Open quantum system

Definition

Let $\mathcal{H}_S$ and $\mathcal{H}_E$ be complex Hilbert spaces for a system $S$ and an environment $E$. Let $H$ be a self-adjoint operator on the tensor product $\mathcal{H}_S \otimes \mathcal{H}_E$, the total Hamiltonian. Assume that it has the form $$H = H_S \otimes I_E + I_S \otimes H_E + H_I,$$ where $H_S$ and $H_E$ act on $\mathcal{H}_S$ and $\mathcal{H}_E$, and $H_I$ is the interaction. Let $\rho(0)$ be a density operator on $\mathcal{H}_S \otimes \mathcal{H}_E$ and $U(t) = e^{-itH/\hbar}$. The system $S$ is then an open quantum system. Its state at time $t$ is the reduced density operator $$\rho_S(t) = \tr_E\big(U(t)\rho(0)U(t)^*\big),$$ where $\tr_E$ is the partial trace over $\mathcal{H}_E$. The special case with $H_I = 0$ and $\rho(0) = \rho_S(0) \otimes \rho_E(0)$ is a closed system, with $\rho_S(t) = e^{-itH_S/\hbar}\rho_S(0)e^{itH_S/\hbar}$.