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-Itô-Segal isomorphism between the Boson Fock reservoir space Γ(L 2 (R + ) ⊗ (C n ⊕ C n )) and the Hilbert space L 2 (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 (J. Phys. A 167, 315 (1992)). This approach also yields an explicit solution of the GisinPercival equation, in terms of the Hudson-Parthasarathy unitary process and a radomized Weyl displacement process. Irreversible dynamics of system density operators described by the wellknown Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

Poisson stochastic master equation unravelings and the measurement problem: A quantum stochastic calculus perspective

“…Since the pioneering GRW work, stochastic collapse models have moved on to CSL and even relativistic equations [6]. CSL uses a version of the Gisin-Percival equation which has already been explored from the perspective of HP evolution by Parthasarathy, Usha Devi, [23], Barchielli and Belavkin [1,4], the latter also deriving Poisson unravellings to slightly different ends with Barchielli [1] and Staszewski [5]. It may also be possible to derive the relativistic models from a unitary evolution in a similar manner.…”

“…Parthasarathy and Usha Devi [23], who realized such a program in the form of a stochastic differential equation driven by Wiener processes. In our case, they are replaced by Poisson processes.…”

“…The procedure outlined above has been realized, using the isomorphism of the Fock space with the L 2 space over the Wiener space by Parthasarathy and Usha Devi [23], although previous work was done by Belavkin, Barchielli, and Staszewski [1,4,5]. In this paper we focus on the Poisson representation.…”

The Gisin-Percival stochastic Schrödinger equation from standard quantum filtering theory

“…We mention that a recent result by Parthasarathy and Usha Devi [20] shows how to derive the Gisin-Percival equation from a quantum stochastic evolution using an appropriate Girsanov transformation. In fact, their scheme uses the same construction as that appearing in our simplest representation of the Gisin-Percival equations in terms of the Belavkin and the , see Subsection II C.…”

“…The author wishes to thank Hendra Nurdin for several useful discussions, and for pointing out preprint [20], as well as several discussions with K.R. Parthasarathy and A.R.…”