K_{y} = ∣ y ⟩ ⟨ y ∣

Λ (X) = \sum_{y} K_{y} X K_{y}^{†}, \sum_{y} K_{y}^{†} K_{y} = \sum_{y} ∣ y ⟩ ⟨ y ∣ = I .

For any ancillary Hilbert space

R

M

R \otimes H

(id_{R} \otimes Λ) (M) = \sum_{y} (I_{R} \otimes K_{y}) M (I_{R} \otimes K_{y}^{†}) \geq 0.

This verifies complete positivity directly, not merely positivity on unextended states. Cyclicity of the trace gives

Tr Λ (X) = Tr [X \sum_{y} K_{y}^{†} K_{y}] = Tr X

. Thus $Λ$ is a CPTP map, the completely dephasing channel in the given basis.

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