## Appendix A Calculation of Asymptotic Secret Key Rate  
We assume that [PERSON] performs the Gaussian collective attack to achieve the useful results, and the information obtained is limited to the Holevo range \(S(E:y)\). The definition of secret key rate in the case of reverse reconciliation under collective attacks can be given by  
\[K_{a}=P_{0}(\beta I(x:y)-S(E:y)), \tag{1}\]  
where \(\beta\) is the reconciliation efficiency, \(I(x:y)\) is the Shannon mutual information between [PERSON] and [PERSON], and \(S(E:y)\) is the Holevo bound of the mutual information between [PERSON] and [PERSON]. The mutual information \(I(x:y)\) for homodyne detection is derived as  
\[I(x:y)=\frac{1}{2}\log_{2}\frac{V+\chi_{tot}}{1+\chi_{tot}}, \tag{2}\]  
where \(\chi_{tot}\) represents the total noise given by \(\chi_{tot}=\chi_{line}+\chi_{hom(het)}/T\) and \(\chi_{line}\) represents the total channel-added noise given by \(\chi_{line}=1/T+\varepsilon-1\). The transmission efficiency \(T\) is calculated as \(T=10^{-\mu L/10}\), where \(\mu=0.2\) dB/km is the loss coefficient for the standard optical fibers and \(L\) is the length of the fiber optics. For homodyne detection, the detection-added noise referred to [PERSON]'s input can be given by  
\[\chi_{hom}=\frac{(1-\eta)+\
u_{el}}{\eta}. \tag{3}\]  
Note that in the case of heterodyne detection, we have,  
\[\chi_{het}=\frac{1+(1-\eta)+2\
u_{el}}{\eta}. \tag{4}\]  
After [PERSON] applies homodyne or heterodyne measurement, [PERSON] purifies the whole system so that the mutual information between [PERSON] and [PERSON] can be expressed as  
\[\begin{split} S(E:y)&=S(E)-S(E|y)\\ &=S(xy)-S(x|y),\end{split} \tag{5}\]  
where the first term \(S(xy)\) is a function of the symplectic eigenvalues \(\lambda_{1,2}\) of \(\gamma_{AB_{2}}\), which is given by  
\[S(xy)=G\left(\frac{\lambda_{1}-1}{2}\right)+G\left(\frac{\lambda_{2}-1}{2} \right), \tag{6}\]where \(G(x)=(x+1)\log_{2}(x+1)-x\log_{2}x\) and \(\lambda_{1,2}\) are symplectic eigenvalues of covariance matrix  
\[\gamma_{AB_{2}}=\begin{bmatrix}(\tau V_{A}+1)\mathbb{I}&\sqrt{T}Z_{N}^{\prime} \sigma_{z}\\ \sqrt{T}Z_{N}^{\prime}\sigma_{z}&T(\tau V_{A}+1+\chi_{line})\mathbb{I}\end{bmatrix}. \tag{10}\]  
Consequently, we have  
\[\lambda_{1,2}=\sqrt{\frac{1}{2}\left(A\pm\sqrt{A^{2}-4B}\right)}, \tag{11}\]  
where \(A\) and \(B\) are parameters given by  
\[A=\text{detfl}_{\text{A}}+\text{detfl}_{\text{B}_{2}}+2\text{dete}_{\text{AB}_ {2}},\text{B}=\text{detfl}_{\text{AB}_{2}}. \tag{12}\]  
The second term \(S(x|y)\) is a function of the symplectic eigenvalues \(\lambda_{3,4}\) of the convariance matrix of [PERSON]'s mode after [PERSON] performs homodyne (heterodyne) detection, which is given by  
\[S(x|y)=G\left(\frac{\lambda_{3}-1}{2}\right)+G\left(\frac{\lambda_{4}-1}{2} \right), \tag{13}\]  
with the symplectic eigenvalues  
\[\lambda_{3,4}=\sqrt{\frac{1}{2}\left(C\pm\sqrt{C^{2}-4D}\right)}. \tag{14}\]  
For homodyne detection, we have  
\[\begin{split} C_{hom}&=\frac{A\chi_{hom}+V\sqrt{B}+T ^{\prime}}{T(V+\chi_{tot})},\\ D_{hom}&=\sqrt{B}\frac{V+\sqrt{B}\chi_{hom}}{T(V+ \chi_{tot})},\end{split} \tag{15}\]  
with \(T^{\prime}=T(V+\chi_{line})\). For heterodyne detection, we have  
\[\begin{split} C_{het}&=\frac{A\chi_{het}^{2}+B+1+2 \chi_{het}(V\sqrt{B}+T^{\prime})+2 TZ_{8}^{2}}{T^{2}(V+\chi_{tot})^{2}},\\ D_{het}&=\left(\frac{V+\sqrt{B}\chi_{het}}{T(V+ \chi_{tot})}\right)^{2}.\end{split} \tag{16}\]