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 initially described by four real numbers: two real and two imaginary components. However, there are two constraints that reduce this count.

• First, from quantum mechanics, the total probability of the system must be one:

or equivalently $\parallel |\psi ⟩{\parallel }^2=1$. This normalization condition removes one degree of freedom, leaving only three independent real parameters.

• Second, quantum states are only physically meaningful up to an overall global phase factor $e^{i\gamma }$, since such a phase does not affect measurement probabilities. This allows us to choose $|\psi ⟩$ such that the coefficient of $|0⟩$ is real and non-negative. This removes one more degree of freedom, leaving only two independent real parameters.

With these constraints, the state $|\psi ⟩$ can be written in the following form:

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

These two parameters, $\theta$ and $\varphi$, define a point on the surface of a unit sphere, just like latitude and longitude define a point on Earth. Thus, the set of all pure qubit states forms a two-dimensional sphere, known as the Bloch sphere.

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 $s=\sqrt{r}$.