quantum repeater, the rate of secret key that can be distilled from a pure-loss bosonic channel (lossy optical fiber or a free-space channel) decreases exponentially with increasing distance [@TGW14Nat].

Later, this bound was improved to establish that the secret-key-agreement capacity of a pure-loss bosonic channel of transmissivity $\eta\in(0,1)$ is equal to $-\log_{2}(1-\eta)$. This bound was claimed in [@PLOB15] and rigorously proven in [@WTB16]. In particular, let $P_{\mathcal{L}_{\eta}}^{\leftrightarrow}(n,\varepsilon)$ denote the highest rate at which $\varepsilon$-close-to-ideal secret key can be distilled by making $n$ invocations of a pure-loss channel $\mathcal{L}_{\eta}$ of transmissivity $\eta$, along with the assistance of public classical communication [@WTB16]. In [@WTB16], $P_{\mathcal{L}_{\eta}}^{\leftrightarrow
}(n,\varepsilon)$ is called the non-asymptotic secret-key-agreement capacity of the channel $\mathcal{L}_{\eta}$. One of the results of [@WTB16] is the following fundamental upper bound:$$P_{\mathcal{L}_{\eta}}^{\leftrightarrow}(n,\varepsilon)\leq-\log_{2}(1-\eta)+\frac{C(\varepsilon)}{n}, \label{eq:SC-bound}$$ where $C(\varepsilon)=\log_{2}6+2\log_{2}(\left[  1+\varepsilon\right]
/\left[  1-\varepsilon\right]  )$. The bound in is known as a strong converse bound because it converges to the secret-key-agreement capacity $-\log_{2}(1-\eta)$ in the limit as $n\rightarrow\infty$. We suspect that there is little room for improvement of the bound in and discuss this point further in Appendix \[sec:disp-rev-coh-info\]. The bound in is to be contrasted with the following weak-converse bound:$$P_{\mathcal{L}_{\eta}}^{\leftrightarrow}(n,\varepsilon)\leq\frac
{1}{1-\varepsilon}\left[  -\log_{2}(1-\eta)+\frac{h_{2}(\varepsilon)}{n}\right]  , \label{eq:WC-bound}$$ which follows as a direct consequence of 