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Bibliographic Details
Main Author:	McMahon, David M. (Author)
Format:	Book
Language:	English
Published:	Hoboken, N.J. Wiley-Interscience 2008
Subjects:	Quantum computers Quantencomputer Lehrbuch
Online Access:	Inhaltsverzeichnis Beschreibung für Leser Beschreibung für Leser Inhaltsverzeichnis
Physical Description:	XVIII, 332 S.
ISBN:	9780470096994

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adam_text	CONTENTS Preface xvii A BRIEF INTRODUCTION TO INFORMATION THEORY 1 Classical Information 1 Information Content in a Signal 2 Entropy and Shannon s Information Theory 3 Probability Basics 7 Example 1.1 8 Solution 8 Exercises 8 QUBITS AND QUANTUM STATES 11 The Qubit 11 Example 2.1 13 Solution 13 Vector Spaces 14 Example 2.2 16 Solution 17 Linear Combinations of Vectors 17 Example 2.3 18 Solution 18 Uniqueness of a Spanning Set 19 Basis and Dimension 20 Inner Products 21 Example 2.4 22 Solution 23 Example 2.5 24 Solution 24 Orthonormality 24 Gram-Schmidt Orthogonalization 26 Example 2.6 26 Solution 26 VI CONTENTS Bra-Ket Formalism 28 Example 2.7 29 Solution 29 The Cauchy-Schwartz and Triangle Inequalities 31 Example 2.8 32 Solution 32 Example 2.9 33 Solution 34 Summary 35 Exercises 36 MATRICES AND OPERATORS 39 Observables 40 The Pauli Operators 40 Outer Products 41 Example 3.1 41 Solution 41 You Try It 42 The Closure Relation 42 Representations of Operators Using Matrices 42 Outer Products and Matrix Representations 43 You Try It 44 Matrix Representation of Operators in Two-Dimensional Spaces 44 Example 3.2 44 Solution 44 You Try It 45 Definition: The Pauli Matrices 45 Example 3.3 45 Solution 45 Hermitian, Unitary, and Normal Operators 46 Example 3.4 47 Solution 47 You Try It 47 Definition: Hermitian Operator 47 Definition: Unitary Operator 48 Definition: Normal Operator 48 Eigenvalues and Eigenvectors 48 The Characteristic Equation 49 Example 3.5 49 Solution 49 You Try It 50 Example 3.6 50 Solution 50 Spectral Decomposition 53 Example 3.7 53 CONTENTS VU Solution 54 The Trace of an Operator 54 Example 3.8 54 Solution 54 Example 3.9 55 Solution 55 Important Properties of the Trace 56 Example 3.10 56 Solution 56 Example 3.11 57 Solution 57 The Expectation Value of an Operator 57 Example 3.12 57 Solution 58 Example 3.13 58 Solution 59 Functions of Operators 59 Unitary Transformations 60 Example 3.14 61 Solution 61 Projection Operators 62 Example 3.15 63 Solution 63 You Try It 63 Example 3.16 65 Solution 65 Positive Operators 66 Commutator Algebra 66 Example 3.17 67 Solution 67 The Heisenberg Uncertainty Principle 68 Polar Decomposition and Singular Values 69 Example 3.18 69 Solution 70 The Postulates of Quantum Mechanics 70 Postulate 1: The State of a System 70 Postulate 2: Observable Quantities Represented by Operators 70 Postulate 3: Measurements 70 Postulate 4: Time Evolution of the System 71 Exercises 71 TENSOR PRODUCTS 73 Representing Composite States in Quantum Mechanics 74 Example 4.1 74 Solution 74 Viii CONTENTS Example 4.2 75 Solution 75 Computing Inner Products 76 Example 4.3 76 Solution 76 You Try It 76 Example 4.4 77 Solution 77 You Try It 77 Example 4.5 77 Solution 77 You Try It 77 Tensor Products of Column Vectors 78 Example 4.6 78 Solution 78 You Try It 78 Operators and Tensor Products 79 Example 4.7 79 Solution 79 You Try It 79 Example 4.8 80 Solution 80 Example 4.9 80 Solution 81 Example 4.10 81 Solution 81 You