## 3 Low-Dimensional Reconciliation
### Noise on the Data  
This section reveals the mathematical description of the noise vector of the Gaussian quantum channel \(\mathcal{N}_{2}\) and its impacts on [PERSON]'s raw data and [PERSON]'s received secret key. We also can exploit that in the evaluation of the noise vector, only the second channel use \(\mathcal{N}_{2}\) has to be taken in to consideration in the error correction. The \(d\)-dimensional _noise vector_\(\Delta_{j}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}_{d}\in \mathbb{R}^{d}\) of the Gaussian channel \(\mathcal{N}_{2}\) on the \(j\)-th \(\mathbf{X}^{\prime}_{j}\) is a Gaussian random vector defined as:  
\[\Delta_{j}=\mathbf{X}^{\prime}_{j}-\mathbf{X}_{j}=\Big{\{}\Delta_{j,0},\ldots,\Delta_{j,d-1}\Big{\}}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2} \Big{)}_{d}\in\mathbb{R}^{d}, \tag{55}\]  
where \(\Delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}\in \mathbb{R}\) identifies the Gaussian noise on the \(i\)-th unit \(X_{i}\) of \(\mathbf{X}^{\prime}_{j}\) as:  
\[\Delta_{j,i}=X^{\prime}_{j,i}-X_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{ N}_{2}}^{2}\Big{)}\in\mathbb{R}. \tag{56}\]  
The noise vector \(\Delta_{j}\) is added to [PERSON]'s \(\mathbf{X}_{j}\), hence [PERSON]'s noisy \(\mathbf{X}^{\prime}_{j}\) is:  
\[\mathbf{X}^{\prime}_{j}=\mathbf{X}_{j}+\Delta_{j}\in\mathbb{R}^{d}. \tag{57}\]  
In terms of raw-data vector units, the Gaussian noise vector \(\Delta_{j,i}\) is described as follows:  
\[X^{\prime}_{j,i}=X_{j,i}+\Delta_{j,i}\in\mathbb{R}, \tag{58}\]and (57) can be rewritten as:  
\[ \begin{array}{ll}\mathbf{X}^{\prime}_{j}&=\Big{\{}X^{\prime}_{j,0},\ldots,X^{ \prime}_{j,d-1}\Big{\}}\\ &=\Big{\{}X_{j,0}+\Delta_{j,0},\ldots,X_{j,d-1}+\Delta_{j,d-1}\Big{\}}.\end{array}  \tag{59}\]  
In the scalar reconciliation, the error-correction is performed on the level of unit sums \(U^{\prime}_{j}\ =\ \sum_{i=0}^{d-1}U^{\prime}_{j,i}\) in \(\mathbb{R}\) as follows. [PERSON] receives the \(d\)-dimensional \(C\big{(}\mathbf{X}^{\prime}_{j}\big{)}\mathbf{U}_{j}\) from [PERSON], from which she obtains \(C\big{(}X^{\prime}_{j}\big{)}U_{j}\) (see (35)) and divides it by her \(C\big{(}X_{j}\big{)}\) (see (34)). The effect of Gaussian noise [9] results in a distorted secret \(U^{\prime}_{j}\in\mathbb{R}\) as:  
\[U^{\prime}_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}=\frac{\sum_{i=0}^{d-1}C\big{(} X^{\prime}_{j,i}\big{)}U_{j,i}}{\sum_{i=0}^{d-1}C\big{(}X_{j,i}\big{)}}=\sum_{i=0}^ {d-1}U_{j,i}+\sum_{i=0}^{d-1}\delta_{j,i}=U_{j}+\delta_{j}\in\mathbb{R}, \tag{60}\]  
where \(\delta_{j,i}\) is the noise on \(U_{j,i}\) (for a plausible example, see Section 5):  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]  
where \(\sigma_{\delta_{j,i}}^{2}\) is the variance of the distribution of \(\delta_{j,i}\), while \(C\big{(}\Delta_{j,i}\big{)}\) is the noise of the CDF-transformed raw data units:  
\[C\big{(}\Delta_{j,i}\big{)}=C\big{(}X^{\prime}_{j,i}\big{)}-C\big{(}X_{j,i} \big{)}\in\mathbb{R}, \tag{62}\]  
where \(C\big{(}\Delta_{j,i}\big{)}\in\mathbb{N}\Big{(}0,\sigma_{C(\Delta_{j,i})}^{2} \Big{)}\), and \(C\big{(}\Delta_{j}\big{)}=C\big{(}X^{\prime}_{j}\big{)}-C\big{(}X_{j}\big{)} \in\mathbb{R}^{d}\), with a distribution of \(\mathbb{N}\Big{(}0,\sigma_{C(\Delta_{j})}^{2}\Big{)}_{\dot{d}}\).