Lectures on Quantum Information

Transcription

1 Lectures on Quantum Information Edited by Dagmar Bruß and Gerd Leuchs BICENTENNIAL BICENTEN NIAL WILEY-VCH Verlag GmbH & Co. KGaA

2 Preface List of Contributors XVII XIX I Classical Information Theory 1 1 Classical Information Theory and Classical Error Correction (M. Grassl) Introduction Basics of Classical Information Theory Abstract communication System The discrete noiseless Channel The discrete noisy Channel Linear Block Codes Repetition code Finitefields Generator and parity check matrix Hamming codes Further Aspects 15 References 16 2 Computational Complexity (5. Mertens) Basics Algorithms and Time Complexity Tractable Trails: The Class P Intractable Itineraries: The class NP Coloring graphs Logical truth Reductions and NP-completeness P vs. NP Optimization Complexity Zoo 34 References 35 Lectures on Quantum Information. Edited by D. Bruß and G. Leuchs Copyright 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 VI Contents II Foundation of Quantum Information Theory 37 3 Discrete Quantum States versus Continuous Variables (J. Eisen) Introduction Finite-dimensional quantum Systems Quantum states Quantum Operations Continuous-variables Phase space Gaussian states Gaussian unitaries Gaussian Channels Gaussian measurements Non-Gaussian Operations 51 References 52 4 Approximate Quantum Cloning (D. Bruß and C. Macchiavellö) Introduction The No-Cloning Theorem State-Dependent Cloning Phase Covariant Cloning Universal Cloning The case of qubits Higher dimensions Entanglement structure Asymmetrie Cloning Probabilistic Cloning Experimental Quantum Cloning Summary and Outlook 69 References 70 5 Channels and Maps (M. Keyl and R. F. Werner) Introduction Completely Positive Maps The Jamiolkowski Isomorphism The Stinespring Dilation Theorem Classical Systems as a Special Case Examples The ideal quantum Channel The depolarizing Channel Entanglement breaking Channels Covariant Channels 84 References 86

4 VII 6 Quantum Algorithms (/. Kempe) Introduction Precursors Deutsch's algorithm Deutsch-Josza algorithm Simon's algorithm Shor's Factoring Algorithm Reduction from factoring to period finding Implementation of the QFT Shor's algorithm for period finding Grover's Algorithm Other Algorithms The hidden subgroup problem Search algorithms Other algorithms Recent Developments Quantum walks Adiabatic quantum algorithms 100 References Quantum Error Correction (M. Grasst) Introduction Quantum Channels Using Classical Error-Correcting Codes Negative results: the quantum repetition code Positive results: a simple three-qubit code Shor's nine-qubit code Steane's seven-qubit code and CSS codes The five-qubit code and stabilizer codes Further Aspects 118 References 119 III Theoryof Entanglement The Separability versus Entanglement Problem (A. Sen(De), U. Sen, M. Lewenstein, anda. Sempera) Introduction Bipartite Pure States: Schmidt Decomposition Bipartite Mixed States: Separable and Entangled States Operational Entanglement Criteria Partial transposition Majorization 127

5 VIII Contents 8.5 Nonoperational Entanglement Criteria Entanglement witnesses Positive maps Bell Inequalities Classification of Bipartite States with Respect to Quantum Dense Coding The Holevo bound Capacity of quantum dense coding Further Reading: Multipartite States 142 References Entanglement Theory with Continuous Variables (P. van Loock) Introduction Phase-Space Description Entanglement of Gaussian States Gaussian states Gaussian Operations Pure entangled Gaussian states Mixed entangled Gaussian states and inseparability criteria More on Gaussian Entanglement 157 References Entanglement Measures (M B. Plenio and S. S. Virmani) Introduction Manipulation of Single Systems Manipulation in die Asymptotic Limit Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems Examples of Axiomatic Entanglement Measures 169 References Purification and Distillation (W. Dür and H. -J. Briegel) Introduction Pure States Bipartite Systems Multipartite Systems Distillability and Bound Entanglement in Bipartite Systems Distillable entanglement and yield Criteria for entanglement distillation Bipartite Entanglement Distillation Protocols Filtering protocol Recurrence protocols N» M protocols, hashing, and breeding 190

