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


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