Classical and Quantum Information Theory for the Physicist

Define the following inner products on the space of complex matrices of appropriate size:Note that the sizes of the matrices in both the cases will generally differ. For example if K is the partial trace operation T r 2 on n C ×n ⊗ m C ×m then in the first case, Y, X will be in n C ×n , while in the second case, they will be in nm C ×nm . Now, let K be the partial trace operation as defined here. Then forThis means that relative to the inner products K(σ) and σ on n C ×n and nm C ×nm respectively, the adjoint of K σ,r is given by K σ, \* r whereThus, when K is theT r 2 operation (partial trace), thenwhere K is the partial trace operation. This means that for any matrix X,)-H(Y S (k)|Y S (j), j ≤ k-1, W T , W T , X S (k), X S (k)) This book deals with certain important problems in Classical and Quantum Information Theory Quantum Information Theory, A Selection of Matrix Inequalities Stochastic Filtering Theory Applied to Electromagnetic Fields and Strings Wigner-distributions in Quantum Mechanics Quantization of Classical Field Theories Statistical Signal Processing Quantum Field Theory, Quantum Statistics, Gravity, Stochastic Fields and Information Problems in Information Theory It will be very helpful for students of Undergraduate and Postgraduate Courses in Electronics, Communication and Signal Processing. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan). This book deals with certain important problems in Classical and Quantum Information Theory and will be very helpful for students of Undergraduate and Postgraduate Courses in Electronics, Communication and Signal Processing. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Brief Contents 6 Table of Contents 8 Preface 12 1. Quantum Information Theory, A Selection of Matrix Inequalities 20 1.1 Monotonicity of Quantum Relative Renyi Entropy 20 1.2 Problems 22 2. Stochastic Filtering Theory Applied to Electromagnetic Fields and Strings 24 2.1 M.Tech Dissertation Topics 24 2.2 Estimating the Time Varying Permittivity and Permeability of a Region of Space Using Nonlinear Stochastic Filtering Theory 24 2.3 Estimating the Time Varying Permittivity and Permeability of a Region of Space Using Nonlinear Stochastic Filtering Theory 25 2.4 Study Project: Reduction of Supersymmetry Breaking by Feedback 31 3. Wigner-distributions in Quantum Mechanics 32 3.1 Quantum Fokker-Planck Equation in theWigner Domain 32 3.2 The Noiseless Quantum Fokker-Planck Equation or Equivalently, the Liouville-Schrodinger- Von-Neumann-equation in the Wigner Domain 36 3.3 Construction of the Quantum Fokker-Planck Equation for a Specific Choice of the Lindblad Operator 38 3.4 Problems in Quantum Corrections to Classical Theories in Probability Theory and in Mechanics with Other Specific Choices of the Lindblad Operator 40 3.5 Belavkin filter for the Wigner Distribution Function 41 3.6 Superstring Coupled to Gravitino Ensures Local Supersymmetry 45 4. Undergraduate and Postgraduate Courses in Electronics, Communication and Signal Processing 46 5. Quantization of Classical Field Theories, Examples 48 5.1 Quantization of Fluid Dynamics in a Curved Space-time Background Using Lagrange Multiplier Functions 48 5.2 d-dimensional Harmonic Oscillator with Electric Field Forcing 50 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 52 5.4 Quantum Filtering for the Gravitational Field Interacting with the Electromagnetic Field 52 5.5 Quantum Filtering for the Gravitational Field Interacting with the Electromagnetic Field 55 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 60 5.7 Quantum Neural Network Using a Single Harmonic Oscillator Perturbed by an Electric Field 62 6. Statistical Signal Processing 66 6.1 Statistical Signal Processing: Long Test 66 6.2 Quantum EKF 69 6.3 Lie Brackets in Quantum Mechanics in Terms of the Wigner Transform of Observables 71 6.4 Simulation of a Class of Markov Processes in Continuous and Discrete Time with Applications to Solving Partial Differential Equations 73 6.5 Gravitational Radiation 73 6.6 Measuring the Gravitational Radiation Using Quantum Mechanical Receivers 81 7. Some More Concepts and Results in Quantum Information Theory 86 7.1 Fidelity Between Two States ρ, σ 86 7.2 An Identity Regarding Fidelity 87 7.3 Adaptive Probability Density Tracking Using the Quantum Master Equation 88 7.4 Quantum Neural Networks Based on Superstring Theory 89 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 92 7.6 Applied Linear Algebra 95 8. Quantum Field Theory, Quantum Statistics, Gravity, Stochastic Fields and Information 100 8.1 Rate Distortion Theory for Ergodic Sources 