Composition of quantum channels ($\{A_k\},\{B_j\}\mapsto \{B_j{A}_k\}$)

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 $B_j{A}_k$. Its completeness relation follows from ${\sum }_{j,k}{A}_k^{†}{B}_j^{†}{B}_j{A}_k=I$. Tensoring either map with the identity on an arbitrary ancilla also shows directly that complete positivity is preserved by function composition.