therein, in addition to the teleportation reduction argument discussed in Section \[sec:tp-simulation\]. The second inequality follows from the definition in , with $\sigma_{AB}$ being an arbitrary separable state. Thus, any resource state for the teleportation simulation of a channel can be used to give an upper bound on its non-asymptotic secret-key-agreement capacity. In particular, if $\omega_{AB}$ and $\sigma_{AB}$ are faithful quantum Gaussian states of finite energy such that $\omega_{AB}\neq\sigma_{AB}$, then the conditions $D(\omega_{AB}\Vert\sigma_{AB}),V(\omega_{AB}\Vert\sigma_{AB}),T(\omega_{AB}\Vert\sigma
_{AB})<\infty$ and $V(\omega_{AB}\Vert\sigma_{AB})>0$ hold, such that applies and we find that$$\begin{gathered}
P_{\mathcal{N}}^{\leftrightarrow}(n,\varepsilon)\leq D(\omega_{AB}\Vert
\sigma_{AB})+\sqrt{\frac{V(\omega_{AB}\Vert\sigma_{AB})}{n}}\Phi
^{-1}(\varepsilon)\label{eq:tp-simul-bound}\\
+O\!\left(  \frac{\log n}{n}\right)  .\end{gathered}$$ The quantities $D(\omega_{AB}\Vert\sigma_{AB})$ and $V(\omega_{AB}\Vert
\sigma_{AB})$ are finite for faithful quantum Gaussian states of finite energy, which holds by inspecting and , and in Appendix \[sec:finiteness\], we argue that the quantity $T(\omega_{AB}\Vert\sigma_{AB})$ is finite as well.

Note that both and can be derived from . The point of deviation in the two derivations is that it is possible, on the one hand, to invoke the Berry–Esseen theorem [@KS10; @S11] in order to arrive at , due to the results of [@LMGA17] and our arguments in Appendices \[sec:AEP-hypo\] and \[sec:finiteness\]. That is, [@LMGA17] showed how to perform teleportation simulation of a single-mode phase-insensitive thermal bosonic channel using a finite-energy 