## 2 Materials and Methods  
### Effects of Modulation Indices Mismatch  
As it was stated earlier, this section will be based on the main approaches developed in [7; 20]. It should be noted that we investigate an idealized model of phase modulation. In this model we disregard single photon detector dark counts, synchronization, and filtration systems imperfections. Thus, we simplify the model and hence we explicitly consider the influence of only optical fiber characteristics on the developed quantum bit error rate (QBER); it is obvious that additional imperfections introduce higher error rates. Electro-optical phase modulator reallocates the energy between the interacting modes in such a way that a multimode state after the modulator can be expressed as  
\[\left|\psi\left(\phi_{A}\right)\right\rangle=\bigotimes_{j=-S}^{S}\left|\alpha_{ j}\right\rangle_{j}=\bigotimes_{j=-S}^{S}\left|\alpha_{0}d_{0j}^{S}(M)e^{-i\phi_{A}j} \right\rangle_{j} \tag{1}\]  
where \(\phi_{A}\) is the phase of radio-frequency modulating signal, \(\alpha_{j}\) is the amplitude of a coherent state at each mode after modulation, \(\alpha_{0}\) is the initial amplitude of single-mode coherent state that enters modulator, \(d_{0j}^{S}\) is the Wigner d-function commonly used throughout quantum angular momentum theory. Its argument is dependent on modulation index m as follows (approximately, taking into account large \(S\)):  
\[M=\frac{m}{S} \tag{2}\]  
After the signal is transmitted through the fiber and then modulated for the second time the multimode state has the form:  
\[\left|\psi_{B}\left(\phi_{A},\phi_{B}\right)\right\rangle=\bigotimes_{j=-S}^{S }\left|\sqrt{\eta(L)}\alpha_{0}e^{-i\left(\theta_{0}+\theta_{1}j\right)}d_{0j }^{S}\left(M^{\prime}\right)\right\rangle_{j} \tag{3}\]  
where \(\eta(L)\) is optical system transmission coefficient dependent on the length of optical fiber, \(\theta_{0}\) is phase obtained by transmission, \(\theta_{1}\) is common phase, introduced by modulator (later on we disregard this phase). A new d-function argument is defined by the following relation considering the perfect case [41]:  
\[\cos\left(M^{\prime}\right)=\cos^{2}(M)-\sin^{2}(M)\cos\left(\phi_{A}-\phi_{B}\right) \tag{4}\]  
If \(\phi_{A}-\phi_{B}=0\) then the d-function argument is doubled \(M^{\prime}=2M\). Meanwhile if \(\phi_{A}-\phi_{B}=\pi\) then the d-function argument turns zero \(M^{\prime}=0\). This case of \(d_{0j}^{S}(0)=\delta_{0j}\) corresponds to total energy allocation back to central mode leaving vacuum states on the side bands. It looks like some kind of interference. In general the expression stated above can be rewritten as follows:  
\[\cos\left(M^{\prime}\right)=\cos(M)\cos(M\sigma)-\sin(M)\sin(M\sigma)\cos\left( \phi_{A}-\phi_{B}\right) \tag{5}\]  
where \(\sigma\) corresponds to the mismatch coefficient of d-function arguments and hence of modulation indices (due to their proportionality). As it can be seen from the above, the signal's mean photon number that is incident on the single photon detector for the time \(\Delta T\) can be obtained as a sum of mean photon number of each spectral component. We want to acknowledge that spectral filtration is needed in order to select the signal of interest on the side bands (central frequency does not contain any phase information). Thus, the optical power of signal is proportional to the following expression:  
\[n_{ph}\left(\phi_{A},\phi_{B}\right)=\sum_{j\
eq 0}\left|\sqrt{\eta(L)}\alpha_{0 }e^{-i\theta_{0j}}d_{0j}^{S}\left(M^{\prime}\right)\right|^{2}=\mu_{0}\eta(L )\left(1-\left|d_{00}^{S}\left(M^{\prime}\right)\right|^{2}\right) \tag{6}\]  
And now let us investigate the expressions stated earlier with regard to several approximations. Firstly, standard fiber electro-optical modulators are used for the SCW QKD; so the following approximations are valid in the limit \(S\rightarrow\infty\) and (from Equation (2)) \(M\to 0\). Consequently, the following approximations may be introduced:  
\[