## 2 Materials and Methods
### QBER Model  
It should be noted that the choice of polarization basis \((H,V)\) is determined in such a way that one of the basis vectors is parallel to the OF dichroism axis (i.e., making relaxation superoperator diagonal). However, modulators may be aligned in the other basis \((H^{\prime},V^{\prime})\), we assume both of them are aligned by the \(H^{\prime}\) direction. Then initial Jones vector of the state at the entrance to the first modulator is in the basis \((H^{\prime},V^{\prime})\) and takes the form  
\[\left( \begin{array}{c}\alpha_{H^{\prime}}=1\\ \alpha_{V^{\prime}}=0\end{array} \right) \tag{19}\]  
We can establish the relation between two vectors:  
\[\left( \begin{array}{c}\alpha_{H}\\ \alpha_{V}\end{array} \right)=\left( \begin{array}{cc}\cos(\theta)&-\sin( \theta)\\ \sin(\theta)&\cos(\theta)\end{array} \right)\left( \begin{array}{c}\alpha_{H ^{\prime}}=1\\ \alpha_{V^{\prime}}=0\end{array} \right)=\left( \begin{array}{c}\cos(\theta) \\ \sin(\theta)\end{array} \right) \tag{20}\]  
where \(\theta\) is the angle between \((H^{\prime},V^{\prime})\) and \((H,V)\) bases. The system of dynamic equations introduced in the previous section takes into account various physical phenomena. It is also necessary to determine the amount of optical power corresponding to each polarization mode. We want to acknowledge that the term \(\frac{(\omega_{H}-\omega_{V})}{2}\) stands for birefringence of OF. This parameter in general depends not only on physical characteristics of OF, but also on stress caused by curls, bends, torsions, etc. Thus, this parameter may vary significantly. In order to take this fact into account we assume that the system may contain a set of wave plates before the receiver's block to balance out and hence disregard the birefringence. We can express assumptions stated above simply as \(\frac{(\omega_{H}-\omega_{V})}{2}=0\) and \(S_{3}(0)=S_{3}(t)=0\). Hence, there is a two-dimensional solution in the \((S_{1},S_{2})\) plain. Then optical power corresponding to each polarization mode can be defined as  
\[P_{H}=\frac{(S_{0}+S_{1})}{2},\quad P_{V}=\frac{(S_{0}-S_{1})}{2} \tag{21}\]  
We substitute the terms defined in (21) into the system (18) to find the solution:  
\[ \begin{array}{l}P_{H}(t,\mu)=P_{H}(0,\mu)e^{-\gamma_{H}t}+\frac{n_{T}}{2} \left(1-e^{-\gamma_{H}t}\right)\\ P_{V}(t,\mu)=P_{V}(0,\mu)e^{-\gamma_{V}t}+\frac{n_{T}}{2}\left(1-e^{-\gamma_{ V}t}\right)\end{array}  \tag{22}\]  
with  
\[P_{H}(0,\mu)=\mu\left|\alpha_{H}\right|^{2},\quad P_{V}(0,\mu)=\mu\left|\alpha _{V}\right|^{2} \tag{23}\]  
where \(\mu\) is the mean photon number of the sidebands, \(\alpha_{H}\) and \(\alpha_{V}\) are Jones vector components in the corresponding basis. It should be taken into account that expressions in the Equation (22) have the form of power, not amplitude, while performing the reverse bases transform:\[ \begin{array}{l}P_{H^{\prime}}(t,\mu)=P_{H}(t,\mu)\cos^{2}(\theta)+P_{V}(t,\mu) \sin^{2}(\theta)+\sqrt{P_{H}(t,\mu)P_{V}(t,\mu)}\sin(2\theta)\\ P_{V^{\prime}}(t,\mu)=P_{V}(t,\mu)\cos^{2}(\theta)+P_{H}(t,\mu)\sin^{2}(\theta) -\sqrt{P_{H}(t,\mu)P_{V}(t,\mu)}\sin(2\theta)\end{array}  \tag{24}\]  
Here we define the following domain \(0\leq\theta\leq\frac{\pi}{2}\). Initial state is aligned by \(H^{\prime}\) then it is modulated by \(m_{H^{\prime}}\). However, due to propagation there might be non-zero \(P_{V^{\prime}}(t,\mu)\) component that should be modulated second time with different modulation index \(m_{V^{\prime}}\) (that depends on signal polarization). Thus we introduce \(\sigma=\frac{m_{V^{\prime}}}{m_{H^{\prime}}}\). The second modulation leads to the increase or reduction in mean photon number depending on modulation signal phase difference as well as on the modulation indices difference, according to the Equation (11). There are four possible cases (it is implied that the equipment possesses no imperfections, in order for us to estimate the contribution from polarization divergence only):  
1.