## 4 Theorems and Proofs
Since \(x\) is a scalar and in (99) the term \(\frac{1}{2}(\mathbf{A}+\mathbf{B})\) is a constant, [PERSON] introduces vector \(\chi\in\mathbf{v}\) as follows:  
\[\chi\equiv\mathbf{U}^{\prime}_{j}-\frac{1}{2}(\mathbf{A}+\mathbf{B})=x( \mathbf{A}-\mathbf{B})+\overset{\rightarrow}{\delta}_{j}. \tag{102}\]  
She also draws an orthogonal matrix \(\mathbf{M}\), which contains \(\mathbf{d}\) and the orthogonal directions \(\mathbf{n}_{1},\ldots,\mathbf{n}_{l}\), with unit norm as:  
\[\mathbf{M}=\left( \begin{array}{c}\mathbf{d}\\ \mathbf{n}_{1}\\ \mathbf{n}_{2}\\ \vdots\\ \mathbf{n}_{l}\end{array} \right). \tag{103}\]  
By multiplying \(\mathbf{M}\) with \(\chi\) leads to:  
\[\mathbf{M}\chi=\left( \begin{array}{c}x\|\mathbf{A}-\mathbf{B}\|\\ 0\\ 0\\ \vdots\\ 0\end{array} \right)+\mathbf{M}\overset{\rightarrow}{\delta}_{j}. \tag{104}\]  
From (104), it clearly follows that only \(x\|\mathbf{A}-\mathbf{B}\|\) and the first component of \(\mathbf{M}\overset{\rightarrow}{\delta}_{j}\) have relevance in the error-correction process, because all of the other components are orthogonal to \(\mathbf{d}\)[15]. Since the evolution of \(\mathbf{d}\) is a trivial process on [PERSON]'s side, the received \(\mathbf{U}^{\prime}_{j}\) can be projected by \(\mathcal{P}\) onto the direction of \(\mathbf{d}\), since all of the valuable information, including the real noise, is carried only by this direction. The projection \(\mathcal{P}\) on \(\mathbf{U}^{\prime}_{j}\) is made by \(\mathbf{d}^{T}\chi\), which then results in:  
\[ \begin{array}{rl}\mathcal{P}\big{(}\mathbf{U}^{\prime}_{j}\big{)}&=\mathbf{ d}^{T}\chi\\ &=\Big{(}\frac{\mathbf{A}-\mathbf{B}}{\|\mathbf{A}-\mathbf{B}\|}\Big{)}^{T}(x( \mathbf{A}-\mathbf{B})+\overset{\rightarrow}{\delta}_{j})\\ &=\mathbf{d}^{T}\Big{(}\mathbf{U}^{\prime}_{j}-\frac{1}{2}(\mathbf{A}+ \mathbf{B})\Big{)}.\end{array}  \tag{105}\]  
The projected vector \(\mathcal{P}\big{(}\mathbf{U}^{\prime}_{j}\big{)}\) is analogous to the scalar representation \(U_{j}=\sum_{i=0}^{d-1}U_{j,i}\) in \(\mathbb{R}\), and makes it possible to correct the noise in the scalar space \(\mathbb{R}\). The received \(U^{\prime}_{j}=U_{j}+\delta_{j}\) hasmean \(\mu_{a}=a\) or \(\mu_{b}=b\), and the decision boundary is \(\frac{\mu_{a}+\mu_{b}}{2}\), which defines a separator in \(\mathbb{R}\). According to the previously obtained calculations, (104) can be rewritten as follows:  
\[\mathbf{M}\chi=\left( \begin{array}{c}x\sqrt{\sum_{i=0}^{d-1}\left(A_{i}-B_{i} \right)^{2}}\\ 0\\ 0\\ \vdots\\ 0\end{array} \right)+\delta_{j}. \tag{106}\]  
As follows, only the first component of \(\mathbf{M}\overset{\rightarrow}{\delta}_{j}\) has relevance in the error-correction, which in particular coincidences with the scalar quantity \(\delta_{j}=\sum_{i=0}^{d-1}\delta_{j,i}=\frac{\sum_{i=0}^{d-1}C\left(\Delta_{ ij}\right)l_{ij}}{\sum_{i=0}^{d-1}C\left(X_{ij}\right)}\) shown in (79). Putting the pieces together, \(\mathcal{P}\left(\mathbf{U}_{j}^{\prime}\right)\) is evaluated as:  
\[\mathcal{P}\left(\mathbf{U}_{j}^{\prime}\right)=x\sqrt{\sum_{i=0}^{d-1}\left(A _{i}-B_{i}\right)^{2}}+\sum_{i=0}^{d-1}\delta_{j,i} \tag{107}\]  
which contains all of the sufficient information for the error correction in \(\mathbb{R}\), which completes the proof. In Theorem 2, the error probability of scalar reconciliation is proposed in an exact form. **Theorem 2. **_The error probability \(\Pr(error)=Q\left(\frac{\left|a-b\right|}{2}\frac{1}{\eta}\right)\) of scalar reconciliation depends only on \(\left|a-b\right|\), where \(Q\left(\frac{\left|a-b\right|}{2}\frac{1}{\eta}\right)=\Pr\left(\frac{\left|a -b\right|}{2}\frac{1}{\eta}<\xi\right)\) is the Q-function (tail function), \(g\) is a standard Gaussian random variable \(g\in\mathbb{N}(0,1)\), and \(\eta=\sqrt{\sigma_{kj}^{2}}=\sqrt{\sum_{i=0}^{d-1}\sigma_{kj,i}^{2}}\) is the standard deviation of the Gaussian noise \(\delta_{j}\). The \(\Pr(error)\) exponentially converges to zero for any \(\left|a-b\right|>2\eta\)._  
Proof.