Stochastic Processes in Classical and Quantum Physics and Engineering

275 Pages·2023·9.175 MB·English

by  Harish Parthasarathy

Stochastic Processes in Classical and

Quantum Physics and Engineering

Stochastic Processes in Classical and

Quantum Physics and Engineering

Harish Parthasarathy
Professor

Electronics & Communication Engineering

Netaji Subhas Institute of Technology (NSIT)

New Delhi, Delhi-110078
First published 2023
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ISBN: 9781032405391 (hbk)
ISBN: 9781032405414 (pbk)
ISBN: 9781003353553 (ebk)
DOI: 10.4324/9781003353553
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2
Preface

Preface
This book covers a wide range of problems involving the applications of
stochastic processes,stochasticcalculus,largedeviation theory,group represen­
tation theory and quantum statistics to diverse fields in dynamical systems, elec­
tromagnetics, statistical signal processing, quantum information theory, quan­
tum neural network theory, quantum filtering theory, quantum electrodynam­
ics, quantum general relativity, string theory, problems in biology and classical
and quantum fluid dynamics. The selection of the problems has been based
on courses taught by the author to undergraduates and postgraduates in elec­
tronics and communication engineering. The theory of stochastic processes has
been appliedtoproblemsofsystemidentificationbased ontime series modelsof
the process concerned to derive time and order recursive parameter estimation
followed by statistical performance analysis. The theory of matrices has been
applied to problems such as quantum hypothesis testing between two states in­
cluding a proof of the quantum Stein Lemma involving asymptotic formulas for
the error probabilities involved in discriminating between a sequence of ordered
pairs of tensor product states. This problem is the quantum generalization of
the large deviation result regarding discriminating between to probability dis­
tributions from iid measurements. The error rates are obtained in terms of the
quantum relative entropy between the two states. Classical neural networks are
used to match an input­output signal pair. This means that the weights of the
network are trained so that it takes as input the true system input and repro­
ducesagood approximaitionof thesystemoutput. Butif the inputandoutput
signals are stochastic processes characterized by probability densities and not
by signal values, then the network must take as input the input probability
distribution and reproduce a good approximation of the output probability dis­
tribution. This is a very hard task since a probability distribution is specified
by an infinite number of values. A quantum neural network solves this problem
easily since Schrodinger’s equation evolves a wave function whose magnitude
square is always a probability density and hence by controlling the potential
in this equation using the input pdf (probability distribution function) and the
errorbetween thedesired outputpdfand thatgeneratedbySchrodinger’sequa­
tion, we are able to track a time varying output pdf successfully during the
training process. This has applications to the problem of transforming speech
datato EEGdata. Speechischaracterizedby apdfandsoisEEG.Further the
speech pdf is correlated to the EEG pdf because if the brain acquires a disease,
the EEGsignal willget distorted andcorrespondingly thespeech signal willalso
getslurred. Thus, the quantum neural network based on Schrodinger’s equation
can exploit this correlation to get a fairly good approximation of the EEG pdf
giventhespeechpdf. Theseissueshavebeendiscussedinthisbook. Otherprob­
lems include a proof of the celebrated Lieb’s inequality which is used in proving
convexity and concavity properties of various kinds of quantum entropies and
quantum relative entropies that are used in establishing coding theorems in
quantum information theory like what is the maximum rate at which classical
information can be transmitted over a Cq channel so that a decoder with error
probability converging to zero can be constructed. Some of the inequalities dis­
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cussed in this book find applications to more advanced problems in quantum
information theory that have not been discussed here like how to generate a se­
quenceof maximally entangled stateshaving maximaldimensionasymptotically
from a sequence of tensor product states using local quantum operations with
one way or two way classical communication using a sequence of TPCP maps
or conversely, how to generate a sequence tensor product states optimally from
a sequence of maximally entangled states by a sequence of TPCP maps with
the maximally entangled states having minimal dimension. Or how to gener­
ate a sequence of maximal shared randomness between two users a sequence of
quantum operations on a sequence of maximally entangled states. The perfor­
mance of such algorithms are characterized by the limit of the logarithm of the
length divided by the number of times that the tensor product is taken. These
quantities are related to the notion of information rate. Or, the maximal rate at
which informationcanbe transmitted fromone user toanotherwhen they share
a partially entangled state. This book also covers problems involving quantiza­
tion of the action functionals in general relativity and expressing the resulting
approximateHamiltonianasperturbations toharmonicoscillatorHamiltonians.
It also covers topics such as writing down the Schrodinger equation for a mixed
state of a particle moving in a potential coupled to a noisy bath in terms of
the Wigner distribution function for position and momentum. The utility of
this lies in the fact that the position and momentum observables in quantum
mechanics do not commute and hence by the Heisenberg uncertainty principle,
theydo nothaveajoint probabilitydensity unlike theclassical caseof aparticle
moving in a potential with fluctuation and dissipation forces present where we
haveaFokker­Planck equationfor thejointpdfof thepositionand momentum.
The Wigner distribution being complex, is not a joint probability distribution
function but its marginals are and it is the closest that one can come in quan­
tum mechanics for making an analogy with classical joint probabilities and we
show in this book that its dynamics is the same a that of a classical Fokker­
Planck equation but with small quantum correction terms that are represented
by a power series in Planck’s constant. We then also apply this formalism to
Belavkin’s quantum filtering theory based on the Hudson­Parthasarathy quan­
tumstochasticcalculusby showing that when thisequationforaparticlemoving
in a potential with damping and noise is described in terms of the Wigner dis­
tribution function, then it just the same as the Kushner­Kallianpur stochastic
filter butwith quantumcorrection termsexpressed asapowerseries in Planck’s
constant. This book also covers some aspects of classical large deviation theory
especially in the derivation of the rate function of the empirical distribution of
a Markov process and applied to the Brownian motion and the Poisson process.
This problem has applications to stochastic control where we desire to control
the probability of deviation of the empirical distribution of the output process
of a stochastic system from a desired probability distribution by incorporating
control terms in the stochastic differential equation. This book also discusses
some aspects of string theory especially the problem of how field equations of
physics get modified when string theory is taken into account and conformal
invariance is imposed. How supersymmetry breaking stochastic terms in the
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Lagrangian can be reduced by feedback stochastic control etc. Some problem
deal with modeling blood flow, breathing mechanisms etc using stochastic or­
dinary and partial differential equations and how to control them by applying
large deviation theory. One of the problems deals with analysis of the motion
of a magnet through a tube with a coil wound around it in a graviational field
taking intoaccount electromagnetic induction and back reaction. This problem
is result of discussions with one of my colleagues. The book discusses quan­
tization of this motion when the electron and electromagnetic fields are also
quantized. In short, this book will be useful to research workers in applied
mathematics, physicsandquantuminformationtheorywhowishtoexplorethe
rangeofapplicationsofclassicalandquantumstochasticstoproblemsofphysics
and engineering.
Table of Contents Author
chapter 1
StabilitytheoryofnonlinearsystemsbasedonTaylorapproximationsofthe
nonlinearity
1.1Stochasticperturbationofnonlineardynamicalsystems:Linearizedanal­
ysis
1.2 Large deviation analysis of the stability problem
1.3LDPforYang­MillsgaugefieldsandLDPforthequantizedmetrictensor
of space­time
1.4 Strings in background fields
chapter 2
2.1 String theory and large deviations
2.2 Evaluating quantum string amplitudes using functional integration
2.3 LDP for Poisson fields
2.4 LDP for strings perturbed by mixture of Gaussian and Poisson noise
chapter 3
ordinary, partial and stochastic differential equation models for biological
systems,quantum gravity, Hawking radiation, quantum fluids,statistical signal
processing, group representations, large deviations in statistical physics
3.1 A lecture on biological aspects of virus attacking the immune system
3.2 More on quantum gravity
3.3 Large deviation properties of virus combat with antibodies
3.4 Large deviations problems in quantum fluid dynamics
3.5 Lecture schedule for statistical signal processing
3.6 Basic concepts in group theory and group representation theory
3.7 Some remarks on quantum gravity using the ADM action
3.8 Diffusion exit problem
3.9 Large deviations for the Boltzmann kinetic transport equation
3.10OrbitalintegralsforSL(2,R)withapplicationstostatisticalimagepro­
cessing.
3.11 Loop Quantum Gravity (LQG)
vii

