Belavkin equation

In quantum probability, the Belavkin equation, also known as Belavkin-Schrödinger equation, quantum filtering equation, stochastic master equation, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988. Unlike the Schrödinger equation, which describes the deterministic evolution of the wavefunction ${\displaystyle \psi (t)}$ of a closed system (without interaction), the Belavkin equation describes the stochastic evolution of a random wavefunction ${\displaystyle \psi (t,\omega )}$ of an open quantum system interacting with an observer:

Here, ${\displaystyle L}$ is a self-adjoint operator (or a column vector of operators) of the system coupled to the external field, ${\displaystyle H}$ is the Hamiltonian, ${\displaystyle i={\sqrt {-1}}}$ is the imaginary unit, ${\displaystyle \hbar }$ is the Planck constant, and ${\displaystyle y(t)=\int _{0}^{t}dy(r)}$ is a stochastic process representing the measurement noise that is a martingale with independent increments with respect to the input probability measure ${\displaystyle P(0,d\omega )=\|\psi (0,\omega )\|^{2}\mu (d\omega )}$. Note that this noise has dependent increments with respect to the output probability measure ${\displaystyle P(t,d\omega )=\|\psi (t,\omega )\|^{2}\mu (d\omega )}$ representing the output innovation process (the observation). For ${\displaystyle L=0}$, the equation becomes the standard Schrödinger equation.

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