## 5 Numerical Evidence and Noise Model
### Noise Analysis
#### 5.2.2 Noise on the Random Secret  
This example demonstrates that the noise \(\delta_{j}=\frac{\sum_{i=0}^{d-1}C\big{(}\Delta_{j}\big{)}U_{ij}}{\sum_{i=0}^{d -1}C\big{(}X_{ij}\big{)}}\) on the secret \(U^{\prime}_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}\) is inherited from the Gaussian quantum channel and by applying the Central Limit Theorem (CLT), the noise of the logical binary channel can be approximated by a Gaussian random variable \(\delta_{j}=\sum_{i=0}^{d-1}\delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\delta_{ j}}^{2}\big{)}\).  
Let \(N=1000\) units, the amount of sample raw data units \(X_{j,i}\), \(X^{\prime}_{j,i}\). The quantity \(C\big{(}\Delta_{j}\big{)}=C\big{(}\mathbf{X}^{\prime}_{j}\big{)}-C\big{(} \mathbf{X}_{j}\big{)}\) measures the difference of \(C\big{(}\mathbf{X}^{\prime}_{j}\big{)}\) and \(C\big{(}\mathbf{X}_{j}\big{)}\), i.e., the noise of Bob's  
Figure 11: (**a**) The distribution of the noisy raw data units on Bob’s side; (**b**) The CDF-transformed raw data units have uniform distribution in \([0,1]\).  
Figure 10: The distribution of the units of the noise vector of the Gaussian quantum channel. The noise affects the combined state in the phase space and the resulting raw data units on Bob’s side.  
CDF-transformed data. Let \(\mathbf{X}_{j}\in\mathbb{N}\big{(}0,\sigma_{X}^{2}=100\big{)}\) and \(\mathbf{X}^{\prime}_{j}\in\mathbb{N}\big{(}0,\sigma_{X^{\prime}}^{2}=104\big{)}\). The example uses an \(d=16\) dimensional approximation.  
The distribution of the error \(C\big{(}\Delta_{j,i}\big{)}\) of the CDF-transformed raw data units \(C\big{(}X^{\prime}_{j,i}\big{)}\), \(C\big{(}X_{j,i}\big{)}\) are depicted in Figure 12.  
In Figure 13a the ratio \(C\big{(}X^{\prime}_{j,i}\big{)}\big{/}C\big{(}X_{j,i}\big{)}\) of the CDF-transformed units is shown in Figure 13a. In the ideal (noise-free) case the ratio equals to 1. In Figure 13b the distribution of the quantity \(C\big{(}\Delta_{j,i}\big{)}\big{/}C\big{(}X_{j,i}\big{)}\) is shown.  
In Figure 14a the distribution of noise \(\delta_{j,i}\) on units \(U^{\prime}_{j,i}\) is shown, assuming that [PERSON] selects \(U_{j,i}\in\{-400/16,400/16\}\).  
In Figure 14b the distribution of \(\delta_{j}\) on \(U^{\prime}_{j}\), using \(U_{j}=\sum_{i=0}^{d-1}U_{j,i}\in\{-400,400\}\) is depicted. The distribution of \(\delta_{j}\) is given by the formula of \(\mathbb{N}\big{(}0,\sigma_{\delta_{j}}^{2}\big{)}\), and the approximation of the binary Gaussian logical channel is justified by the CLT and the Lyapunov-condition.  
Figure 13: (**a**) The distribution of the ratio of the raw data level noise and [PERSON]’s CDF-transformed raw data units. It equals to 1 for a noise-free case; (**b**) The distribution of quantity \(C\big{(}\Delta_{j,i}\big{)}\big{/}C\big{(}X_{j,i}\big{)}\).  
The results make it possible to achieve a high-precision conversion of the physical Gaussian quantum channel into a logical binary Gaussian channel. Precisely, only an approximation is possible by the logical layer manipulations, which gets closer to perfect as \(d\rightarrow\infty\). At \(d=16\) the approximation is almost perfect, and the noise on \(U^{\prime}{}_{j}=\sum_{i=0}^{d-1}U^{\prime}{}_{j,i}\) is a real Gaussian noise \(\mathbb{N}\left(0,\sigma_{\delta_{j}}^{2}\right)\).