Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 4 / a / Solution

For Hermitian $X$, $\Vert X{\Vert }_1={\max }_{-I\le H\le I}\mathrm{Tr}(HX)$. The adjoint of a trace-preserving completely positive map is unital and positive, so $-I\le H\le I$ implies $-I\le {\Lambda }^{\ast }(H)\le I$. ThereforeHence trace distance is contractive under quantum channels.