In quantum computing, a qubit, which stands for "quantum bit," is the fundamental unit of quantum information. Unlike classical bits, which can only exist in one of two states (0 or 1) at any given time, qubits can exist in a superposition of both states simultaneously.
In a superposition, a qubit can represent a linear combination of the 0 and 1 states, often denoted as α|0⟩ + β|1⟩, where α and β are complex numbers and |0⟩ and |1⟩ are the basis states corresponding to the classical states 0 and 1, respectively.
The total number of states that a qubit can take on or exist as simultaneously corresponds to the number of possible complex combinations of α and β that satisfy the normalization condition, |α|^2 + |β|^2 = 1. Since α and β are complex numbers, they each have two degrees of freedom (real and imaginary parts), resulting in a total of four degrees of freedom.
Therefore, a qubit can exist in an infinite number of states due to the continuous range of possible values for α and β. However, when considering only normalized states, the number of distinct states is limited. By fixing the normalization condition, the number of distinct states reduces to three. These states form a three-dimensional unit sphere in the complex plane, known as the Bloch sphere.
The three distinct states on the Bloch sphere are often referred to as the computational basis states or the cardinal states. They are:
- |0⟩ state (corresponding to α=1, β=0)
- |1⟩ state (corresponding to α=0, β=1)
- Superposition state (corresponding to α and β having non-zero values, such as α=1/√2, β=1/√2)
These three states can be considered as orthogonal basis vectors on the Bloch sphere, and any qubit state can be expressed as a linear combination of these basis states.