# 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