Hopf fibration is a connection between state vector representation of a qubit $\lvert\psi\rangle\in\mathbb{C}^2$ and points on the Bloch sphere $(x, y, z)\in\mathbb{R}^3$.

Most compactly, the Hopf map $\pi$ can be written in the following way using Dirac notation.

$$\pi:\mathbb{C}^2\rightarrow\mathbb{R}^3\qquad \pi(\lvert\psi\rangle) = \langle\psi\lvert\sigma_i\lvert\psi\rangle $$

Explicit coordinates on the Bloch sphere: $$ x = 2\mathfrak{Re}(a^*b)$$ $$ y = 2\mathfrak{Im}(a^*b)$$ $$ z = |a|^2 - |b|^2$$