In operator language, a density operator for a system is a positive semi-definite, Hermitian operator of trace one acting on the Hilbert space of the system. This definition can be motivated by considering a situation where each pure state | ψ ψ j ⟩ ⟩ is prepared with probability p j , describing an ensemble of pure states. The probability of obtaining projective measurement result m when using projectors Π Π m is given by p ( m ) = ∑ ∑ j p j ⟨ ψ ψ j | Π Π m | ψ ψ j ⟩ = tr ⁡ ⁡ [ Π Π m ( ∑ ∑ j p j | ψ ψ j ⟩ ⟨ ψ ψ j | ) ] , which makes the density operator, defined as ρ ρ = ∑ ∑ j p j | ψ ψ j ⟩ ⟨ ψ ψ j | , a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, Hermitian, and has trace one. Conversely, it follows from the spectral theorem that every operator with these properties can be written as ∑ ∑ j p j | ψ ψ j ⟩ ⟨ ψ ψ j | for some states | ψ ψ j ⟩ and coefficients p j that are non-negative and add up to one. However, this representation will not be unique, as shown by the Schrödinger–HJW theorem.