Bloch sphere

The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors $|0⟩$ and $|1⟩$, respectively, which in turn might correspond, for example, to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful.

Given an orthonormal basis, any pure state $|\psi ⟩$ of a two-level quantum system can be written as a superposition of the basis vectors $|0⟩$ and $|1⟩$, where the coefficient of (or contribution from) each of the two basis vectors is a complex number. This means that the state is described by four real numbers. However, only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directly measurable), so that there is redundancy in this description. We can take the coefficient of $|0⟩$ to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one:

or equivalently $\parallel |\psi ⟩{\parallel }^2=1$.
Given this constraint, we can write $|\psi ⟩$ using the following representation:

where $0\le \theta \le \pi$ and $0\le \varphi <2\pi$.

The representation is always unique, because, even though the value of $\varphi$ is not unique when $|\psi ⟩$ is one of the states $|0⟩$ or $|1⟩$, the point represented by $\theta$ and $\varphi$ is unique.

The parameters $\theta$ and $\varphi$, re-interpreted in spherical coordinates as respectively the colatitude with respect to the z-axis and the longitude with respect to the x-axis, specify a point

on the unit sphere in ${\mathbb{R}}^3$.

Flag and flagpole interpretation

If we do not restrict to the case

we can still use this visualization by using a kind of "flag-flagpole space". In this case what we have a is a type of spinor, and we have from this video:

Although maybe it is better this other visualization ("An introduction to spinors", by A. Steane, calibre):

with the spinor given by

with $s=\sqrt{r}$.