# Continuous-Variable Quantum Key Distribution Robust Against Polarization-Dependent Loss
## 5 Performance Analysis  
The four-state protocol has higher data reconciliation efficiency, and thus can be used for lengthening the transmission distance of the CVQKD system [35; 36]. [PERSON] sends a random coherent state \(|\alpha_{k}\rangle\)=\(|\alpha\exp[i(2k+1)/4]\rangle\), \(\forall\ k\in\{0,1,2,3\}\), to [PERSON], where \(\alpha\) is a real number that can be optimized to maximize the security code rate. In what follows, we show the secure communication code rate of the four-state protocol. In order to facilitate the security analysis, it is often used to consider the equivalent preparation-measurement scheme. Due to the nonideal modulation in the preparation-measurement scheme and the noise of the light source, the two-mode entangled state produced by [PERSON] is not a pure coherent state, but a mixed state with mixed noise. Due to the symmetry, it is assumed that the mixed state has the same noise as the input of the \(X\) component and the \(P\) component. The noise will not be used by [PERSON] and can be introduced as a neutral Charlie. Considering the existence of the neutral Charlie, the equivalent entanglement scheme of the case becomes the tripartite subsystems, involving [PERSON], [PERSON], and [PERSON], which constitutes a pure entangled state. Assuming the attacker is strong enough, we can replace the actual channel between [PERSON] and [PERSON] with a perfect channel and pure the state \(\rho_{ABC}\). So, the total system \(|\psi_{ABC}\rangle\) is a pure state. If [PERSON] can use the state in [PERSON]'s hand, she can get more information. Therefore, the lowest minimum of the security key rate can be obtained. \[K=\beta I(a:b)-\chi(b:EC), \tag{8}\]  
where \(\beta\) is the key extraction efficiency and \(I(a:b)\) is the mutual information between [PERSON] and [PERSON]. For the binary symmetric system, \(I(a:b)\) can be completely decided by the SNR of Bob [37] (more details in Appendix B):  
\[I_{a:b}=1-h(e_{ab}), \tag{9}\]  
where \(h(p)=-p\log_{2}(p)-(1-p)\log_{2}(1-p)\) is the binary entropy function and  
\[e_{ab}=1-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\sqrt{\text{SNR}}}e^{-\frac{x^{ 2}}{2}}\text{d}x. \tag{10}\]  
Combining with the above-mentioned PPC scheme, we can obtain the decreased error rate, as shown in Figure 7, where the dashed line shows the transitional four-state protocol and the solid line indicates the PPC-based protocol. The error rate changes with SNR for \(\triangle SNR=3\ dB\). It can be observed that the pairwise coded signals achieve a lower error rate to the same SNR extent. In Figure 8, we show the relationship between SNR and \(I(a:b)\) for four different four-state protocols. The blue line represents the traditional protocol, where we use the most basic four-state protocol. The orange line shows the PDM-based protocol, the amount of \(I(a:b)\) has nearly doubled. The purple line represents the PDM-PPC-based protocol, where we use PPC scheme on the basis of PDM-based protocol. Under the same SNR, the PDM-PPC-based protocol can obtain the maximum \(I(a:b)\). Subsequently, the Holevo boundary \(\chi_{b:EC}\) is defined as  
\[\chi_{b:EC}=S(\rho_{EC})-\int p(b)S\rho_{EC}^{b}\text{d}b. \tag{11}\]  
Before [PERSON]'s measurement, the state \(|\psi_{ABC}\rangle\) is a pure state, resulting in \(S(\rho_{EC})=S(\rho_{AB})\). After [PERSON]'s measurement, the state \(\rho_{AEC}\  
Consequently, the lowest minimum of the secret key rate can be calculated as  
\[K=\beta I(a:b)-S(\rho_{AB})+S(\rho_{A}^{b}). \tag{13}\]  
Assume that the channel transmittance between [PERSON] and [PERSON] is \(T_{0}\) and the excess noise is \(\varepsilon_{0}\). When [PERSON] sends \(\rho_{B0}\) to [PERSON], it evolves to be \(\rho_{B}\), and the covariance matrix \(\gamma_{AB}\) of the \(\rho_{AB}\) is derived as  
\[\gamma_{AB}= \begin{bmatrix}(V_{M}+1)I&\sqrt{T_{0}}Z\sigma_{z}\\ \sqrt{T_{0}}Z\sigma_{z}&[T_{0}(V_{M}+\varepsilon_{0}+\delta\varepsilon)+1]I \end{bmatrix}