Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let | Ψ Ψ ⟩ ⟩ be a pure entangled state in the composite Hilbert space H 1 ⊗ ⊗ H 2 . The probability of obtaining measurement result m when measuring projectors Π Π m on the Hilbert space H 1 alone is given by p ( m ) = ⟨ Ψ Ψ | ( Π Π m ⊗ ⊗ I ) | Ψ Ψ ⟩ = tr ⁡ ⁡ [ Π Π m ( tr 2 ⁡ ⁡ | Ψ Ψ ⟩ ⟨ Ψ Ψ | ) ] , where tr 2 denotes the partial trace over the Hilbert space H 2 . This makes the operator ρ ρ = tr 2 ⁡ ⁡ | Ψ Ψ ⟩ ⟨ Ψ Ψ | a convenient tool to calculate the probabilities of these local measurements. It is known as the reduced density matrix of | Ψ Ψ ⟩ ⟩ on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the Schrödinger–HJW theorem implies that all density operators can be written as tr 2 ⁡ ⁡ | Ψ Ψ ⟩ ⟨ Ψ Ψ | for some state | Ψ Ψ ⟩ .