= Solution
The map $\Lambda$ is a <completely positive map> when $\Lambda\otimes\operatorname{id}_m$ maps positive operators to positive operators for every $m$. In a basis $\{|i\rangle\}$ define the unnormalized <Choi matrix>
$$
J(\Lambda)=\sum_{i,j}|i\rangle\langle j|\otimes\Lambda(|i\rangle\langle j|).
$$
If $\Lambda$ is completely positive, $J(\Lambda)=(\operatorname{id}\otimes\Lambda)(|\Omega\rangle\langle\Omega|)\geq0$, where $|\Omega\rangle=\sum_i|ii\rangle$. Conversely, decompose $J=\sum_k|a_k\rangle\langle a_k|$ and reshape each vector into an operator $A_k$. The Choi reconstruction formula gives $\Lambda(X)=\sum_kA_kXA_k^\dagger$, which is completely positive. Thus
$$
\boxed{\Lambda\text{ is completely positive}\iff J(\Lambda)\geq0}.
$$
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