6 IX 11.5 Distillability and Bound Entanglement in Multipartite Systems n-party distillability m-party distillability with respect to coarser partitions Bound entanglement in multipartite Systems Entanglement Purification Protocols in Multipartite Systems Graph states Recurrence protocol Hashing protocol Entanglement purification of nonstabilizer states Distillability with Noisy Apparatus Distillable entanglement and yield Errormodel Bipartite recurrence protocols Multipartite recurrence protocols Hashing protocols Applications of Entanglement Purification Quantum communication and cryptography Secure State distribution Quantum error correction Quantum computation Summary and Conclusions 205 References Bound Entanglement {Pawel Horodecki) Introduction Distillation of Quantum Entanglement: Repetition Bipartite entanglement distillation Multipartite entanglement distillation Bound Entanglement Bipartite Case Bound entanglement the phenomenon Bound entanglement and entanglement measures. Asymptotic irreversibility Which states are bound entangled? Applications in Single copy case Applications in asymptotic regime Bound Entanglement: Multipartite Case Which multipartite states are bound entangled? Activation effects Remote quantum Information concentration Violaüon of Bell inequalities and communication complexity reduction Feedback to classical theory: multipartite bound information and its activation Bound entanglement and multiparty quantum Channels 230

7 12.5 Further Reading: Continuous Variables 230 References Multiparticle Entanglement (/. Eisert and D. Gross) Introduction Pure States Classifying entanglement of Single specimens Asymptotic manipulation of multiparticle quantum states Mixed States Classifying mixed State entanglement Methodsofdetection Quantifying Multiparticle Entanglement Stabilizer States and Graph States Applications of Multiparticle Entangled States 249 References 250 IV Quantum Communication Quantum Teleportation (L. C. Dävila Römern and N. Korolkova) Introduction Setting up the problem and the role of entanglement A template for quantum teleportation Efficiency and fidelity Experimental Realization The first quantum teleportation experiment Further experiments Continuous Variables Concept and Extension 263 References Theory of Quantum Key Distribution (QKD) (N. Lütkenhaus) Introduction Classical Background to QKD Ideal QKD Idealized QKD in noisy environment Realistic QKD in noisy and lossy environment Improved Schemes Improvements in Public Discussion Conclusion 283 References 283

8 XI 16 Quantum Communication Experiments with Discrete Variables (H. Weinfurter) Aunt Martha Quantum Cryptography Faint pulse QKD Entanglement-Based QKD Single Photon QKD Entanglement-Based Quantum Communication Quantum Dense Coding Error Correction Conclusion 295 References Continuous Variable Quantum Communication (U. L. Andersen and G. Leuchs) Introduction Continuous Variable Quantum Systems Tools for State Manipulation Gaussian transformations Homodyne detection and feed forward Non-Gaussian transformations Quantum Communication Protocols Quantum dense coding Quantum key distribution Long distance communication 308 References 310 V Quantum Computing: Concepts Requirements for a Quantum Computer (A. EkertandA. Kay) Classical World of Bits and Probabilities Parallel composition = tensor products Sequential composition = matrix products Logically Impossible Operations? Quantum World of Probability Amplitudes Interference Revisited Tools ofthe Trade Quantum states Unitary Operations Quantum measurements Composite Systems Density Operators Quantum Circuits Economy of resources 342

9 XII Contents Computations Summary Probabilistic Quantum Computation and Linear Optical Realizations (N. Lütkenhaus) Introduction Gottesman/Chuang Trick Optical Background Optical qubits Linear Optics Framework Knill-Laflamme-Milburn (KLM) scheme Extension of Gottesman-Chuang trick Implementation with linear optics Offline probabilistic gates 356 References One-way Quantum Computation {D.E. Browne andh.j. Briegel) Introduction Cluster states and graph states Single-qubit measurements and rotations Simple examples Connecting one-way pattems - arbitrary single-qubit Operations Graph states as a resource Two-qubit gates Cluster-state quantum Computing Beyond quantum circuit Simulation Stabilizer formalism A logical Heisenberg picture Dynamical variables on a stabilizer sub-space One-way pattems in the stabilizer formalism Pauli measurements Pauli measurements and the Clifford group Non-Pauli measurements Diagonal unitaries Gate pattems beyond the Standard network model -CD-decomposition Implementations Optical lattices Linear optics and cavity QED Recent developments Outlook 376 References 378