Brief Contents

1. Stability Theory of Nonlinear Systems Based on Taylor
Approximations of the Nonlinearity 1–10
2. String Theory and Large Deviations 11–14
3. Ordinary, Partial and Stochastic Differential Equation Models for
Biological Systems, Quantum Gravity, Hawking Radiation,
Quantum Fluids, Statistical Signal Processing,
Group Representations, Large Deviations in Statistical Physics 15–60
4. Research Topics on Representations 61–74
5. Some Problems in Quantum Information Theory 75–86
6. Lectures on Statistical Signal Processing 87–106
7. Some Aspects of Stochastic Processes 107–126
8. Problems in Electromagnetics and Gravitation 127–128
9. Some More Aspects of Quantum Information Theory 129–140
10. Lecture Plan for Statistical Signal Processing 141–182
11. My Commentary on the Ashtavakra Gita 183–186
12 LDP-UKF Based Controller 187–192
13. Abstracts for a Book on Quantum Signal Processing Using
Quantum Field Theory 193–200
14. Quantum Neural Networks and Learning Using Mixed State
Dynamics for Open Quantum Systems 201–208
15. Quantum Gate Design, Magnet Motion, Quantum Filtering,
Wigner Distributions 209–238
16. Large Deviations for Markov Chains, Quantum Master Equation,
Wigner Distribution for the Belavkin Filter 239–248
17. String and Field Theory with Large Deviations, Stochastic
Algorithm Convergence 249–260
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