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arXiv:2510.04967 · Journal of Mathematical Physics

Quantum filtering at finite temperature

John Edward Gough

Published Aug 1, 2026• 0 citations
arXivHTMLPDF

Abstract

We pose and solve the problem of quantum filtering based on continuous-in-time quadrature measurements (homodyning) for the case where the quantum process is in a thermal state. The standard construction of quantum filters involves the determination of the conditional expectation onto the von Neumann algebra generated by the measured observables - with the non-demolition principle telling us to restrict the domain (the observables to be estimated) to the commutant of the algebra. The finite-temperature case, however, has additional structure: we use the Araki–Woods representation for the measured quadratures, but the Tomita–Takesaki theory tells us that there exists a separate, commuting representation and therefore the commutant will have a richer structure than encountered in the Fock vacuum case. We apply this to the question of quantum trajectories to the Davies–Fulling–Unruh model. Here, the two representations are interpreted as the fields in the right and left Rindler wedges.

DOI
10.1063/5.0305163

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We pose and solve the problem of quantum filtering based on continuous-in-time quadrature measurements (homodyning) for the case where the quantum process is in a thermal state. The standard construction of quantum filters involves the determination of the conditional expectation onto the von Neumann algebra generated by the measured observables - with the non-demolition principle telling us to restrict the domain (the observables to be estimated) to the commutant of the algebra. The finite-temperature case, however, has additional structure: we use the Araki–Woods representation for the measured quadratures, but the Tomita–Takesaki theory tells us that there exists a separate, commuting representation and therefore the commutant will have a richer structure than encountered in the Fock vacuum case. We apply this to the question of quantum trajectories to the Davies–Fulling–Unruh model. Here, the two representations are interpreted as the fields in the right and left Rindler wedges.
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.
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.
This manuscript presents an advanced generalization of the Belavkin quantum filter, applying it to Boson and Fermion noise via the Hudson-Parthasarathy quantum stochastic calculus. It focuses on estimating quantum system observables and states in noisy conditions, highlighting the transition from non-commutative to commutative observables post-filtering. The work introduces an algorithm for constructing quantum filter coefficients by solving an infinite series of linear algebraic equations, enabling real-time filtering in the observable domain. This development offers significant insights into quantum noise analysis and state estimation, with potential applications in quantum computing and information processing. The manuscript also explores the integration of classical noisy measurements with quantum non-demolition measurements, creating a hybrid estimation framework that enhances quantum filtering capabilities. The work has potential applications in estimating the time varying state of a quantum gravitational wave from measurements on electromagnetic noise fields with which the wave interacts.
Abstract The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter’s state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task. In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability. This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.
In this paper, we first derive the discrete time recursive stochastic filtering equations for a classical Markov process when noisy measurements of the process are made with the measurement noise at each discrete sample forms an iid Gaussian or non-Gaussian process. The filter involves deriving general formulae for the conditional mean process given measurements up to that time and also simultaneously recursive formulae for the conditional moments of the estimation error process. In the next section, we build upon the method of John Gough et al., to construct a supersymmetric quantum stochastic filter, i.e. the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson–Parthasarathy noisy Schrödinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, i.e. [Formula: see text]. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the Canonical Anti-commutation Relations (CAR). We prove that when a superposition of Bosonic quantum Brownian motion and Bosonic counting process along with Fermionic counting process is measured through the Hudson–Parthasarathy noisy Schrödinger system with both Bosonic and supersymmetric noise, then it satisfies the non-demolition property and hence conditional expectations can be defined. The construction of the quantum filter here involves taking into consideration all the positive integer powers of the output measurement noise differential and we give an algorithm for computing these powers based on the quantum Ito’s formula. The method for deriving a countably infinite number of linear equations for the filter coefficients is based on the reference probability method of Gough et al. The method of constructing supersymmetric noise for driving the HPS equation is based on the work of Timothy Eyre. The main result of this paper is to derive an infinite system of linear equations for the quantum filter coefficients thereby yielding a real-time implementable filter for estimating system observables or equivalently, the system mixed state from the most general form of non-demolition measurements comprising Bosonic continuous and counting noise plus Fermionic counting noise.
Simulation of nonlinear stochastic master equations generally requires trajectory-level evolution of large Hilbert-space density matrices, making strongly nonlinear continuously monitored systems computationally challenging. Here we identify an exactly solvable class centered on polynomial Hamiltonians of commuting quadratures with linear damping and homodyne measurement of those same quadratures. For initial states with a Gaussian measured-quadrature marginal and polynomial conditional-momentum sections, we derive an exact finite stochastic representation through any fixed momentum order. The result extends to interacting multimode systems, arbitrary measurement efficiency, and finite thermal occupation. Its computational cost is linear in trajectory length and independent of Hilbert-space dimension, requiring no Fock-space truncation, Gaussian approximation, or approximate moment closure. As an application, we derive an exact causal filter for monitored cubic-phase gates that uses the homodyne record to determine the variance-minimizing momentum displacement. Numerical simulations validate the representation against direct stochastic-master-equation evolution, demonstrate substantial speedups over converged Fock-space propagation, and show that this record-conditioned recentering can efficiently recover nonlinear squeezing otherwise lost through trajectory averaging.