Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 1 / b / i / Solution

Choose $|\psi ⟩=|0⟩$. Since $|0⟩=(|{\varphi }_0⟩+|{\varphi }_2⟩)/\sqrt{2}$, the probability amplitudes in the measurement basis are $(1/\sqrt{2},0,1/\sqrt{2})$. Their squared moduli give $(1/2,0,1/2)$ by the Born rule.