## 2 System Description  
In this section, we describe the GMCS protocol over the satellite-mediated links based on its entanglement-based (EB) representation for the convenience of security analysis. Note that one usually uses the equivalent prepare-measure (PM) scheme instead of the EB scheme for the experimental setup because the PM scheme uses 'virtual' entanglement without requiring a real entangled state [29]. However, [PERSON] is assumed to hold the purification of the system between [PERSON] and [PERSON] when one uses the EB scheme for security analysis. Therefore, the EB scheme is not convenient for security analysis with a trusted-loss or trusted-noise detector. In our manuscript, we assume an untrusted detector, where both the detector loss and setup noise are controlled by [PERSON]. The security analysis with a trusted-loss or trusted-noise detector can be found in [30], which is based on the PM scheme. Moreover, the notation used for noise analysis is the equivalent number of thermal photons due to it being more appropriate for free-space channels compared to excess noise [30]. In addition, the wavelength of the protocol is set to \(\lambda=800\) nm. A higher frequency case, such as microwave frequency, exhibits higher loss and increased noise [30] and holds potential feasibility for short-range communication or inter-satellite communication, which are beyond the scope of this work. Figure 1a shows the EB scheme of the GMCS protocol. [PERSON] first prepares an Einstein-Podolsky-Rosen (EPR) state as the entangled source, which can be characterized by its covariance matrix [31]  
\[\mathrm{V}_{AB}=\left[ \begin{array}{cc}\mu\mathrm{I}&\sqrt{\mu^{2}-1} \mathrm{Z}\\ \sqrt{\mu^{2}-1}\mathrm{Z}&\mu\mathrm{I}\end{array} \right], \tag{1}\]  
where \(\mathrm{Z}=\mathrm{diag}(1,-1)\), \(\mathrm{I}=\mathrm{diag}(1,1)\), and the signal variance \(\mu=\sigma_{x}^{2}+1\), with a modulation variance of \(\sigma_{x}^{2}\). Then, mode \(A\) of the EPR state is kept locally for heterodyne detection while another mode, \(B\), is sent to [PERSON] over the satellite-mediated (fading) link, which is characterized by the random transmittance \(\eta_{\mathrm{ch}}\) and the mean number of background thermal photons \(\bar{n}_{B}\). Note that [PERSON] performing heterodyne detection on mode \(A\) will collapse mode \(B\) into a coherent state, which reveals the equivalence between the PM scheme and the EB scheme [29]. In terms of the Glauber-Sudarshan P function, the evolution of the quantum state in a fading link can be described by [32]  
\[P_{\mathrm{out}}(\alpha)=\int_{0}^{1}d\eta_{\mathrm{ch}}\mathcal{P}(\eta_{ \mathrm{ch}})\frac{1}{\eta_{\mathrm{ch}}}P_{\mathrm{in}}(\frac{\alpha}{\sqrt{ \eta_{\mathrm{ch}}}}), \tag{2}\]  
where \(\mathcal{P}(\eta_{\mathrm{ch}})\) is the PDT, and \(P_{\mathrm{in}}(\alpha)\) and \(P_{\mathrm{out}}(\alpha)\) are the P functions of the input state and output state, respectively. According to Equation (2), the evolution of the quantum state is characterized by the statistics \(\mathcal{P}(\eta_{\mathrm{ch}})\). Therefore, we treat the fading link using the statistical mean as in [22], where the link is divided into \(K\) stable sub-links, with a constant transmittance \(\{\eta_{\mathrm{ch}}^{j}\}_{j=1,\ldots,K}\) (\(0\leq\eta_{\mathrm{ch}}^{j}\leq 1\)) and corresponding probability \(\{\mathcal{P}^{j}\}_{j=1,\ldots,K}\) (note that \(\Sigma_{j=1}^{K}\mathcal{P}^{j}=1\)). Then, the arriving quantum state is a mixture of the results in each sub-link, weighted by corresponding probability, whose total covariance matrix is given by  
\[\mathrm{V}_{AB_{1}}=\left[ \begin{array}{cc}\mu\mathrm{I}&\langle\sqrt{\tau} \rangle\sqrt{\mu^{2}-1}\mathrm{Z}\\ \langle\sqrt{\tau}\rangle\sqrt{\mu^{2}-1}\mathrm{Z}&\left[\langle\tau\rangle (\mu-1)+2\bar{n}+1\right]\mathrm{I}\end{array}