## 2 System Model
### Uniform Distribution in the Spherical Space  
In this section we review the background of the multidimensional approaches, and the properties of Gaussian random vectors in the spherical space. The multidimensional reconciliation processes for CVQKD were not implementable without the use of spherical codes and a high-dimensional spherical space. First, let us clarify how a \(d\)-dimensional Gaussian random vector is formulated in the framework of a two-way CVQKD protocol. The outcoming beam from [PERSON] (and Bob) can be regarded as a collection of Gaussian random variables. A standard Gaussian random variable \(g\in\mathbb{N}(0,1)\in\mathbb{R}\) is a real variable selected from a Gaussian distribution. A standard Gaussian variable \(g\in\mathbb{N}(0,1)\) has probability density function [15; 18]:  
\[f(g)=\frac{1}{\sqrt{2\pi}}e^{\frac{-g^{2}}{2}}. \tag{12}\]  
A non-standard Gaussian random variable \(g^{*}\in\mathbb{N}\big{(}\mu,\sigma^{2}\big{)}\in\mathbb{R}\) with nonzero mean \(\mu\
eq 0\), and variance \(\sigma^{2}\), can be expressed from \(g\in\mathbb{N}(0,1)\) as \(g^{*}=g\sigma+\mu\). A non-standard Gaussian random variable \(g^{*}\) has probability density function:  
\[f(g^{*})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{\frac{-(g^{*}-\mu)^{2}}{2\sigma^{2 }}}. \tag{13}\]  
In [PERSON]'s raw data, a \(d\)-dimensional Gaussian vector  
\[\mathbf{X}_{j}=\left(X_{j,0},\ldots,X_{j,d-1}\right)^{T}\in\mathbb{N}\Big{(}0,\sigma_{X}^{2}\Big{)}_{d}\in\mathbb{R}^{d} \tag{14}\]  
is a collection of \(d\) independent Gaussian random variables \(X_{j,0},\ldots,X_{j,d-1}\), where each \(X_{j,i}\) is a real variable \(\mathbb{R}\) drawn from a Gaussian random distribution \(\mathbb{N}(0,\sigma_{X}^{2})\). [PERSON]'s Gaussian vector is referred by \(\mathbf{X}_{j}\in\mathbb{N}(0,\sigma_{X}^{2})_{d}\in\mathbb{R}^{d}\), and its noisy version at [PERSON]'s side is denoted by \(\mathbf{X}^{\prime}_{j}\in\mathbb{N}(0,\sigma_{X^{\prime}}^{2})_{d}\in \mathbb{R}^{d}\). The values of [PERSON]'s units are affected by the Gaussian noise that arises from the quantum channel. First, let us evaluate why the normalized vector structure has importance in the multidimensional scenario. The normalized \(d\)-dimensional Gaussian vectors change the probability distribution from Gaussian random to uniform on the \(d\)-dimensional unit sphere, \(\Gamma^{d-1}\). It has a relevance, since only uniform distribution is allowed in the reconciliation phase. The result clearly follows from the [PERSON] law [18], the application of [PERSON]'s formula [19], [PERSON]'s conjecture [22], and [PERSON]'s result [23], which are connected to the contributions of spherical coding [24]. We formulate \(d\)-length blocks \(\mathbf{X}^{\prime}_{j}=\left(\mathbf{X}^{\prime}_{j,0},\ldots,\mathbf{X}^{ \prime}_{j,d-1}\right)^{T}\in\mathbb{N}\big{(}0,\sigma_{\mathbf{X}^{\prime}} ^{2}\big{)}_{d}\in\mathbb{R}^{d}\), where \(\mathbf{X}^{\prime}_{j,i}\in\mathbb{N}\big{(}0,\sigma_{\mathbf{X}^{\prime}}^{ 2}\big{)}\in\mathbb{R}\), for \(i\in[d]\). The \(d\)-length Gaussian random vector \(\mathbf{X}^{\prime}_{j}\) has norm \(\left\|\mathbf{X}^{\prime}_{j}\right\|\), mean  
\[\mathbb{E}\big{[}\|\mathbf{X}^{\prime}_{j}\|\big{]}=\sigma_{\mathbf{X}^{ \prime}}\sqrt{d-\frac{1}{2}} \tag{15}\]  
and variance  
\[\mathrm{var}\big{[}\|\mathbf{X}^{\prime}_{j}\|\big{]}\leq\frac{\sigma_{ \mathbf{X}^{\prime}}^{2}}{2}. \tag{16}\]  
We step further from this point. Since the variance of \(\mathbf{X}^{\prime}_{j}\) is not unit, the covariance matrix \(\mathfrak{E}\big{(}\mathbf{X}^{\prime}_{j}\big{)}\) is not equal to identity, but the random units \(\mathbf{X}^{\prime}_{j,i}\) are uncorrelated, so \(\mathfrak{E}\big{(}\mathbf{X}^{\prime}_{j}\big{)}\) is diagonal.