In order to achieve the same Lindblad dynamics from a mixed quantum-classical method such as MFE, the coefficients c 1 and c 0 (and {c j }) must be evolved in a way such that their time evolution corresponds to equations 28a through 29d. The density matrix elements of equations 28a through 29d cannot, however, be directly replaced with the usual products of the coefficients c 1 and c 0 (and {c j }). This is because this density matrix, in general, represents a mixed quantum state due to the fact that Lindblad dynamics describe an interaction with an external environment. The density matrix of a mixed quantum state cannot, in general, be represented by the density matrix of a single pure state as c 0 |ψ 0 ⟩ + c 1 |ψ 1 ⟩. Thus, alternative approaches must be used to perform the time evolution of c 1 and c 0 with a combined result that satisfies Eq. 28a through Eq. 29d.