More mathematically; if a state describing two subsystems A and B | Ψ Ψ A B ⟩ ⟩ = | ϕ ϕ A ⟩ ⟩ | ϕ ϕ B ⟩ ⟩ is a separable state, then the reduced density matrix ρ ρ A = Tr B ⁡ ⁡ | Ψ Ψ A B ⟩ ⟩ ⟨ ⟨ Ψ Ψ A B | = | ϕ ϕ A ⟩ ⟩ ⟨ ⟨ ϕ ϕ A | is a pure state. Thus, the entropy of the state is zero. Similarly, the density matrix of B would also have 0 entropy. A reduced density matrix having a non-zero entropy is therefore a signal of the existence of entanglement in the system.