which in turn leads to the improved upper bound in .

The relative entropy of entanglement of a bipartite state $\rho_{AB}$ is defined as follows [@VP98]:$$E_{R}(A;B)_{\rho}=\inf_{\sigma_{AB}\in\operatorname{SEP}(A:B)}D(\rho_{AB}\Vert\sigma_{AB}),$$ where $\operatorname{SEP}(A\!:\!B)$ denotes the set of separable (unentangled) states [@W89]. Analogously, we have the $\varepsilon$-relative entropy of entanglement [@BD11]:$$E_{R}^{\varepsilon}(A;B)_{\rho}=\inf_{\sigma_{AB}\in\operatorname{SEP}(A:B)}D_{H}^{\varepsilon}(\rho_{AB}\Vert\sigma_{AB}).
\label{eq:eps-rel-entr-enta}$$ For a two-mode Gaussian state $\rho_{AB}$ in standard form, one can always choose the separable state $\sigma_{AB}^{\prime}$ to be in standard form with the same values for $a$ and $b$ but with $c$ chosen to saturate the inequality in , such that $c=c_{\text{sep}}$ [@PLOB15]. By definition, for this suboptimal choice, we have that$$\begin{aligned}
E_{R}(A;B)_{\rho}  &  \leq D(\rho_{AB}\Vert\sigma_{AB}^{\prime}),\\
E_{R}^{\varepsilon}(A;B)_{\rho}  &  \leq D_{H}^{\varepsilon}(\rho_{AB}\Vert\sigma_{AB}^{\prime}),\end{aligned}$$ and this is the choice made in [@PLOB15; @WTB16] to arrive at various upper bounds on secret-key-agreement capacity. In what follows, we refer to $D(\rho_{AB}\Vert\sigma_{AB}^{\prime})$ as the suboptimal relative entropy of entanglement of $\rho_{AB}$.

In [@WTB16 Eq. (4.34)], the following bound was established on the non-asymptotic secret-key-agreement capacity of a channel $\mathcal{N}$ that is teleportation simulable with associated resource state $\omega_{AB}$:$$\label{eqn:hyptest}P_{\mathcal{N}}^{\leftrightarrow}(n,\varepsilon)\leq
\frac{1}{n}E_{R}^{\varepsilon}(A^{n};B^{n})_{\omega^{\otimes n}}\leq\frac
{1}{n}D_{H}^{\varepsilon}(\omega_{AB}^{\otimes n}\Vert\sigma_{AB}^{\otimes
n}).$$ The argument for the first inequality critically relies upon the connection between secret-key-agreement protocols and private-state distillation protocols established in [@HHHO05; @HHHO09] and some other results contained 