 Solution

The map $\Lambda$ is a completely positive map when $\Lambda \otimes {\mathrm{id}}_m$ maps positive operators to positive operators for every $m$. In a basis $\{|i⟩\}$ define the unnormalized Choi matrixIf $\Lambda$ is completely positive, $J(\Lambda )=(\mathrm{id}\otimes \Lambda )(|\Omega ⟩⟨\Omega |)\ge 0$, where $|\Omega ⟩={\sum }_i|ii⟩$. Conversely, decompose $J={\sum }_k|a_k⟩⟨a_k|$ and reshape each vector into an operator $A_k$. The Choi reconstruction formula gives $\Lambda (X)={\sum }_k{A}_{k}X{A}_k^{†}$, which is completely positive. Thus