Solution
= Solution
For every ancillary system,
$$
(\Lambda\otimes\operatorname{id})(X)
=\sum_k(A_k\otimes I)X(A_k^\dagger\otimes I)\geq0
$$
when $X\geq0$, proving complete positivity. Cyclicity of trace gives
$$
\operatorname{Tr}\Lambda(\rho)
=\operatorname{Tr}\left(\rho\sum_kA_k^\dagger A_k\right).
$$
This equals $\operatorname{Tr}\rho$ for every $\rho$ exactly when
$$
\boxed{\sum_kA_k^\dagger A_k=I},
$$
the trace-preserving condition in the <Kraus representation>.