Solution
= Solution
The finite-dimensional <Stinespring dilation> theorem states that every <quantum channel> $\Lambda:\mathcal B(H_A)\to\mathcal B(H_B)$ has an environment $H_E$ and an <isometry> $V:H_A\to H_B\otimes H_E$ such that
$$
\boxed{\Lambda(\rho)=\operatorname{Tr}_E(V\rho V^\dagger)}.
$$
Conversely, every map of this form is completely positive and trace preserving. From <Kraus operators> $A_j$, one may take $V=\sum_jA_j\otimes|j\rangle_E$.