Quantum Covariance and Filtering

TL;DR

This paper provides a tutorial on quantum filtering, emphasizing quantum covariances and conditional expectations within the quantum probabilistic framework, bridging classical and quantum filtering theories.

Contribution

It introduces a comprehensive exposition of quantum filtering equations using von Neumann algebras, highlighting the role of quantum covariances and conditional expectations.

Findings

Clarifies the quantum filtering equation derivation

Highlights the importance of quantum covariances

Connects classical and quantum filtering concepts

Abstract

We give a tutorial exposition of the analogue of the filtering equation for quantum systems focusing on the quantum probabilistic framework and developing the ideas from the classical theory. Quantum covariances and conditional expectations on von Neumann algebras play an essential part in the presentation.