# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## 3 Modified Protocol with Noiseless Attenuation  
Due to the limited size of the quantum satellite, an enhanced source preparation scheme at the ground station becomes a feasible solution. Motivated by characteristics of the source preparation [23; 24; 25], we apply a suitable NA operation before transmission for performance improvement (see Figure 1). In this section, the NA-based source preparation is realized via the ZPC operation, which can be implemented by mixing zero photon at an asymmetrical beam splitter (ABS) and then post-selecting the ABS output by conditional detection of zero photon (see Figure 3). For the TMSV state, the catalyzed mode (mode B) is sent to one of the input ports of ABS with transmittance \(T_{A}\), while a zero-photon ancillary Fock state incidents on another port for interference. In an ideal case, the auxiliary zero-photon state will not be destroyed at all and the photon number resolving detector (PNRD) at the corresponding output ports can register zero photon. Therefore, the ZPC-involved attenuation happens to be a zero-energy-input attenuation that does not cause the consumption of external energy. Mathematically, the ZPC process can be regarded as an equivalent operation \(\hat{O}_{0}\) given by  
\[\hat{O}_{0} =\frac{\partial^{m}}{m!\partial\chi^{m}}\left\{\frac{T_{A}^{m}}{ 1-\chi}\left(\frac{\sqrt{T}_{A}-\chi/\sqrt{T}_{A}}{1-\chi}\right)^{\dagger b} \right\}_{m=0,\chi=0}, \tag{8}\]  
where \(b^{\dagger}(b)\) denotes the addition (annihilation) photon operation of input mode, \(m\) is the number of photons detected by PNRD when the ZPC process is successful. Obviously, \(m=0\) in the ZPC process as the description above.  
When [PERSON] performs heterodyne detection on mode A, the transmission mode B will project to a coherent state [27]. It means that the ZPC operation on TMSV is equivalent to that acts on a coherent state, which can be written as  
\[\hat{O}_{0}\mid\alpha\rangle=(\sqrt{T_{A}})^{\dagger b}\mid\alpha\rangle=\mid \sqrt{T_{A}}\alpha\rangle, \tag{9}\]  
Figure 3: The ZPC operation for the NA-based source preparation. ABS: asymmetrical beam splitter. PNRD: photon number resolving detector.  
where \(\mid a\rangle\) denotes the coherent state. We find that the ZPC operation happens to be an NA process with attenuation coefficient \(g=\sqrt{T_{A}}\). In addition, the ZPC operation acting on a coherent state is a probability process with success probability  
\[P=\frac{2}{1+T_{A}+RV}, \tag{10}\]  
where \(R=1-T_{A}\). Figure 4 shows the relationship between ABS transmittance \(T_{A}\) and attenuation coefficient \(g\) with the corresponding success probability \(P\). We find that both attenuation coefficient and success probability saw an upward trend with the increasing transmittance.  
We note that the covariance matrix of the ZPC-involved attenuation state can be derived by its characteristic function (CF), as shown in Appendix D. The form of the covariance matrix can be expressed as  
\[\gamma_{AB_{1}}=\left[\begin{array}{cc}X\Pi&Y\sigma_{z}\\ Y\sigma_{z}&X\Pi\end{array}\right], \tag{11}\]  
where the factors X and Y are listed as follows  
\[X=\frac{2V-RV+R}{1+T_{A}+RV}, \tag{12}\]  
\[Y=\frac{2\sqrt{T_{A}(V^{2}-1)}}{1+T_{A}+RV}. \tag{13}\]  
Then, the attenuation state transfers the uplink channel for quadrature measurement. The final covariance matrix of arriving attenuation state is given by  
\[\gamma_{AB_{2}}=\left[\begin{array}{cc}X\Pi&\langle\sqrt{T}\rangle Y\sigma_{ z}\\ \langle\sqrt{T}\rangle Y\sigma_{z}&\langle T\rangle(X+\chi_{\rm line})\Pi \end{array}\right]. \tag{14}\]