Quantum states are represented using 2-dimensional vector spaces over the complex number field C, specifically C². This representation is crucial for expressing the state of a qubit. As discussed previously in several sections, we will reiterate it here. For instance, in the case of a spin-1/2 particle or a quantum bit (qubit), there are two

• basis states: spin up ∣↑⟩ and spin down ∣↓⟩.

These basis states can be written as a linear combination representing any quantum state:

– ψ = α↑ ∣↑⟩ + α↓ ∣↓⟩

Here, α↑​ and α↓ are complex numbers that serve as amplitudes forming the superposition of the quantum state. The absolute squares of these coefficients provide the probabilities of finding the particle in the up or down spin states during measurement. We have discussed this condition before, but we will emphasize it again here.

|α↑|² + |α↓|² represent the probabilities of measuring the spin as up or down, respectively, and the sum of these probabilities must equal 1. The expression below illustrates this:

– |α↑|² + |α↓|² = 1

This equality signifies that the norm of superposition states is unity, and these states can be expressed as unit vectors. This is a special case.