Classical channel capacity $C(\mathcal N)$ of a quantum channel $\mathcal N$ is defined to be a rate below which any classical communication can succeed with an arbitrarily small error.
Specifically, given a quantum channel $\mathcal N$ and any small number $\epsilon$, there exists $n$, $\mathcal E$ and $\mathcal R$ such that $\mathcal R \circ \mathcal N^{\otimes n} \circ \mathcal E$ is $\epsilon$-close to a completely dephasing channel $\mathcal D^{\otimes \alpha n}$ if and only if $\alpha < C(\mathcal N)$. Here $\mathcal D(\cdot) = |0\rangle\langle 0|(\cdot)|0 \rangle\langle 0| + |1 \rangle\langle 1|(\cdot)|1 \rangle\langle 1|$. This is called the HSW theorem (see e.g., this text book).
My question is, fixing a constant $\alpha < C(\mathcal N)$, is it always true that $\epsilon$ can be exponentially small in $n$ (choosing some optimal $\mathcal E$ and $\mathcal R$ for each $n$)? It seems to be true as when I searched for this problem online I found some statements like "As for error exponent (large deviation) analyses, these result in an error that vanishes exponentially with n but the difference between the rate and the capacity is a constant term." (https://arxiv.org/abs/2112.07167) and "It is well-known that when communicating by block codes over a discrete memoryless channel at rate below the capacity, the error probability goes to zero exponentially in the block length" (https://arxiv.org/abs/1502.02987). However, I never really found any exact proof of this fact. Anyone has any idea where I should look?
