a teleportation simulation with negative gain $g$, as discussed in Section \[sec:tp-simulation\], is critical for us to solve these constraints by numerical search. With these finite-energy states in hand, we then numerically compute the relative entropy variance in and can apply the bound in .

Results {#sec:results}
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In Figure \[fig:results\], we plot upper bounds on the asymptotic secret-key-agreement capacity of the thermal channel given by (dashed line) and upper bounds on the non-asymptotic secret-key-agreement capacity given by  (solid line) versus the number of channel uses. It is important to stress that the latter bound is only an approximation (known as the normal approximation) if $n$ is not sufficiently large (i.e., $n$ should be proportional to $1/\varepsilon^{2}$ in order for the bounds to really apply). At the same time, many prior works have shown that the normal approximation is an excellent approximation for non-asymptotic capacities even for small $n$ [@polyanskiy10; @P10; @MW12; @P13; @TBR15]. In each case, we choose the key-quality parameter $\varepsilon$ to be $10^{-10}$, in accordance with the same conservative value chosen in [@TCGR12]. In the plots, we select $\eta\in(0,1)$, (hence the corresponding distance $L$) and the thermal mean photon number $N_{B}>0$ as indicated above each figure. The 