Definition 41.3.3 (Completely Positive Map).label Let $A, B$ be $C^{*}$-algebras with $A$ being unital, $S \subset A$ be an operator system, $\phi: S \to B$ be a linear map, and $n \in \natp$, then $\phi$ is $n$-positive if $\phi_{n}: M_{n}(S) \to M_{n}(B)$ is positive. The map $\phi$ is completely positive if $\phi$ is $n$-positive for all $n \in \natp$.