Proposition: Receiver-only factorization

A channel from a complete finite system $R\otimes B$ to an output can be implemented using only $R$ if its adjoint factors as

for a UCP map $K^{\ast }$ on $R$. The decoder ${\mathcal{D}}_{12}$ has this factorization when extended to the complete encoder output.

Proof. The factorization is the adjoint of $\Phi =K\circ {\mathrm{Tr}}_B$, which uses only the $R$ snapshot. The decoder's operations and Kraus maps act on $S,R_1,R_2$ only; their complete-system adjoints tensor with $I_B$.