Qubits

Burton Rosenberg

Representation on the Bloch sphere

A qubit is a minimal quantum state space, consisting of all superpositions of 1 and 0. The state space is, mathematically speaking, a point on the complex projective line. That is, a pair of complex numbers x and y, not both zero, normalized so that the vector (x,y) has unit length, and disregarding any common phase.

   (x e^iθ,y e^iθ) = (x,y)  θ real
   |(x,y)| = 1

This can be visualized as a point on the surface of a 3-sphere in R³. If the sphere is made unit diameter, then the distance of a point from the north and south poles, d_N, d_S, will satisfy,

   d_N² + d_S² = 1

Hence d_N is the magnitude of x, and d_S is the magnitude of y. A common phase does not matter, in the sense that it cannot ever be measured. It has no significance in reality and is therefore ignored in the mathematics. The relative phase of x and y, however, determine the coherence of the true and false components of the superposition, and whether on combining they demonstrate constructive or destructive interference. Note that in this picture a point at either poles has no relative phaser since only one component is non-zero. The representation is called the Bloch sphere.

The point is written as a complex 2-vector. Two orthonormal vectors are chosen as a basis and are denoted |a> and |b>. The superposition is written,

     c_a |a> + c_b |b>
     s.t. |c_a|² + |c_b|² = 1

Classical and Quantum Superposition

This viewpoint is still underdevelopment. See vol. 3, Feynmann 11-10 - 11-11 for better statement.

Forces (classical) are superposed, but the meaning is very different. Two forces add vectorially and act as a single force of direction and magnitude of the vector sum. The force can be resolved vectorially and there is a projection of the force along each component vector of the superposition.

Quantum states cannot be observed otherwise then along a single vector, the set of possible vectors defined by the measurement. Although unobserved a quantum state can be partially in several states, upon measurement the quantum must decide, and jump wholly into one state or the other. Consider the qubit in state,

    c₀ |0⟩ + c₁ |1⟩

Whereas a classical superposition would, on each measurement, project onto the |0⟩ and |1⟩ directions, a quantum superposition will, on measurement, become aligned with the |0⟩ direction with probability |c₀|² or aligned with the |1⟩ direction with probability |c₁|². Classical behavior is recovered by considering many quantum particles assuming on measurement one or the other direction, probabilistically and independently.

An experiment with polarization

A photon is a quantum of excitation of the quantized electromagnetic field. (In measurement, it is localized and indivisible.; in calculation, it follows multiple paths and interferes with itself.) It has an energy proportional to wave frequency (E=hν) and a wavelength inverse to the frequency (νλ=c). (λ has dimensions of seconds (or meters), ν and E inverse seconds. Everything else is dimensionless.) Its electric and magnetic fields are orthogonal to each other and to its direction of motion. The angle of the electric field is the polarization.

Consider two basis to describe linear polarization, |0^o⟩ and |90^o⟩ and |-45^o⟩ and |+45^o⟩, representing 0, 90, -45 and 45 degree polarizations.

Place a photon in |0^o⟩ polarization by passing it through a polarizing substance.
Attempt to pass this photon through a |90^o⟩ polarized filter. Since it has a c₉₀ = 0, it will never jump to this polarization and the photon will not pass.
Now interpose between the 0 and 90 degree filters a filter at 45 degrees. Expressed in the -45, 45 basis, the photon is in a superposition,
```
    (1/sqrt(2)) |45^o⟩ + (1/sqrt(2)) |-45^o⟩
```
On passing by the filter it will choose with probability 1/2 to be aligned 45 degrees, or with probability 1/2 to be aligned with -45 degrees.
It will now pass 50% of the time through the 90 degree polarization filter. Half the photons will now pass.

Note that when the photon's polarization was re-expressed, it is now mathematically equivalent to a photon which "can't make up its mind" whether to be 45 or -45 degree polarized, even though it appeared to have a definite polarization along the bisector of these two directions.

Last Update: 27 aug 03