## 2 Propagation Modeling and Key Rate Estimation  
### Propagation Modeling  
In some practical applications, if one is to implement a QKD over a free-space channel between a satellite platform and a ground station, the turbulence may cause the beam depolarization and the rotation of polarization angle, as Figure 1 shows. This causes continuous phase modulations on the optical beam, and thereby leads to random refraction and diffraction effects, imposing distortions on the optical beam as it propagates through the atmospheric channel. To model the propagation properties of quantum signals in a free-space channel, we suppose that a coherent Gaussian Schell-model (GSM) is assumed as the quantum signal beam [22], which is generated by the Alice located in the source plane \(z=0\) and propagates along the transmission direction \(z=0\) into the detection Bob (the received plane \(z>0\)). In our model, the use of the elements of cross-spectral density matrix (CSDM) in the x and y coordinates \(W_{ij}(\mathbf{\rho_{1}},\mathbf{\rho_{2}};0),\ i=x,y\) is exploited to show the changes in the polarization of quantum light propagation, which can be expressed as [23]  
\[ \begin{array}{ll}W_{ij}(\mathbf{r_{1}},\mathbf{r_{2}};0)&=\left\langle E_{i }(\mathbf{r_{1}};0)E_{j}^{*}(\mathbf{r_{1}};0)\right\rangle\\ &=A_{i}A_{j}\exp[-(\frac{\mathbf{r_{1}^{2}}}{4w_{0i}^{2}}+\frac{\mathbf{r_{2}^ {2}}}{4w_{0i}^{2}})]B_{ij}\exp(-\frac{|\mathbf{r_{1}}-\mathbf{r_{2}}|^{2}}{2 \delta_{ij}^{2}}),i,j=x,y\end{array}  \tag{1}\]  
Figure 1: The rotation of polarization angle \(\theta\) as the beam propagates. Here, \(E_{i}\) and \(E_{j}\) are, respectively, the components of the random optical field in the \(x\) and \(y\) directions, and the asterisk represents the complex conjugate. \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) denote a pair of points with arbitrary transverse position vectors in the source plane \(z=0\), \(\langle\ \ \rangle\) denotes the ensemble average. In addition, \(A_{i}(A_{j})=1\) and \(w_{0i}(w_{0j})\) are the amplitudes and the waist radius of beam in the \(x\) and \(y\) directions; The coefficient \(B_{ij}=1\), \(i=j\); \(\big{|}B_{ij}\big{|}\leq 1\), \(i\
eq j\) is the spectral degree of coherence. Without any loss of generality, \(\delta_{ij}\) is the coherence length of the source and, therein, \(\delta_{xx}(\delta_{yy})\) is the waist radius of the GSM beam in the \(x/y\) direction, respectively. Subsequently, when the beam propagates along the transmission direction z from the Alice (source plane \(z=0\)) into Bob (received detection plane \(z>0\)) in atmospheric turbulence channel, the cross-spectral density matrix at z plane can be expressed by the extended Huygens-Fresnel principle [24]  
\[ \begin{array}{ll}W_{ij}(\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2};z)&=A_{i }A_{j}\Big{(}\frac{k}{2\pi z}\Big{)}^{2}\!\!\int\!\!\!\!\!\int\exp[-(\frac{ \boldsymbol{r}_{1}^{2}}{4w_{0i}^{2}}+\frac{\boldsymbol{r}_{2}^{2}}{4w_{0i}^{2 }})]\mu_{ij}(\mathbf{r}_{1}-\mathbf{r}_{2})\\ &\times\exp[-\frac{k}{2\pi}(\left|\boldsymbol{\rho}_{1}-\mathbf{r}_{1}\right|^ {2}-\left|\boldsymbol{\rho}_{2}-\mathbf{r}_{2}\right|^{2})]\\ &\times\exp[\psi(\boldsymbol{\rho}_{1},\mathbf{r}_{1})+\psi^{*}(\boldsymbol{ \rho}_{2},\mathbf{r}_{2})]\big{)}_{\rm at}\mathbf{r}_{1}\mathbf{d}\mathbf{r}_{ 2}\end{array}  \tag{2}\]  
where \(k\) is the optical wave number related to the wavelength \(\lambda\) by \(k=2\pi/\lambda\). \(\psi(\boldsymbol{\rho}_{1},\mathbf{r}_{1})\) describes the effects of the atmospheric turbulence on the propagation of a spherical wave due to the atmospheric turbulence from the point \((\mathbf{r},0)\) to \((\boldsymbol{\rho},z)\).