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Unital map

Mapping preserving identity

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Source updated July 18, 2025

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

$\phi (\rho )=\sum _{i}E_{i}\rho E_{i}^{\dagger }.$

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

$\sum _{i}E_{i}E_{i}^{\dagger }=I.$

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This article is adapted from the Wikipedia article “Unital map”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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