Try It 82 Example 4.11 82 Solution 82 You Try It 82 Tensor Products of Matrices 83 Example 4.12 83 Solution 83 You Try It 84 Exercises 84 THE DENSITY OPERATOR 85 The Density Operator for a Pure State 86 Definition: Density Operator for a Pure State 87 Definition: Using the Density Operator to Find the Expectation Value 88 Example 5.1 88 Solution 89 You Try It 89 Time Evolution of the Density Operator 90 Definition: Time Evolution of the Density Operator 91 CONTENTS IX The Density Operator for a Mixed State 91 Key Properties of a Density Operator 92 Example 5.2 93 Solution 93 Expectation Values 95 Probability of Obtaining a Given Measurement Result 95 Example 5.3 96 Solution 96 You Try It 96 Example 5.4 96 Solution 97 You Try It 98 You Try It 99 You Try It 99 Characterizing Mixed States 99 Example 5.5 100 Solution 100 Example 5.6 102 Solution 103 You Try It 103 Example 5.7 103 Solution 104 Example 5.8 105 Solution 105 Example 5.9 106 Solution 106 You Try It 108 Probability of Finding an Element of the Ensemble in a Given State 108 Example 5.10 109 Solution 109 Completely Mixed States 111 The Partial Trace and the Reduced Density Operator 111 You Try It 113 Example 5.11 114 Solution 114 The Density Operator and the Bloch Vector 115 Example 5.12 116 Solution 116 Exercises 117 6 QUANTUM MEASUREMENT THEORY 121 Distinguishing Quantum States and Measurement 121 Projective Measurements 123 Example 6.1 125 Solution 126 CONTENTS Example 6.2 128 Solution 129 You Try It 130 Example 6.3 130 Solution 130 Measurements on Composite Systems 132 Example 6.4 132 Solution 132 Example 6.5 133 Solution 134 Example 6.6 135 Solution 135 You Try It 136 Example 6.7 136 Solution 137 You Try It 138 Example 6.8 138 Solution 138 Generalized Measurements 139 Example 6.9 140 Solution 140 Example 6.10 140 Solution 140 Positive Operator-Valued Measures 141 Example 6.11 141 Solution 142 Example 6.12 142 Solution 143 Example 6.13 143 Solution 144 Exercises 145 7 ENTANGLEMENT 147 Bell s Theorem 151 Bipartite Systems and the Bell Basis 155 Example 7.1 157 Solution 157 When Is a State Entangled? 157 Example 7.2 158 Solution 158 Example 7.3 158 Solution 158 Example 7.4 159 Solution 159 You Try It 162 CONTENTS XI You Try It 162 The Pauli Representation 162 Example 7.5 162 Solution 162 Example 7.6 163 Solution 163 Entanglement Fidelity 166 Using Bell States For Density Operator Representation 166 Example 7.7 167 Solution 167 Schmidt Decomposition 168 Example 7.8 168 Solution 168 Example 7.9 169 Solution 169 Purification 169 Exercises 170 8 QUANTUM GATES AND CIRCUITS 173 Classical Logic Gates 173 You Try It 175 Single-Qubit Gates 176 Example 8.1 178 Solution 178 You Try It 179 Example 8.2 179 Solution 180 More Single-Qubit Gates 180 You Try It 181 Example 8.3 181 Solution 181 Example 8.4 182 Solution 182 You Try It 183 Exponentiation 183 Example 8.5 183 Solution 183 You Try It 184 The Z-Y Decomposition 185 Basic Quantum Circuit Diagrams 185 Controlled Gates 186 Example 8.6 187 Solution 188 Example 8.7 188 Solution 188 XII CONTENTS Example 8.8 190 Solution 190 Example 8.9 191 Solution 192 Gate Decomposition 192 Exercises 195 QUANTUM ALGORITHMS 197 Hadamard Gates 198 Example 9.1 200 Solution 201 The Phase Gate 201 Matrix Representation of Serial and Parallel Operations 201 Quantum Interference 202 Quantum Parallelism and Function Evaluation 203 Deutsch-Jozsa Algorithm 207 Example 9.2 208 Solution 208 Example 9.3 209 Solution 