From quantum stochastic differential equations to Gisin-Percival state diffusion

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 $\{\mathbf{B}(t),t\ge 0\}$, 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.

28 pages, one pdf figure. An error in the multiplying factor in Eq. (102) corrected. To appear in Journal of Mathematical Physics

References in corpus (1)

• Continuous quantum measurement and Itô formalism

Cited by in corpus (3)

• Poisson stochastic master equation unravellings and the measurement problem: a quantum stochastic calculus perspective
• Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models
• The Gisin-Percival Stochastic Schrödinger Equation from Standard Quantum Filtering Theory