where ĤS is the system Hamiltonian, ÎS is the identity in the system Hilbert space H S , ĤE is the environment Hamiltonian, ÎE is the identity in the environment Hilbert space H E , and ĤI is the interaction Hamiltonian between the system and the environment. Note that the partitioning of the system and environment is not unique. Any part of the total system that we do not want to explicitly simulate or is difficult to simulate can be treated as the environment. The time evolution of the density matrix ρT describing a quantum state of the entire system plus the environment is governed by the following quantum Liouville equation