Quantum Information

Quantum information processing, a vibrant field of research at the intersection of physics, mathematics and computer science, has seen tremendous progress over the last couple of years, both with respect to the experimental realization of stable qubits (the quantum analog to bits in conventional computing) and the fundamental understanding, development and implementation of quantum-based algorithms. Consequently, proof-of-concept studies have unambiguously shown that quantum computers are possible and have the potential to dramatically exceed the capabilities of today's conventional computers. This comprehensive handbook on the rapidly advancing field presents the foundations of quantum information, taking into account the current state of research and development, and the emerging and already realized applications in quantum technology. It thus covers all current concepts in quantum computing, both theoretical and experimental, before moving on to the latest implementations of quantum computing and communication protocols. It is a must-have for experimental and theoretical physicists in academia and industry as well as for materials scientists, engineers and computer scientists who work in the field of quantum information.

Autorentext
Dagmar Bru? graduated at RWTH University Aachen, Germany, and received her PhD in theoretical particle physics from the University of Heidelberg in 1994. As a research fellow at the University of Oxford she became interested in quantum information. Another European fellowship at ISI Torino, Italy, followed. While being a research assistant at the University of Hannover she completed her habilitation. Since 2004 Professor Bru? has been holding a chair at the Institute of Theoretical Physics at Heinrich-Heine-University Dusseldorf, Germany. Her research pertains to theoretical aspects of quantum information processing.

Gerd Leuchs studied physics and mathematics at the University of Cologne, Germany, and received his Ph.D. in 1978. After two years at the University of Colorado in Boulder, USA, he headed the German gravitational wave detection group from 1985 to 1989. He became technical director at Nanomach AG in Switzerland. Since 1994 Professor Leuchs has been holding the chair for optics at the University of Erlangen-Nuremberg, Germany. In 2009 he was a founding director of the Max Planck Institute for the Science of Light. He is visiting professor at the University of Ottawa. His fields of research span the range from modern aspects of classical optics to quantum optics and quantum information.

Klappentext

This comprehensive textbook on the rapidly advancing field introduces readers to the fundamental concepts of information theory and quantum entanglement, taking into account the current state of research and development. It thus covers all current concepts in quantum computing, both theoretical and experimental, before moving on to the latest implementations of quantum computing and communication protocols. It contains problems and exercises and is therefore ideally suited for students and lecturers in physics and informatics, as well as experimental and theoretical physicists in academia and industry who work in the field of quantum information processing.

The second edition incorporates important recent developments such as quantum metrology, quantum correlations beyond entanglement, and advances in quantum computing with solid state devices.

Inhalt

Preface to the New Edition xvii

Preface to Lectures on Quantum Information (2006) xix

Part I Classical Information Theory 1

**1 Classical Information Theory and Classical Error Correction 3
**Markus Grassl

1.1 Introduction 3

1.2 Basics of Classical Information Theory 3

1.3 Linear Block Codes 10

1.4 Further Aspects 16

References 16

**2 Computational Complexity 19
**Stephan Mertens

2.1 Basics 19

2.2 Algorithms and Time Complexity 21

2.3 Tractable Trails: The Class P 22

2.4 Intractable Itineraries: The Class NP 24

2.5 Reductions and NP-Completeness 29

2.6 P Versus NP 31

2.7 Optimization 34

2.8 Complexity Zoo 37

References 37

Part II Foundations of Quantum Information Theory 39

**3 Discrete Quantum States versus Continuous Variables 41
**Jens Eisert

3.1 Introduction 41

3.2 Finite-Dimensional Quantum Systems 42

3.3 Continuous-Variables 45

References 53

**4 Approximate Quantum Cloning 55
**Dagmar Bruß and Chiara Macchiavello

4.1 Introduction 55

4.2 The No-Cloning Theorem 56

4.3 State-Dependent Cloning 57

4.4 Phase-Covariant Cloning 63

4.5 Universal Cloning 65

4.6 Asymmetric Cloning 69

4.7 Probabilistic Cloning 70

4.8 Experimental Quantum Cloning 70

4.9 Summary and Outlook 71

Exercises 72

References 73

**5 Channels and Maps 75
**M. Keyl and R. F.Werner

5.1 Introduction 75

5.2 Completely Positive Maps 75

5.3 The ChoiJamiolkowski Isomorphism 78

5.4 The Stinespring Dilation Theorem 80

5.5 Classical Systems as a Special Case 83

5.6 Channels with Memory 84

5.7 Examples 86

Problems 89

References 90

**6 Quantum Algorithms 91
**Julia Kempe

6.1 Introduction 91

6.2 Precursors 93

6.3 Shor's Factoring Algorithm 97

6.4 Grover's Algorithm 100

6.5 Other Algorithms 101

6.6 Recent Developments 103

Exercises 105

References 106

**7 Quantum Error Correction 111
**Markus Grassl

7.1 Introduction 111

7.2 Quantum Channels 111

7.3 Using Classical Error-Correcting Codes 115

7.4 Further Aspects 124

References 124

Part III Theory of Entanglement 127

**8 The Separability versus Entanglement Problem 129
**Sreetama Das, Titas Chanda,Maciej Lewenstein, Anna Sanpera, Aditi Sen De, and Ujjwal Sen

8.1 Introduction 129

8.2 Bipartite Pure States: Schmidt Decomposition 130

8.3 Bipartite Mixed States: Separable and Entangled States 131

8.4 Operational Entanglement Criteria 132

8.5 Non-operational Entanglement Criteria 141

8.5.1 Technical Preface 141

8.6 Bell Inequalities 149

8.7 Quantification of Entanglement 152

8.8 Classification of Bipartite States with Respect to Quantum Dense Coding 158

8.9 Multipartite States 162

Exercises 167

Acknowledgments 168

References 169

**9 Quantum Discord and Nonclassical Correlations Beyond Entanglement 175
**Gerardo Adesso, Marco Cianciaruso, and Thomas R. Bromley

9.1 Introduction 175

9.2 Quantumness Versus Classicality (of Correlations) 176

9.3 Quantifying Quantum Correlations Quantum Discord 180

9.4 Interpreting Quantum Correlations Local Broadcasting 184 9.5 Alternative Characterizations of Quantum Correlations 186<...