unital map

{{Short description|Mapping preserving identity}}{{No inline|date=May 2025}}

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

:$\backslash phi\; (\; I\; )\; =\; I.$

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

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

:$\backslash phi\; (\; \backslash rho\; )\; =\; \backslash sum\_i\; E\_i\; \backslash rho\; E\_i^\backslash dagger.$

(The $E\_i$ are the Kraus operators associated with $\backslash phi$). In this case, the unital condition can be expressed as

:$\backslash sum\_i\; E\_i\; E\_i\; ^\backslash dagger=\; I.$