## 4 Theorems and Proofs  
First, we show that [PERSON]'s noisy secret can be corrected in the \(\mathbf{v}\) vector space of \(\mathbb{R}^{d}\) by using an error-correction rule based on the apparatus provided by the maximum-likelihood decision [15; 16; 17; 18; 19; 24; 25], which renders unnecessary the use of the spherical space of \(\Gamma^{d-1}\). **Proposition 1** (Vector reconciliation of correlated Gaussian variables). : _The Gaussian noise \(\delta_{j}\) on the received vector \(\mathbf{U}_{j}^{\prime}\in\mathbb{R}^{d}:\left\{U_{j,0}^{\prime},\ldots,U_{j,d -1}^{\prime}\right\}\) can be corrected in the vector space \(\mathbf{v}\) of \(\mathbb{R}^{d}\)._  
Figure 4: The real Gaussian noise of the quantum channel \(\mathcal{N}_{2}\) causes a rotation and rescaled vector in the combined phase space \(\mathcal{S}_{A\times B}\) (\(x\): position quadrature, \(p\): momentum quadrature). The magnitude \(\|\Delta_{j,i}\|\) of the noise vector \(\Delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}\) is not preserved, since the noise characteristic describes an ellipse in the combined phase space. Proof. : First, [PERSON] selects the \(d\)-dimensional vector \(\mathbf{U}_{j}\in\left\{U_{j,0},\ldots,U_{j,d-1}\right\}\in\mathbb{R}^{d}\) where \(\sum_{i=0}^{d-1}U_{j,i}=a\) or \(\sum_{i=0}^{d-1}U_{j,i}=b\), \(U_{j,i}\in\mathcal{U}\) and sends \(C\big{(}\mathbf{X}^{\prime}_{j}\big{)}\mathbf{U}_{j}\) over the classical channel. [PERSON] uses her CDF-transformed raw data \(C\big{(}\mathbf{X}_{j}\big{)}=\left\{C\big{(}X_{j,0}\big{)},\ldots,C\big{(}X_{ j,d-1}\big{)}\right\}\) to obtain \(\mathbf{U}^{\prime}_{j}\in\mathbb{R}^{d}\). Since [PERSON] knows \(a\), \(b\) and \(d\), she can draw two vectors \(\mathbf{A}=\left(A_{0},\ldots,A_{d-1}\right)^{T}\in\mathbb{R}^{d}\), with norm \(\|\mathbf{A}\|=\sqrt{\sum_{i=0}^{d-1}\left(A_{i}\right)^{2}}\), where \(\left\{\sum_{i=0}^{d-1}A_{i}=a\right\}\), \(A_{i}\in\mathcal{U}\) and \(\mathbf{B}=\left(B_{0},\ldots,B_{d-1}\right)^{T}\in\mathbb{R}^{d}\), with \(\|\mathbf{B}\|=\sqrt{\sum_{i=0}^{d-1}\left(B_{i}\right)^{2}}\), where \(\left\{\sum_{i=0}^{d-1}B_{i}=b\right\}\), \(B_{i}\in\mathcal{U}\). She then corrects the noise on \(\mathbf{U}^{\prime}_{j}\) by the following error-correction rule [15; 16; 17; 18; 19]:  
\[\mathbf{U}_{j}=\mathbf{A}:\|\mathbf{U}^{\prime}_{j}-\mathbf{A}\|<\|\mathbf{U} ^{\prime}_{j}-\mathbf{B}\|, \tag{77}\]  
\[\mathbf{U}_{j}=\mathbf{B}:\|\mathbf{U}^{\prime}_{j}-\mathbf{A}\|>\|\mathbf{U} ^{\prime}_{j}-\mathbf{B}\|, \tag{78}\]  
where the quantity \(\|\mathbf{U}^{\prime}_{j}-\mathbf{U}_{j}\|\), \(\mathbf{U}_{j}\in\{\mathbf{A},\mathbf{B}\}\) is evaluated as  
\[ \begin{array}{ll}\|\mathbf{U}^{\prime}_{j}-\mathbf{U}_{j}\|&=\sqrt{\sum_{i= 0}^{d-1}\left(U^{\prime}_{j,i}-U_{j,i}\right)^{2}}=\sqrt{\sum_{i=0}^{d-1} \left(\frac{u_{j,i}}{c\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{i,j}\big{)}\right) ^{2}}\\ &=\sqrt{\sum_{i=0}^{d-1}\left(\delta_{j,i}\right)^{2}}=\|\overset{\rightarrow}{ \delta_{j}}\|,\end{array}  \tag{79}\]  
which precisely coincidences with the norm of the Gaussian noise in (67). However, since [PERSON] does not know [PERSON]'s \(U_{j,i}\), in (79) an additional noise, \(\Upsilon_{j}\), also brings up, i.e., \(\|\mathbf{U}^{\prime}_{j}-\mathbf{U}_{j}\|=\|\delta_{j}+\Upsilon_{j}\|\). The noise vector \(\overset{\rightarrow}{\Upsilon_{j}}\) with expected variance \(\sigma_{\overset{\rightarrow}{\Upsilon_{j}}}^{2}\) is independent from the real noise on \(U^{\prime}_{j,i}\). This problem will be resolved in Theorem 1 and will be shown that this quantity completely vanishes from the picture. [PERSON] receives the \(d\)-dimensional vectors \(\mathbf{U}^{\prime}_{j}\in\left\{U^{\prime}_{j,0},\ldots,U^{\prime}_{j,d-1} \right\}\in\mathbb{R}^{d}\), and corrects \(\mathbf{U}^{\prime}_{j}\) into \(\mathbf{U}_{j}\) and then from the components she rebuilds the full key \(\mathbf{K}=\left(U_{0},\ldots,U_{(N/d)-1}\right)^{T}\in\mathbb{R}^{N/d}\).