Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 4 / d / ii

Daisy's random choice of X or Z agrees with Alice's independent random basis with probability $1/2$. Agreement gives the correct label with certainty and disagreement gives it with probability $1/2$. Hencethe same result as a fixed Z measurement. Randomizing her basis does not supply the missing information about Alice's choice. These are the probabilities for the two specified strategies, rather than an assertion that they are optimal among all possible quantum measurements.