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

Choose $|\psi ⟩=|{\varphi }_2⟩$. Its inner products with the three measurement vectors are $(0,0,1)$, so the Born rule gives certainty for the outcome labelled 2 and zero probability for the other two outcomes. This uses the measurement basis, rather than assuming that its label 2 means the initial state must be the computational vector $|2⟩$.