# Continuous-Variable Quantum Key Distribution Robust Against Polarization-Dependent Loss
## 5 Performance Analysis
, \tag{14}\]  
where \(V_{M}=2\alpha^{2}\) is the modulation variance of Alice, \(I\) is the identity matrix, \(Z\) is the correlation function, source noise is \(\delta\varepsilon\), and \(\sigma_{z}=diag(1,-1)\). The covariance matrix of the output of the state \(\rho_{AB}\) after the phase shift operation \(U(\delta\phi)\) is then derived as  
\[\gamma_{\delta\phi}= \begin{bmatrix}(V_{M}+1)I&Y\\ Y&[T_{0}(V_{M}+\varepsilon_{0}+\delta\varepsilon)+1]I\end{bmatrix} , \tag{15}\]  
Figure 8: The \(I(a:b)\) with SNR for \(\triangle SNR=3\ dB\). Figure 7: The resulting error rate with SNR for \(\triangle SNR=3\ dB\). where  
\[Y= \begin{bmatrix}\sqrt{T_{0}}Zcos\delta\phi&-\sqrt{T_{0}}Zsin\delta\phi\\ -\sqrt{T_{0}}Zsin\delta\phi&-\sqrt{T_{0}}Zcos\delta\phi\end{bmatrix} . \tag{16}\]  
Then, the state affected by the phase noise is a classical mixture of states with random phase shifts [7]  
\[\rho^{\prime}_{AB}=\int(I_{A}\otimes U_{B}(\delta\phi))\rho_{AB}(I_{A}\otimes U _{B}(\delta\phi))p(\delta\phi)d\delta\phi, \tag{17}\]  
which corresponds to the covariance matrix  
\[\gamma_{AB}= \begin{bmatrix}(V_{M}+1)I&\sqrt{kT_{0}}Z\sigma_{z}\\ \sqrt{kT_{0}}Z\sigma_{z}&[T_{0}(V_{M}+\varepsilon_{0}+\delta\varepsilon)+1]I \end{bmatrix} , \tag{18}\]  
where we assumed that the distribution \(\delta\phi\) is symmetric. More specifically, \(\delta\phi\) satisfies \(\int p(\delta\phi)sin\delta\phi d\delta\phi=0\) and \(k=(\int p(\delta\phi)cos\delta\phi d\delta\phi)^{2}=(E[cos\delta\phi])^{2}\), where \(E[X]\) denotes the expectation of the random variable \(X\). Therefore, the equivalent transmittance \(T_{0}^{k}\) and the excess noise \(\varepsilon_{0}^{k}\) of quantum channel after PPC scheme can be expressed as  
\[T_{0}^{k} =kT_{0}, \tag{19}\] \[\varepsilon_{0}^{k} =[\varepsilon_{0}+(1-k)V_{M}]/k.\]  
Subsequently, the excess noise \(\varepsilon_{pdl}\) due to lack of orthogonality is given by  
\[\varepsilon_{pdl}=\varepsilon_{0}^{k}-\varepsilon_{0}=(1-k)(\varepsilon_{0}+V _{M})/k. \tag{20}\]  
According to the optimality of Gaussian attacks, when the covariance matrix is equal, \(\rho_{AB}\) is the Gaussian state, leading to the minimum \(K\). For the Gaussian modulation protocol, when the channel transmittance is \(T\) and the excess noise is \(\varepsilon\), the mixed state covariance matrix of [PERSON] and Bob is given by  
\[\gamma_{AB}^{G}= \begin{bmatrix}(V_{M}+1)I&\sqrt{kT_{0}}Z_{EPR}\sigma_{z}\\ \sqrt{kT_{0}}Z_{EPR}\sigma_{z}&[T_{0}(V_{M}+\varepsilon)+1]I\end{bmatrix} . \tag{21}\]  
To make \(\gamma_{AB}^{G}=\gamma_{AB}\), we obtain  
\[T=\sqrt{kT_{0}}\frac{Z^{2}}{Z_{EPR}^{2}},\ \varepsilon=\frac{Z_{EPR}^{2}}{Z^{2}} (V_{M}+\varepsilon_{0}+\delta\varepsilon)-V_{M}. \tag{22}\]  
Therefore, for the discrete modulation scheme with a modulation variance \(V_{M}\), source noise \(\delta\varepsilon\), channel transmittance \(T_{0}\), and excess noise \(\varepsilon_{0}\), the lowest minimum of the secret key rate can be obtained by the secret key rate of the Gaussian modulation scheme. The transmittance \(T\) and excess noise \(\varepsilon\) of the equivalent Gaussian modulation scheme are derived by (15). Combining with (6), the secret key rate can be obtained. And the main parameters of the CVQKD protocol in the security analysis are shown in Table 1.  
In Figure 9, we demonstrate the secret key rate as a function of modulation variance \(V_{M}\) or excess noise \(\varepsilon\) with transmission distance \(d=100 km\), \(\forall\in\{0.003,0.005,0.008,0,01\}\). We can achieve the maximum secret key rate for \(V_{M}=0.35\) in condition of the variational excess noise. In Figure 10, we show the secret key rate as a function of transmission distance, where the dotted line represents the original four-state protocol, the dashed line denotes the PDM-based scheme with discrete modulation, and the solid line is the PDM-based scheme of the four-state protocol with PPC. We find that the secret key rate of the multiplexing scheme is two times higher than that of the conventional four-state protocol.