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 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.

arXiv:2510.04967Journal of Mathematical PhysicsAug 1, 20260 citations

Stable quantum filters with scattering phenomena

Woakil Uddin Ahamed, C. Kambhampati

Machine Intelligence ResearchApr 1, 20082 citations

Fundamentals of Classical and Supersymmetric Quantum Stochastic Filtering Theory

Harish Parthasarathy

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 upto 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 etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, $dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*$ . Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the 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 Schrodinger 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 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.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 upto 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 etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, $dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*$. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the 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 Schrodinger 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 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.

QeiosMay 17, 20241 citation

dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*

. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the 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 Schrodinger 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 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.