Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 60 / 3 / ii / Solution

Set $K_y=|y⟩⟨y|$. The map is in Kraus representation:For any ancillary Hilbert space $R$ and positive operator $M$ on $R\otimes \mathcal{H}$,This verifies complete positivity directly, not merely positivity on unextended states. Cyclicity of the trace gives $\mathrm{Tr}\Lambda (X)=\mathrm{Tr}[X{\sum }_y{K}_y^{†}{K}_y]=\mathrm{Tr}X$. Thus $\Lambda$ is a CPTP map, the completely dephasing channel in the given basis.