Mathematically each quantum mechanical system is associated with a separable complex Hilbert space H . A pure state of a quantum system is represented by a non-zero vector ψ ψ in H . As the vectors ψ ψ and λ λ ψ ψ (with λ λ ∈ ∈ C ) represent the same state, the level of the quantum system corresponds to the dimension of the Hilbert space and pure states can be represented as equivalence classes, or, rays in a projective Hilbert space P ( H n ) = C P n − − 1 . For a two-dimensional Hilbert space, the space of all such states is the complex projective line C P 1 . This is the Bloch sphere, which can be mapped to the Riemann sphere.