Let { | ψ ψ n ⟩ ⟩ } be a set of orthonormal states (i.e., ⟨ ⟨ ψ ψ m | ψ ψ n ⟩ ⟩ = δ δ m , n ) that form a complete eigenbasis for each of the two compatible observables A and B represented by the self-adjoint operators A ^ ^ and B ^ ^ with corresponding (real-valued) eigenvalues { a n } and { b n } , respectively. This implies that