## 5 Numerical Evidence and Noise Model  
### Reconciliation Characteristics  
In this section, we analyze the performance of the proposed reconciliation for Gaussian modulation, in terms of secret key rates (bits/pulse) and distances. The excess noise \(\mathfrak{N}\) of the Gaussian quantum channel is expressed as  
\[\mathfrak{N}=\Big{(}\sigma_{\omega_{E}}^{2}-1\Big{)}(1-T)T^{-1}, \tag{117}\]  
Figure 5: The error probability of the scalar reconciliation process. It converges exponentially to zero as \(|a-b|>2\eta\). where \(T\) is the transmission, and \(\sigma_{\omega_{E}}^{2}\) is [PERSON]'s modulation variance [1]. Assuming reconciliation efficiency \(0\leq\beta\leq 1\), the key rate can be rewritten as  
\[R=\beta I(A:B)-\chi(B:E), \tag{118}\]  
where \(I(A:B)\) is the mutual information between [PERSON] and [PERSON], while \(\chi(B:E)\) is the Holevo information between [PERSON] and [PERSON], respectively, with relation  
\[\chi(B:E)<\chi(A:E), \tag{119}\]  
where \(\chi(A:E)\) is the Holevo information between [PERSON] and [PERSON] at a direct reconciliation [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. At a given SNR, the mutual information of [PERSON] and [PERSON] is [1; 2; 3; 4; 5; 6; 7; 8]  
\[\chi(A:B)\geq 1/2\,\log_{2}(1+{\rm SNR}), \tag{120}\]  
where  
\[{\rm SNR}=\sigma_{\phi}^{2}\Big{/}\sigma_{N_{2}}^{2}, \tag{121}\]  
where \(\sigma_{\phi}^{2}\) is the transmit signal's variance, \(\sigma_{N_{2}}^{2}\) is the variance of \(\mathcal{N}_{2}\), which has parameters that can be calculated from \(T\) and \(\mathfrak{N}\). In Figure 6a the \(d\sigma_{\delta_{j}}^{2}\) quantities of the converted logical binary Gaussian channel for various dimensions are shown. As depicted by the red line, the Lyapunov-condition can be exploited to get variance  
\[\lim_{N/d\to\infty}d{\rm var}\Big{[}\delta_{0...N/d}\,\Big{]}={\rm var}\Big{[} \delta_{0...N/d}\,\Big{]}\approx\Big{(}\sigma_{N_{2}}^{2}\Big{)}_{d} \tag{122}\]  
for arbitrary \(d\) to maximize the SNR value,  
\[{\rm SNR}=\sigma_{X}^{2}\Big{/}\sigma_{\delta_{j}}^{2} \tag{123}\]  
of the converted logical channel. As depicted in Figure 6b, for \(d\to\infty\), the efficiency converges to one, \(\beta\to 1\), because the noise perfectly converges to a zero-mean Gaussian random variable. Figure 6: **(a)** The Signal-to-Noise Ratio (SNR) of the resulting logical binary channel is maximized by the Lyapunov-condition (red line). It makes possible to convert the physical Gaussian quantum channel to a logical channel with the same noise variance for arbitrary \(d\). For the blue line the Lyapunov-condition is not satisfied. **(b)** The capacity of the logical channel for various dimensions. At low SNRs the capacity of the physical Gaussian quantum channel (dashed line) coincidences with the capacity of the binary Gaussian channel (red). For \(d=16\), the capacity of the logical channel is very close to the capacity of a binary Gaussian channel, and at low SNRs it perfectly coincidences with the capacity of the Gaussian quantum channel. The reconciliation efficiency at \(d=16\) is \(\beta=0.97\). The curves for lower \(d\)-s do not exist because the resulting logical channels are not Gaussian, since the Lyapunov-condition is not satisfied in the low-regimes. The numerical analysis uses a two-way CVQKD protocol, with homodyne measurements. The parameters are as follows. Excess noise \(\mathfrak{N}=0.015\), \(T=0.8\), variance \(\sigma_{X}^{2}=1.06\), channel correlation \(n_{C}=0.5\), which parameter describes the correlation of the Gaussian attacks of [PERSON] in the range of \(0\leq n_{C}\leq 1\)[7; 8]. (_Note_: If \(n_{C}=0\), there is no correlation between her attacks of \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\)[7; 8]). In Figure 7 the SNR of the logical binary Gaussian are depicted for various dimensions. The performance of scalar reconciliation is summarized in Figure 8. The performance of the simulated protocol without scalar reconciliation with reconciliation efficiency \(\beta=0.9\), is depicted by the blue curve [7; 8].