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by Gerd Leuchs, Dagmar Bruss
Quantum Information, 2 Volume Set, 2nd Edition
Cover
Preface to the New Edition
Preface to Lectures on Quantum Information (2006)
Volume 1
Part I: Classical Information Theory
1 Classical Information Theory and Classical Error Correction
1.1 Introduction
1.2 Basics of Classical Information Theory
1.3 Linear Block Codes
1.4 Further Aspects
References
2 Computational Complexity
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‐Completeness
2.6 P Versus NP
2.7 Optimization
2.8 Complexity Zoo
References
Part II: Foundations of Quantum Information Theory
3 Discrete Quantum States versus Continuous Variables
3.1 Introduction
3.2 Finite‐Dimensional Quantum Systems
3.3 Continuous‐Variables
References
4 Approximate Quantum Cloning
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
5.1 Introduction
5.2 Completely Positive Maps
5.3 The Choi–Jamiolkowski Isomorphism
5.4 The Stinespring Dilation Theorem
5.5 Classical Systems as a Special Case
5.6 Channels with Memory
5.7 Examples
References
6 Quantum Algorithms
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
7.1 Introduction
7.2 Quantum Channels
7.3 Using Classical Error‐Correcting Codes
7.4 Further Aspects
References
Part III: Theory of Entanglement
8 The Separability versus Entanglement Problem
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 Non‐operational Entanglement Criteria
