The theorem states that the vacuum state | Ω Ω ⟩ ⟩ is a cyclic vector for the field algebra A ( O ) corresponding to any open set O in Minkowski space. That is, any state | ψ ψ ⟩ ⟩ can be approximated to arbitrary precision by acting on the vacuum with an operator selected from the local algebra, even for | ψ ψ ⟩ ⟩ that contain excitations arbitrarily far away in space. In this sense, states created by applying elements of the local algebra to the vacuum state are not localized to the region O .