bloch sphere

See qubits. Any qubit can be written as $e^{i γ} (\cos \frac{θ}{2} | 0 ⟩ + e^{i ϕ} \sin \frac{θ}{2} | 1 ⟩)$ where $0 \leq θ \leq π$ is the azimuthal angle and $0 \leq ϕ \leq 2 π$ is the longitude

1. complex projective line

Any quantum state can be identified by two quantum numbers $α$ and $β$ : $α | 0 ⟩ + β | 1 ⟩$ . For any quantum state, the global phase is not physically observable. That is, we can multiply both $α$ and $β$ by the same $λ = e^{i θ}$ and the system would be the same.

The complex projective line $C P^{1}$ is the set of all 1D subspaces in $C P^{2}$ . In other words, $C P^{1}$ is the quotient group of the equivalence relation where $(a, b) \equiv (λ a, λ b)$ . There is a bijection between the complex projective line $C P^{1}$ and the 2-sphere $S^{2}$ see hopf

Then, any $(a, b) \in C^{2}$ (note not just the ones with unit magnitude), you can get the corresponding vector in the Bloch sphere as: $(a, b) \leftrightarrow (\begin{matrix} 2 R e (a^{*} b) \\ 2 I m (a^{*} b) \\ | a |^{2} - | b |^{2} \end{matrix})$

see formula