The time evolution of a closed quantum system is described by the
Schrödinger equation, i\dot{\psi} = H\psi \qquad \psi_t = U_t \psi_0 , For more than one parameter, such as in an entangled state or a classical ensemble of quantum states, the density matrix is instead used. For the density matrix, the time evolution is given by the
von Neumann equation, \dot{\rho} = -i[H,\rho] \qquad \rho_t = U_t \rho_0 U_t^{\dagger} However, this still describes a closed system. Instead, the system is described by the evolution law \rho' = \Lambda \rho = \sum_{\alpha} K_{\alpha} \rho K_{\alpha}^{\dagger} , \quad\textrm{where}\quad \sum_{\alpha} K_{\alpha}K_{\alpha}^{\dagger} = I This is not yet a quantum master equation since it is not a differential equation. In the Markovian approximation, this law gives: \frac{d\rho}{dt} = \mathcal{L}\rho \qquad \rho_t = e^{t\mathcal{L}}\,\rho_0 , which is a master equation for \rho in the Markovian approximation. ==See also==