Quantum theory is the soul of theoretical physics. It is not just a theory of specific physical systems, but rather a new framework with universal applicability. This book shows how we can reconstruct the theory from six information-theoretical principles, by rebuilding the quantum rules from the bottom up. Step by step, the reader will learn how to master the counterintuitive aspects of the quantum world, and how to efficiently reconstruct quantum information protocols from first principles. Using intuitive graphical notation to represent equations, and with shorter and more efficient derivations, the theory can be understood and assimilated with exceptional ease. Offering a radically new perspective on the field, the book contains an efficient course of quantum theory and quantum information for undergraduates. It is aimed at researchers, professionals, and students in physics, computer science and philosophy, as well as the curious outsider seeking a deeper understanding of the theory.

Preface xiii

Acknowledgments xv

1 Introduction 1

1.1 The Quest for Principles: von Neumann 2

1.2 Quantum Information Resurrects the Quest 3

1.3 Quantum Theory as an OPT 4

1.4 The Principles 5

Part I The Status Quo 9

2 Quantum Theory from hilbert Spaces 11

2.1 Primitive notions 12

2.2 Hilbert-space Postulates for Quantum Theory 14

2.3 Density Matrices and POVMs 16

2.4 Causality, Convex Structure, Discriminability 20

2.5 Quantum States 25

2.6 Entangled Quantum States and effects 30

2.7 Compression 33

2.8 Quantum Transformations 35

2.9 Classical Theory as a Restriction of Quantum 49

2.10 Purification 50

2.11 Quantum No Cloning 59

2.12 The von Neumann Postulate: Do We Need It? 61

2.13 Quantum Teleportation 63

2.14 Inverting Transformations 66

2.15 Summary 73

Notes 73

Appendix 2. 1 Polar Decomposition 78

Appendix 2.2 The Golden Rule for Quantum Extensions 79

Problems 80

Solutions to Selected Problems and Exercises 85

Part II The Informational Approach 109

3 The Framework 111

3.1 The Operational Language 111

3.2 Operational Probabilistic Theory 115

3.3 States and Effects 116

3.4 Transformations 118

3.5 Coarse-graining and Refinement 119

3.6 Operational Distance Between States 120

3.7 Operational Distances for Transformations and Effects 122

3.8 Summary 124

Notes 124

Problems 126

Solutions to Selected Problems and Exercises 126

4 The New Principles 130

4.1 Atomicity of Composition 130

4.2 Perfect Discriminability 132

4.3 Ideal Compression 133

4.4 A Preview of the Three Main Principles 135

4.5 Summary 137

Notes 138

5 Causal Theories 139

5.1 Causality: From Cinderella to Principle 139

5.2 No Signaling from the Future 141

5.3 Conditioning 142

5.4 A Unique Wastebasket 143

5.5 No Signaling at a Distance 150

5.6 Causality and Space-Time 151

5.7 Theories without Causality 151

5.8 Summary 154

Notes 155

Solutions to Selected Problems and Exercises 156

6 Theories with Local Discriminability 157

6.1 Entanglement and Holism 157

6.2 The Principle 159

6.3 Reconciling Holism with Reductionism 160

6.4 Consequences of Local Discriminability 163

6.5 Different Degrees of Holism 163

6.6 Summary 165

Notes 165

Problems 166

Solutions to Selected Problems and Exercises 167

7 The purification Principle 168

7.1 A Distinctive and Fundamental Trait 168

7.2 The Purification Principle 171

7.3 Entanglement 172

7.4 Reversible Transformations and Twirling 173

7.5 Steering 175

7.6 Process Tomography 176

7.7 No Information Without Disturbance 177

7.8 Teleportation 178

7.9 A Reversible Picture of an Irreversible World 179

7.10 Displacing the Von Neumanns Cut 181

7.11 The State-transformation Isomorphism 182

7.12 Everything Not Forbidden is Allowed 184

7. 13 Purification in a nutshell 186

7.14 Summary 188

Notes 188

Appendix 7. 1 Caratheodory's Theorem 189

Solutions to Selected Problems and Exercises 190

Part Ill Quantum Information Without Hilbert Spaces 191

8 Encoding Information 193

8.1 Processing Data= Processing Entanglement 193

8.2 Ideal Encodings 195

8.3 Ideal Compression 198

8.4 The Minimal Purification 199

8.5 Sending Information Through a Noisy Channel 200

8.6 The Condition for Error Correction 201

8.7 Summary 204

Solutions to Selected Problems and exercises 204

9 Three No-go Theorems 206

9.1 No Cloning 207

9.2 No Programming 209

9.3 No Bit Commitment 211

9. 4 Summary 215

Notes 215

Solutions to Selected Problems and Exercises 217

10 Perfectly Discriminable States 218

10.1 Perfect Discriminability of States 218

10.2 No Disturbance Without Information 219

10.3 Perfect Discriminability Implies No Disturbance 222

10.4 Orthogonality 225

10.5 Maximal Sets of Perfectly Discriminable States 226

10.6 Summary 227

Solutions to Selected Problems and Exercises 228

11 Identifying Pure States 229

11.1 The State Identification Task 229

11.2 Only One Pure State for Each Atomic Effect 230

11.3 Every Pure State can be Identified 231

11. 4 For a Pure State, Only One Atomic Effect 232

11.5 The Dagger 233

11.6 Transposing States 235

11.7 Transposing Effects 237

11.8 Playing with Transposition 239

11.9 Summary 241

12 Diagonalization 242

12.1 Conjugate Systems and Conjugate States 242

12.2 A Most Fundamental Result 244

12. 3 The Informational Dimension 246

12.4 The Informational Dimension of a Face 247

12.5 Diagonalizing States 248

12.6 Diagonalizing effects 251

12.7 Operational Versions of the Spectral Theorem 252

12.8 Operational Version of the Schmidt Decomposition 253

12.9 Summary 255

Solutions to selected Problems and Exercises 255

Part IV Quantum Theory from the Principles 257

13 Condusive Teleportation 259

13.1 The Task 259

13.2 The Causality Bound 261

13.3 Achieving the Causality bound 263

13.4 The Local Discriminability Bound 264

13.5 Achieving the Local Discriminability bound 266

13.6 The Origin of the hilbert space 267

13.7 Isotropic States and Effects 268

13.8 Summary 272

Appendix 13.1 Unitary and Orthogonal Representations 272

14 The Qubit 274

14.1 Two-dimensional Systems 274

14.2 Summary 278

Solutions to Selected Problems and Exercises 278

15 Projections 280

15.1 Orthogonal Complements 280

15.2 Orthogonal Faces 282

15.3 Projections 285

15. 4 Projection of a Pure State on Two Orthogonal Faces 292

15.5 Summary 296

Solutions to Selected Problems and Exercises 296

16 The Superposition Principle 297

16.1 The Superposition Principle 297

16.2 Completeness for Purification 298

16.3 Equivalence of Systems with Equal Dimension 299

16.4 Reversible Operations of Perfectly Discriminable Pure States 299

16.5 Summary 300

17 Derivation of Quantum Theory 301

17. 1 The basis 301

17.2 Matrix Representation of States and Effects 304

17.3 Representation of Two-qubit systems 310

17.4 Positive matrices 320

17.5 Quantum Theory in Finite Dimensions 324

17.6 Summary 327

Solutions to Selected Problems and Exercises 327

References 329

Index 338

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