distance $L$ can be related to the transmissivity $\eta$ of the thermal channel as $\eta
=\exp[-L/L_{0}]$, where $L_{0}$ is the fiber attenuation length [@RGRKVRHWE17]. In the plots, we consider $L_{0}=0.542$ km [@RGRKVRHWE17]. The thermal mean photon number $N_{B}$ relevant in experimental contexts, whenever thermal noise is due exclusively to dark counts, is given by the dark counts per second times the integration period $t_{\mathrm{int}}$. In the plots, the lowest $N_{B}$ we consider corresponds to a dark count rate of $10$ per second and $t_{\mathrm{int}}=30$ ns [RGRKVRHWE17]{}. For completeness, we also consider higher values of $N_{B}$, which could occur due to excessive background thermal radiation or tampering by an eavesdropper.

As noted in the introduction of our paper, these upper bounds can be interpreted to serve as benchmarks for quantum repeaters [@L15]. That is, the upper bounds on secret-key-agreement capacity hold for any protocol that uses the channel and LOCC but is not allowed to use a quantum repeater. As such, exceeding these upper bounds constitutes a demonstration of a quantum repeater [@L15]. What our results indicate is that the previous upper bounds from [@PLOB15; @WTB16] on the asymptotic secret-key-agreement capacity are too pessimistic of a benchmark for protocols that are 