Bloch Sphere

BLOCH SPHERE

Concept

The Bloch sphere is a geometric representation of qubit states in terms of points on the surface of a unit sphere. Many operations on single qubits that are commonly used in quantum information processing can be neatly described with the Block sphere picture.

Construction

The Bloch sphere is an imaginary sphere (similar to the sphere constructed in density of states) and has a unit radius. (i.e., r=1). Fig.4.3 shows a qubit represented as a Bloch sphere. The arrow on the sphere represents the state of the qubit.

Its north and south poles are selected to represent the basis states |0> and |1>, respectively; the other locations are superpositions of |0> and |1>. While the state of a classical bit can either be the north or the south pole of the equator, a qubit can be any point on the sphere.

The Bloch sphere representation

The Bloch sphere allows the state of a qubit to be represented using unit spherical coordinates with polar angle θ and the azimuth angle ϕ.

The Bloch sphere representation of a qubit is

 |Ψ>=cos(θ/2)| 0 > +e^iϕsin(θ/2)| 1>         …………(1)

Where

0≤ θ≤π and 0≤ϕ≤2π

Here, the normalization constraint is |Ψ|² =1

.. Equation (1) shall be written as

i.e,  |Ψ|² = |cos(θ/2)|² + |sin(θ/2)|² = 1

Note that

(i) When θ = 0 then, |Ψ > = |0> and

(ii) When θ = π then, |Ψ > = |1> regardless of ϕ.

Explanation

In the Bloch sphere representation a qubit shall not only be in either the north or the south pole of the sphere but also in states that are blend of these two states.

In other words, a qubit can exist in multiple states simultaneously. This is basically the essence of the principle of superposition that happens because of the wave nature of subatomic particles.

Qubits can also work with the overlap of both 0 and 1 states. A quantum state in super position can be written as a linear combination of |0> and |1>

 |Ψ >= α|0 >+ β|1>                 …………(2)

Where |Ψ> is the state of the qubit and |0> and |1> are the computational basis states.

The coefficients α and β are complex numbers and are called as probability amplitudes.

If a is the probability amplitude of 0 state, then the probability of qubit being in 0 state is

 αα* = |α|²

Where α* is the complex conjugate of α.

However, |α|² + |β|² = 1

The quantum state Ψ in equation (2) can be written as a unit column vector in a two dimensional complex plane spanned by the two basis states.

Example

Let us consider a qubit with a two basis states, viz, |0> and 1> as shown in Fig. 4.4.

From Fig. 4.4 We can see that the vectors |0> and 1> are orthogonal.

Here,

(i) A qubit with state |0> is represented by the column vector 

 i.e.,|0>=        ........ (3)

(ii) A qubit with state |1> is represented by the column vector 

 i.e. | 1 >=        ........ (4)

Thus, by substituting equation (3) and equation (4) in equation (2) we get

 Ψ= α + β

In other words, an arbitary qubit state is represented by the vector 

Inference

(i) As discussed, classical bit can only be in a single state, whereas, a qubit cannot only be in one of the two discrete states, it can also exist simultaneously in a blend of some of these states.

(ii) The proportions of |0> and |1> in the blended states need not be equal and can be arbitary.

(iii) Thus an infinite number of possible combinations of |0> and |1> is possible in a qubit, provided, the constraint |α|² + |β|² =1 is satisfied.

Conclusion

Thus, we can conclude that, it is possible to store a vast amount of information on a single qubit but it is impossible to retrieve the information. When the value in a qubit is measured, it returns |0> with probability α² or it returns |1> with probability β², and then the qubit assumes the state just returned.