8.6 Bell Inequalities
8.7 Quantification of Entanglement
8.8 Classification of Bipartite States with Respect to Quantum Dense Coding
8.9 Multipartite States
Acknowledgments
References
9 Quantum Discord and Nonclassical Correlations Beyond Entanglement
9.1 Introduction
9.2 Quantumness Versus Classicality (of Correlations)
9.3 Quantifying Quantum Correlations – Quantum Discord
9.4 Interpreting Quantum Correlations – Local Broadcasting
9.5 Alternative Characterizations of Quantum Correlations
9.6 General Desiderata for Measures of Quantum Correlations
9.7 Outlook
References
10 Entanglement Theory with Continuous Variables
10.1 Introduction
10.2 Phase‐Space Description
10.3 Entanglement of Gaussian States
10.4 More on Gaussian Entanglement
References
11 Entanglement Measures
11.1 Introduction
11.2 Manipulation of Single Systems
11.3 Manipulation in the Asymptotic Limit
11.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems
11.5 Examples of Axiomatic Entanglement Measures
Acknowledgments
References
12 Purification and Distillation
12.1 Introduction
12.2 Pure States
12.3 Distillability and Bound Entanglement in Bipartite Systems
12.4 Bipartite Entanglement Distillation Protocols
12.5 Distillability and Bound Entanglement in Multipartite Systems
12.6 Entanglement Purification Protocols in Multipartite Systems
12.7 Distillability with Noisy Apparatus
12.8 Applications of Entanglement Purification
12.9 Summary and Conclusions
Acknowledgments
References
13 Bound Entanglement
13.1 Introduction
13.2 Distillation of Quantum Entanglement: Repetition
13.3 Bound Entanglement – Bipartite Case
13.4 Bound Entanglement: Multipartite Case
13.5 Further Reading: Continuous Variables
Exercises
References
14 Multipartite Entanglement
14.1 Introduction
14.2 General Theory
14.3 Important Classes of Multipartite states
14.4 Specialized Topics
Acknowledgments
References
Part IV: Quantum Communication
15 Quantum Teleportation
15.1 Introduction
15.2 Quantum Teleportation Protocol
15.3 Implementations
References
16 Theory of Quantum Key Distribution (QKD)
16.1 Introduction
16.2 Classical Background to QKD
16.3 Ideal QKD
16.4 Idealized QKD in Noisy Environment
16.5 Realistic QKD in Noisy and Lossy Environment
16.6 Improved Schemes
16.7 Improvements in Public Discussion
16.8 Conclusion
References
17 Quantum Communication Experiments with Discrete Variables
17.1 Aunt Martha
17.2 Quantum Cryptography
17.3 Entanglement‐Based Quantum Communication
17.4 Conclusion
References
18 Continuous Variable Quantum Communication with Gaussian States
18.1 Introduction
18.2 Continuous‐Variable Quantum Systems
18.3 Tools for State Manipulation
18.4 Quantum Communication Protocols
References
Volume 2
Part V: Quantum Computing: Concepts
19 Requirements for a Quantum Computer
19.1 Classical World of Bits and Probabilities
19.2 Logically Impossible Operations?
19.3 Quantum World of Probability Amplitudes
19.4 Interference Revisited
19.5 Tools of the Trade
19.6 Composite Systems
19.7 Quantum Circuits
19.8 Summary
20 Probabilistic Quantum Computation and Linear Optical Realizations
20.1 Introduction
20.2 Gottesman/Chuang Trick
20.3 Optical Background
20.4 Knill–Laflamme–Milburn (KLM) Scheme
References
21 One‐Way Quantum Computation
21.1 Introduction
21.2 Simple Examples
21.3 Beyond Quantum Circuit Simulation
21.4 Implementations
21.5 Recent Developments
21.6 Outlook
Acknowledgments
Exercises
References
22 Holonomic Quantum Computation
22.1 Geometric Phase and Holonomy
22.2 Application to Quantum Computation
References
Part VI: Quantum Computing: Implementations
23 Quantum Computing with Cold Ions and Atoms: Theory
23.1 Introduction
23.2 Trapped Ions
23.3 Trapped Neutral Atoms
References
24 Quantum Computing Experiments with Cold Trapped Ions
24.1 Introduction to Trapped‐Ion Quantum Computing
24.2 Paul Traps
24.3 Ion Crystals and Normal Modes
24.4 Trap Technology
Acknowledgements
References
25 Quantum Computing with Solid‐State Systems
25.1 Introduction
25.2 Concepts
25.3 Electron Spin Qubits
25.4 Superconducting Qubits
References
26 Time‐Multiplexed Networks for Quantum Optics
26.1 Introduction
26.2 Multiplexing
26.3 Photon‐Number‐Resolving Detection with Time Multiplexing
26.4 Quantum Walks in Time
26.5 Conclusion
References
27 A Brief on Quantum Systems Theory and Control Engineering
27.1 Introduction
27.2 Systems Theory of Closed Quantum Systems
27.3 Toward a Systems Theory for Open Quantum Systems
27.4 Relation to Numerical Optimal Control
