## 3 Low-Dimensional Reconciliation
### Achieving the Uniform Distribution
#### 3.2.2 Central Limit Theorem  
In the multidimensional case, the precision of the approximation of the logical binary Gaussian channel (i.e., the quality of the physical-logical channel conversion) was quantified by the Dirac distribution [9; 10; 11]. Since in the scalar reconciliation the spherical space is eliminated, a different solution was needed to analyze the accuracy of the conversion between the physical-logical Gaussian channels. Our answer for the problem is the Central Limit Theorem [26; 43; 44; 45] and a mathematical result from the 19 th century--the so-called Lyapunov-condition [26; 45]. The accuracy of the physical-logical conversion of scalar reconciliation can be maximized, and it can be made in arbitrary high dimensions, as it is being stated in Lemma 1. **Lemma 1**. : _The noise variance of the converted logical binary Gaussian channel asymptotically coincidences with the noise variance of the physical quantum channel, which allows to reach the theoretical maximum of the capacity of the converted logical binary channel._  
Proof. : Let \(X_{j,i}\in\mathbb{R}\) and \(X^{\prime}{}_{j,i}\in\mathbb{R}\) the \(j\)-th units of [PERSON]'s and [PERSON]'s raw data, respectively. For a \(d\)-dimensional vector \(\mathbf{U}_{j}=\left(U^{\prime}{}_{j,0},\ldots,U^{\prime}{}_{j,d-1}\right)^{ \text{T}}\), the sum of the independent noise \(\left\{\delta_{j,0},\ldots,\delta_{j,d-1}\right\}\) units on the secret noisy key units \(U^{\prime}{}_{j,i}=U_{j,i}+\delta_{j,i}\) will approximate a zero-meanGaussian random variable with mean \(\mathbb{E}\left[\delta_{j,i}\right]=\mu_{\delta_{j,i}}=0\), noise variance \(\text{var}\left[\delta_{j,i}\right]=\sigma_{\delta_{j,i}}^{2}\) (see Sections 3.1 and 3.3 for a detailed derivation) as follows:  
\[ \begin{array}{l}\mathbf{CLT}:\frac{1}{\sqrt{\sum_{i=0}^{d-1}\sigma_{\delta_{j,i}}^{2}}}\delta_{j}=\frac{1}{\sqrt{\sum_{i=0}^{d-1}\sigma_{\delta_{j,i}}^{2}} }\left(\sum_{i=0}^{d-1}\delta_{j,i}\right)\rightarrow\mathbb{N}\left(0,1 \right)_{d}\\ \delta_{j}=\begin{pmatrix}\sum_{i=0}^{d-1}\delta_{j,i}\end{pmatrix}\to \mathbb{N}\left(0,\sum_{i=0}^{d-1}\sigma_{\delta_{j,i}}^{2}\right)=\mathbb{N} \left(0,\sigma_{\delta_{j,i}}^{2}\right)_{d}.\end{array}  \tag{43}\]  
To show that (43) holds for the \(d\)-dimensional noise parameter \(\delta_{j}\), we exploit the Lyapunov-condition [26]. Applying the standard mathematical description of the Lyapunov condition [45], let \(\mathfrak{L}>0\), then  
\[\lim_{d\rightarrow\infty}\frac{1}{\left(\sqrt{\sum_{i=0}^{d-1}\sigma_{\delta_ {j,i}}^{2}}\right)^{2+\mathfrak{L}}}\sum_{i=0}^{d-1}\mathbb{E}\left[\left| \delta_{j,i}\right|^{2+\mathfrak{L}}\right]=0 \tag{44}\]  
is satisfied for any \(d\rightarrow\infty\), by theory. As follows, the noise on \(\mathbf{U}_{j}\in\mathbb{R}^{d}\) will converge to  
\[\delta_{j}=\left(\sum_{i=0}^{d-1}\delta_{j,i}\right)\in\mathbb{N}\left(0, \sigma_{\delta_{j}}^{2}\right)_{d}, \tag{45}\]  
and the resulting logical channel will be equivalent to a logical binary Gaussian channel with noise variance \(\sigma_{\delta_{j}}^{2}\). By the same argumentation, the variance of the resulting logical binary Gaussian channel will converge to the variance of the physical Gaussian quantum channel \(\sigma_{\mathcal{N}_{2}}^{2}\) for \(N\rightarrow\infty\). Let again \(\mathfrak{L}>0\), and \(d\) is an appropriate dimension for which (44) is satisfied, and let the expected variance of \(\delta_{j}\) is \(\text{var}\left[\delta_{j}\right]=\sigma_{\mathcal{N}_{2}}^{2}\). Then  
\[\lim_{N\rightarrow\infty}\frac{1}{\left(\sqrt{\sum_{j=0}^{\left(N/d\right)-1 }\sigma_{\mathcal{N}_{2}}^{2}}\right)^{2+\mathfrak{L}}}\sum\
olimits_{j=0}^{ \left(N/d\right)-1}\mathbb{E}\left[\left|\delta_{j}\right|^{2+\mathfrak{L}} \right]=0, \tag{46}\]  
is satisfied by theory, from which  
\[