100 8.2 Problems 105 8.3 Simulation of Time Varying Joint Probability 106 8.4 An application of the Radiatively Corrected Propagator to Quantum Neural Network Theory Densities Using Yang-Mills Gauge Theories 108 8.5 An Experiment Involving the Measurement of Newton’s Gravitational Constant G 110 8.6 Extending the Fluctuation-Dissipation Theorem 111 8.7 A discrete Poisson Collision Approach to Brownian Motion 111 8.8 The Born-Oppenheimer Program 113 8.9 The Superposition Principle for Wave Functions of the Curved Space-time Metric Field Could Lead to Contradictions and what are the Fundamental Difficulties in Developing a Background Independent Theory of Quantum Gravity 115 8.10 Attempts to Detect Gravitational Waves from Rotating Pulsars and Sudden Burst of a Star Using Crystal Detectors 115 8.11 Sketch of the Proof of Shannon’s Coding Theorems 116 8.12 The Notion of a Field Operator or Rather an Operator Valued Field 118 8.13 Group Theoretic Pattern Recognition 121 8.14 Controlling the Probability Distribution in Functional Space of the Klein-Gordon Field Using a Field Dependent Potential 123 8.15 Quantum Processing of Classical Image Fields Using a Classical Neural Network 124 8.16 Entropy and Supersymmetry 124 9. Problems in Information Theory 128 9.1 Problems in Quantum Neural Networks 158 9.2 MATLAB Simulation Exercises in Statistical Signal Processing 160 9.3 Problems in Information Theory 162 9.4 Problems in Quantum Neural Networks 164 9.5 Quantum Gaussian States and Their Transformations 165 10. Lecture Plan for Information Theory, Sanov’s Theorem, Quantum Hypothesis Testing and State Transmission, Quantum Entanglement, Quantum Security 172 10.1 Lecture Plan 172 10.2 A problem in Information Theory 174 10.3 Types and Sanov’s Theorem 176 10.4 Quantum Stein’s Theorem 178 10.5 Problems in Statistical Image Processing 180 10.6 A Remark on Quantum State Transmission 183 10.7 An Example of a Cq Channel 184 10.8 Quantum State Transformation Using Entangled States 186 10.9 Generation of Entangled States from Tensor Product States 187 10.10 Security in Quantum Communication from Eavesdroppers 187 10.11 Abstract on Internet of Things in Electromagnetics and Quantum Mechanics 188 11. More Problems in Classical and Quantum Information Theory 190 11.1 Problems 190 11.2 Examples of Cq data Transmission 202 12. Information Transmission and Compression with Distortion, Ergodic Theorem, Quantum Blackhole Physics 208 12.1 Examples of Cq data Transmission 208 12.2 The Shannon-Mcmillan-Breiman Theorem 210 12.3 Entropy Pumped by the Bath into a Quantum System as Measured by An Observer Making Noisy Non-demolition Measurements 212 12.4 Prove the Joint Convexity of the Relative Entropy Between two Probability Distributions Along the Following Steps 215 12.5 Quantum Blackhole Physics and the Amount of Information Pumped by the Quantum Gravitating Blackhole Into a System of Other Elementary Particles 216 12.6 Direct Part of the Capacity Theorem for Relay Channels 217 12.7 An Entropy Inequality 221 12.8 Entropy Pumped by a Random Electromagnetic Field and Bath Noise Into an Electron 221 12.9 Some Problems in the Detection and Transmission of Electromagnetic Signals and Image Fields Using Quantum Communication Techniques 222 12.10 The Degraded Broadcast Channel 227 12.11 Rate Distortion with Side Information 229 12.12 Proof of the Stein Theorem in Classical Hypothesis Testing 232 12.3 Source Coding with Side Information 234 12.14 Some Problems on Random Segmentation of Image Fields 236 12.15 The Shannon Code 240 12.16 Some Control Problems Involving the Theory of Large Deviations 243 13. Examination Problems in Classical Information Theory 246 13.1 Converse Part of the Achievability Result for a Multiterminal Network 255 13.2 More Examination Problems in Information Theory 257 classical,information,theory;,electromagnetic;,fields;,ergodic,theorem;,matrix,inequalities;,neural,network;,quantum,blackhole,physics;,quantum,entanglement;,quantum,field,theory;,quantum,gravity;,quantum,hypothesis,testing;,quantum,information,theory;,quantum,mechanics;,quantum,security;,quantum,statistics;,Sanov’s,theorem;,state,transmission;,statistical,signal,processing;,stochastic,filtering,theory;,Wigner-distributions classical information theory,electromagnetic,fields,ergodic theorem,matrix inequalities,neural network,quantum blackhole physics,quantum entanglement,quantum field theory,quantum gravity,quantum hypothesis testing,quantum information theory,quantum mechanics,quantum security,quantum statistics,Sanov’s theorem,state transmission,statistical signal processing,stochastic filtering theory,Wigner-distributions