27.5 Outlook on Infinite‐Dimensional Systems
27.6 Conclusion
Acknowledgments
References
28 Quantum Computing Implemented via Optimal Control: Application to Spin and Pseudospin Systems
28.1 Introduction
28.2 From Controllable Spin Systems to Suitable Molecules
28.3 Scalability
28.4 Algorithmic Platform for Quantum Control Systems
28.5 Applied Quantum Control
28.6 Worked Example: Unitary Controls for Classifying Knots by NMR
28.7 Conclusions
Acknowledgments
References
Part VII: Quantum Interfaces and Memories
29 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons
29.1 Introduction
29.2 Microwave Cavity Quantum Electrodynamics
29.3 Optical Cavity Quantum Electrodynamics
29.4 Conclusions and Outlook
References
30 Quantum Repeater
30.1 Introduction
30.2 Concept of the Quantum Repeater
30.3 Proposals for Experimental Realization
30.4 Summary and Conclusions
Acknowledgments
References
31 Quantum Interface Between Light and Atomic Ensembles
31.1 Introduction
31.2 Off‐Resonant Interaction of Light with Atomic Ensemble
31.3 Entanglement of Two Atomic Clouds
31.4 Quantum Memory for Light
31.5 Multiple Passage Protocols
31.6 Atoms‐Light Teleportation and Entanglement Swapping
31.7 Quantum Cloning into Atomic Memory
31.8 Summary
Acknowledgment
References
32 Echo‐Based Quantum Memory
32.1 Overview of Photon Echo Techniques
32.2 Platforms for Echo‐Based Quantum Memory
32.3 Characterization
32.4 Demonstrations
32.5 Outlook
References
33 Quantum Electrodynamics of a Qubit
33.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity
33.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal
References
34 Elementary Multiphoton Processes in Multimode Scenarios
34.1 A Generic Quantum Electrodynamical Model
34.2 The Multiphoton Path Representation
34.3 Examples
34.4 Conclusion
Appendix Evaluation of the Field Commutator
References
Part VIII: Towards Quantum Technology Applications
35 Quantum Interferometry with Gaussian States
35.1 Introduction
35.2 The Interferometer
35.3 Interferometer with Coherent States of Light
35.4 Interferometer with Squeezed States of Light
35.5 Fundamental Limits
35.6 Summary and Discussion
Problems
References
36 Quantum Logic‐Enabled Spectroscopy
36.1 Introduction
36.2 Trapping and Doppler Cooling of a Two‐Ion Crystal
36.3 Coherent Atom–Light Interaction and State Manipulation
36.4 Quantum Logic Spectroscopy for Optical Clocks
36.5 Photon Recoil Spectroscopy
36.6 Quantum Logic with Molecular Ions
36.7 Nonclassical States for Spectroscopy
36.8 Future Directions
Acknowledgments
References
37 Quantum Imaging
37.1 Introduction
37.2 The Quantum Laser Pointer
37.3 Manipulation of Spatial Quantum Noise
37.4 Two‐Photon Imaging
37.5 Other Topics in Quantum Imaging
37.6 Conclusion and Perspectives
Acknowledgment
References
38 Quantum Frequency Combs
38.1 Introduction
38.2 Parametric Down Conversion of a Frequency Comb
38.3 Experiment
38.4 Experimental Results
38.5 Application to Quantum Information Processing
38.6 Application to Quantum Metrology
38.7 Conclusion
Acknowledgment
References
Index
End User License Agreement
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Prev
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Title Page
Table of Contents
Cover
Preface to the New Edition
Preface to Lectures on Quantum Information (2006)
Volume 1
Part I: Classical Information Theory
1 Classical Information Theory and Classical Error Correction
1.1 Introduction
1.2 Basics of Classical Information Theory
1.3 Linear Block Codes
1.4 Further Aspects
References
2 Computational Complexity
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
Versus
NP
2.7 Optimization
2.8 Complexity Zoo
References
Part II: Foundations of Quantum Information Theory
3 Discrete Quantum States versus Continuous Variables
3.1 Introduction
3.2 Finite‐Dimensional Quantum Systems
3.3 Continuous‐Variables
References
4 Approximate Quantum Cloning
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
5.1 Introduction
5.2 Completely Positive Maps
5.3 The Choi–Jamiolkowski Isomorphism
5.4 The Stinespring Dilation Theorem
5.5 Classical Systems as a Special Case
5.6 Channels with Memory
5.7 Examples
References
6 Quantum Algorithms
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
7.1 Introduction
7.2 Quantum Channels
7.3 Using Classical Error‐Correcting Codes
7.4 Further Aspects
References
Part III: Theory of Entanglement
8 The Separability versus Entanglement Problem
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 Non‐operational Entanglement Criteria