209 Quantum Fourier Transform 211 Phase Estimation 213 Shor s Algorithm 216 Quantum Searching and Graver s Algorithm 218 Exercises 221 10 APPLICATIONS OF ENTANGLEMENT: TELEPORTATION AND SUPERDENSE CODING 225 Teleportation 226 Teleportation Step 1 : Alice and Bob Share an Entangled Pair of Particles 226 Teleportation Step 2: Alice Applies a CNOT Gate 226 Teleportation Step 3: Alice Applies a Hadamard Gate 227 Teleportation Step 4: Alice Measures Her Pair 227 Teleportation Step 5: Alice Contacts Bob on a Classical Communi¬ cations Channel and Tells Him Her Measurement Result 228 The Peres Partial Transposition Condition 229 Example 10.1 229 Solution 230 Example 10.2 230 Solution 231 Example 10.3 232 Solution 232 Entanglement Swapping 234 Superdense Coding 236 CONTENTS Xiü Example 10.4 237 Solution 237 Exercises 238 11 QUANTUM CRYPTOGRAPHY 239 A Brief Overview of RSA Encryption 241 Example 11.1 242 Solution 242 Basic Quantum Cryptography 243 Example 11.2 245 Solution 245 An Example Attack: The Controlled NOT Attack 246 The B92 Protocol 247 The E91 Protocol (Ekert) 248 Exercises 249 12 QUANTUM NOISE AND ERROR CORRECTION 251 Single-Qubit Errors 252 Quantum Operations and Krauss Operators 254 Example 1 2.1 255 Solution 255 Example 12.2 257 Solution 257 Example 12.3 259 Solution 259 The Depolarization Channel 260 The Bit Flip and Phase Flip Channels 261 Amplitude Damping 262 Example 12.4 265 Solution 265 Phase Damping 270 Example 12.5 271 Solution 271 Quantum Error Correction 272 Exercises 277 13 TOOLS OF QUANTUM INFORMATION THEORY 279 The No-Cloning Theorem 279 Trace Distance 28 і Example 13.1 282 Solution 282 You Try It 283 Example 13.2 283 xjv CONTENTS Solution 284 Example 13.3 285 Solution 285 Fidelity 286 Example 13.4 287 Solution 288 Example 13.5 289 Solution 289 Example 13.6 289 Solution 289 Example 13.7 290 Solution 290 Entanglement of Formation and Concurrence 291 Example 13.8 291 Solution 292 Example 13.9 293 Solution 293 Example 13.10 294 Solution 294 Example 13.11 295 Solution 295 You Try It 296 Information Content and Entropy 296 Example 13.12 298 Solution 298 Example 13.13 299 Solution 299 Example 13.14 299 Solution 299 Example 13.15 300 Solution 300 Example 13.16 301 Solution 301 Example 13.17 302 Solution 302 Exercises 303 14 ADIABATIC QUANTUM COMPUTATION 305 Example 14.1 307 Solution 307 Adiabatic Processes 308 Example 14.2 308 Solution 309 Adiabatic Quantum Computing 310 Example 14.3 310 CONTENTS XV Solution 310 Exercises 313 15 CLUSTER STATE QUANTUM COMPUTING 315 Cluster States 316 Cluster State Preparation 316 Example 15.1 317 Solution 317 Adjacency Matrices 319 Stabilizer States 320 Aside: Entanglement Witness 322 Cluster State Processing 324 Example 15.2 326 Exercises 326 References 329 Index 331
adam_txt	CONTENTS Preface xvii A BRIEF INTRODUCTION TO INFORMATION THEORY 1 Classical Information 1 Information Content in a Signal 2 Entropy and Shannon's Information Theory 3 Probability Basics 7 Example 1.1 8 Solution 8 Exercises 8 QUBITS AND QUANTUM STATES 11 The Qubit 11 Example 2.1 13 Solution 13 Vector Spaces 14 Example 2.2 16 Solution 17 Linear Combinations of Vectors 17 Example 2.3 18 Solution 18 Uniqueness of a Spanning Set 19 Basis and Dimension 20 Inner Products 21 Example 2.4 22 Solution 23 Example 2.5 24 Solution 24 Orthonormality 24 Gram-Schmidt Orthogonalization 26 Example 2.6 26 Solution 26 VI CONTENTS Bra-Ket Formalism 28 Example 2.7 29 Solution 29 The