Quantum Filtering for Squeezed Noise Inputs
John Gough, Dylon Rees
TL;DR
This work extends quantum filtering to open quantum systems driven by general quasi-free (including squeezed) bosonic inputs, using balanced Bogoliubov transformations and Araki–Woods representations. By exploiting Tomita–Takesaki theory and a quantum reference-probability approach, it derives a representation-independent quantum filter for quadrature (homodyne) measurements, culminating in a Kallianpur–Striebel-type formula with a Kushner–Stratonovich update and a well-defined innovations process. The framework recasts squeezed-noise input into a non-Fock stochastic calculus setting, enabling explicit filtering equations that reduce to known thermal-filter results in the appropriate limit. The results broaden quantum estimation tools for nonclassical light and connect to foundational topics in quantum optics and relativistic quantum field theory (e.g., Unruh–DeWitt detectors and Hawking radiation analogues).
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.