Quantum projection filter for a highly nonlinear model in cavity QED

Abstract

Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case). One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a strongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.

Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds

“…Several authors have noticed analogies between the filtering SPDEs hinted at above and the evolution equations in quantum physics, see for example [26]. Moreover, the related projection filter developed by D. Brigo and co-authors has been applied to quantum electrodynamics, see for example [21]. The SPDE case driven by rough paths such as dY is of particular interest because it combines the geometry in the state space for X and Y and the geometry in the space of probability measures associated with X conditional on Y 's history.…”

Low-dimensional manifolds for exact representation of open quantum systems

“…In either case, it is essential to represent the SSE in the Stratonovich picture as this enables us to carry out stochastic projections [39]. Contrary to the usual notation we will always take XdY to indicate a Stratonovich stochastic differential as opposed to an Ito differential.…”

Adaptive low-rank approximation and denoised Monte Carlo approach for high-dimensional Lindblad equations

“…The remaining question is to derive a stochastic dynamics on a wave function |ψ LR such that E (|ψ LR ψ LR |) = ρ LR , given that the dynamics of ρ LR is known since easy to compute via (U, σ) solutions of (2) and (3). For this purpose, it is a natural idea to seek |ψ LR under the form |ψ LR = U |ν where the reduced wave function |ν ∈ R m is a stochastic process to be determined.…”

“…(color online) Comparison between the full-rank trajectory t → ρ(t) illustrated on figure 1 with the adaptive rank trajectory t → ρLR(t) governed by(2)(3). Top plot: the solid curve corresponds to Tr ((ρ − ρLR) 2 ) versus time with scale on the left.…”