Cauchy-Schwartz and Triangle Inequalities 31 Example 2.8 32 Solution 32 Example 2.9 33 Solution 34 Summary 35 Exercises 36 MATRICES AND OPERATORS 39 Observables 40 The Pauli Operators 40 Outer Products 41 Example 3.1 41 Solution 41 You Try It 42 The Closure Relation 42 Representations of Operators Using Matrices 42 Outer Products and Matrix Representations 43 You Try It 44 Matrix Representation of Operators in Two-Dimensional Spaces 44 Example 3.2 44 Solution 44 You Try It 45 Definition: The Pauli Matrices 45 Example 3.3 45 Solution 45 Hermitian, Unitary, and Normal Operators 46 Example 3.4 47 Solution 47 You Try It 47 Definition: Hermitian Operator 47 Definition: Unitary Operator 48 Definition: Normal Operator 48 Eigenvalues and Eigenvectors 48 The Characteristic Equation 49 Example 3.5 49 Solution 49 You Try It 50 Example 3.6 50 Solution 50 Spectral Decomposition 53 Example 3.7 53 CONTENTS VU Solution 54 The Trace of an Operator 54 Example 3.8 54 Solution 54 Example 3.9 55 Solution 55 Important Properties of the Trace 56 Example 3.10 56 Solution 56 Example 3.11 57 Solution 57 The Expectation Value of an Operator 57 Example 3.12 57 Solution 58 Example 3.13 58 Solution 59 Functions of Operators 59 Unitary Transformations 60 Example 3.14 61 Solution 61 Projection Operators 62 Example 3.15 63 Solution 63 You Try It 63 Example 3.16 65 Solution 65 Positive Operators 66 Commutator Algebra 66 Example 3.17 67 Solution 67 The Heisenberg Uncertainty Principle 68 Polar Decomposition and Singular Values 69 Example 3.18 69 Solution 70 The Postulates of Quantum Mechanics 70 Postulate 1: The State of a System 70 Postulate 2: Observable Quantities Represented by Operators 70 Postulate 3: Measurements 70 Postulate 4: Time Evolution of the System 71 Exercises 71 TENSOR PRODUCTS 73 Representing Composite States in Quantum Mechanics 74 Example 4.1 74 Solution 74 Viii CONTENTS Example 4.2 75 Solution 75 Computing Inner Products 76 Example 4.3 76 Solution 76 You Try It 76 Example 4.4 77 Solution 77 You Try It 77 Example 4.5 77 Solution 77 You Try It 77 Tensor Products of Column Vectors 78 Example 4.6 78 Solution 78 You Try It 78 Operators and Tensor Products 79 Example 4.7 79 Solution 79 You Try It 79 Example 4.8 80 Solution 80 Example 4.9 80 Solution 81 Example 4.10 81 Solution 81 You Try It 82 Example 4.11 82 Solution 82 You Try It 82 Tensor Products of Matrices 83 Example 4.12 83 Solution 83 You Try It 84 Exercises 84 THE DENSITY OPERATOR 85 The Density Operator for a Pure State 86 Definition: Density Operator for a Pure State 87 Definition: Using the Density Operator to Find the Expectation Value 88 Example 5.1 88 Solution 89 You Try It 89 Time Evolution of the Density Operator 90 Definition: Time Evolution of the Density Operator 91 CONTENTS IX The Density Operator for a Mixed State 91 Key Properties of a Density Operator 92 Example 5.2 93 Solution 93 Expectation Values 95 Probability of Obtaining a Given Measurement Result 95 Example 5.3 96 Solution 96 You Try It 96 Example 5.4 96 Solution 97 You Try It 98 You Try It 99 You Try It 99 Characterizing Mixed States 99 Example 5.5 100 Solution 100 Example 5.6 102 Solution 103 You Try It 103 Example 5.7 103 Solution 104 Example 5.8 105 Solution 105 Example 5.9 106 Solution 106 You Try It 108 Probability of Finding an Element of the Ensemble in a Given State 108 Example 5.10 109 Solution 109 Completely Mixed States 111 The Partial Trace and the Reduced