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 paper starts with a basic formulation of a distributed parameter transmission line with memristor per unit length in series and also memristor per unit length in parallel apart from the other standard distributed parameters. The result of having memristors distributed along the line is a nonlinear effect in the form of coupling between charge and current and coupling between charge and voltage. These nonlinear effects result in the generation of an infinite number of higher harmonic components along the line even when the input voltage is of a fixed frequency or a finite superposition of voltages of different frequencies. Taking into account random white noise line loading effects, we formulate the resulting line equations based on KCL and KVL in the form of an infinite dimensional nonlinear stochastic differential equation. We study the problem of quantizing the transmission line dynamics using the well known Hudson-Parthasarathy theory of quantum stochastic differential equations derived from the Hudson-Parthasarathy noisy Schrodinger equation constructed from an appropriate transmission line Hamiltonian and Lindblad operators. We then conclude by discussing the effect of quantum stochastic noise and quantum stochastic supersymmetric noise on the action functional of a field theory and how such a model can be used to obtain corrections to the quantum effective action functional or more precisely, to the TPCP map that evolves an initially mixed state of the field to another mixed state after a finite duration. Some remarks on training and testing of quantum neural networks based on the Belavkin quantum filter are presented. This involves constructing a quantum evolution of a mixed state in such a way that the evolving pdf of a system observable will be obtained from the conditional expectation of the noisy quantum state given non-demolition measurements that are strongly correlated with actually measured signals.
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.
The aim of this article is to introduce into quantum field theory,
Zn×Zn 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
n2 kinds graded by
Zn×Zn rather than by
Z2. Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the
Zn×Zn graded tensor product by means of a bi-character constructed from a primitive
nth 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
Zn×Zn for appropriately defined graded quantum stochastic processes. Using the
Zn×Zn 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
Zn×Zn 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.
We consider the propagation of the electromagnetic field within an optical fibre. The field consists of a classical component and a quantum component. The classical component is large and it interacts with the matter within the fibre comprising of atoms that are vibrating, ie, phonons. This interaction causes scattering of the classical field component that interacts with the quantum component of the field, thereby altering the state of the quantum field. After constructing this model, we propose two methods for reducing this "classical-photon-phonon interaction noise". The first method is based on an optimal control algorithm wherein we generate a "control potential" in such a way that after incorporating this potential in the Schrodinger dynamics of the state the state tracks as closely as possible a desired "noiseless state". The second approach is based on Belavkin’s quantum filtering theory wherein we model the classical photons as a quantum Bosonic white noise process and by taking non-demolition measurements on the system, we generate on a real-time basis, an estimate of the evolving state which by application of a control potential is made to track a desired state. Analysis of the spectrum of the transmitted quantum electromagnetic field is carried out by using a quantum stochastic differential model for the noise based on the Hudson-Parthasarathy quantum stochastic calculus. Further, the basic ideas of Cq channel capacities in quantum information theory are used to analyze the rate of information transmission through this optical fibre quantum mechanical channel. Transmission of other kinds of particles like non-Abelian matter and gauge particles are also discussed and the role of superstring theory in building a foundation for corrections to the Yang-Mills action is also discussed. We also highlight the use of the quantum effective action of a field when it interacts with other fields and with random current sources. The quantum effective action, obtained using Feynman’s path integral methods for fields provides a firm foundation for corrections to the classical action by quantum effects and enables one to describe the quantum corrections to the field equations accurately.
We use supersymmetry to enlarge the dimension of the Hilbert space on which the unitary evolution of the state of the quantum fields acts. We discuss how to control the unitary evolution or TPCP maps generated by the quantum evolution of the fields by controlling the vacuum expectations of other fields in the theory. This amounts to breaking supersymmetry using controlled vacuum expectation values of the other fields. The evolution of the wave functional or TPCP maps obtained by tracing out over other fields is based on the Feynman path integral formula for the fields. By using the methods of quantum stochastic filtering, we estimate the evolving state of the fields from non-demolition noise measurements and then design a family of TPCP maps evolving in time whose outputs match the estimated evolving state. In this way, we are able to simulate the evolution of the state of the quantum noisy fields. Direct matching of the designed TPCP map to output the evolving system state is not possible since there is no way by which we can determine the exact evolving state; we can only estimate it using non-demolition measurements. The family of designed TPCP maps can be based on using a simulated master equation with unknown parameters incorporated into the Hamiltonian and the other Lindblad operators, chosen so as to match the state outputted by the quantum filter.