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

Assume Alice independently chooses X or Z with equal probabilities and that bit labels $0,1$ identify the corresponding positive and negative eigenstates. If Daisy measures Z, she reads a Z-encoded bit correctly with probability one and an X-encoded bit correctly with probability one half. ThereforeThe probability to read the entire independently encoded $n$-bit check number correctly by this strategy is $(3/4{)}^n$.

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.