# Performance Analysis of Continuous-Variable Quantum Key Distribution with Multi-Core Fiber
## Appendix A The Secret Key Rate Calculation of the One-Way Protocol  
In this section we give the secret key rate calculation of the one-way protocol. The equivalent entanglement-based (EB) description is used to for the security analysis. We first come to the transmitted LO scheme. The mutual information of [PERSON] and [PERSON] can be written as [17]  
\[I_{AB}=\frac{1}{2}log_{2}\frac{V+\chi_{tot}}{1+\chi_{tot}} \tag{1}\]  
with  
\[\chi_{tot}=\chi_{line}+\chi_{h}, \tag{2}\] \[\chi_{line}=\frac{1-T_{c}}{T_{c}}+\epsilon_{0},\] (3) \[\chi_{h}=\frac{(1-\eta)+v_{el}}{\eta}, \tag{4}\]  
where \(V=V_{A}+1\) and \(V_{A}\) is the modulation variance of [PERSON], \(T_{c}=\eta_{F}T\). The Holevo bound in the information between [PERSON] and [PERSON] can be given by  
\[\chi_{BE}=\sum_{i=1}^{2}G\left(\frac{\lambda_{i}-1}{2}\right)-\sum_{i=3}^{5}G \left(\frac{\lambda_{i}-1}{2}\right), \tag{5}\]  
where \(G(x)=(x+1)log_{2}(x+1)-xlog_{2}x\). The symplectic eigenvalues \(\lambda_{1,2}\) and \(\lambda_{3,4}\) are given by  
\[\lambda_{1,2}^{2} = \frac{1}{2}[A\pm\sqrt{A^{2}-4B}], \tag{6}\] \[\lambda_{3,4}^{2} = \frac{1}{2}[C\pm\sqrt{C^{2}-4D}], \tag{7}\]with  
\[A = V^{2}(1-2T_{c})+2T_{c}+(T_{c})^{2}(V+\chi_{line}), \tag{100}\] \[B = T_{c}^{2}(V\chi_{line})^{2},\] (101) \[C = \frac{A\chi_{h}+V\sqrt{B}+T_{c}(V+\chi_{line})}{T_{c}(V+\chi_{tot})},\] (102) \[D = \sqrt{B}\frac{V+\sqrt{B}\chi_{h}}{T_{c}(V+\chi_{tot})}. \tag{103}\]  
The symplectic eigenvalue \(\lambda_{5}=1\).  
Then, for the locally generated LO scheme, the covariance matrix of the final Gaussian state \(\rho_{AB}^{G}\) shared between [PERSON] and [PERSON] can be written as [8; 39]  
\[\gamma_{AB}^{G}=\left(\begin{array}{cc}V_{a}\mathbb{I}&\mathbb{C}\sigma_{Z} \\ \mathbb{C}\sigma_{Z}&V_{b}\mathbb{I}\end{array}\right), \tag{104}\]  
where \(\mathbb{I}\) is diag(1, 1), \(\sigma_{Z}\) is diag(1, -1), \(V_{a}=V_{A}+1\), \(V_{b}=T_{e}(V+\chi)\), \(\mathbb{C}=\overline{\cos\varphi}\sqrt{T_{e}(V_{a}^{2}-1)}\) for the locally generated LO scheme. Here \(T_{e}=\eta_{F}\eta T\), \(\chi\) is the channel noise and is equal to \((1-T_{e})/T_{e}+v_{el}/T_{e}+\epsilon_{0}\). \(\overline{\cos\varphi}=\int_{-\pi}^{\pi}d\varphi\mathcal{P}(\varphi)cos\varphi\) and \(\mathcal{P}(\varphi)\) is the probability distribution of the phase estimation error \(\varphi\). Therefore, \(I_{AB}\) can be written as  
\[I_{AB}=\frac{1}{2}\log_{2}\left(\frac{V^{\prime}}{V_{a|b}}\right) \tag{105}\]  
with \(V^{\prime}=(V_{a}+1)/2\) and \(V_{a|b}=V^{\prime}-\mathbb{C}^{2}/2V_{b}\). The Holevo bound \(\chi_{BE}\) can be given by  
\[\chi_{BE}=G\left(\frac{\lambda_{1}^{\prime}-1}{2}\right)+G\left(\frac{\lambda _{2}^{\prime}-1}{2}\right)-G\left(\frac{\lambda_{3}^{\prime}-1}{2}\right). \tag{106}\]  
The eigenvalues \(\lambda_{1}^{\prime}\) and \(\lambda_{2}^{\prime}\) are given by  
\[\lambda_{1,2}^{\prime 2}=\frac{1}{2}\left(\Delta\pm\sqrt{\Delta^{2}-4\mathbb{D} ^{2}}\right) \tag{107}\]  
with \(\Delta=V_{a}^{2}+V_{b}^{2}-2\mathbb{C}^{2}\) and \(\mathbb{D}=V_{a}V_{b}-\mathbb{C}^{2}\). The square of symplectic eigenvalue \(\lambda_{3}^{\prime}\) reads:  
\[\lambda_{3}^{\prime 2}=V_{a}(V_{a}-\frac{\mathbb{C}^{2}}{V_{b}}). \tag{108}\]  
Note, \(\mathbb{C}^{2}\) can be written as [8; 24]  
\[\mathbb{C}^{2}=T_{e}(V_{a}^{2}-1)\overline{cos\phi}^{2}=T_{e}(V_{a}^{2}-1)(1-V _{\hat{\theta}}) \tag{109}\]  
with \(V_{\theta}=(\chi+1)/V_{R}+\delta_{R}/T_{e}V_{R}\), where \(\delta_{R}=1\) for single-reference-pulse mode, \(V_{R}\) is the amplitude of weak reference pulse.