Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 58 / 3 / a

For a pure qubit $a|0⟩+b|1⟩$, the Bloch vector is $(2\mathrm{Re}(a^{\ast }b),2\mathrm{Im}(a^{\ast }b),|a{|}^2-|b{|}^2)$. Substitution givesBoth lie on the unit Bloch sphere, at the same azimuth and opposite heights. They are reflections across its equatorial plane, not antipodal vectors. Choose $\theta =\pi /3$ and $\phi =\pi /4$, so their coordinates are $(\sqrt{6}/4,\sqrt{6}/4,\pm 1/2)$. The following original sketch shows those vectors and their common meridian: