Unital map explained

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 ^{\dagger.} _{i}

(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 ^{\dagger =} _{i} I.

References

• Book: Paulsen, Vern I.. Completely bounded maps and operator algebras. 2002. Cambridge University Press. 0-511-06103-X. Cambridge. 228110971.