P_{\mathcal{L}_{\eta,N_{B}}}^{\leftrightarrow}(n,\varepsilon)\leq-\log
_{2}(\left[  1-\eta\right]  \eta^{N_{B}})-g(N_{B})\label{eq:SC-thermal}\\
+\sqrt{\frac{2V_{\eta,N_{B}}}{n\left(  1-\varepsilon\right)  }}+\frac
{C(\varepsilon)}{n},\end{gathered}$$ where$$V_{\eta,N_{B}}=N_{B}(N_{B}+1)\log_{2}^{2}(\eta\left[  N_{B}+1\right]  /N_{B}),$$ and the following weak-converse bound is a direct consequence of [@WTB16 Section 8] and [@WR12; @MW12] (explained also in Appendix \[sec:weak-converse-bnds\]):$$\begin{gathered}
P_{\mathcal{L}_{\eta,N_{B}}}^{\leftrightarrow}(n,\varepsilon)\leq
\label{eq:WC-thermal}\\
\frac{1}{1-\varepsilon}\left[  -\log_{2}(\left[  1-\eta\right]  \eta^{N_{B}})-g(N_{B})+\frac{h_{2}(\varepsilon)}{n}\right]  .\end{gathered}$$ Again, the value of these bounds is that they apply for any finite number $n$ of channel uses and key-quality parameter $\varepsilon$. However, by inspecting , we see that the order $1/\sqrt{n}$ and lower terms are strictly positive.

The main contribution of the present paper is to improve the bound in in such a way that the order $1/\sqrt{n}$ term is negative whenever $\varepsilon<1/2$, representing the backoff from capacity incurred by using the channel a finite number of times while allowing for non-zero error. In fact, we find the following improved bound for several realistic values of $\eta$ and $N_{B}$:$$\begin{gathered}
P_{\mathcal{L}_{\eta,N_{B}}}^{\leftrightarrow}(n,\varepsilon)\leq-\log
_{2}(\left[  1-\eta\right]  \eta^{N_{B}})-g(N_{B})\label{eq:new-bound}\\
+\sqrt{\frac{V_{\eta,N_{B}}^{\prime}}{n}}\Phi^{-1}(\varepsilon)+\frac{O(\log
n)}{n},\end{gathered}$$ where $V_{\eta,N_{B}}^{\prime}$ is a channel-dependent parameter that we discuss later and $\Phi^{-1}$ denotes the inverse of the cumulative normal distribution function (see ), for which we have that $\Phi^{-1}(\varepsilon)<0$ whenever $\varepsilon<1/2$. We should note that the bound in  applies only for $n$ sufficiently large (such that $n$ is proportional to $1/\varepsilon^{2}$), as it relies on 