Mathematical Representation of Qubits

The mathematical description of a qubit state requires the use of complex vector spaces and linear algebra. A general qubit state can be written as:|ψ⟩ = α|0⟩ + β|1⟩

"Psi is equal to alpha ket zero plus beta ket one."

where α and β are complex numbers called probability amplitudes, and |0⟩ and |1⟩ represent the computational basis states corresponding to classical 0 and 1 respectively [3]. The vertical bars and angle brackets constitute Dirac notation, which we will explore in detail in a later section.The probability amplitudes α and β must satisfy the normalization condition:

|α|² + |β|² = 1

The absolute square of alpha plus the absolute square of beta equals one.

This constraint ensures that the total probability of measuring the qubit in any state equals 1, maintaining the fundamental requirement of probability theory. The squared magnitudes |α|² (absolute square of alpha)and |β|² (  absolute square of beta) represent the probabilities of measuring the qubit in states |0⟩ and |1⟩ respectively when a measurement is performed [4].