As understood from the Stern-Gerlach experiment, different unit vectors in C² can represent the same quantum spin state. For instance, when describing the state of a particle, different mathematical representations can be used, but they all denote the same physical state. Let us discuss this concept further.

For example, the spin states of a particle can be represented as follows:
(1/√2) |↑⟩ + (1/√2) |↓⟩

This expression represents a superposition state, where both the spin-up and spin-down states are equally probable. It is a special type of quantum state. The coefficients used here include square roots, but typically, the numbers used for qubits are complex numbers. This means that they consist not only of real numbers but also include imaginary components such as i (the imaginary unit).

The above expression can also be written as:
i (1/√2) |↑⟩ + (1/√2) |↓⟩

This expression also represents the same state because only a global phase factor has been introduced. If all terms are multiplied by the same complex factor, the resulting vector still represents the same physical state. However, using different phase factors results in obtaining different states.

In conclusion, quantum states are referred to as “rays” in vector space, and these rays can be altered without changing the physical properties of the states. However, specific phase factors are important for understanding the differences between these states.