Wolfram Alpha:

```Qubit
-----

The bit is the basic unit of information used by
computers to represent information.  The bit can
take the value of 0 or 1.  The Qubit is similar to
the bit but can also be a superposition of both the
0 and 1 states.  Thus,

|ψ> = α|0> + β|1>

where α and β are probability amplitudes that
can be complex numbers, i.e. α = eiθα.

The Qubit can also be represented in terms of the
BLOCH SPHERE.

|ψ> = α|0> + β|1>

= rαeiθα|0> + rβeiθβ|1>

= rα|0> + rβei(θβ - θα)|1>

= cos(θ/2)|0> + eiφsin(θ/2)|1>

Where rα2 + rβ2 = r = 1.

A pure Qubit state can be represented by any point
on the surface.
-       -
|ψ> = (1/√2)|0> + (1/√2)|1> => ρ = | 1/2 1/2 |
| 1/2 1/2 |
-       -
= cos(θ/2)|0> + exp(iφ)sin(θ/2)

θ = 0 => |ψ> = |0>

θ = π => |ψ> = |1>

θ = π/2, φ = 0 => |ψ> = (1/√2)|0> + (1/√2)|0>

ρ can also be also be written in the Pauli basis as:

ρ = aI + bσx + cσy + dσz}

-   -       -   -       -    -       -    -
= a| 1 0 | + rx| 0 1 | + ry| 0 -i | + rz| 1  0 |
| 0 1 |     | 1 0 |     | i  0 |     | 0 -1 |
-   -       -   -       -    -       -    -

Where rx2 + ry2 + rz2 = 1

-                  -
= (1/2)|  1 + rz   rx - iry |
| rx + iry   1 - rz  |
-                  -

Therefore, rx = 1, ry = rz = 0

-       -
ρ = | 1/2 1/2 |
| 1/2 1/2 |
-       -

It is possible to put the Qubit in a mixed state,
a statistical combination of different pure states.
Mixed states can be represented by points inside
the Bloch sphere.  A mixed Qubit state has three
degrees of freedom: the angles φ and θ, as well
as the length, r, of the vector that represents
the mixed state.  In this case rx2 + ry2 + rz2 ≤ 1
and,

|ψ> = r|0> + (1 - r)|1>
```