= Capacity of a disjoint sum of channels
{title2=$C=\log_2\sum_i2^{C_i}$}
Combine finitely many <discrete memoryless channels> with disjoint input <alphabets> and disjoint output <alphabets> into a block-diagonal <Communication channel>[channel]. The block label $B$ is determined by both input and output. If it has law $\alpha$, the <mutual information> is at most $H(\alpha)+\sum_i\alpha_iC_i$, attained by using each block's capacity-achieving conditional input law. Put $S=\sum_i2^{C_i}$ and $r_i=2^{C_i}/S$. The expression equals $\log_2S-d(\alpha,r)$, so <Gibbs inequality> proves the displayed <Channel capacity>[capacity], attained for block <probabilities> $\alpha_i=r_i$. Even blocks of <Channel capacity>[capacity] zero contribute distinguishable block choices; three disjoint zero-capacity blocks give <Channel capacity>[capacity] $\log_2 3$.
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