# From quantum stochastic differential equations to Gisin-Percival state diffusion

## TL;DR

This paper derives a nonlinear stochastic Schrödinger equation from quantum stochastic differential equations, linking quantum state diffusion to classical Brownian motion and providing explicit solutions, thus connecting quantum and classical stochastic processes.

## Contribution

It introduces a novel derivation of the Gisin-Percival state diffusion equation from quantum stochastic calculus using the Wiener-Ito-Segal isomorphism.

## Key findings

- Explicit solution of Gisin-Percival equation using Hudson-Parthasarathy process
- Connection established between quantum state diffusion and classical Brownian motion
- Unraveling of Lindblad dynamics through quantum trajectories

## Abstract

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space $\Gamma(L^2(\mathbb{R}_+)\otimes (\mathbb{C}^{n}\oplus \mathbb{C}^{n}))$ and the Hilbert space $L^2(\mu)$, where $\mu$ is the Wiener probability measure of a complex $n$-dimensional vector-valued standard Brownian motion $\{\mathbf{B}(t), t\geq 0\}$, we derive a non-linear stochastic Schrodinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion $\mathbf{B}$. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation (J. Phys. A, 167, 315 (1992)). This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a radomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.