Density Operator 111 You Try It 113 Example 5.11 114 Solution 114 The Density Operator and the Bloch Vector 115 Example 5.12 116 Solution 116 Exercises 117 6 QUANTUM MEASUREMENT THEORY 121 Distinguishing Quantum States and Measurement 121 Projective Measurements 123 Example 6.1 125 Solution 126 CONTENTS Example 6.2 128 Solution 129 You Try It 130 Example 6.3 130 Solution 130 Measurements on Composite Systems 132 Example 6.4 132 Solution 132 Example 6.5 133 Solution 134 Example 6.6 135 Solution 135 You Try It 136 Example 6.7 136 Solution 137 You Try It 138 Example 6.8 138 Solution 138 Generalized Measurements 139 Example 6.9 140 Solution 140 Example 6.10 140 Solution 140 Positive Operator-Valued Measures 141 Example 6.11 141 Solution 142 Example 6.12 142 Solution 143 Example 6.13 143 Solution 144 Exercises 145 7 ENTANGLEMENT 147 Bell's Theorem 151 Bipartite Systems and the Bell Basis 155 Example 7.1 157 Solution 157 When Is a State Entangled? 157 Example 7.2 158 Solution 158 Example 7.3 158 Solution 158 Example 7.4 159 Solution 159 You Try It 162 CONTENTS XI You Try It 162 The Pauli Representation 162 Example 7.5 162 Solution 162 Example 7.6 163 Solution 163 Entanglement Fidelity 166 Using Bell States For Density Operator Representation 166 Example 7.7 167 Solution 167 Schmidt Decomposition 168 Example 7.8 168 Solution 168 Example 7.9 169 Solution 169 Purification 169 Exercises 170 8 QUANTUM GATES AND CIRCUITS 173 Classical Logic Gates 173 You Try It 175 Single-Qubit Gates 176 Example 8.1 178 Solution 178 You Try It 179 Example 8.2 179 Solution 180 More Single-Qubit Gates 180 You Try It 181 Example 8.3 181 Solution 181 Example 8.4 182 Solution 182 You Try It 183 Exponentiation 183 Example 8.5 183 Solution 183 You Try It 184 The Z-Y Decomposition 185 Basic Quantum Circuit Diagrams 185 Controlled Gates 186 Example 8.6 187 Solution 188 Example 8.7 188 Solution 188 XII CONTENTS Example 8.8 190 Solution 190 Example 8.9 191 Solution 192 Gate Decomposition 192 Exercises 195 QUANTUM ALGORITHMS 197 Hadamard Gates 198 Example 9.1 200 Solution 201 The Phase Gate 201 Matrix Representation of Serial and Parallel Operations 201 Quantum Interference 202 Quantum Parallelism and Function Evaluation 203 Deutsch-Jozsa Algorithm 207 Example 9.2 208 Solution 208 Example 9.3 209 Solution 209 Quantum Fourier Transform 211 Phase Estimation 213 Shor's Algorithm 216 Quantum Searching and Graver's Algorithm 218 Exercises 221 10 APPLICATIONS OF ENTANGLEMENT: TELEPORTATION AND SUPERDENSE CODING 225 Teleportation 226 Teleportation Step 1 : Alice and Bob Share an Entangled Pair of Particles 226 Teleportation Step 2: Alice Applies a CNOT Gate 226 Teleportation Step 3: Alice Applies a Hadamard Gate 227 Teleportation Step 4: Alice Measures Her Pair 227 Teleportation Step 5: Alice Contacts Bob on a Classical Communi¬ cations Channel and Tells Him Her Measurement Result 228 The Peres Partial Transposition Condition 229 Example 10.1 229 Solution 230 Example 10.2 230 Solution 231 Example 10.3 232 Solution 232 Entanglement Swapping 234 Superdense Coding 236 CONTENTS Xiü Example 10.4 237 Solution 237 Exercises 238 11 QUANTUM CRYPTOGRAPHY 239 A Brief Overview of RSA Encryption 241 Example 11.1 242 Solution 242 Basic Quantum Cryptography 243 Example 11.2 245 Solution 245 An Example Attack: The Controlled NOT Attack 246 The B92 Protocol 247 The E91 Protocol (Ekert) 248 Exercises 249 12 QUANTUM NOISE AND ERROR CORRECTION 251 