Quantum Information, 2 Volume Set

Dagmar Bruß 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 Bruß has been holding a chair at the Institute of Theoretical Physics at Heinrich-Heine-University Düsseldorf, 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.

• Preface to the New Edition xviiPreface to Lectures on Quantum Information (2006) xixPart I Classical Information Theory 11 Classical Information Theory and Classical Error Correction 3Markus Grassl1.1 Introduction 31.2 Basics of Classical Information Theory 31.3 Linear Block Codes 101.4 Further Aspects 16References 162 Computational Complexity 19Stephan Mertens2.1 Basics 192.2 Algorithms and Time Complexity 212.3 Tractable Trails: The Class P 222.4 Intractable Itineraries: The Class NP 242.5 Reductions and NP-Completeness 292.6 P Versus NP 312.7 Optimization 342.8 Complexity Zoo 37References 37Part II Foundations of Quantum Information Theory 393 Discrete Quantum States versus Continuous Variables 41Jens Eisert3.1 Introduction 413.2 Finite-Dimensional Quantum Systems 423.3 Continuous-Variables 45References 534 Approximate Quantum Cloning 55Dagmar Bruß and Chiara Macchiavello4.1 Introduction 554.2 The No-Cloning Theorem 564.3 State-Dependent Cloning 574.4 Phase-Covariant Cloning 634.5 Universal Cloning 654.6 Asymmetric Cloning 694.7 Probabilistic Cloning 704.8 Experimental Quantum Cloning 704.9 Summary and Outlook 71Exercises 72References 735 Channels and Maps 75M. Keyl and R. F.Werner5.1 Introduction 755.2 Completely Positive Maps 755.3 The Choi–Jamiolkowski Isomorphism 785.4 The Stinespring Dilation Theorem 805.5 Classical Systems as a Special Case 835.6 Channels with Memory 845.7 Examples 86Problems 89References 906 Quantum Algorithms 91Julia Kempe6.1 Introduction 916.2 Precursors 936.3 Shor’s Factoring Algorithm 976.4 Grover’s Algorithm 1006.5 Other Algorithms 1016.6 Recent Developments 103Exercises 105References 1067 Quantum Error Correction 111Markus Grassl7.1 Introduction 1117.2 Quantum Channels 1117.3 Using Classical Error-Correcting Codes 1157.4 Further Aspects 124References 124Part III Theory of Entanglement 1278 The Separability versus Entanglement Problem 129Sreetama Das, Titas Chanda,Maciej Lewenstein, Anna Sanpera, Aditi Sen De, and Ujjwal Sen8.1 Introduction 1298.2 Bipartite Pure States: Schmidt Decomposition 1308.3 Bipartite Mixed States: Separable and Entangled States 1318.4 Operational Entanglement Criteria 1328.5 Non-operational Entanglement Criteria 1418.5.1 Technical Preface 1418.6 Bell Inequalities 1498.7 Quantification of Entanglement 1528.8 Classification of Bipartite States with Respect to Quantum Dense Coding 1588.9 Multipartite States 162Exercises 167Acknowledgments 168References 1699 Quantum Discord and Nonclassical Correlations Beyond Entanglement 175Gerardo Adesso, Marco Cianciaruso, and Thomas R. Bromley9.1 Introduction 1759.2 Quantumness Versus Classicality (of Correlations) 1769.3 Quantifying Quantum Correlations – Quantum Discord 1809.4 Interpreting Quantum Correlations – Local Broadcasting 1849.5 Alternative Characterizations of Quantum Correlations 1869.6 General Desiderata for Measures of Quantum Correlations 1909.7 Outlook 191Exercises 191References 19210 Entanglement Theory with Continuous Variables 195Peter van Loock and Evgeny Shchukin10.1 Introduction 19510.2 Phase-Space Description 19710.3 Entanglement of Gaussian States 19710.4 More on Gaussian Entanglement 209Exercises 211References 21211 Entanglement Measures 215Martin B. Plenio and Shashank S. Virmani11.1 Introduction 21511.2 Manipulation of Single Systems 21711.3 Manipulation in the Asymptotic Limit 21811.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems 22111.5 Examples of Axiomatic Entanglement Measures 224Acknowledgments 228References 22812 Purification and Distillation 231Wolfgang Dür and Hans-J. Briegel12.1 Introduction 23112.2 Pure States 23312.3 Distillability and Bound Entanglement in Bipartite Systems 23512.4 Bipartite Entanglement Distillation Protocols 23912.5 Distillability and Bound Entanglement in Multipartite Systems 24712.6 Entanglement Purification Protocols in Multipartite Systems 24812.7 Distillability with Noisy Apparatus 25212.8 Applications of Entanglement Purification 25712.9 Summary and Conclusions 260Acknowledgments 261References 26113 Bound Entanglement 265Paweł Horodecki13.1 Introduction 26513.2 Distillation of Quantum Entanglement: Repetition 26513.3 Bound Entanglement – Bipartite Case 26913.4 Bound Entanglement: Multipartite Case 28213.5 Further Reading: Continuous Variables 287Exercises 287References 28814 Multipartite Entanglement 293Michael Walter, David Gross, and Jens Eisert14.1 Introduction 29314.2 General Theory 29414.3 Important Classes of Multipartite states 31014.4 Specialized Topics 316Acknowledgments 321References 321Part IV Quantum Communication 33115 Quantum Teleportation 333Natalia Korolkova15.1 Introduction 33315.2 Quantum Teleportation Protocol 33415.3 Implementations 340References 34916 Theory of Quantum Key Distribution (QKD) 353Norbert Lütkenhaus16.1 Introduction 35316.2 Classical Background to QKD 35316.3 Ideal QKD 35416.4 Idealized QKD in Noisy Environment 35716.5 Realistic QKD in Noisy and Lossy Environment 36016.6 Improved Schemes 36316.7 Improvements in Public Discussion 36416.8 Conclusion 365References 36517 Quantum Communication Experiments with Discrete Variables 369Harald Weinfurter17.1 Aunt Martha 36917.2 Quantum Cryptography 36917.3 Entanglement-Based Quantum Communication 37517.4 Conclusion 379References 37918 Continuous Variable Quantum Communication with Gaussian States 383Ulrik L. Andersen and Gerd Leuchs18.1 Introduction 38318.2 Continuous-Variable Quantum Systems 38418.3 Tools for State Manipulation 38618.4 Quantum Communication Protocols 391Exercises 397References 397Part V Quantum Computing: Concepts 40119 Requirements for a Quantum Computer 403Artur Ekert and Alastair Kay19.1 Classical World of Bits and Probabilities 40319.2 Logically Impossible Operations? 40819.3 Quantum World of Probability Amplitudes 41019.4 Interference Revisited 41419.5 Tools of the Trade 41619.6 Composite Systems 42219.7 Quantum Circuits 42819.8 Summary 433Exercises 43320 Probabilistic Quantum Computation and Linear Optical Realizations 437Norbert Lütkenhaus20.1 Introduction 43720.2 Gottesman/Chuang Trick 43820.3 Optical Background 43920.4 Knill–Laflamme–Milburn (KLM) Scheme 441References 44621 One-Way Quantum Computation 449Dan Browne and Hans Briegel21.1 Introduction 44921.2 Simple Examples 45121.3 Beyond Quantum Circuit Simulation 45521.4 Implementations 46521.5 Recent Developments 46621.6 Outlook 469Acknowledgments 469Exercises 469References 47022 Holonomic Quantum Computation 475Angelo C. M. Carollo and Vlatko Vedral22.1 Geometric Phase and Holonomy 47522.2 Application to Quantum Computation 479References 480Part VI Quantum Computing: Implementations 48323 Quantum Computing with Cold Ions and Atoms: Theory 485Dieter Jaksch, Juan José García-Ripoll, Juan Ignacio Cirac, and Peter Zoller23.1 Introduction 48523.2 Trapped Ions 48523.3 Trapped Neutral Atoms 495References 51524 Quantum Computing Experiments with Cold Trapped Ions 519Ferdinand Schmidt-Kaler and Ulrich Poschinger24.1 Introduction to Trapped-Ion Quantum Computing 51924.2 Paul Traps 52224.3 Ion Crystals and Normal Modes 52624.4 Trap Technology 529Acknowledgements 547References 54725 Quantum Computing with Solid-State Systems 553Guido Burkard and Daniel Loss25.1 Introduction 55325.2 Concepts 55425.3 Electron Spin Qubits 56325.4 Superconducting Qubits 575References 58326 Time-Multiplexed Networks for Quantum Optics 587Sonja Barkhofen, Linda Sansoni and Christine Silberhorn26.1 Introduction 58726.2 Multiplexing 58826.3 Photon-Number-Resolving Detection with Time Multiplexing 58926.4 Quantum Walks in Time 59226.5 Conclusion 600References 60127 A Brief on Quantum Systems Theory and Control Engineering 607Thomas Schulte-Herbrüggen, Robert Zeier,Michael Keyl, and Gunther Dirr27.1 Introduction 60727.2 Systems Theory of Closed Quantum Systems 60927.3 Toward a