Up to this point, we have frequently emphasized that qubits are mathematically represented on a two-dimensional vector space and that vectors in this space are in an inner product relationship with each other. The fundamental reason for this is due to a deep theorem in mathematical structures known as Frobenius’s theorem. Frobenius’s theorem states that the fields valid for two-dimensional spaces are limited only to real numbers (ℝ) or complex numbers (ℂ). If multiplication is to be commutative, Frobenius’s theorem offers another alternative: Hamilton quaternions (H). Here, the expression “if multiplication is to be commutative” emphasizes that the order of multiplication of two numbers should not affect the result.

– a × b = b × a

However, in the development of quantum theory, complex numbers (ℂ) have been accepted as the preferred field. Among the reasons for this preference is that complex numbers allow phases and amplitudes in quantum mechanics to be expressed more appropriately. Jauch, despite the widespread adoption of the complex number field ℂ, has noted that alternatives like real numbers and quaternions can also be considered, providing references to the literature on this subject. Quantum theory is defined within a vector space (ℂ²) developed over the complex number field ℂ, and superposition states are written as follows:

– ψ = α↑ |↑⟩ + α↓ |↓⟩

Here, α↑ ∈ ℂ and α↓ ∈ ℂ, so these components belong to the complex number field, and their total norm equals 1. This situation is accepted as a unit vector according to the inner product in the ℂ² space.