## 3 Low-Dimensional Reconciliation
### Noise on the Data
The error-corrected \(U_{j}\) can be expressed from the noisy \(U^{\prime}_{j,i}\), as follows:  
\[U_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}-\sum_{i=0}^{d-1}\varsigma_{j,i}=U_{j}- \varsigma_{j}\in\mathbb{R}, \tag{63}\]  
where \(\varsigma_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\varsigma_{j,i}}^{2}\Big{)}\) characterizes the same amount of noise as (61), i.e., and \(\varsigma_{j,i}=\delta_{j,i}\), however it is evaluated from the noisy raw-data units \(U^{\prime}_{j,i}\), \(C\big{(}X^{\prime}_{j,i}\big{)}\) as:  
\[\varsigma_{j,i}=\frac{U^{\prime}_{j,i}}{C\big{(}X^{\prime}_{j,i}\big{)}}C\big{(} \Delta_{j,i}\big{)}\in\mathbb{R}, \tag{64}\]  
with \(\varsigma_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\varsigma_{j,i}}^{2}\Big{)}\). The \(d\)-dimensional vector \(\mathbf{U}^{\prime}_{j}\in\mathbb{R}^{d}\) can be expressed as:  
\[\mathbf{U}^{\prime}_{j}=\mathbf{U}_{j}+\overset{\rightarrow}{\delta}_{j}\in \mathbb{R}^{d}, \tag{65}\]  
where the noise vector \(\overset{\rightarrow}{\delta}_{j}=\Big{\{}\delta_{j,0},\ldots,\delta_{j,d-1} \Big{\}}\in\mathbb{R}^{d}\) is as follows:  
\[\overset{\rightarrow}{\delta}_{j}=\frac{\mathbf{U}_{j}}{C\big{(}\mathbf{X}_{j }\big{)}}C\big{(}\Delta_{j}\big{)}\in\mathbb{N}\Big{(}0,\sigma_{\delta_{j}}^{ 2}\Big{)}_{d}=\mathbb{N}\Big{(}0,\sigma_{\delta_{j,0},\ldots,\delta_{j,d-1}}^{2 }\Big{)}. \tag{66}\]  
According to the CLT, the sum of independent noise on units \(U^{\prime}_{j,i}\) in \(\mathbf{U}^{\prime}_{j}\in\mathbb{R}^{d}\) is evaluated by a Gaussian random variable as:  
\[\delta_{j}=\sum_{i=0}^{d-1}\delta_{j,i}=\frac{\sum_{i=0}^{d-1}C\big{(}\Delta_{ j,i}\big{)}U_{j,i}}{\sum_{i=0}^{d-1}C\big{(}X_{j,i}\big{)}}\in\mathbb{N}\Big{(}0, \sigma_{\delta_{j}}^{2}=\sum_{i=0}^{d-1}\sigma_{\delta_{j,i}}^{2}\Big{)}. \tag{67}\]The \(d\)-dimensional vector \(\mathbf{U}_{j}\in\mathbb{R}^{d}\) can be expressed as  
\[\mathbf{U}_{j}=\mathbf{U}^{\prime}_{j}-\overset{\rightarrow}{\xi}_{j}\in \mathbb{R}^{d}, \tag{68}\]  
and the noise vector \(\overset{\rightarrow}{\xi}_{j}=\left\{\xi_{j,0},\ldots,\xi_{j,d-1}\right\} \in\mathbb{R}^{d}\) is as follows:  
\[\overset{\rightarrow}{\xi}_{j}=\frac{\mathbf{U}^{\prime}_{j}}{C\big{(} \mathbf{X}_{j}\big{)}+C\big{(}\Delta_{j}\big{)}}C\big{(}\Delta_{j}\big{)}\in \mathbb{N}\bigg{(}0,\sigma_{\xi_{j}}^{2}\bigg{)}_{d}. \tag{69}\]  
The sum of independent noise on units \(U^{\prime}_{j,i}\) of \(\mathbf{U}^{\prime}_{j}\in\mathbb{R}^{d}\) can also be identified as:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]0  
From the physical properties of a Gaussian quantum channel [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11], we know exactly what happens during the transmission of the coherent combined signal from [PERSON] to [PERSON]. The noise on \(X^{\prime}_{j,i}\) has a non-standard Gaussian random distribution \(\Delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{N_{2}}^{2}\Big{)}\). We have to analyze in detail the properties of the noise vector. The noise vector \(\Delta_{j}\in\mathbb{N}\Big{(}0,\sigma_{N_{2}}^{2}\Big{)}_{d}\in\mathbb{R}^{d}\) of \(\mathcal{N}_{2}\) that generates the noisy \(\mathbf{X}^{\prime}_{j}\) from \(\mathbf{X}_{j}\) is characterized, as follows. First we decompose the noise vector \(\Delta_{j}\) into its components:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]1  
where matrix \(\mathbf{A}_{j}\) represents a linear transformation in \(\mathbb{R}^{d}\), while \(\Lambda_{j}\) is a the standard Gaussian noise vector \(\Lambda_{j}\in\mathbb{N}\big{(}0,1\big{)}_{d}\in\mathbb{R}^{d}\). The probability density function of \(\Lambda_{j}\) is:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]2  
where \(\|\Lambda_{j}\|=\sqrt{\Lambda_{j,0}^{2}+\ldots+\Lambda_{j,d-1}^{2}}\) is magnitude, in other words, the Euclidean distance from the origin to \(\Lambda_{j}\in\mathbb{R}^{d}\).