c & b
\end{bmatrix}
\oplus\begin{bmatrix}
a & -c\\
-c & b
\end{bmatrix}
. \label{eq:CM-standard-form}$$ The symplectic diagonalization of the covariance matrix $V$ simplifies as well [@SIS04]:$$V=S\left(  D\oplus D\right)  S^{T},$$ where$$\begin{aligned}
S  &  =\left(  I_{2}\oplus\sigma_{Z}\right)  S_{0}^{\oplus2}\left(
I_{2}\oplus\sigma_{Z}\right)  ,\\
S_{0}  &  =\begin{bmatrix}
\omega_{+} & \omega_{-}\\
\omega_{-} & \omega_{+}\end{bmatrix}
,\qquad\omega_{\pm} =\sqrt{\frac{a+b\pm\sqrt{y}}{2\sqrt{y}}},\\
D  &  =\begin{bmatrix}
\nu_{-} & 0\\
0 & \nu_{+}\end{bmatrix}
,\qquad\nu_{\pm} =\left[  \sqrt{y}\pm\left(  b-a\right)  \right]  /2,\\
y  &  =\left(  a+b\right)  ^{2}-4c^{2},\end{aligned}$$ and $\sigma_{Z}$ denotes the standard Pauli $Z$ matrix. Given a two-mode state with covariance matrix in standard form as in , it is a separable state if$$c\leq c_{\text{sep}}\equiv\sqrt{\left(  a-1\right)  \left(  b-1\right)  },
\label{eq:c_sep}$$ which can be determined from the condition given in [@AI05 Eq. (14)]. We return to this condition when we discuss the relative entropy of entanglement for quantum Gaussian states.

A quantum Gaussian channel is one that preserves Gaussian states [@CEGH08; @adesso14; @S17]. The action of a quantum Gaussian channel on an input state $\rho$ is characterized by two matrices $X$ and $Y$, which transform the covariance matrix $V^{\rho}$ of $\rho$ as follows:$$V^{\rho}\rightarrow XV^{\rho}X^{T}+Y, \label{eq:Gaussian-channel}$$ where $X^{T}$ is the transpose of the matrix $X$. In this formalism, the thermal channel $\mathcal{L}_{\eta,N_{B}}$ with transmissivity $\eta\in(0,1)$ and thermal mean photon number $N_{B}>0$ 