# Continuous-Variable Quantum Key Distribution Robust Against Polarization-Dependent Loss
## 4 Polarization-Pairwise Coding Scheme  
On account of the polarization states whose relative orientations are random and time dependent, the overall instantaneous PDL varies randomly, both in frequency and in time. As a result, it is a corresponding randomization of the received optical signal noise ratio (OSNR). In what follows, we propose a PPC scheme to mitigate the involved PDL, leading to the increased secret key rate of the PDM-based CVQKD system. As shown in Figure 4, The continuous light emitted by the light source is divided into two groups of light carriers with mutually orthogonal polarization through the polarization splitter. Then, the random code of quantum random number generator (QRNG) is modulated into the corresponding driving signal through the level generator and modulator driver. Two Stokes parameter encoders are the key to the realization of phase modulators in each polarization direction. Finally, the optical signals of these two groups of orthogonal polarization are synthesized into one group using PBC, so as to obtain the optical signals. As shown in Figure 5a, when interleaving the real (I) and imaginary (Q) components of both polarizations, only the I or Q of each polarization suffers the worse SNR. Whereas the other signal component has a better SNR, resulting in  
Figure 3: (**a**) The effect of PDL on the components of the electric. (**b**) Orthogonality of two polarizations with increase of \(\mu\). the improved performance of both polarizations. The PPC scheme is used for improving the CVQKD system without encoding processing in a wide range of worst cases of PDL. The polarization-pairwise pre-coding scheme is shown in Figure 5. Two polarizations are mapped to quadrature amplitude modulation (QAM) symbols \(X_{n}=X_{0}+P_{0}j\) and \(Y_{n}=X_{1}+P_{1}j\), respectively. After a constant phase shift, \(\theta\), is applied to the related symbols, we obtain \(X_{\theta,n}\) and \(Y_{\theta,n}\), respectively. The optimal rotation angle, \(\theta_{\text{opt}}\), can be derived analytically for QPSK [26]:  
\[\theta_{opt}=\left\{ \begin{array}{cc}\pi/4,&\lambda\leq 3\\ \tan^{-1}[(\lambda-1)-\sqrt{(\lambda-1)^{2}-\lambda}],&\lambda>3,\end{array} \right. \tag{5}\]  
where \(\lambda=\bigtriangleup\)SNR defines the OSNR difference induced by PDL between two polarizations by \(\bigtriangleup\)SNR = OSNRgood/OSNRbad, and this rotation angle is used for minimizing the error rate \(e_{ab}\) for a  
Figure 4: (Color online.) Polarization-division-multiplexing (PDM)-quadrature phase shift modulation (QPSK) optical transmitter. Figure 5: (Color online.) The polarization-pairwise coding (PPC)-based CVQKD system with PDM-QPSK. (**a**) Transmitter polarization-pairwise pre-coding. (**b**) Receiver polarization-pairwise decoding. given \(\triangle\)SNR between two polarizations. After the angular rotation, I/Q component interleaving is used for generating signals for two polarizations. The processing can also be described as  
\[TX_{n} =\Re(X_{\theta})+\Re(Y_{\theta}) \tag{6}\] \[=X_{\mathbb{C}}^{0}\cos\theta-P_{\mathbb{C}}^{0}\sin\theta+(X_{ \mathbb{C}}^{1}\cos\theta-P_{\mathbb{C}}^{1}\sin\theta)j,\] \[TY_{n} =\Im(X_{\theta})+\Im(Y_{\theta})\] \[=X_{\mathbb{C}}^{0}\sin\theta-P_{\mathbb{C}}^{0}\cos\theta+(X_{ \mathbb{C}}^{1}\sin\theta-P_{\mathbb{C}}^{1}\cos\theta)j,\]  
where \(\Re\) and \(\Im\) denote the I and Q parts of signals, respectively. Figure 4a indicates the PPC-based CVQKD system with PDM-QPSK for the rotation angle \(\theta_{opt}=45^{\circ}\). The I/Q interleaving and angular rotation can be rewritten as  
\[ \begin{bmatrix}\Re(TX_{n})&\Im(TX_{n})\\ \Re(TY_{n})&\Im(TY_{n})\end{bmatrix} = \begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}X_{\mathbb{C}}^{0}&X_{ \mathbb{C}}^{1}\\ P_{\mathbb{C}}^{0}&P_{\mathbb{C}}^{1}\end{bmatrix} . \tag{7}\]  
We note that the above-derived transform matrix is orthogonal, and, hence, such a coding scheme will not involve extra sensitivity penalty in comparison to conventional QAM modulation without PDL.