## 2 System Model
### Physical Coding
If she measured in the position quadrature basis she obtains:  
\[X_{i}=x_{A,i}+x^{\prime}_{B,i}, \tag{4}\]  
or, if she used the momentum quadrature basis she gets  
\[X_{i}=p_{A,i}+p^{\prime}_{B,i}. \tag{5}\]  
The second mode of the combined signal in \(\mathcal{S}_{A\times B}\) is transmitted directly back to [PERSON] over the noisy channel \(\mathcal{N}_{2}\), given as:  
\[\left|\varphi_{i}\right\rangle=\left|x_{A,i}-x^{\prime}_{B,i}+i\Big{(}p_{A,i} -p^{\prime}_{B,i}\Big{)}\right\rangle\in\mathcal{CN}\Big{(}0,\sigma_{\varphi_ {i}}^{2}\Big{)}\in\mathcal{S}_{A\times B}, \tag{6}\]  
with \(\mathbb{N}\Big{(}0,2\sigma_{\omega}^{2}+\sigma_{\mathcal{N}_{1}}^{2}\Big{)}\) Gaussian random quadratures, and \(\sigma_{\varphi_{i}}^{2}=\mathbb{E}\Big{[}\left|\varphi_{i}\right|^{2}\Big{]}\). The Gaussian noise of the quantum channel \(\mathcal{N}_{2}\) defines a noise vector \(\Delta_{i}\in\mathcal{CN}\Big{(}0,\sigma_{\Delta_{i}}^{2}\Big{)}\in\mathcal{ S}_{A\times B}\), with noise components \(\Delta_{x_{i}}\in\mathbb{N}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}\), \(\Delta_{p_{i}}\in\mathcal{CN}\Big{(}0,\sigma_{\mathcal{N}_{2}}^{2}\Big{)}\), which results in the noisy state \(\left|\xi_{i}\right\rangle\in\mathcal{S}_{A\times B}\) as follows:  
\[\left|\xi_{i}\right\rangle=\left|\varphi_{i}\right\rangle+\Delta_{i}=\left|x^{ \prime}_{A,i}-x^{\prime}_{B,i}+i\Big{(}p^{\prime}_{A,i}-p^{\prime\prime}_{B,i} \Big{)}\right\rangle\in\mathcal{CN}\Big{(}0,\sigma_{\xi_{i}}^{2}\Big{)}\in \mathcal{S}_{A\times B}, \tag{7}\]  
with \(\mathbb{N}\Big{(}0,2\sigma_{\omega}^{2}+\sigma_{\mathcal{N}_{1}}^{2}+\sigma_{ \mathcal{N}_{2}}^{2}\Big{)}\) distributed Gaussian random quadratures, and \(\sigma_{\xi_{i}}^{2}=\mathbb{E}\Big{[}\left|\xi_{i}\right|^{2}\Big{]}\), where \(x^{\prime}_{A,i}\), \(p^{\prime}_{A,i}\) are [PERSON]'s noisy quadratures modified by \(\mathcal{N}_{2}\), while \(x^{\prime\prime}_{B,i}\), \(p^{\prime\prime}_{B,i}\) are [PERSON]'s noisy quadratures modified by \(\mathcal{N}_{2}\). In the next phase, [PERSON] applies a random quadrature measurement \(M_{2}\) (assumed to be homodyne) and gets block \(Y_{i}\). If he used a position quadrature basis, he gets  
\[{Y^{\prime}}_{i}={x^{\prime}}_{A,i}-{x^{\prime\prime}}_{B,i}, \tag{8}\]  
and for the momentum quadrature basis he obtains:  
\[{Y^{\prime}}_{i}={p^{\prime}}_{A,i}-{p^{\prime\prime}}_{B,i}. \tag{9}\]  
[PERSON], calibrating his resulted block \({Y_{i}}^{\prime}\) by \(2{x^{\prime\prime}}_{B,i}\) or \(2{p^{\prime\prime}}_{B,i}\) (depending on the used quadrature measurement), gets back the noisy version \({X^{\prime}}_{i}\) of [PERSON]'s raw data unit \(X_{i}\) as:  
\[{X_{i}}^{\prime}={Y^{\prime}}_{i}+2{x^{\prime\prime}}_{B,i}={x^{\prime}}_{A,i} -{x^{\prime\prime}}_{B,i}+2{x^{\prime\prime}}_{B,i}={x^{\prime}}_{A,i}+{x^{ \prime\prime}}_{B,i} \tag{10}\]  
and  
\[{X_{i}}^{\prime}={Y^{\prime}}_{i}+2{p^{\prime\prime}}_{B,i}={p^{\prime}}_{A,i }-{p^{\prime\prime}}_{B,i}+2{p^{\prime\prime}}_{B,i}={p^{\prime}}_{A,i}+{p^{ \prime\prime}}_{B,i}. \tag{11}\]  
which is referred as [PERSON]'s raw data unit. The nature of the of error of the quantum channel will be characterized in detail in Section 4, however, at this point, we can surmise that the noise of the quantum channel is analogous to the addition of a non-standard Gaussian random noise vector \(\Delta_{i}\) to [PERSON]'s raw data block \(X_{i}\). [PERSON]'s and [PERSON]'s modes in the combined phase space \(\mathcal{S}_{A\times B}\) right after being outputted from the BS are \(|\varphi_{i}\rangle\) and \(|\varphi_{i}\rangle\), as shown in Figure 2. [PERSON] obtains the first mode of the beam, \(|\varphi_{i}\rangle\), the second mode \(|\varphi_{i}\rangle\) is sent back to [PERSON]. The noise that exists in \(\mathcal{S}_{A\times B}\) arises from the modulation noise \(\partial\in\mathcal{CN}\big{(}0,\sigma_{\partial}^{2}\big{)}\) (already included in the quadrature distributions) and the two channel uses, \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\).