## 2 Materials and Methods
### Analysis of the Polarization Dynamics  
We would like to investigate the Liouville equation as in [39] that describes non-unitary dynamics of quantum state \(\hat{\rho}_{0}\) with regard to chosen polarization basis \((H,V)\):  
\[\begin{cases}\frac{\partial\hat{\rho}(t)}{\partial t}=-i[\hat{H},\hat{\rho}(t )]+\hat{\Gamma}\hat{\rho}(t)\\ \hat{\rho}(t)|_{t=0}=\hat{\rho}_{0}\end{cases} \tag{12}\]  
where Hamiltonian is as follows:  
\[\hat{H}=\sum_{i\in H,V}\omega_{i}\left(\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{ 1}{2}\right) \tag{13}\]  
where \(\omega_{i}\) is the optical signal frequency (with respect to basis), \(\hat{a}_{i}\) (\(\hat{a}_{i}^{\dagger}\)) is the annihilation (creation) operator of a polarization mode. The relaxation superoperator that describes the thermalization process acting on the state's density operator is as follows:  
\[\hat{\Gamma}\hat{\rho}=\sum_{i\in H,V}\frac{-\gamma_{i}}{2}\left(\left(n_{T}+ 1\right)\left(\hat{a}_{i}^{\dagger}\hat{a}_{i}\hat{\rho}+\hat{\rho}\hat{a}_{i }^{\dagger}\hat{a}_{i}-2\hat{a}_{i}\hat{\rho}\hat{a}_{i}^{\dagger}\right)+n_{ T}\left(\hat{a}_{i}\hat{a}_{i}^{\dagger}\hat{\rho}+\hat{\rho}\hat{a}_{i}\hat{a}_{i}^{ \dagger}-2\hat{a}_{i}^{\dagger}\hat{\rho}\hat{a}_{i}\right)\right) \tag{14}\]  
where \(\gamma_{i}\) is the thermalization rate (with respect to basis), \(n_{T}\) is the mean number of thermal photons, which equals  
\[n_{T}=\frac{1}{e^{\frac{\hbar\omega_{i}}{kT}}-1} \tag{15}\]  
where \(\hbar\) is reduced [PERSON]'s constant, \(k\) is [PERSON]'s constant, \(T\) is environmental temperature (here we assume that both polarization modes have almost the same value of \(n_{T}\) since it is small, e.g., \(n_{T}\approx 10^{-13}\) for 1.55 \(\upmu\)m and 300 K).  
We introduce the Stokes parameters in order to estimate the degree of polarization of a weak coherent state (considering a single mode) after being transmitted through optical fiber. The Stokes parameters can be expressed as mean values of the following operators [42]:  
\[\begin{array}{l}\hat{S}_{0}=\hat{a}_{H}^{\dagger}\hat{a}_{H}+\hat{a}_{V}^{ \dagger}\hat{a}_{V}\\ \hat{S}_{1}=\hat{a}_{H}^{\dagger}\hat{a}_{H}-\hat{a}_{V}^{\dagger}\hat{a}_{V} \\ \hat{S}_{2}=\hat{a}_{H}^{\dagger}\hat{a}_{V}+\hat{a}_{V}^{\dagger}\hat{a}_{H} \\ \hat{S}_{3}=-i\left(\hat{a}_{H}^{\dagger}\hat{a}_{V}-\hat{a}_{V}^{\dagger}\hat{ a}_{H}\right)\end{array} \tag{16}\]  
Then we may act by any of the stated above operators on the Equation (12) and apply the trace operation:\[\text{Tr}(\hat{S}\frac{\partial\hat{\rho}(t)}{\partial t})=-i\,\text{Tr}(\hat{S}[ \hat{H},\hat{\rho}(t)])+\text{Tr}(\hat{S}\hat{\Gamma}\hat{\rho}(t)) \tag{17}\]  
Thus, after some algebraic actions, we obtain a system of equations describing the dynamics of the Stokes parameters (here we assume that the bandwidth of the signal spectrum is rather narrow, comparable to several GHz, and, as a consequence, optical fiber characteristics are almost equal to its different frequency components):  
\[\begin{array}{l}\frac{\partial S_{0}}{\partial t}=-\frac{(\gamma_{H}+\gamma_ {V})}{2}S_{0}-\frac{(\gamma_{H}-\gamma_{V})}{2}S_{1}+n_{T}\left(\gamma_{H}+ \gamma_{V}\right)\\ \frac{\partial S_{1}}{\partial t}=-\frac{(\gamma_{H}-\gamma_{V})}{2}S_{0}- \frac{(\gamma_{H}+\gamma_{V})}{2}S_{1}+n_{T}\left(\gamma_{H}-\gamma_{V}\right) \\ \frac{\partial S_{2}}{\partial t}=-\frac{(\gamma_{H}+\gamma_{V})}{2}S_{2}- \frac{(\omega_{H}-\omega_{V})}{2}S_{3}\\ \frac{\partial S_{3}}{\partial t}=\frac{(\omega_{H}-\omega_{V})}{2}S_{2}- \frac{(\gamma_{H}+\gamma_{V})}{2}S_{3}\end{array} \tag{18}\]