Quantum Filtering for Squeezed Noise Inputs

TL;DR

This paper develops a quantum filtering theory for systems driven by squeezed noise inputs, extending previous models to more general quantum states using advanced mathematical frameworks.

Contribution

It introduces a new quantum filter for quadrature measurements with squeezed inputs, utilizing Bogoliubov transformations and Tomita-Takesaki theory for a more general quantum noise model.

Findings

Derived the quantum filter for squeezed noise inputs.

Extended previous models from thermal to general quasi-free states.

Utilized advanced mathematical tools for filter construction.

Abstract

We derive the quantum filter for a quantum open system undergoing quadrature measurements (homodyning) where the input field is in a general quasi-free state. This extends previous work for thermal input noise and allows for squeezed inputs. We introduce a convenient class of Bogoliubov transformations which we refer to as balanced and formulate the quantum stochastic model with squeezed noise as an Araki-Woods type representation. We make an essential use of the Tomita-Takesaki theory to construct the commutant of the C*-algebra describing the inputs and obtain the filtering equations using the quantum reference probability technique. The derived quantum filter must be independent of the choice of representation and this is achieved by fixing an independent quadrature in the commutant algebra.