Relationship between the Itô–Schrödinger and Hudson–Parthasarathy Equations

Abstract—The Hudson–Parthasarathy equation and the Itô–Schrödinger equation (known also as the Belavkin equation) describe a Markov approximation of the dynamics of open quantum systems. The former is a stochastic version of the classical Heisenberg equation, while the latter is a stochastic version of the classical Schrödinger equation (but this analogy is not complete). Two versions of stochastic Heisenberg equations are considered, one of which uses a white noise operator constructed from (non–self-adjoint) birth and death operators and the other uses a white noise operator constructed from (self-adjoint) coordinate and momentum operators.