8.6 Bell Inequalities
8.7 Quantification of Entanglement
8.8 Classification of Bipartite States with Respect to Quantum Dense Coding
8.9 Multipartite States
Acknowledgments
References
9 Quantum Discord and Nonclassical Correlations Beyond Entanglement
9.1 Introduction
9.2 Quantumness Versus Classicality (of Correlations)
9.3 Quantifying Quantum Correlations – Quantum Discord
9.4 Interpreting Quantum Correlations – Local Broadcasting
9.5 Alternative Characterizations of Quantum Correlations
9.6 General Desiderata for Measures of Quantum Correlations
9.7 Outlook
References
10 Entanglement Theory with Continuous Variables
10.1 Introduction
10.2 Phase‐Space Description
10.3 Entanglement of Gaussian States
10.4 More on Gaussian Entanglement
References
11 Entanglement Measures
11.1 Introduction
11.2 Manipulation of Single Systems
11.3 Manipulation in the Asymptotic Limit
11.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems
11.5 Examples of Axiomatic Entanglement Measures
Acknowledgments
References
12 Purification and Distillation
12.1 Introduction
12.2 Pure States
12.3 Distillability and Bound Entanglement in Bipartite Systems
12.4 Bipartite Entanglement Distillation Protocols
12.5 Distillability and Bound Entanglement in Multipartite Systems
12.6 Entanglement Purification Protocols in Multipartite Systems
12.7 Distillability with Noisy Apparatus
12.8 Applications of Entanglement Purification
12.9 Summary and Conclusions
Acknowledgments
References
13 Bound Entanglement
13.1 Introduction
13.2 Distillation of Quantum Entanglement: Repetition
13.3 Bound Entanglement – Bipartite Case
13.4 Bound Entanglement: Multipartite Case
13.5 Further Reading: Continuous Variables
Exercises
References
14 Multipartite Entanglement
14.1 Introduction
14.2 General Theory
14.3 Important Classes of Multipartite states
14.4 Specialized Topics
Acknowledgments
References
Part IV: Quantum Communication
15 Quantum Teleportation
15.1 Introduction
15.2 Quantum Teleportation Protocol
15.3 Implementations
References
16 Theory of Quantum Key Distribution (QKD)
16.1 Introduction
16.2 Classical Background to QKD
16.3 Ideal QKD
16.4 Idealized QKD in Noisy Environment
16.5 Realistic QKD in Noisy and Lossy Environment
16.6 Improved Schemes
16.7 Improvements in Public Discussion
16.8 Conclusion
References
17 Quantum Communication Experiments with Discrete Variables
17.1 Aunt Martha
17.2 Quantum Cryptography
17.3 Entanglement‐Based Quantum Communication
17.4 Conclusion
References
18 Continuous Variable Quantum Communication with Gaussian States
18.1 Introduction
18.2 Continuous‐Variable Quantum Systems
18.3 Tools for State Manipulation
18.4 Quantum Communication Protocols
References
Volume 2
Part V: Quantum Computing: Concepts
19 Requirements for a Quantum Computer
19.1 Classical World of Bits and Probabilities
19.2 Logically Impossible Operations?
19.3 Quantum World of Probability Amplitudes
19.4 Interference Revisited
19.5 Tools of the Trade
19.6 Composite Systems
19.7 Quantum Circuits
19.8 Summary
20 Probabilistic Quantum Computation and Linear Optical Realizations
20.1 Introduction
20.2 Gottesman/Chuang Trick
20.3 Optical Background
20.4 Knill–Laflamme–Milburn (KLM) Scheme
References
21 One‐Way Quantum Computation
21.1 Introduction
21.2 Simple Examples
21.3 Beyond Quantum Circuit Simulation
21.4 Implementations
21.5 Recent Developments
21.6 Outlook
Acknowledgments
Exercises
References
22 Holonomic Quantum Computation
22.1 Geometric Phase and Holonomy
22.2 Application to Quantum Computation
References
Part VI: Quantum Computing: Implementations
23 Quantum Computing with Cold Ions and Atoms: Theory
23.1 Introduction
23.2 Trapped Ions
23.3 Trapped Neutral Atoms
References
24 Quantum Computing Experiments with Cold Trapped Ions
24.1 Introduction to Trapped‐Ion Quantum Computing
24.2 Paul Traps
24.3 Ion Crystals and Normal Modes
24.4 Trap Technology
Acknowledgements
References
25 Quantum Computing with Solid‐State Systems
25.1 Introduction
25.2 Concepts
25.3 Electron Spin Qubits
25.4 Superconducting Qubits
References
26 Time‐Multiplexed Networks for Quantum Optics
26.1 Introduction
26.2 Multiplexing
26.3 Photon‐Number‐Resolving Detection with Time Multiplexing
26.4 Quantum Walks in Time
26.5 Conclusion
References
27 A Brief on Quantum Systems Theory and Control Engineering
27.1 Introduction
27.2 Systems Theory of Closed Quantum Systems
27.3 Toward a Systems Theory for Open Quantum Systems
27.4 Relation to Numerical Optimal Control
27.5 Outlook on Infinite‐Dimensional Systems
27.6 Conclusion
Acknowledgments
References
28 Quantum Computing Implemented via Optimal Control: Application to Spin and Pseudospin Systems
28.1 Introduction
28.2 From Controllable Spin Systems to Suitable Molecules
28.3 Scalability
28.4 Algorithmic Platform for Quantum Control Systems
