambda_{1,2}^{2}=\frac{1}{2}[A\pm\sqrt{A^{2}-4B}], \tag{7}\]  
where [8]  
\[ \begin{array}{l}A=V^{2}(1-2<\sqrt{T}>^{2})+2<\sqrt{T}>^{2}+<T>^{2}(V+1/<T>- 1+\varepsilon)^{2}\\ B=[V^{2}\textit{var}(\sqrt{T})+<\sqrt{T}>^{2}+<T>(V/<T>-1+\varepsilon)]^{2}, \end{array}  \tag{8}\]  
\(\textit{var}(\sqrt{T})=<T>-<\sqrt{T}>^{2}\) is the variance of \(\sqrt{T}\), \(\lambda_{3,4,5}\) are the symplectic eigenvalues of covariance matrix \(\gamma^{m^{B}}_{AFG}\), \(\gamma^{m^{B}}_{AFG}=\gamma_{AFG}-\gamma_{h}\). Furthermore, \(\gamma_{h}=\sigma^{T}_{AFG_{2}}H\sigma_{AFG_{2}}\). For heterodyne \(H^{\textit{let}}=(\gamma_{B_{3}}+I)^{-1}\)[8]. \[\gamma_{AFG_{3}}=\left( \begin{array}{cc}\gamma_{AFG}&\sigma^{T}_{AFG_{2}}\\ \sigma_{AFG_{2}}&\gamma_{B_{3}}\end{array} \right). \tag{9}\]When we simplify the results of equation \(\gamma_{AFG}^{m^{B}}=\gamma_{AFG}-\gamma_{h}\), the elements of Equation (4) can be defined as [8]  
\[ \begin{array}{l}a=V\\ b=<\sqrt{T}>\sqrt{V^{2}-1}\\ c=<T>(V+1/<T>-1+\varepsilon),\end{array}  \tag{10}\]  
then we can obtain [8]  
\[\gamma_{AFG}=\left(