Unital map

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

$\phi (I)=I.$

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

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

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

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

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