28.5 Applied Quantum Control
28.6 Worked Example: Unitary Controls for Classifying Knots by NMR
28.7 Conclusions
Acknowledgments
References
Part VII: Quantum Interfaces and Memories
29 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons
29.1 Introduction
29.2 Microwave Cavity Quantum Electrodynamics
29.3 Optical Cavity Quantum Electrodynamics
29.4 Conclusions and Outlook
References
30 Quantum Repeater
30.1 Introduction
30.2 Concept of the Quantum Repeater
30.3 Proposals for Experimental Realization
30.4 Summary and Conclusions
Acknowledgments
References
31 Quantum Interface Between Light and Atomic Ensembles
31.1 Introduction
31.2 Off‐Resonant Interaction of Light with Atomic Ensemble
31.3 Entanglement of Two Atomic Clouds
31.4 Quantum Memory for Light
31.5 Multiple Passage Protocols
31.6 Atoms‐Light Teleportation and Entanglement Swapping
31.7 Quantum Cloning into Atomic Memory
31.8 Summary
Acknowledgment
References
32 Echo‐Based Quantum Memory
32.1 Overview of Photon Echo Techniques
32.2 Platforms for Echo‐Based Quantum Memory
32.3 Characterization
32.4 Demonstrations
32.5 Outlook
References
33 Quantum Electrodynamics of a Qubit
33.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity
33.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal
References
34 Elementary Multiphoton Processes in Multimode Scenarios
34.1 A Generic Quantum Electrodynamical Model
34.2 The Multiphoton Path Representation
34.3 Examples
34.4 Conclusion
Appendix Evaluation of the Field Commutator
References
Part VIII: Towards Quantum Technology Applications
35 Quantum Interferometry with Gaussian States
35.1 Introduction
35.2 The Interferometer
35.3 Interferometer with Coherent States of Light
35.4 Interferometer with Squeezed States of Light
35.5 Fundamental Limits
35.6 Summary and Discussion
Problems
References
36 Quantum Logic‐Enabled Spectroscopy
36.1 Introduction
36.2 Trapping and Doppler Cooling of a Two‐Ion Crystal
36.3 Coherent Atom–Light Interaction and State Manipulation
36.4 Quantum Logic Spectroscopy for Optical Clocks
36.5 Photon Recoil Spectroscopy
36.6 Quantum Logic with Molecular Ions
36.7 Nonclassical States for Spectroscopy
36.8 Future Directions
Acknowledgments
References
37 Quantum Imaging
37.1 Introduction
37.2 The Quantum Laser Pointer
37.3 Manipulation of Spatial Quantum Noise
37.4 Two‐Photon Imaging
37.5 Other Topics in Quantum Imaging
37.6 Conclusion and Perspectives
Acknowledgment
References
38 Quantum Frequency Combs
38.1 Introduction
38.2 Parametric Down Conversion of a Frequency Comb
38.3 Experiment
38.4 Experimental Results
38.5 Application to Quantum Information Processing
38.6 Application to Quantum Metrology
38.7 Conclusion
Acknowledgment
References
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 The extended Euclidean algorithm (see (4)).
Table 1.2 Standard array for decoding the code
, the dual of a binary Hamming ...
Chapter 7
Table 7.1 Orthogonal decomposition corresponding to the three‐qubit code correct...
Chapter 27
Table 27.1 Bilinear quantum control systems.
Table 27.2 Heisenberg–
spin chains with a single control on one end (or both) ...
Table 27.3 Ising‐
spin chains with joint controls on all the qubits locally ca...
Table 27.4 Controllability results for several two‐level atoms in a cavity as de...
Chapter 35
Table 35.1 Summary of various scenarios for driving a quantum interferometer and...
List of Illustrations
Chapter 1
Figure 1.1 Schematic diagram of a general communication system.
Figure 1.2 Schematic representation of a discrete memoryless channel. A...
Figure 1.3 Relationship between entropy and mutual information.
Figure 1.4 The binary symmetric channel (BSC) and its generalization, t...
Figure 1.5 Geometry of the codewords. Any sphere of radius
around a...
Chapter 2
Figure 2.1 The seven bridges of Königsberg, as drawn in Euler's paper f...
Figure 2.2 Sir Hamilton's Icosian game: Find a route along the edges of...
Figure 2.3 Example of the execution history of a nondeterministic algor...
Figure 2.4 The Petersen graph (a) with a proper 3‐coloring. The cart‐wh...
Figure 2.5 Three tentative maps of
NP
. We can rule out (b), and it is v...
Figure 2.6 A weighted graph and its minimum spanning tree (bold edges) ...
Chapter 4
Figure 4.1 Vectors and angles for cloning of two nonorthogonal states. ...
Figure 4.2 Fidelity for each output copy of the state‐dependent cloner ...
Figure 4.3 Geometrical disposition of two pairs of orthogonal states.
Figure 4.4 Optimal fidelity for cloning two pairs of orthogonal states,...
Figure 4.5 Concatenation of an
cloner with a state estimation of th...
Chapter 5
Figure 5.1 Graphical representation of a channel, transforming the syst...
Figure 5.2 Channels can be applied to subsystems even if the overall sy...
Figure 5.3 If
is an effect (i.e. a yes/no measurement) and
a ch...
