Unital map

Template:Short description Template:Unreferenced In abstract algebra, a unital map on a C*-algebra is a map $ϕ$ which preserves the identity element:

ϕ (I) = I .

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

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

ϕ (ρ) = \sum_{i} E_{i} ρ E_{i}^{†} .

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

\sum_{i} E_{i} E_{i}^{†} = I .

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