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.
Starting from the Gisin-Percival state diffusion equation for the pure state trajectory of a composite bipartite quantum system and exploiting the purification of a mixed state via its Schmidt decomposition, we write the diffusion equation for the quantum trajectory of the mixed state of a subsystem
S of the bipartite system, when the initial state in
S is mixed. Denoting the diffused state of the system
S at time
t by
ρt(B) for each
t≥0, where
B is the underlying complex
n-dimensional vector-valued Brownian motion process and using It{ô} calculus, along with an induction procedure, we arrive at the stochastic differential of the scalar-valued moment process
Tr[ρtm(B)],m=2,3,… in terms of
dB and
dt. This shows that each of the processes
{Tr[ρtm(B)],t≥0} admits a Doob-Meyer decomposition as the sum of a martingale
Mt(m)(B) and a non-negative increasing process
St(m)(B). This ensures the existence of
t→∞limTr[ρtm(B)] almost surely with respect to the Wiener probability measure
μ of the Brownian motion
B, for each
m=2,3,…. In particular, when
S is a finite level system, the spectrum and therefore the entropy of
ρt(B) converge almost surely to a limit as
t→∞. In the Appendix, by employing probabilistic means, we prove a technical result which implies the almost sure convergence of the spectrum for countably infinite level systems.