Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint? Let \(\mathcal N_\eta\) be the single-mode pure-loss channel with transmissivity \(0<\eta<1\), defined in the Heisenberg picture by
\begin{equation}
\hat b=\sqrt{\eta}\,\hat a+\sqrt{1-\eta}\,\hat e,
\tag{1}
\end{equation}
where the environment mode \(E\) is in the vacuum. In an \(n\)-use code, the channel in Eq. (1) is used to transmit one of \(M\) input states \(\rho_m^{A^n}\), with a decoding POVM \(\{\Lambda_m^{B^n}\}_{m=1}^M\). Write \(\overline\rho_{A^n}:=M^{-1}\sum_m\rho_m^{A^n}\) and let \(\Pi_{\lceil nN_S\rceil}\) project onto the \(n\)-mode subspace of total photon number at most \(\lceil nN_S\rceil\). For fixed \(N_S>0\), \(\varepsilon\in(0,1)\), and \(c>0\), impose
\begin{equation}
\frac1M\sum_{m=1}^M
\operatorname{Tr}\!\left[
\Lambda_m\mathcal N_\eta^{\otimes n}(\rho_m)
\right]
\geq1-\varepsilon,
\qquad
\operatorname{Tr}\!\left[
\Pi_{\lceil nN_S\rceil}\overline\rho_{A^n}
\right]
\geq1-\delta_n,
\qquad
0\leq\delta_n\leq2^{-cn}.
\tag{2}
\end{equation}
Let \(M^*_{\rm occ}(n,\eta,N_S,\varepsilon,c)\) be the largest \(M\) satisfying Eq. (2). Define the thermal entropy and its entropy variance by
\begin{equation}
g(x):=(x+1)\log_2(x+1)-x\log_2x,
\qquad
v(x):=x(x+1)
\left[\log_2(x+1)-\log_2x\right]^2,
\tag{3}
\end{equation}
where \(0\log_2 0:=0\). With the functions in Eq. (3), is the following upper bound valid as \(n\to\infty\)?
\begin{equation}
\log_2 M^*_{\rm occ}(n,\eta,N_S,\varepsilon,c)
\leq ng(\eta N_S)
+\sqrt{n\,v(\eta N_S)}\,\Phi^{-1}(\varepsilon)
+O(\log n),
\tag{4}
\end{equation}
Here \(\Phi^{-1}\) in Eq. (4) is the inverse standard-normal cumulative distribution function, and the implicit constant may depend on \(\eta,N_S,\varepsilon,\) and \(c\), but not on \(n\).