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# Hopf fibration
**Hopf fibration** is a map connecting the state vector representation of a qubit $\lvert\psi\rangle \in \mathbb{C}^2$ to a point on the [[bloch-sphere|Bloch sphere]] $(x, y, z) \in \mathbb{R}^3$. It explains why a qubit — an object living in the complex space $\mathbb{C}^2$ — can be visualized as a point on a real three-dimensional sphere.

A qubit is, in a nutshell, a pair of complex numbers $a, b \in \mathbb{C}$ subject to $|a|^2 + |b|^2 = 1$. A normalized vector in $\mathbb{C}^2$ lives on the 3-sphere $S^3$ (a sphere in 4-dimensional space). However, the qubit's global phase is physically unobservable, so the true state space is $S^3/U(1) \cong S^2$ — the 2-sphere, which is the Bloch sphere. The Hopf fibration $\pi: S^3 \to S^2$ is exactly this projection. Each point on $S^2$ is the image of a circle's worth of points on $S^3$ (one full $U(1)$ orbit).

## The Hopf map
The Hopf map $\pi$ can be written compactly using the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$.

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

For a qubit $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$, the three Bloch sphere coordinates are:

$$x = 2\,\mathfrak{Re}(a^*b), \qquad y = 2\,\mathfrak{Im}(a^*b), \qquad z = |a|^2 - |b|^2$$

One can verify that $x^2 + y^2 + z^2 = (|a|^2 + |b|^2)^2 = 1$, so every normalized qubit maps to a point on the unit sphere.

## Fiber structure
The "fibration" part of the Hopf fibration refers to the fiber over each point on $S^2$. For a fixed Bloch sphere point $(x, y, z)$, there is a whole circle $S^1$ of qubit states in $\mathbb{C}^2$ that all map to it — they differ only by a global phase $e^{i\phi}$. This circle is the fiber, and all such fibers are pairwise linked (they are unknots on $S^3$ that are all linked with each other), which is the topologically remarkable feature of the Hopf fibration discovered by Heinz Hopf in 1931.