Unital map

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

$\varphi (I)=I.$

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

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

$\varphi (\rho )=\sum _i{E}_i\rho E_i^{†}.$

(The $E_i$ are the Kraus operators associated with $\varphi$). In this case, the unital condition can be expressed as

$\sum _i{E}_i{E}_i^{†}=I.$

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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