bloch sphere

See qubits. Any qubit can be written as \[e^{i\gamma} \left(\cos\frac{\theta}{2}\ket{0} + e^{i\phi}\sin\frac{\theta}{2}\ket{1} \right)\] where \(0 \leq \theta \leq \pi\) is the azimuthal angle and \(0 \leq \phi \leq 2\pi\) is the longitude

1. complex projective line

Any quantum state can be identified by two quantum numbers \(\alpha\) and \(\beta\): \(\alpha\ket{0} + \beta\ket{1}\). For any quantum state, the global phase is not physically observable. That is, we can multiply both \(\alpha\) and \(\beta\) by the same \(\lambda = e^{i\theta}\) and the system would be the same.

The complex projective line \(CP^1\) is the set of all 1D subspaces in \(CP^2\). In other words, \(CP^1\) is the quotient group of the equivalence relation where \((a,b) \equiv (\lambda a, \lambda b)\). There is a bijection between the complex projective line \(CP^1\) 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: \[(a,b) \leftrightarrow \begin{pmatrix} 2Re(a^* b) \\ 2Im(a^* b) \\ |a|^2 - |b|^2 \end{pmatrix} \]

see formula