## 3 Low-Dimensional Reconciliation
### Noise on the Data
This type of noise exhibits different behavior than the real Gaussian noise of a quantum channel, and it is characterized by the same magnitude \(\|\Lambda_{j}\|\) in every direction. This property is connected to the standard Gaussian random noise, and it cannot be applied in a realistic CVQKD scenario, because it does not properly describe the noise characteristic of the quantum channel. The probability density function of \(\Delta_{j}\in\mathbb{R}^{d}\) is:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]3  
where \(\mathbf{A}_{j}\mathbf{A}_{j}^{T}\) stands for the \(\mathfrak{C}(\Delta_{j})\) covariance matrix of \(\Delta_{j}\), and it analogous of \(\sigma_{N_{2}}^{2}\), i.e., in a more precise form \(\mathfrak{C}\big{(}\Delta_{j}\big{)}=\mathbb{E}\Big{(}\Delta_{j}\Delta_{j}^{T} \Big{)}=\mathbf{A}_{j}\mathbf{A}_{j}^{T}\). The noise on the units \(X^{\prime}_{j,i}\) of \(\mathbf{X}^{\prime}_{j}\) at [PERSON]'s side arises from the quantum-level transmission of the combined phase space states \(|\phi_{j,i}\rangle\in\mathcal{S}_{A\times B}\), and vectors \(\Lambda_{j}\in\mathbb{N}(0,1)_{d}\) and \(\Delta_{j}\in\mathbb{N}\Big{(}0,\sigma_{N_{2}}^{2}\Big{)}_{d}\) is built up by \(d\) components, \(\Lambda_{j,i}\in\mathbb{N}(0,1)\in\mathbb{R}\) and \(\Delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{N_{2}}^{2}\Big{)}\in\mathbb{R}\). The error \(\Delta_{j,i}\) on the \(i\)-th unit \(X^{\prime}_{j,i}\) is as follows:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]4where \(\mathbf{A}_{j,i}\) is a linear transformation that scales \(\Lambda_{j,i}\). The probability density function of \(\Lambda_{j,i}\) is:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]5  
where \(\|\Lambda_{j,i}\|=\sqrt{\Lambda_{j,i}^{2}}\) is the magnitude of \(\Lambda_{j,i}\). The probability density function of \(\Delta_{j,i}\) is:  
\[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]6  
where \(\mathbf{A}_{j,i}\mathbf{A}_{j,i}^{T}=\mathbb{E}\Big{(}\Delta_{j,i}\Delta_{j,i }^{T}\Big{)}=\mathfrak{C}\big{(}\Delta_{j,i}\big{)}\). From \(\Lambda_{j,i}\) and \(\Delta_{j,i}\), the correction of [PERSON]'s noisy secret \(\mathbf{U}_{j}\) can be approached by the units \(\left\{U_{j,0}^{\prime},\ldots,U_{j,d-1}^{\prime}\right\}\), because the noise of \(\mathcal{N}_{2}\) is survived in the raw data level and lives also on \(U_{j,i}^{\prime}\), but in a modified form, see (61). Let us denote by \(\left|\phi_{j,i}\right.\rangle\), the phase-space representation of [PERSON]'s noise-free raw data unit \(X_{j,i}\) given by (6), and by \(\left|\xi_{j,i}\right.\rangle\) the noisy raw data unit \(X_{j,i}^{\prime}\) of [PERSON], from (7). (State \(\left|\phi_{j,i}\right.\rangle\) is the second mode of the combined beam, while \(\left|\xi_{j,i}\right.\rangle\) is its noisy version). The effect of the real Gaussian noise of the quantum channel is shown in \[\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}, \tag{61}\]7 The noise vector \(\Delta_{j}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}_{d}\in \mathcal{S}_{A\times B}\) of the quantum channel is a non-standard Gaussian random vector, which distorts the density. The circles of \(\Lambda_{j,i}\in\mathbb{N}(0,1)\) are scaled by \(\mathbf{A}_{j,i}\), resulting in ellipses. The magnitude \(\|\Delta_{j,i}\|\) of \(\Delta_{j,i}\) is not preserved in all directions, which leads to a different density. The \(x\) and \(p\) quadratures of \(\left|\phi_{j,i}\right.\rangle\in\mathcal{S}_{A\times B}\) are modified by \(\Delta_{x}\) and \(\Delta_{p}\) in \(\left|\xi_{j,i}\right.\rangle\in\mathcal{S}_{A\times B}\).