= Composition of quantum channels
{title2=$\{A_k\},\{B_j\}\ \mapsto\ \{B_jA_k\}$}
The <function composition> of two <quantum channels> is another <quantum channel>. If their <Kraus representations> use $A_k$ and $B_j$, the composed map uses $B_jA_k$. Its completeness relation follows from $\sum_{j,k}A_k^\dagger B_j^\dagger B_jA_k=I$. Tensoring either map with the identity on an arbitrary <ancilla> also shows directly that <complete positivity> is preserved by <function composition>.
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