Unital map

Short description: Mapping preserving identity

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

[math]\displaystyle{ \phi ( I ) = I. }[/math]

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

If [math]\displaystyle{ \phi }[/math] is completely positive, it can always be represented as

[math]\displaystyle{ \phi ( \rho ) = \sum_i E_i \rho E_i^\dagger. }[/math]

(The [math]\displaystyle{ E_i }[/math] are the Kraus operators associated with [math]\displaystyle{ \phi }[/math]). In this case, the unital condition can be expressed as

[math]\displaystyle{ \sum_i E_i E_i ^\dagger= I. }[/math]