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.
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.
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(R+)⊗(Cn⊕Cn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B(t),t≥0}, we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.
Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.
Remark: By adding a non-commutative noise term to the position field in the Lagrangian density for the metric tensor of general relativity, we transform commuting position fields into non-commuting position fields. This means that the accessible position field becomes non-accessible after adding a noise term to it. This passage from accessible to inaccessible fields enables us to formulate the Hudson-Parthasarathy noisy Schrodinger dynamics that describes the joint evolution of system and bath as a unitary evolution. Even though the density operator of the field state alone (ie the system state) obeys the Lindblad master equation) that is non-unitary when we trace out over the bath, the joint system and bath state satisfies a unitary evolution. The intepretation is that the number of particles in the system alone is not conserved since particles can be exchanged between system and bath, yet the overall number of particles in the system and bath is conserved because of the joint unitary evolution. Thus although we have added noncommutative noise terms to the position fields of the system, thereby passing over from accessible fields to inaccessible fields, we have gained something in this process, namely the field enlarged by noise satisfies unitary evolution. This fact can be regarded as a fundamental aspect of the transformation from accessible to inaccessible variables.
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.