Classical and Quantum Information Theory for the Physicist

263 Pages·2022·9.032 MB·English

by  Harish Parthasarathy

Classical and Quantum Information
Theory for the Physicist

Classical and Quantum Information
Theory for the Physicist
Harish Parthasarathy
Professor
Electronics & Communication Engineering
Netaji Subhas Institute of Technology (NSIT)
New Delhi, Delhi-110078
Firstpublished2023
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Brief Contents
Preface vii–xiv
1. Quantum Information Theory, A Selection of Matrix Inequalities 1–4
2. Stochastic Filtering Theory Applied to Electromagnetic
Fields and Strings 5–12
3. Wigner-distributions in Quantum Mechanics 13–26
4. Undergraduate and Postgraduate Courses in Electronics,
Communication and Signal Processing 27–28
5. Quantization of Classical Field Theories, Examples 29–46
6. Statistical Signal Processing 47–66
7. Some More Concepts and Results in Quantum
Information Theory 67–80
8. Quantum Field Theory, Quantum Statistics, Gravity,
Stochastic Fields and Information 81–108
9. Problems in Information Theory 109–152
10. Lecture Plan for Information Theory, Sanov’s Theorem,
Quantum Hypothesis Testing and State Transmission,
Quantum Entanglement, Quantum Security 153–170
11. More Problems in Classical and Quantum Information Theory 171–188
12. Information Transmission and Compression with Distortion,
Ergodic Theorem, Quantum Blackhole Physics 189–226
13. Examination Problems in Classical Information Theory 227–244
v

