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

In an orthonormal computational basis, the squared norm is the sum of the squared moduli of the coefficients:Thus the vector is normalized and represents a pure state of a qubit. The imaginary part contributes positively to the squared modulus; squaring the coefficient without complex conjugation would give the wrong norm.

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.