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arXiv:2601.12564

Quantum Filtering for Squeezed Noise Inputs

John Gough, Dylon Rees

quant-phmath-phSubmitted Jan 18, 2026• Announced Jan 20, 2026• 0 citations• Revised Apr 30, 2026 (v2)
v1 · Jan 18, 2026v2 · Apr 30, 2026
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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.

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20 pages, one figure

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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.
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.
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 We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize the statistics of photons in the output field are given.
In our work, we show how, for a network of bosonic modes, canonical commutation relations constrain the coefficients relating input and internal modes. Based on these constraints, we derive a lower bound on the total steady-state squeezing achievable in reservoir-engineered (dissipative) squeezing schemes, quantified by the sum of mode-optimal quadrature variances normalized to its corresponding input variance. The bound follows solely from canonical commutation relations and stability, and is saturated in the strong-coupling limit at 1. Furthermore, we show that adding independent parametric driving terms for each mode changes the quantum noise-gain balance and yields a distinct optimum bound, approaching 1/2. In addition, we show how these constraints allow us to reformulate the Duan inseparability criterion for a three-mode bosonic system in terms of a single parameter-dependent figure of merit. Our results apply directly to current electromechanical and nanomechanical experiments and indicate that the two-mode bounds can be experimentally approached even at room temperature.