An Introduction to Quantum Filtering

Abstract

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

Single photon quantum filtering using non-Markovian embeddings

“…The dynamics of the system will be described, using the quantum stochastic calculus [10,[19][20][21][22]. Quantum stochastic integrals are defined in terms of fundamental field operators B(t), B * (t) and L(t) ([19, ch.…”

“…When light interacts with a quantum system, information about the system is contained in the scattered light (output) and this may be used to monitor or control the system. The problem of extracting information from continuous measurement of the scattered light is a problem of quantum filtering [6][7][8][9][10][11][12][13]; however, this has tended to consider inputs only in a vacuum or other Gaussian state, with quadrature or counting measurements. The purpose of this paper is to solve a quantum filtering problem for systems driven by fields in single photon states.…”

Coherently tracking the covariance matrix of an open quantum system

“…However, research suggests the situation becomes much more complicated when there is no measurement step. It has been proven the BelavkinKalman filter fails in the presence of a fully quantum non-commutative output signal [38][39][40] and furthermore measurement-based Kalman filters are challenging to be realized efficiently with quantum hardware [41]. The CMT observer is the first coherent method of providing an estimate of the full quantum state of a plant.…”

Principles and applications of quantum control engineering

“…Linear models arise for systems with canonical coordinates for which the triple (S , L, H ) leads to a linear system of Heisenberg-Langevin equations and input/output relations: this happens when H is quadratic, L is linear and S is independent of the canonical coordinates. In this case, the performance specification becomes tractable and one may generalize some of the known results for classical linear systems for linear quadratic Gaussian (LQG) problems and H ∞ control, see the surveys [46][47][48] for more details. While the robust control problem turns out to be tractable in the quantum case, unfortunately the optimal LQG controller need not correspond to a genuine quantum input-system-output model.…”