## 2 System Model
### Physical Coding  
In the following description we give a considerable view of the coding of two-way CVQKD, focusing on the contributions of information theory. Let us denote the quadratures of the \(i\)-th signal \(S_{Alice,i}\) in the phase space \(\mathcal{S}_{A}\) by \(x_{A,i},p_{A,i}\), and the quadratures of [PERSON]'s signal \(S_{Bob,i}\) in the phase space \(\mathcal{S}_{B}\) by \(x_{B,i},p_{B,i}\), where \(x_{A,i},p_{A,i}\in\mathbb{N}\big{(}0,\sigma_{\omega}^{2}\big{)}\) and \(x_{B,i},p_{B,i}\in\mathbb{N}\big{(}0,\sigma_{\omega}^{2}\big{)}\) are drawn from a Gaussian random distribution with mean \(\mu=0\), and variance \(\sigma_{\omega}^{2}\), where \(\sigma_{\omega}^{2}\) is the modulation variance [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. The coherent states \(S_{Alice,i}=\left|x_{A,i}+ip_{A,i}\right\rangle\in\mathcal{S}_{A}\) and \(S_{Bob,i}=\left|x_{B,i}+ip_{B,i}\right\rangle\in\mathcal{S}_{B}\) are encoded by Gaussian modulation with dedicated centers \((x_{A,i},p_{A,i})\in\mathcal{S}_{A}\) and \((x_{B,i},p_{B,i})\in\mathcal{S}_{B}\), respectively (_Note_: Each \(S_{i}\) define a zero-mean, circular symmetric complex Gaussian random variable \(\mathcal{CN}\Big{(}0,\sigma_{S_{i}}^{2}\Big{)}\) with variance \(\sigma_{S_{i}}^{2}=\mathbb{E}\Big{[}\left|S_{i}\right|^{2}\Big{]}\) in the phase space \(\mathcal{S}\), with i.i.d. real and imaginary components \(x_{i},p_{i}\in\mathbb{N}\big{(}0,\sigma_{\omega}^{2}\big{)}\), thus \(\sigma_{S_{i}}^{2}=2\sigma_{\omega}^{2}\). The squared magnitude \(\left|S_{i}\right|^{2}\) is exponentially distributed with density \(f\Big{(}\left|S_{i}\right|^{2}\Big{)}=1\Big{/}\sigma_{S_{i}}^{2}\)\(\exp\Big{(}-\left|S_{i}\right|^{2}\Big{/}\sigma_{S_{i}}^{2}\Big{)}, \left|S_{i}\right|^{2}\geq 0\). The two beams are correlated at Alice's BS, which results in a combined signal in the combined phase space \(\mathcal{S}_{A\times B}\). The modulation noise \(\partial\in\mathcal{CN}\big{(}0,\sigma_{\partial}^{2}\big{)}\), is precisely centered around \((x_{A,i}+x_{B,i},p_{A,i}+p_{B,i})\in\mathcal{S}_{A\times B}\) and \((x_{A,i}-x_{B,i},p_{A,i}-p_{B,i})\in\mathcal{S}_{A\times B}\) in \(\mathcal{S}_{A\times B}\). After the two beams \(S_{Alice,i}\) and \(S^{\prime}_{Bob,i}\) are correlated at a BS at [PERSON]'s side, where \(S^{\prime}_{Bob,i}\) is the noisy version of \(S_{Bob,i}\), [PERSON] applies a random quadrature measurement \(M_{1}\) on the first mode of the beam, while the second mode is transmitted back to [PERSON] over quantum channel \(\mathcal{N}_{2}\). [PERSON]'s state in the combined phase space \(\mathcal{S}_{A\times B}\) is as follows:  
\[\left|\varphi_{i}\right\rangle=\left|x_{A,i}+x^{\prime}_{B,i}+i\Big{(}p_{A,i} +p^{\prime}_{B,i}\Big{)}\right\rangle\in\mathcal{CN}\Big{(}0,\sigma_{\varphi_ {i}}^{2}\Big{)}\in\mathcal{S}_{A\times B}, \tag{3}\]  
with Gaussian random quadrature components \(\mathbb{N}\Big{(}0,2\sigma_{\omega}^{2}+\sigma_{\mathcal{N}_{1}}^{2}\Big{)}\), where \(2\sigma_{\omega}^{2}\) is the cumulated modulation variance, \(\sigma_{\mathcal{N}_{1}}^{2}\) is the variance of \(\mathcal{N}_{1}\), \(x^{\prime}_{B,i}\), \(p^{\prime}_{B,i}\) are [PERSON]'s noisy quadratures modified by \(\mathcal{N}_{1}\), while \(\sigma_{\varphi_{i}}^{2}=\mathbb{E}\Big{[}\left|\varphi_{i}\right|^{2}\Big{]}\). Assuming a homodyne measurement \(M_{1}\), [PERSON] gets an \(X_{i}\) unit of her raw data, which is a binary string.