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

With $P_i=|y_i⟩⟨y_i|$, the channel has Kraus form $Y(\rho )={\sum }_i{P}_i\rho P_i$ and ${\sum }_i{P}_i^{†}{P}_i=I$, so it is completely positive and trace preserving. It measures in the $y$ basis and prepares the observed basis state, hence is a measure-and-prepare channel. If $p_i=⟨y_i|\rho |y_i⟩$, then $Y(\rho )={\sum }_i{p}_i{P}_i$ andMoreover $\log Y(\rho )={\sum }_i(\log p_i)P_i$ on its support, so