to solve an inverse problem regarding teleportation simulation. Given values of $\tau$ and $y$ corresponding to a physical channel different from the pure-loss channel, the authors of [@LMGA17] proved that there exists a finite-energy, two-mode Gaussian state in standard form satisfying , having its smaller symplectic eigenvalue equal to one, and having its logarithmic negativity equal to that of the Choi state of the channel. It should be stressed that the states found in [@LMGA17] have an analytical form, which has to do with the form of the above constraints.

Information quantities and bounds for secret-key-agreement protocols
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The basic information quantities that we need in this paper are the quantum relative entropy [@U62; @Lindblad1973], the relative entropy variance [@li12; @TH12], and the hypothesis testing relative entropy [@BD10; @WR12]. For two states $\rho$ and $\sigma$ defined on a separable Hilbert space with the following spectral decompositions:$$\begin{aligned}
\rho &  =\sum_{x}\lambda_{x}|\phi_{x}\rangle\langle\phi_{x}|,\label{eq:spec-decomp-rho}\\
\sigma &  =\sum_{y}\mu_{y}|\psi_{y}\rangle\langle\psi_{y}|,
\label{eq:spec-decomp-sigma}$$ the quantum relative entropy $D(\rho\Vert\sigma)$ [@Lindblad1973] and the relative entropy variance $V(\rho\Vert\sigma)$ [@li12; @TH12] are defined as $$\begin{aligned}
D(\rho\Vert\sigma)  &  =\sum_{x,y}\left\vert \left\langle \psi_{y}|\phi
_{x}\right\rangle \right\vert ^{2}\lambda_{x}\log_{2}(\lambda_{x}/\mu_{y}),\\
V(\rho\Vert\sigma)  &  =\sum_{x,y}\left\vert \left\langle \psi_{y}|\phi
_{x}\right\rangle \right\vert ^{2}\lambda_{x}\left[  \log_{2}(\lambda_{x}/\mu_{y})-D(\rho\Vert\sigma)\right]  ^{2}.\end{aligned}$$

For quantum Gaussian states, the quantities $D(\rho\Vert\sigma)$ [@PhysRevA.71.062320], [@PLOB15] and $V(\rho\Vert\sigma)$ 