is called a thermal channel and denoted by $\mathcal{L}_{\eta,N_{B}}$ (also called thermal-lossy channel, as in [@RGRKVRHWE17]). This added thermal noise is often called excess noise [@NH04; @LDTG05], which can serve as a simple model of tampering by an eavesdropper. Additionally, there are realistic effects in communication schemes, such as dark counts of photon detectors that can be modeled as arising from thermal photons in the environment [@S09; @RGRKVRHWE17]. As such, it is an important goal to establish upper bounds on the secret-key-agreement capacity of the thermal channel in order to assess the performance of practical secret-key-agreement protocols, and the main contribution of the present paper is to establish upper bounds on the non-asymptotic secret-key-agreement capacity $P_{\mathcal{L}_{\eta,N_{B}}}^{\leftrightarrow}(n,\varepsilon)$ of the thermal channel $\mathcal{L}_{\eta,N_{B}}$, which improve upon the prior known bounds from [@PLOB15; @WTB16] in certain regimes.

Prior works established that$$-\log_{2}(\left[  1-\eta\right]  \eta^{N_{B}})-g(N_{B})
\label{eq:rel-ent-thermal}$$ is an upper bound on the secret-key-agreement capacity of a thermal channel $\mathcal{L}_{\eta,N_{B}}$ with transmissivity $\eta\in(0,1)$ and thermal mean photon number $N_{B}>0$. This bound was claimed in [@PLOB15] and rigorously proven in [@WTB16]. In this expression,$$g(N_{B})=(N_{B}+1)\log_{2}(N_{B}+1)-N_{B}\log_{2}N_{B}$$ is the entropy of a thermal state of mean photon number $N_{B}$. In particular, the following bound was given in [@WTB16 Section 8]$$\begin{gathered}
