In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition.

Let ${\displaystyle A}$ and ${\displaystyle B}$ be C*-algebras. A linear map ${\displaystyle \phi :A\to B}$ is called a positive map if ${\displaystyle \phi }$ maps positive elements to positive elements: ${\displaystyle a\geq 0\implies \phi (a)\geq 0}$.

Any linear map ${\displaystyle \phi :A\to B}$ induces another map

in a natural way. If ${\displaystyle \mathbb {C} ^{k\times k}\otimes A}$ is identified with the C*-algebra ${\displaystyle A^{k\times k}}$ of ${\displaystyle k\times k}$-matrices with entries in ${\displaystyle A}$, then ${\displaystyle {\textrm {id}}\otimes \phi }$ acts as

We then say ${\displaystyle \phi }$ is k-positive if ${\displaystyle {\textrm {id}}_{\mathbb {C} ^{k\times k}}\otimes \phi }$ is a positive map and completely positive if ${\displaystyle \phi }$ is k-positive for all k.