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

$\sum_i E_i E_i ^\dagger= I. \,$