The aim of this article is to introduce into quantum field theory, (\Bbb Z_n\times\Bbb Z_n) graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of (n^2) kinds graded by (\Bbb Z_n\times\Bbb Z_n) rather than by (\Bbb Z_2). Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the (\Bbb Z_n\times\Bbb Z_n) graded tensor product by means of a bi-character constructed from a primitive (n^{th}) root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by (\Bbb Z_n\times\Bbb Z_n) for appropriately defined graded quantum stochastic processes. Using the (\Bbb Z_n\times\Bbb Z_n) graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of (\Bbb Z_n\times\Bbb Z_n) graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. The addition of higher degree terms in the string field action is then based on the condition that the action should satisfy the quantum master equation for invariance of the matrix elements under the choice of the gauge fixing functional. We suggest methods by which quantum noise fields can also be taken into account in string field theories. The entire aim of adding quantum noise to a quantum field theory is motivated by the fact that the resulting noisy Hamiltonian should describe a Hudson-Parthasarathy noisy Schrodinger equation with unitary evolution on the joint System-Bath Hilbert space so that after tracing out over the bath, one obtains a quantum dynamical semigroup of TPCP maps describing system evolution alone for open quantum systems. For noisy quantum field theories, finally, we explain how to compute propagator corrections caused by 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)ΛdA,dJ∗=(−1)ΛdA∗. 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)ΛdA,dJ∗=(−1)ΛdA∗. 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 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.
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.
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 book introduces the vast subject of supersymmetry along with many specific examples of engineering applications, for example: The design of quantum unitary gates using supersymmetric actions Bosonic and Fermionic noise in quantum systems using the Hudson-Parthasarathy quantum stochastic calculus Superstring theory applied to the quantum mechanics of neurons and supersymmetric quantum filtering theory which can, for example, be used to filter out the noise in a cavity resonator electromagnetic field produced by the presence electrons and positrons in a bath surrounding it Simplified versions of super-Yang-Mills theory with gauge and gaugino fields, both transforming under the adjoint representation of the gauge group and elementary super-gravity models have also been introduced All through the book, emphasis is laid upon exploiting the supersymmetry existing in the nature of Boson-Fermion exchange in designing engineering systems like quantum computers and analyzing the performance of systems in the presence of supersymmetric quantum noise.