Detailed Contents
Preface vii–xiv
1. Quantum Information Theory, A Selection of Matrix Inequalities 1–4
1.1 Monotonicity of Quantum Relative Renyi Entropy 1
1.2 Problems 3
2. Stochastic Filtering Theory Applied to Electromagnetic
Fields and Strings 5–12
2.1 M.Tech Dissertation Topics 5
2.2 Estimating the Time Varying Permittivity and Permeability of
a Region of Space Using Nonlinear Stochastic Filtering Theory 5
2.3 Estimating the Time Varying Permittivity and Permeability of
a Region of Space Using Nonlinear Stochastic Filtering Theory 6
2.4 Study Project: Reduction of Supersymmetry Breaking by Feedback 12
3. Wigner-distributions in Quantum Mechanics 13–26
3.1 Quantum Fokker-Planck Equation in theWigner Domain 13
3.2 The Noiseless Quantum Fokker-Planck Equation or Equivalently,
the Liouville-Schrodinger- Von-Neumann-equation in
the Wigner Domain 17
3.3 Construction of the Quantum Fokker-Planck Equation for
a S(cid:83)(cid:72)(cid:70)(cid:76)(cid:191)(cid:70)(cid:3)Choice of the Lindblad Operator 19
3.4 Problems in Quantum Corrections to Classical Theories in
Probability Theory and in Mechanics with Other S(cid:83)(cid:72)(cid:70)(cid:76)(cid:191)(cid:70)(cid:3)
Choices of the Lindblad Operator 21
(cid:3) (cid:22)(cid:17)(cid:24)(cid:3) (cid:37)(cid:72)(cid:79)(cid:68)(cid:89)(cid:78)(cid:76)(cid:81)(cid:3)(cid:191)(cid:79)(cid:87)(cid:72)(cid:85)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:58)(cid:76)(cid:74)(cid:81)(cid:72)(cid:85)(cid:3)Distribution Function 22
3.6 Superstring Coupled to Gravitino Ensures Local Supersymmetry 26
4. Undergraduate and Postgraduate Courses in Electronics,
Communication and Signal Processing 27–28
5. Quantization of Classical Field Theories, Examples 29–46
5.1 Quantization of Fluid Dynamics in a Curved Space-time
Background Using Lagrange Multiplier Functions 29
5.2 d-dimensional Harmonic Oscillator with Electric Field Forcing 31
5.3 A Problem: Design a Quantum Neural Network Based on Matching
the Diagonal Slice of the Density Operator to a Given Probability
Density Function 33
vii
5.4 Quantum Filtering for the Gravitational Field Interacting with
the Electromagnetic Field 33
5.5 Quantum Filtering for the Gravitational Field Interacting with
the Electromagnetic Field 36
5.6 Harmonic Oscillator with Time Varying Electric Field and
Lindblad Noise with Lindblad Operators Being Linear in
the Creation and Annihilation Operators, Transforms a Gaussian
State into Another After Time t 41
5.7 Quantum Neural Network Using a Single Harmonic Oscillator
Perturbed by an Electric Field 43
6. Statistical Signal Processing 47–66
6.1 Statistical Signal Processing: Long Test 47
6.2 Quantum EKF 50
6.3 Lie Brackets in Quantum Mechanics in Terms of the Wigner
Transform of Observables 52
6.4 Simulation of a Class of Markov Processes in Continuous and
Discrete Time with Applications to Solving Partial D(cid:76)(cid:3445)(cid:72)(cid:85)(cid:72)(cid:81)(cid:87)(cid:76)(cid:68)(cid:79)
Equations 54
6.5 Gravitational Radiation 54
6.6 Measuring the Gravitational Radiation Using Quantum
Mechanical Receivers 62
7. Some More Concepts and Results in Quantum
Information Theory 67–80
7.1 Fidelity Between Two States (cid:85)(cid:15)(cid:3)(cid:86) 67
7.2 An Identity Regarding Fidelity 68
7.3 Adaptive Probability Density Tracking Using the Quantum
Master Equation 69
7.4 Quantum Neural Networks Based on Superstring Theory 70
7.5 Designing a Quantum Neural Network for Tracking a Multivariate
pdf Based on Perturbing a Multidimensional Harmonic Oscillator
Hamiltonian by an An-harmonic Potential 73
7.6 Applied Linear Algebra 76
8. Quantum Field Theory, Quantum Statistics, Gravity,
Stochastic Fields and Information 81–108
8.1 Rate Distortion Theory for Ergodic Sources 81
8.2 Problems 86
8.3 Simulation of Time Varying Joint Probability 87
viii
8.4 An application of the Radiatively Corrected Propagator to Quantum
Neural Network Theory Densities Using Yang-Mills Gauge Theories 89
8.5 An Experiment Involving the Measurement of Newton’s
Gravitational Constant G 91
8.6 Extending the Fluctuation-Dissipation Theorem 92
8.7 A discrete Poisson Collision Approach to Brownian Motion 92
8.8 The Born-Oppenheimer Program 94
8.9 The Superposition Principle for Wave Functions of the Curved
Space-time Metric Field Could Lead to Contradictions and what
are the Fundamental D(cid:76)(cid:3446)(cid:70)(cid:88)(cid:79)(cid:87)(cid:76)(cid:72)(cid:86)(cid:3)(cid:76)(cid:81)(cid:3)Developing a Background
Independent Theory of Quantum Gravity 96
8.10 Attempts to Detect Gravitational Waves from Rotating Pulsars
and Sudden Burst of a Star Using Crystal Detectors 96
8.11 Sketch of the Proof of Shannon’s Coding Theorems 97
8.12 The Notion of a Field Operator or Rather an Operator Valued Field 99
8.13 Group Theoretic Pattern Recognition 102
8.14 Controlling the Probability Distribution in Functional Space of
the Klein-Gordon Field Using a Field Dependent Potential 104
8.15 Quantum Processing of Classical Image Fields Using
a Classical Neural Network 105
8.16 Entropy and Supersymmetry 105
9. Problems in Information Theory 109–152
9.1 Problems in Quantum Neural Networks 139
9.2 MATLAB Simulation Exercises in Statistical Signal Processing 141
9.3 Problems in Information Theory 143
9.4 Problems in Quantum Neural Networks 145
9.5 Quantum Gaussian States and Their Transformations 146
10. Lecture Plan for Information Theory, Sanov’s Theorem,
Quantum Hypothesis Testing and State Transmission,
Quantum Entanglement, Quantum Security 153–170
10.1 Lecture Plan 153
10.2 A problem in Information Theory 155
10.3 Types and Sanov’s Theorem 157
10.4 Quantum Stein’s Theorem 159
10.5 Problems in Statistical Image Processing 161
10.6 A Remark on Quantum State Transmission 164
10.7 An Example of a Cq Channel 165
10.8 Quantum State Transformation Using Entangled States 167
ix