Unital map

In abstract algebra, a unital map on a C*-algebra is a map ${\displaystyle \phi }$ which preserves the identity element:

${\displaystyle \phi (I)=I.}$

This condition appears often in the context of completely positive maps, especially when they represent quantum operations.

If ${\displaystyle \phi }$ is completely positive, it can always be represented as

${\displaystyle \phi (\rho )=\sum _{i}E_{i}\rho E_{i}^{\dagger }.}$

(The ${\displaystyle E_{i}}$ are the Kraus operators associated with ${\displaystyle \phi }$). In this case, the unital condition can be expressed as

${\displaystyle \sum _{i}E_{i}E_{i}^{\dagger }=I.}$