Remark: An attaining ensemble is not a reliable full codebook

When $\chi (\mathcal{A})<1$, the $2^n$ equiprobable phase-pair products do not thereby become $2^n$ reliably distinguishable messages. Their ensemble entropy difference is $n\chi (\mathcal{A})$. Coding uses suitable codebooks and decoding POVMs; it is a different construction.