Even if the combined system is in a pure state and can be described by a wavefunction Ψ Ψ , a subsystem in general cannot be described by a wavefunction. This observation motivated the formalism of density matrices, or density operators, introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau in 1927 and Felix Bloch in 1946. In general, the state of a subsystem is described by the density operator ρ ρ and the expectation value of an observable A by the scalar product ( ρ ρ ⋅ ⋅ A ) = t r { ρ ρ A } . There is no way to know if the combined system is pure from the knowledge of observables of the subsystem alone. In particular, if the combined system has quantum entanglement, the state of the subsystem is not pure.