Unital map

Mapping preserving identity

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

ϕ ( I ) = I . {\displaystyle \phi (I)=I.}

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

If ϕ {\displaystyle \phi } is completely positive, it can always be represented as

ϕ ( ρ ) = ∑ i E i ρ E i † . {\displaystyle \phi (\rho )=\sum _{i}E_{i}\rho E_{i}^{\dagger }.}

(The E i {\displaystyle E_{i}} are the Kraus operators associated with ϕ {\displaystyle \phi } ). In this case, the unital condition can be expressed as

∑ i E i E i † = I . {\displaystyle \sum _{i}E_{i}E_{i}^{\dagger }=I.}

References

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


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