Quantum filtering: a reference probability approach

Abstract

These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.

28 pages; see also math-ph/0511021 and math.OC/0601741

References in corpus (12)

• Feedback control of quantum state reduction
• Adaptive homodyne measurement of optical phase
• Quantum Continual Measurements and a Posteriori Collapse on CCR
• Quantum Kalman Filtering and the Heisenberg Limit in Atomic Magnetometry
• Linear stochastic wave-equations for continuously measured quantum systems
• Risk-Sensitive Optimal Control of Quantum Systems
• Bellman equations for optimal feedback control of qubit states
• Optimal Quantum Filtering and Quantum Feedback Control
• Hamilton-Jacobi-Bellman equations for Quantum Filtering and Control
• Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions
• The Langevin equation from Markovian Quantum Central Limits
• Feynman Diagrams and the Quantum Stochastic Calculus

Cited by in corpus (8)

• On the separation principle of quantum control
• Optimal error tracking via quantum coding and continuous syndrome measurement
• How to estimate past measurement interventions on a quantum system undergoing continuous monitoring
• Non-Markovian Quantum Feedback Networks II: Controlled Flows
• Continuous Measurement and Stochastic Methods in Quantum Optical Systems
• The Gisin-Percival Stochastic Schrödinger Equation from Standard Quantum Filtering Theory
• Stochastic Master Equations in Thermal Environment
• Quantum Stochastic Stability