Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 2 / d / Solution

Write $q_j=⟨z_j|\rho |z_j⟩$, $r_j=⟨z_j|Y(\rho )|z_j⟩$, and $c={\max }_{i,j}|⟨y_i|z_j⟩{|}^2$. Thenand thereforeThe data-processing inequality for quantum relative entropy applied to $Z$ and part c giveCombining them proves