and some open questions.

Preliminaries
=============

Quantum Gaussian states and channels
------------------------------------

The main class of quantum states in which we are interested in this paper are quantum Gaussian states [@adesso14; @S17]. In our brief review, we consider $m$-mode Gaussian states, where $m$ is some fixed positive integer. Let $\hat{x}_{j}$ denote each quadrature operator ($2m$ of them for an $m$-mode state), and let $\hat{x}\equiv\left[  \hat{q}_{1},\ldots,\hat{q}_{m},\hat
{p}_{1},\ldots,\hat{p}_{m}\right]  \equiv\left[  \hat{x}_{1},\ldots,\hat
{x}_{2m}\right]  $ denote the vector of quadrature operators, so that the first $m$ entries correspond to position-quadrature operators and the last $m$ to momentum-quadrature operators. The quadrature operators satisfy the following commutation relations:$$\left[  \hat{x}_{j},\hat{x}_{k}\right]  =i\Omega_{j,k},\quad\mathrm{where}\quad\Omega=\begin{bmatrix}
0 & 1\\
-1 & 0
\end{bmatrix}
\otimes I_{m}\text{,} \label{eq:symplectic-form}$$ and $I_{m}$ is the $m\times m$ identity matrix. We also take the annihilation operator $\hat{a}=\left(  \hat{q}+i\hat{p}\right)  /\sqrt{2}$. Let $\rho$ be a Gaussian state, with the mean-vector entries $\left\langle \hat{x}_{j}\right\rangle ^{\rho}=\mu_{j}^{\rho}$, and let $\mu^{\rho}$ denote the mean vector. The entries of the covariance matrix $V^{\rho}$ of $\rho$ are given by $$V_{j,k}^{\rho}\equiv\left\langle \left\{  \hat{x}_{j}-\mu_{j}^{\rho},\hat
{x}_{k}-\mu_{k}^{\rho}\right\}  \right\rangle ^{\rho}.
\label{eq:covariance-matrices}$$ A $2m\times2m$ matrix $S$ is symplectic if it preserves the symplectic form: $S\Omega S^{T}=\Omega$. According to Williamson’s theorem [@W36], there is a diagonalization of the covariance matrix $V^{\rho}$ of the form, 