Finite cq block codes, average error, and capacity

Fix a finite-input cq channel as in Finite-input memoryless cq channel. For integers $n,M\ge 1$, an $(n,M)$ cq block code consists of a deterministic encoder $m\mapsto x^n(m)\in {\mathcal{X}}^n$ for $m\in [M]:=\{1,\dots ,M\}$ and a finite POVM (Finite POVMs and their outcome law) $({\mathrm{\Lambda }}_m{)}_{m\in [M]}$ on $H_B^{\otimes n}$. Under a uniform message, its average block error is $e_n:=\frac{1}{M}{\sum }_{m=1}^M(1-\mathrm{tr}({\mathrm{\Lambda }}_m{\rho }_{x^n(m)}))\mathrm{.}$ Its rate is $n^{-1}\ln M$ nats per channel use, using the natural-log unit convention of Finite-dimensional von Neumann entropy. A sequence of such codes is reliable if $e_n\to 0$. The classical capacity $C_{\mathrm{c}\mathrm{q}}$ is the supremum of ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{n}^{-1}\ln M_n$ over reliable sequences.

For each transmitted state, the POVM outcome weights sum to one by POVM outcome weights are probabilities, so $e_n$ is the uniform-message block error probability.