Lectures on Quantum Information (Paperback)

Quantum Information Processing is a young and rapidly growing field of research at the intersection of physics, mathematics, and computer science. Its ultimate goal is to harness quantum physics to conceive—and ultimately build—"quantum" computers that would dramatically overtake the capabilities of today's "classical" computers. One example of the power of a quantum computer is its ability to efficiently find the prime factors of a larger integer, thus shaking the supposedly secure foundations of standard encryption schemes.

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. With its series of exercises, this is ideal reading for students and lecturers in physics and informatics, as well as experimental and theoretical physicists, and physicists in industry.

 

Preface.

List of Contributors.

I Classical Information Theory.

1 Classical Information Theory and Classical Error Correction (M. Grassl).

1.1 Introduction.

1.2 Basics of Classical Information Theory.

1.3 Linear Block Codes.

1.4 Further Aspects.

References.

2 Computational Complexity (S. Mertens).

2.1 Basics.

2.2 Algorithms and Time Complexity.

2.3 Tractable Trails: The Class P.

2.4 Intractable Itineraries: The class NP.

2.5 Reductions and NP-completeness.

2.6 P vs. NP.

2.7 Optimization.

2.8 Complexity Zoo.

References.

II Foundation of Quantum Information Theory.

3 Discrete Quantum States versus Continuous Variables (J. Eisert).

3.1 Introduction.

3.2 Finite-dimensional quantum systems.

3.3 Continuous-variables.

References.

4 Approximate Quantum Cloning (D. Bruß and C. Macchiavello).

4.1 Introduction.

4.2 The No-Cloning Theorem.

4.3 State-Dependent Cloning.

4.4 Phase Covariant Cloning.

4.5 Universal Cloning.

4.6 Asymmetric Cloning.

4.7 Probabilistic Cloning.

4.8 Experimental Quantum Cloning.

4.9 Summary and Outlook.

References.

5 Channels and Maps (M. Keyl and R. F. Werner).

5.1 Introduction.

5.2 Completely Positive Maps.

5.3 The Jamiolkowski Isomorphism.

5.4 The Stinespring Dilation Theorem.

5.5 Classical Systems as a Special Case.

5.6 Examples.

References.

6 Quantum Algorithms (J. Kempe).

6.1 Introduction.

6.2 Precursors.

6.3 Shor’s Factoring Algorithm.

6.4 Grover’s Algorithm.

6.5 Other Algorithms.

6.6 Recent Developments.

References.

7 Quantum Error Correction (M. Grassl).

7.1 Introduction.

7.2 Quantum Channels.

7.3 Using Classical Error-Correcting Codes.

7.4 Further Aspects.

References.

III Theory of Entanglement.

8 The Separability versus Entanglement Problem (A. Sen(De), U. Sen, M. Lewenstein, and A. Sanpera).

8.1 Introduction.

8.2 Bipartite Pure States: Schmidt Decomposition.

8.3 Bipartite Mixed States: Separable and Entangled States.

8.4 Operational Entanglement Criteria.

8.5 Nonoperational Entanglement Criteria.

8.6 Bell Inequalities.

8.7 Classification of Bipartite States with Respect to Quantum Dense Coding.

8.8 Further Reading: Multipartite States.

References.

9 Entanglement Theory with Continuous Variables (P. van Loock).

9.1 Introduction.

9.2 Phase-Space Description.

9.3 Entanglement of Gaussian States.

9.4 More on Gaussian Entanglement.

References.

10 Entanglement Measures (M. B. Plenio and S. S. Virmani).

10.1 Introduction.

10.2 Manipulation of Single Systems.

10.3 Manipulation in the Asymptotic Limit.

10.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems.

10.5 Examples of Axiomatic Entanglement Measures.

References.

11 Purification and Distillation (W. Dür and H.-J. Briegel).

11.1 Introduction.

11.2 Pure States.

11.3 Distillability and Bound Entanglement in Bipartite Systems.

11.4 Bipartite Entanglement Distillation Protocols.

11.5 Distillability and Bound Entanglement in Multipartite systems.

11.6 Entanglement Purification Protocols in Multipartite Systems.

11.7 Distillability with Noisy Apparatus.

11.8 Applications of Entanglement Purification.

11.9 Summary and Conclusions.

References.

12 Bound Entanglement (Pawe³ Horodecki).

12.1 Introduction.

12.2 Distillation of Quantum Entanglement: Repetition.

12.3 Bound Entanglement—Bipartite Case.

12.4 Bound Entanglement: Multipartite Case.

12.5 Further Reading: Continuous Variables.

References.

13 Multiparticle Entanglement (J. Eisert and D. Gross).

13.1 Introduction.

13.2 Pure States.

13.3 Mixed States.

13.4 Quantifying Multiparticle Entanglement.

13.5 Stabilizer States and Graph States.

13.6 Applications of Multiparticle Entangled States.

References.

IV Quantum Communication.

14 Quantum Teleportation (L. C. Dávila Romero and N. Korolkova).

14.1 Introduction.

14.2 Experimental Realization.

14.3 Continuous Variables—Concept and Extension.

References.

15 Theory of Quantum Key Distribution (QKD) (N. Lütkenhaus).

15.1 Introduction.

15.2 Classical Background to QKD.

15.3 Ideal QKD.

15.4 Idealized QKD in noisy environment.

15.5 Realistic QKD in noisy and lossy environment.

15.6 Improved Schemes.

15.7 Improvements in Public Discussion.

15.8 Conclusion.

References.

16 Quantum Communication Experiments with Discrete Variables (H. Weinfurter).

16.1 Aunt Martha.

16.2 Quantum Cryptography.

16.3 Entanglement-Based Quantum Communication.

16.4 Conclusion.

References.

17 Continuous Variable Quantum Communication (U. L. Andersen and G. Leuchs).

17.1 Introduction.

17.2 Continuous Variable Quantum Systems.

17.3 Tools for State Manipulation.

17.4 Quantum Communication Protocols.

References.

V Quantum Computing: Concepts.

18 Requirements for a Quantum Computer (A. Ekert and A. Kay).

18.1 Classical World of Bits and Probabilities.

18.2 Logically Impossible Operations?

18.3 Quantum World of Probability Amplitudes.

18.4 Interference Revisited.

18.5 Tools of the Trade.

18.6 Composite Systems.

18.7 Quantum Circuits.

18.8 Summary.

19 Probabilistic Quantum Computation and Linear Optical Realizations (N. Lütkenhaus).

19.1 Introduction.

19.2 Gottesman/Chuang Trick.

19.3 Optical Background.

19.4 Knill–Laflamme–Milburn (KLM) scheme.

References.

20 One-way Quantum Computation (D.E. Browne and H.J. Briegel ).

20.1 Introduction.

20.2 Simple examples.

20.3 Beyond quantum circuit simulation.

20.4 Implementations.

20.5 Recent developments.

20.6 Outlook.

References.

21 Holonomic Quantum Computation (A.C.M. Carollo and Vlatko Vedral).

21.1 Geometric Phase and Holonomy.

21.2 Application toQuantum Computation.

References.

VI Quantum Computing: Implementations.

22 Quantum Computing with Cold Ions and Atoms: Theory (D. Jaksch, J.J. García-Ripoll, J.I. Cirac, and Peter Zoller).

22.1 Introduction.

22.2 Trapped Ions.

22.3 Trapped Neutral Atoms.

References.

23 Quantum Computing Experiments with Cold Trapped Ions (F. Schmidt-Kaler).

23.1 Introduction.

23.2 Paul Traps.

23.3 Ioncrystals and their normal modes.

23.4 Ion–light interaction.

23.5 Levels and Transitions for Typical Qubit Candidates.

23.6 Various Two-Qubit Gates.

23.7 Teleportation.

23.8 Segmented Traps and Future Directions.

References.

24 Quantum Computing with Solid State Systems (G. Burkard and D. Loss).

24.1 Introduction.

24.2 Concepts.

24.3 Electron Spin Qubits.

24.4 Superconducting Qubits.

References.

25 Quantum Computing Implemented via Optimal Control: Theory and Application to Spin and Pseudo-Spin Systems (T. Schulte-Herbrüggen, A.K. Spörl, R. Marx, N. Khaneja, J.M. Myers, A. F. Fahmy, and S. J. Glaser).

25.1 Introduction.

25.2 From Controllable Spin Systems to Suitable Molecules.

25.3 Scalability.

25.4 Control Theory for Spin- and Pseudo-Spin Systems.

25.5 Applied Quantum Control.

25.6 Conclusions.

References.

VII Transfer of Quantum Information Between Different Types of Implementations.

26 Quantum Repeater (W. Dür, H.-J. Briegel, and P. Zoller).

26.1 Introduction.

26.2 Concept of  the quantum repeater.

26.3 Proposals for Experimental Realization.

26.4 Summary and Conclusions.

References.

27 Quantum Interface Between Light and Atomic Ensembles (E. S. Polzik and J. Fiurášek).

27.1 Introduction.

27.2 Off-Resonant Interaction of Light with Atomic Ensemble.

27.3 Entanglement of Two Atomic Clouds.

27.4 Quantum Memory for Light.

27.5 Multiple Passage Protocols.

27.6 Atoms-light teleportation and entanglement swapping.

27.7 Quantum Cloning into Atomic Memory.

27.8 Summary.

References.

28 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons (J.-M. Raimond and G. Rempe).

28.1 Introduction.

28.2 Microwave Cavity Quantum Electrodynamics.

28.3 Optical Cavity Quantum Electrodynamics.

28.4 Conclusions and Outlook.

References.

29 Quantum Electrodynamics of a Qubit (G. Alber and G. M. Nikolopoulos).

29.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity.

29.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal.

References.

VIII Towards Quantum Technology Applications.

30 Quantum Interferometry (O. Glöckl, U. L. Andersen, and G. Leuchs).

30.1 Introduction.

30.2 The Interferometer.

30.3 Interferometer with Coherent States of Light.

30.4 Interferometer with Squeezed States of Light.

30.5 Summary and Discussion.

References.

31 Quantum Imaging (C. Fabre and N. Treps).

31.1 Introduction.

31.2 The Quantum Laser Pointer.

31.3 Manipulation of Spatial Quantum Noise.

31.4 Two-PhotonImaging.

31.5 Other Topics in Quantum Imaging.

31.6 Conclusion and Perspectives.

References.

Index.