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.
This paper explores some of the applications of quantum stochastic calculus (qsc) as founded by R.L.Hudson and K.R.Parthasarathy and the theory of the quantum filter established by V.P.Belavkin based on qsc which was simplified considerably by John Gough, Claude Kostler and others to quantum field theory. Specifically, we first derive some general versions of the Belavkin quantum filter for the case when the Hamiltonian and Lindblad operators in the Hudson-Parthasarathy noisy Schrodinger equation (HPSDE) are functions of a classical Markovian time varying parameter process whose statistics is determined by an infinitesimal generator. Using this generalized Belavkin filter, we show how we can estimate on real time basis not only the evolving mixed state of the system but also the probability density of the Markovian parameter. Our generalized filter enables us to obtain a function
ρ(t,θ) that represents an estimate of the state of the quantum system at time
t as well as the probability density of the parameter
θ at time
t. Specifically, when
X(θ)=∑k=1nfk(θ)Xk is a system observable dependent upon the Markovian
θ, then,
∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of
X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time
tThis paper explores some of the applications of quantum stochastic calculus (qsc) as founded by R.L.Hudson and K.R.Parthasarathy and the theory of the quantum filter established by V.P.Belavkin based on qsc which was simplified considerably by John Gough, Claude Kostler and others to quantum field theory. Specifically, we first derive some general versions of the Belavkin quantum filter for the case when the Hamiltonian and Lindblad operators in the Hudson-Parthasarathy noisy Schrodinger equation (HPSDE) are functions of a classical Markovian time varying parameter process whose statistics is determined by an infinitesimal generator. Using this generalized Belavkin filter, we show how we can estimate on real time basis not only the evolving mixed state of the system but also the probability density of the Markovian parameter. Our generalized filter enables us to obtain a function
ρ(t,θ) that represents an estimate of the state of the quantum system at time
t as well as the probability density of the parameter
θ at time
t. Specifically, when
X(θ)=∑k=1nfk(θ)Xk is a system observable dependent upon the Markovian
θ, then,
∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of
X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time
t. After that, we show how the Weyl-Moyal-Wigner transform can be used to formulate noisy quantum evolutions as well as Belavkin filter equations in a classical language wherein intepretations of quantum observables and states are given in terms of classical complex functions of positions and momenta. Marginals of the Wigner distribution of the state for example represent probabiity densities of positions and momenta but the Wigner distribution of the state does not represent any joint probability density of the positions and momenta, indeed the Heisenberg uncertainty does not allow any such interpretation. We then illustrate how a quantum mechanical theory taking into account quantum noise can be constructed in a natural way by perturbing the electromagnetic field with quantum super-symmetric noise introducing even and odd system operators. We explain how the interaction between the matter supercurrent and the metric superfield in supergravity theories can be perturbed by quantum noise by adding supersymmetric quantum noise to the matter supercurrent. Further, with the hope that Feynman path integrals can be used in quantum noisy field theories to evaluate transition amplitudes of scattering, we propose a Feynnman path integral approach to the solution of the HPQSDE when the system can be described by canonical position and momentum operators. We then describe quantum scattering in the presence of quantum noise in the HPQSDE framework. Here either one or both of the Hamiltonians contains noise and the corresponding joint unitary evolution of system and bath is described by a HPQSDE and we illustrate some methods for computing the scattering wave operators and analyzing their existence using standard quantum scattering inequalities. Finally, we develop some formulae for computing the rate of change of Von-Neumann entropy of the Belavkin filtered quantum state based on combining Lie algebraic methods for evaluating the differential of the matrix exponential map with Ito’s formula for Brownian motion, Poisson and Levy processes and even for Martingales. These formulae that describe rate of entropy increase have potential applications to quantum information and quantum commmunication when we require to measure how much information has been transmitted as a function of time across a quantum channel after filtering out the noise on a real time basis. In the context of quantum communication theory, we illustrate the situation of an optical fibre wherein the information carrying quantum electromagnetic field interacts with a classical electromagnetic field that has been scattered by the phonon lattice and hence contains unknown Markovian parameters. By estimating the Markovian paremeters using our generalized Belvakin filter on a real time basis, we are therefore able to generate a counter Hamiltonian via causing a simulated counter classical field using this estimated parameter to interact with the propagating quantum field and hence mitigate the effect of the interaction of the classical field with the quantum field. One of the novel results of this paper is the exact representation of the Weyl-Moyal-Wigner transform of products of quantum operators that includes Lie brackets between operators to all orders in Planck’s constant. We use this idea to obtain a classical representation of the quantum Boltzmann equation in terms of Wigner densities that are complex valued functions of position and momenta representing the system state. Future work on Wigner transforming the quantum Boltzmann equation for all r-particle marginal states of an N-particle system is in progress. Work on interpreting the Belavkin filter and quantum Boltzmann equation in the Wigner domain in terms of corrections to the classical Fujisaki-Kushner-Kallianpur stochastic filter and the classical Boltzmann equation is also in progress.
Belavkin first constructed non-demolition measurements for such systems using which one could construct the conditional expectation of a Heisenberg observable evolving in the HPS-qsde framework and that this conditional expectation could be computed on a real time basis leading to the Belavkin filter that generalizes the classical Kushner-Kallianpur filter in stochastic filtering theory. Physical meaning to the filtering results in such coherent states can be obtained only after averaging the results by means of a weight function dependent upon the Grassmannian parameters using the Berezin integral. This idea also plays a fundamental role in the theory of quantum antennas wherein the electromagnetic field produced by the current density generated by electrons and positrons in the Dirac picture is a quadratic combination of the Fermionic creation and annihilation operators and hence by applying the retarded potential formula, the electromagnetic field radiated out by such a quantum antenna is also a quadratic functional of the Fermionic creation and annihilation operators.
In this chapter, the author proposes to solve three kinds of related problems related to quantum filtering and control. The first deals with designing standard finite matrix based MATLAB simulation algorithms for the Hudson-Parthasarathy noisy Schrodinger equation which is today accepted as the standard technique for describing the unitary evolution of system and noisy bath with the system getting noisy inputs from the bath in the form of the three kinds of fundamental quantum noise processes. The second deals with simulating the well known Belavkin quantum filter for obtaining estimates of system states observables on a real time basis. The final problem dealt in this work involves computing the evolution of the Von-Neumann entropy of, the Hudson-Parthasarathy noisy Schrodinger system state after tracing out over the bath variables. The Belavkin filter may be simulated both the observable and the state domain either as ordinary non-random matrix differential equations or equivalently as classical stochastic differential equations.
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.
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.