Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 1 / d / ii

Use the preparation and measurement from part (i). RandomReset returns the density matrix ${\rho }_R=(|0⟩⟨0|+|1⟩⟨1|)/2=I/2$, a maximally mixed state. In the $|+⟩,|-⟩$ basis its two outcomes each have probability $1/2$. In contrast, RandomFlip always gives outcome 0.

Guess RandomFlip after 0 and RandomReset after 1. With the supplied equal priors,An outcome 1 identifies the reset action, but outcome 0 does not identify the flip action with certainty. For example, Bayes' theorem gives posterior probability $2/3$ for the flip action after 0. The total success rate already accounts for the reset runs that also produce 0.