In quantum mechanics, particularly quantum information,
the Bloch sphere is an invaluable tool to visualize qubits.
All pure qubit states are represented by a point on the sphere’s surface:
The , and -axes represent the components of a spin-1/2-alike system,
and their extremes are the eigenstates of the Pauli matrices:
Where the latter two pairs are expressed as follows in the conventional -basis:
More generally, every point on the surface of the sphere
describes a pure qubit state in terms of the angles and ,
respectively the elevation and azimuth:
Another way to describe states is the Bloch vector ,
which is simply the -coordinates of a point on the sphere.
Let the radius :
Note that is not actually a qubit state,
but rather a description of one.
The main point of the Bloch vector is that it allows us
to describe the qubit using a density operator:
Where is the Pauli “vector”.
Now, we know that a density matrix represents a pure ensemble
if and only if it is idempotent, i.e. :
You can easily convince yourself that, if ,
we get again, so the state is pure:
Therefore, if the radius , the ensemble is pure,
else if it is mixed.
Another useful property of the Bloch vector
is that the expectation value of the Pauli matrices
are given by the corresponding component of :
This is a consequence of the above form of the density operator .
For example for :
- N. Brunner,
Quantum information theory: lecture notes,
- J.B. Brask,
Quantum information: lecture notes,