It is important to emphasize the difference between a probabilistic mixture (i.e. an ensemble) of quantum states and the superposition of two states. If an ensemble is prepared to have half of its systems in state | ψ ψ 1 ⟩ ⟩ and the other half in | ψ ψ 2 ⟩ ⟩ , it can be described by the density matrix: