From quantum stochastic differential equations to Gisin-Percival state diffusion

Article Type

Research Article

Publication Title

Journal of Mathematical Physics


Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(ℝ+)⊗(ℂn⊕ℂn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion (B(t),t≥0), we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, 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 randomized 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.



Publication Date



Open Access, Green

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