of the resource state on a 50-50 beam splitter. The sender then performs ideal homodyne detection of the position quadrature of the first mode and ideal homodyne detection of the momentum quadrature of the second mode, leading to measurement outcomes $Q_{+}$ and $P_{-}$. The sender communicates these real values over ideal classical communication channels to the receiver, and the receiver performs displacement operations of his mode by $g\sqrt{2}Q_{+}$ and $g\sqrt{2}P_{-}$, for some $g\in\mathbb{R}$. The result of all of these operations is to implement a quantum Gaussian channel of the following form on the input state:$$\begin{aligned}
X  &  =gI_{2},\\
Y  &  =\left[  g^{2}a+2gc+b\right]  I_{2},\end{aligned}$$ where we note the different sign convention from [@LMGA17 Eq. (7)], due to our slightly different convention for the standard form in . If $g>0$, then the channel implemented is a single-mode phase-insensitive Gaussian channel with$$\tau=g^{2},\qquad y=g^{2}a+2gc+b. \label{eq:TP-simulation-1-mode-G}$$ If $g<0$, then one can postprocess the output according to a unitary Gaussian channel with $X=-I_{2}$ and $Y=0$ (a phase flip channel), such that the overall channel is a single-mode phase-insensitive Gaussian channel with $\tau$ and $y$ as in . A generalization of these steps beyond two-mode states is given in [@WPG07].

Where [@LMGA17] departs from prior works is 