Single-Qubit Errors 252 Quantum Operations and Krauss Operators 254 Example 1 2.1 255 Solution 255 Example 12.2 257 Solution 257 Example 12.3 259 Solution 259 The Depolarization Channel 260 The Bit Flip and Phase Flip Channels 261 Amplitude Damping 262 Example 12.4 265 Solution 265 Phase Damping 270 Example 12.5 271 Solution 271 Quantum Error Correction 272 Exercises 277 13 TOOLS OF QUANTUM INFORMATION THEORY 279 The No-Cloning Theorem 279 Trace Distance 28 і Example 13.1 282 Solution 282 You Try It 283 Example 13.2 283 xjv CONTENTS Solution 284 Example 13.3 285 Solution 285 Fidelity 286 Example 13.4 287 Solution 288 Example 13.5 289 Solution 289 Example 13.6 289 Solution 289 Example 13.7 290 Solution 290 Entanglement of Formation and Concurrence 291 Example 13.8 291 Solution 292 Example 13.9 293 Solution 293 Example 13.10 294 Solution 294 Example 13.11 295 Solution 295 You Try It 296 Information Content and Entropy 296 Example 13.12 298 Solution 298 Example 13.13 299 Solution 299 Example 13.14 299 Solution 299 Example 13.15 300 Solution 300 Example 13.16 301 Solution 301 Example 13.17 302 Solution 302 Exercises 303 14 ADIABATIC QUANTUM COMPUTATION 305 Example 14.1 307 Solution 307 Adiabatic Processes 308 Example 14.2 308 Solution 309 Adiabatic Quantum Computing 310 Example 14.3 310 CONTENTS XV Solution 310 Exercises 313 15 CLUSTER STATE QUANTUM COMPUTING 315 Cluster States 316 Cluster State Preparation 316 Example 15.1 317 Solution 317 Adjacency Matrices 319 Stabilizer States 320 Aside: Entanglement Witness 322 Cluster State Processing 324 Example 15.2 326 Exercises 326 References 329 Index 331
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spelling	McMahon, David M. Verfasser aut Quantum computing explained David McMahon Hoboken, N.J. Wiley-Interscience 2008 XVIII, 332 S. txt rdacontent n rdamedia nc rdacarrier Quantum computers Quantencomputer (DE-588)4533372-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantencomputer (DE-588)4533372-5 s DE-604 http://www.loc.gov/catdir/toc/ecip0714/2007013725.html Inhaltsverzeichnis http://www.loc.gov/catdir/enhancements/fy0740/2007013725-b.html Beschreibung für Leser http://www.loc.gov/catdir/enhancements/fy0740/2007013725-d.html Beschreibung für Leser Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016465395&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	McMahon, David M. Quantum computing explained Quantum computers Quantencomputer (DE-588)4533372-5 gnd
subject_GND	(DE-588)4533372-5 (DE-588)4123623-3
title	Quantum computing explained
title_auth	Quantum computing explained
title_exact_search	Quantum computing explained
title_exact_search_txtP	Quantum computing explained
title_full	Quantum computing explained David McMahon
title_fullStr	Quantum computing explained David McMahon
title_full_unstemmed	Quantum computing explained David McMahon
title_short	Quantum computing explained
title_sort	quantum computing explained
topic	Quantum computers Quantencomputer (DE-588)4533372-5 gnd
topic_facet	Quantum computers Quantencomputer Lehrbuch
url	http://www.loc.gov/catdir/toc/ecip0714/2007013725.html http://www.loc.gov/catdir/enhancements/fy0740/2007013725-b.html http://www.loc.gov/catdir/enhancements/fy0740/2007013725-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016465395&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT mcmahondavidm quantumcomputingexplained

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