Exercise 2.12

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.12}$

a) Show that the surface of the Bloch sphere can be parametrized in terms of two real-valued parameters, the angles $\theta$ and $\phi$ illustrated in above Figure. Make sure your parametrization is in one-to-one correspondence with points on the sphere, and therefore single-qubit quantum states, in the range $\theta\in [0, \pi]$ and $\phi\in [0, 2\pi]$ except for the points corresponding to $\ket 0$ and $\ket 1$.

b) What are $\theta$ and $\phi$ for each of the states $\ket +$, $\ket -$, $\ket{\i}$, and $\ket{-\i}$?