10 XIII 21 Holonomic Quantum Computation (A. CM. Carollo and Vlatko Vedral) Geometrie Phase and Holonomy Adiabatic implementation of holonomies Application to Quantum Computation Example 385 References 386 VI Quantum Computing: Implementations Quantum Computing with Cold Ions and Atoms: Theory ( >. Jaksch, J.J. Garcia-Ripoü, J.I. Cirac, and Peter Zoller) Introduction Trapped Ions Motional degrees of freedom Internal degrees of freedom and atom-laser interaction Lamb-Dicke limit and sideband transitions Single-qubit Operations and State measurement The gate Cirac-Zoller ' Optimal gates based on quantum control Trapped Neutral Atoms Optical lattices The (Böse) Hubbard Hamiltonian Loading schemes Quantum Computing in optical lattices 408 References Quantum Computing Experiments with Cold Trapped Ions (F. Schmidt-Kaler) Introduction Paul Traps Stability diagram of dynamic trapping D confinement in a linear Paul trap Ion crystals and their normal modes Lagrangian of the ion motion in the trap Eigenmodes Ion-light interaction Levels and Transitions for Typical Qubit Candidates Various Two-Qubit Gates The Cirac and Zoller scheme Experimental realization of the Cirac and Zoller gate The Sörensen and Mölmer scheme The Jonathan, Plenio, and Knight scheme Geometrie phase shift gates 440

11 The Mintert and Wunderlich gate proposal Gate proposals based on the interaction of ions with a common optical mode Teleportation Segmented Traps and Future Directions 444 References Quantum Computing with Solid State Systems (G. Burkard and D. Loss) Introduction Concepts The exchange coupling Anisotropie exchange Universal QC with the exchange coupling Adiabaticity Electron Spin Qubits Quantum dots Exchange in laterally coupled QDs Semiconductor microeavities Decoherence Superconducting Qubits Regimes of Operation Decoherence, visibility, and leakage 470 References 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) Introduction From Controllable Spin Systems to Suitable Molecules Reachability and controllability Molecular hardware for quantum computation Scalability Scaling problem with pseudo-pure states Approaching pure states Scalable quantum Computing on thermal ensembles Control Theory for Spin- and Pseudo-Spin Systems Applied Quantum Control Regime of fast local controls: the NMR limit Regime of finite local controls: beyond NMR Conclusions Ensemble quantum Computing From gate-complexity to time-complexity by optimal control Beyond NMR spin Systems 496 References 498

12 XV VII Transfer of Quantum Information Between Different Types of Implementations Quantum Repeater (W. Dür, H.-J. Briegel, andp. Zoller) Introduction Concept of the quantum repeater Entanglement purification Connection of elementary pairs Nested purification loops Resources Proposais for Experimental Realization Photons and cavities Atomic ensembles Quantum dots Summary and Conclusions 513 References Quantum Interface Between Light and Atomic Ensembles (E. S. Polzik and J. Fiuräsek) Introduction Off-Resonant Interaction of Light with Atomic Ensemble Entanglement oftwo Atomic Clouds Quantum Memory for Light Multiple Passage Protocols Atoms-light teleportation and entanglement swapping Quantum Cloning into Atomic Memory Summary 534 References Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons (J.-M. Raimond and G. Rempe) Introduction Microwave Cavity Quantum Electrodynamics Optical Cavity Quantum Electrodynamics Conclusions and Outlook 549 References Quantum Electrodynamics of a Qubit (G. Alber and G. M. Nikolopoulos) Quantum Electrodynamics of a Qubit in a Spherical Cavity Themodel Mode structure of the free radiation field in a spherical cavity Dynamics of spontaneous photon emission 559

13 XVI Contents 29.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal Photonic crystals and associated density of states "Photon + atom" bound states Beyond the two-level approximation Exercises 568 References 570 VIII Towards Quantum Technology Applications Quantum Interferometry (O. Glöckl, U. L. Andersen, and G. Leuchs) Introduction The Interferometer Sensitivity Interferometer with Coherent States of Light Geometrical visualization Interferometer with Squeezed States of Light Interferometer operating with a coherent State and a squeezed vacuum state Interferometer operating with two bright squeezed states Interferometer operating with a bright squeezed State and a squeezed vacuum State Summary and Discussion 587 References Quantum Imaging (C. Fahre and N. Treps) Introduction The Quantum Laser Pointer Manipulation of Spatial Quantum Noise Observation of pure spatial quantum correlations in parametric down conversion Noiseless image parametric amplification Two-Photon Imaging Other Topics in Quantum Imaging Conclusion and Perspectives 598 References 598 Index 601