Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 1 / a / ii / Solution

For the Hadamard basis, $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$. The Born rule givesWriting the two computational basis amplitudes as $a,b$, normalization gives $\mathrm{Pr}(\pm )=\frac{1}{2}\pm \mathrm{Re}(a^{\ast }b)$, and $\mathrm{Re}(a^{\ast }b)=\sqrt{3}/8$. HenceThese probabilities sum to one. The cross term is interference between probability amplitudes, so the probabilities cannot be obtained by treating the input as an incoherent mixture of the computational basis alternatives.