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. \,

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