= Solution
Set $K_y=|y\rangle\langle y|$. The map is in <Kraus representation>:
$$
\Lambda(X)=\sum_yK_yXK_y^\dagger,\qquad
\sum_yK_y^\dagger K_y=\sum_y|y\rangle\langle y|=I.
$$
For any ancillary <Hilbert space> $R$ and positive operator $M$ on $R\otimes\mathcal H$,
$$
(\operatorname{id}_R\otimes\Lambda)(M)=\sum_y(I_R\otimes K_y)M(I_R\otimes K_y^\dagger)\geq0.
$$
This verifies complete positivity directly, not merely positivity on unextended states. Cyclicity of the trace gives $\operatorname{Tr}\Lambda(X)=\operatorname{Tr}[X\sum_yK_y^\dagger K_y]=\operatorname{Tr}X$. Thus \b[$\Lambda$ is a <CPTP map>], the <completely dephasing channel> in the given basis.
Back to article page
