## 4 Performance Comparison between Protocols
### Optical Perturbation Statistics  
As we saw, we have a number of parameters characterizing the perturbation per quantum channel: \(\omega\), \(\varphi\), and \(\theta\). As, in general, they are different for [PERSON] and for [PERSON], we have six different quantities. Now, each of these six parameters is a random variable. We assume that the PDF is separable as a product of probability functions of each of the six variables following Gaussian distributions. Even if it is not always valid, it is very reasonable for the case that we are going to study. Moreover, assuming a Gaussian distribution for the phase is something that has already been considered in the literature [40] in a setting similar to ours. It is also reasonable from the point of view of crosstalk [41]. We assume Gaussians centered at zero for \(\delta=\omega_{a},\omega_{b},\varphi_{a},\varphi_{b}\), taking values between 0 and \(2\pi\). We assume that the variance of the distribution is proportional to the propagation length. This way, we can capture the physical fact that the longer the fiber link, the more the perturbation effects accumulate. Furthermore, the specific reason why the proportionality follows \(\sigma\propto\sqrt{L}\) assumes that we are dealing with a random walk process [42]. In other words, for the variables above \(\delta=\omega_{a},\omega_{b},\varphi_{a},\varphi_{b}\), we assume the following PDF:  
\[p(\delta)=\frac{1}{\sqrt{N_{\delta}\gamma_{\delta}L}}\exp\left[-\frac{1}{2} \bigg{(}\frac{\delta^{2}}{\gamma_{\delta}L}\bigg{)}\right], \tag{23}\]  
where \(N_{\delta}\) is a normalization factor and \(\gamma_{\delta}\) is the proportionality constant of the variance with \(L\), which measures, in a certain sense, the strength of the perturbation relative to the length scale. For \(\theta_{a}\) and \(\theta_{b}\), we need to modify such a distribution and take into account the proper Haar measure [30]. This means that the factor \(\sin(\theta)\), with \(\theta\) standing for \(\theta_{a}\) and \(\theta_{b}\), needs to be included in the distribution, with \(\theta\) being confined to the range \([0,\pi]\), that is,  
\[p(\theta)=\frac{\sin\theta}{\sqrt{N_{\theta}\gamma_{\theta}L}}\exp\left[-\frac {1}{2}\left(\frac{\theta^{2}}{\gamma_{\theta}L}\right)\right], \tag{24}\]  
where \(N_{\theta}\) is again another normalization factor. We must stress that the introduction of normalization factors \(N\), beyond the ones that can be computed analytically, allows for computational flexibility, as we can truncate the distribution around values that are found to be highly improbable, i.e., values having negligible contributions to the expected values. This defines a region of interest in which more samples can be allocated, thus optimizing computational resources. In addition to this, note that, in principle, the \(\gamma\)s (and the \(N\)s) are different for each variable. To determine a typical value for each of them, we must link the PDF's variance with experimental data corresponding to codirectional encoding. Note that the off-diagonal terms in Equation (8) are bounded by \(\sin(\theta_{i}/2)\) (\(j=a,b\)). These terms combine the states \(|1_{aI}\rangle\) with \(|1_{aII}\rangle\), so \(\gamma_{\theta_{a}}\) (and similarly \(\gamma_{\theta b}\)) is related with the intercore crosstalk in multicore fibers. This crosstalk decreases with the core pitch, as the overlap between modes of neighbouring cores is reduced. The experimental ratios of power transference range between \(10^{-4}\) and \(10^{-6}\) km\({}^{-1}\)[43] and are obtained by launching light in a core and measuring the power transferred to the other ones at the fiber exit. These values give us a typical value of \(\sin^{2}(\theta/2)\simeq\theta^{2}/4\). Thus, we consider \(\gamma_{\theta a}=\gamma_{\theta b}\simeq 5\times 10^{-5}\) km\({}^{-1}\). Since \(\theta\) is small in Equation (8), \(\varphi\) and \(\omega\) are directly related to the relative phase between the modes of both cores.