## 3 Secret Key Rate in Inter-Satellite Links
### Secret Key Rate  
Now, we will apply inter-satellite links to estimate the SKR of the QSS protocol. According to step 4.6, the final SKR is the minimum of \(\{R_{i}\}_{i=1}^{n}\). We assume that the transmission distance between the dealer ([PERSON]) and the farthest player ([PERSON]) is \(d\) and the other \(n-1\) players are located between them at same intervals. Each player introduces the same amount of noise \(\xi_{0}\). Theoretically, the smallest SKR comes from the farthest player. However, the smallest SKR in a realistic QSS system must be estimated according to the practical data, so it does not necessarily come from the farthest player. Thus, when [PERSON] implement the proposed QSS protocol in practice, she should employ the experimental data to evaluate SKR of each player, and select the smallest one as the SKR of QSS. The asymptotic SKR of the protocol by using reverse reconciliation can be calculated by [38; 39]  
\[R=\beta I_{AB}-\chi_{AE}, \tag{3}\]  
where \(I_{AB}\) is the mutual information between [PERSON] and [PERSON], \(\beta\) is the reconciliation efficiency, and \(\chi_{AE}\) is the Holevo bound information available to the eavesdropper [PERSON] and the other dishonest players on [PERSON]'s measurement. Note that in this protocol, classical information is passed from [PERSON] to the players for reverse reconciliation. The transmittance of the \(i\)-th player in the inter-satellite channel can be calculated by  
\[T_{i}=1-\exp(-2r_{a}^{2}/l_{i}^{2}), \tag{4}\]  
where,  
\[l_{i}=r_{b}\sqrt{1+(\lambda d_{i}/\pi r_{b}^{2})^{2}}, \tag{5}\]  
and \(d_{i}=\frac{n-i+1}{n}d\) is the transmission distance between the dealer and the \(i\)-th player. Thus, when referred to the channel input, the excess noise contributed by the \(i\)-th player, expressed in shot noise units, can be given by [32]  
\[\xi_{i}=\xi_{0}\frac{T_{i}}{T_{1}}. \tag{6}\]  
The excess noise is an additional noise except vacuum noise which is mainly caused by the imperfection of system, e.g., modulation noise, Raman noise, background light, etc. Therefore, the total channel-added noise referred to the channel input can be defined as  
\[\chi_{\text{line}}=\frac{1-T_{1}}{T_{1}}+\sum_{i=1}^{n}\xi_{i}, \tag{7}\]  
where, \((1-T_{1})/T_{1}\) represents the channel loss. The heterodyne detection added noise referred to [PERSON]'s input is given by \[\chi_{\text{het}}\,=\,\frac{(2-\eta+2v_{el})}{\eta}, \tag{8}\]  
where, \(\eta\) denotes the detection efficiency and \(v_{el}\) denotes electronics noise of [PERSON]'s detector. The overall noise referred to the channel input can then be expressed as  
\[\chi_{\text{tot}}=\chi_{\text{line}}+\frac{\chi_{\text{het}}}{T_{1}}. \tag{9}\]  
The mutual information between [PERSON] and [PERSON] can be calculated by [12]  
\[I_{AB}=\text{log}_{2}\,\frac{V+\chi_{\text{tot}}}{1+\chi_{\text{tot}}}, \tag{10}\]  
where, \(V=V_{M}+V_{0}\), \(V_{M}\) is the modulation variance of [PERSON] and \(V_{0}\) is the shot noise given by [12]  
\[V_{0}=2\,\text{$\hbar$}+1, \tag{11}\]  
where,  
\[\text{$\hbar$}=\frac{1}{\exp(\text{$\hbar$}/k_{T}\tau_{B})-1}, \tag{12}\]  
\(f\) is the frequency of quantum signals, \(\hbar\) is [PERSON]'s constant, \(k_{T}\) is [PERSON]'s constant, and \(\tau_{B}\) is the absolute temperature. For reverse reconciliation, [PERSON]'s information is bounded by the Holevo bound \(\chi_{AE}\) which can be calculated by [39]  
\[T_{i}=1-\exp(-2r_{a}^{2}/l_{i}^{2}), \tag{4}\]0  
where \(S(\cdot)\) is the von Neumann entropy. \(\rho_{E}^{x_{A},p_{A}}\) is [PERSON]'s conditional density operator for the states on [PERSON]'s measurement result. \(P(x_{A},p_{A})\) is the measured probability density. \(x_{A}\) and \(p_{A}\) are [PERSON]'s measurement results.