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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 = e^{iθα}.
The Qubit can also be represented in terms of the
BLOCH SPHERE.
|ψ> = α|0> + β|1>
= r_{α}e^{iθα}|0> + r_{β}e^{iθβ}|1>
e^{-iθα}|ψ> = r_{α}|0> + r_{β}e^{i(θβ - θα)}|1>
= cos(θ/2)|0> + e^{iφ}sin(θ/2)|1>
Where r_{α}^{2} + r_{β}^{2} = r^{2} = 1, r_{α} = cos(θ/2), r_{β} = sin(θ/2)
∴ θ = 2cos^{-1}r_{α} = 2sin^{-1}r_{β}.
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)|1>
θ = 0 => |ψ> = |0>
θ = π => |ψ> = |1>
θ = π/2, φ = 0 => |ψ> = (1/√2)|0> + (1/√2)|1>
ρ can also be also be written in the Pauli basis as:
ρ = aI + bσ_{x} + cσ_{y} + dσ_{z}}
- - - - - - - -
= a| 1 0 | + r_{x}| 0 1 | + r_{y}| 0 -i | + r_{z}| 1 0 |
| 0 1 |_{ } | 1 0 |_{ } | i 0 |_{ } | 0 -1 |
- - - - - - - -
Where r_{x}^{2} + r_{y}^{2} + r_{z}^{2} = 1
- -
= (1/2)| 1 + r_{z} r_{x} - ir_{y} |
| r_{x} + ir_{y} 1 - r_{z} |
- -
Therefore, r_{x} = 1, r_{y} = r_{z} = 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 r_{x}^{2} + r_{y}^{2} + r_{z}^{2} ≤ 1
and,
|ψ> = r|0> + (1 - r)|1>