Given an orthonormal basis, any pure state | ψ ψ ⟩ ⟩ of a two-level quantum system can be written as a superposition of the basis vectors | 0 ⟩ ⟩ and | 1 ⟩ ⟩ , where the coefficient of (or contribution from) each of the two basis vectors is a complex number. This means that the state is described by four real numbers. However, only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directly measurable), so that there is redundancy in this description. We can take the coefficient of | 0 ⟩ ⟩ to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.