With ω ω ± ± 2 = ω ω 1 2 + ω ω 2 2 ± ± ω ω 2 2 , the system's pure ground state density matrix is ρ ρ A B = | 0 ⟩ ⟩ ⟨ ⟨ 0 | , which in position basis is ⟨ ⟨ q A , q B | ρ ρ A B | q A ′ , q B ′ ⟩ ⟩ ∝ ∝ exp ⁡ ⁡ ( − − ω ω + ( q A + q B ) 2 / 2 − − ω ω − − ( q A − − q B ) 2 / 2 − − ω ω + ( q A ′ + q B ′ ) 2 / 2 − − ω ω − − ( q A ′ − − q B ′ ) 2 / 2 ) . Then