Systems Theory for Open Quantum Systems 62027.4 Relation to Numerical Optimal Control 62427.5 Outlook on Infinite-Dimensional Systems 62627.6 Conclusion 633Acknowledgments 633Exercises 634References 63528 Quantum Computing Implemented via Optimal Control: Application to Spin and Pseudospin Systems 643Thomas Schulte-Herbrüggen, Andreas Spörl, Raimund Marx, Navin Khaneja, JohnMyers, Amr Fahmy, Samuel Lomonaco, Louis Kauffman, and Steffen Glaser28.1 Introduction 64328.2 From Controllable Spin Systems to Suitable Molecules 64528.3 Scalability 64728.4 Algorithmic Platform for Quantum Control Systems 64928.5 Applied Quantum Control 65128.6 Worked Example: Unitary Controls for Classifying Knots by NMR 65628.7 Conclusions 661Acknowledgments 662Exercises 662References 663Part VII Quantum Interfaces and Memories 66929 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons 671Jean-Michel Raimond and Gerhard Rempe29.1 Introduction 67129.2 Microwave Cavity Quantum Electrodynamics 67229.3 Optical Cavity Quantum Electrodynamics 67729.4 Conclusions and Outlook 683References 68430 Quantum Repeater 691Wolfgang Dür, Hans-J. Briegel, Peter Zoller, and Peter v Loock30.1 Introduction 69130.2 Concept of the Quantum Repeater 69330.3 Proposals for Experimental Realization 69730.4 Summary and Conclusions 699Acknowledgments 699References 69931 Quantum Interface Between Light and Atomic Ensembles 701Eugene S. Polzik and Jaromír Fiurášek31.1 Introduction 70131.2 Off-Resonant Interaction of Light with Atomic Ensemble 70231.3 Entanglement of Two Atomic Clouds 71131.4 Quantum Memory for Light 71231.5 Multiple Passage Protocols 71531.6 Atoms-Light Teleportation and Entanglement Swapping 71831.7 Quantum Cloning into Atomic Memory 72031.8 Summary 721Acknowledgment 721References 72132 Echo-Based Quantum Memory 723G. T. Campbell, K. R. Ferguson, M. J. Sellars, B. C. Buchler, and P. K. Lam32.1 Overview of Photon Echo Techniques 72432.2 Platforms for Echo-Based Quantum Memory 72832.3 Characterization 73132.4 Demonstrations 73432.5 Outlook 736References 73733 Quantum Electrodynamics of a Qubit 741Gernot Alber and Georgios M. Nikolopoulos33.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity 74233.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal 750Exercises 755References 75634 Elementary Multiphoton Processes in Multimode Scenarios 759Nils Trautmann and Gernot Alber34.1 A Generic Quantum Electrodynamical Model 76134.2 The Multiphoton Path Representation 76134.3 Examples 76734.4 Conclusion 772Appendix A: Evaluation of the Field Commutator 773References 774Part VIII Towards Quantum Technology Applications 77735 Quantum Interferometry with Gaussian States 779Ulrik L. Andersen, Oliver Glöckl, Tobias Gehring, and Gerd Leuchs35.1 Introduction 77935.2 The Interferometer 78035.3 Interferometer with Coherent States of Light 78335.4 Interferometer with Squeezed States of Light 78635.5 Fundamental Limits 79235.6 Summary and Discussion 793Problems 795References 79636 Quantum Logic-Enabled Spectroscopy 799Piet O. Schmidt36.1 Introduction 79936.2 Trapping and Doppler Cooling of a Two-Ion Crystal 80036.3 Coherent Atom–Light Interaction and State Manipulation 80236.4 Quantum Logic Spectroscopy for Optical Clocks 80536.5 Photon Recoil Spectroscopy 80936.6 Quantum Logic with Molecular Ions 81536.7 Nonclassical States for Spectroscopy 81936.8 Future Directions 821Acknowledgments 822References 82237 Quantum Imaging 827Claude Fabre and Nicolas Treps37.1 Introduction 82737.2 The Quantum Laser Pointer 82837.3 Manipulation of Spatial Quantum Noise 83037.4 Two-Photon Imaging 83237.5 Other Topics in Quantum Imaging 83337.6 Conclusion and Perspectives 834Acknowledgment 835References 83538 Quantum Frequency Combs 837Claude Fabre and Nicolas Treps38.1 Introduction 83738.2 Parametric Down Conversion of a Frequency Comb 83938.3 Experiment 84038.4 Experimental Results 84338.5 Application to Quantum Information Processing 84938.6 Application to Quantum Metrology 85338.7 Conclusion 854Acknowledgment 855References 855Index 859