Theoretical Statistics and Mathematics Unit, ISI Delhi

From quantum stochastic differential equations to Gisin-Percival state diffusion

by K. R. Parthasarathy and A. R. Usha Devi

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy (Comm. Math. Phys. {\bf 93}, 301 (1984)) and exploiting the Wiener-It{\^o}-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 Schr{\"o}dinger 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 {\bf 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.

isid/ms/2017/02 [fulltext]

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