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

Choose $|{\psi }_0⟩=|+⟩$ and its orthogonal companion $|{\psi }_1⟩=|-⟩$. Since $X|+⟩=|+⟩$, either branch of RandomFlip returns the same pure state. Alice therefore obtains label 0 with certainty, andThe alternative choice $|-⟩$ also works: its sign change under $X$ is only a global phase. The useful property is being a Pauli eigenstate of the Pauli X gate.