Figure 5.4 A noisy channel arising from interaction with the environmen...
Chapter 6
Figure 6.1 A reversible black box for a function
f
: {0
,
1}
n
→
{...
Figure 6.2 Deutsch's circuit.
Figure 6.3 Deutsch–Josza algorithm.
Figure 6.4 Simon's algorithm – quantum Fourier sampling. In our algorit...
Figure 6.5 QFT on ℤ
8
. An element of ℤ
8
is represented in binary notatio...
Figure 6.6 Subroutine in Grover's algorithm.
Chapter 7
Figure 7.1 Modeling the interaction with the environment by a unitary t...
Figure 7.2 Unitary representation of a quantum channel acting on the su...
Figure 7.3 Quantum circuit for encoding one qubit, computing the error ...
Figure 7.4 Quantum circuit for encoding one qubit, computing the error ...
Chapter 8
Figure 8.1 The structure of the state space in light of the partial tra...
Figure 8.2 Schematic picture of the Hahn–Banach theorem. The (unique) u...
Figure 8.3Figure 8.3 Schematic view of the Hilbert space with two states
Figure 8.4 Schematic diagram showing the direction of
,
,
,
...
Figure 8.5 Classification of bipartite quantum states according to thei...
Figure 8.6 Geometric representation of the hierarchy of multipartite en...
Figure 8.7 Different classes of three‐qubit pure states. Two states in ...
Chapter 9
Figure 9.1
Operational interpretations and quantification of quantum co
...
Chapter 12
12.1 Purification curve for the BBPSSW protocol. Gain in output fidelit...
12.2 Maximal reachable fidelity
F
max
and minimal required fidelity
F
...
Chapter 14
Figure 14.1 Quantum states associated with graphs. (a) Defines the
line
...
Chapter 15
Figure 15.1 Quantum teleportation template: cartoon showing how the qua...
Figure 15.2 Quantum teleportation template: timeline. BSM: Bell‐state m...
Figure 15.3 Schematic of Bouwmeester
et al
. experimental setup (2). BS...
Figure 15.4 Teleportation scheme for continuous variables. AQM/PQM, amp...
Chapter 16
Figure 16.1 The two phases of the BB84 protocol.
Figure 16.2 Without authentication of the classical channel, no secure ...
Figure 16.3 In the PNS attack, Eve can guide the signals depending on t...
Figure 16.4 In the strong phase reference scheme, Alice sends a weak co...
Chapter 17
Figure 17.1 Setup of the first quantum cryptography demonstration.
Figure 17.2 Quantum cryptography setups. (a) Unbalanced interferometers...
Figure 17.3 QKD demonstration over 23.4 km between Zugspitze and Westli...
Figure 17.4 (a) Scheme for the efficient transmission of classical info...
Figure 17.5 (a) Scheme for entanglement purification. (b) Detection pro...
Chapter 18
Figure 18.1 Wigner functions of three different pure states: (a) vacuum...
Figure 18.2 Table of Gaussian transformations. BS, beam splitter; PS, p...
Figure 18.3 Schematic diagram of commonly used detector systems as well...
Figure 18.4 Continuous‐variable quantum key distribution based on coher...
Figure 18.5 Implementation of an entangler and an amplifier using alternat...
Chapter 20
Figure 20.1 The teleportation procedure in (a) allows to shift the prob...
Figure 20.2 (a) The definition of the basic single‐qubit operations. (b...
Figure 20.3 (a) The basic trick of Gottesman and Chuang can be extended...
Figure 20.4 A CSIGN gate can be implemented using two nonlinear sign sh...
Figure 20.5 A probabilistic nonlinear sign shift gate can be implemente...
Chapter 21
Figure 21.1 One‐way quantum computation consists of single‐qubit measur...
Figure 21.2 Single‐qubit projective measurements will be represented by...
Figure 21.3 The one‐way graph and measurement patterns for (a) the sing...
Figure 21.4 Any
‐qubit Clifford‐group operation may be implemented (...
Figure 21.5 The full orbit of locally equivalent four‐qubit graph state...
Figure 21.6 The one‐way pattern that implements the unitary “double‐z r...
Figure 21.7 Arbitrary diagonal unitaries may be implemented in a single...
Chapter 22
Figure 22.1 (a) The geometric phase for a single qubit is proportional ...
Chapter 23
Figure 23.1 Coupling of the atom + trap levels according to the Hamilto...
Figure 23.2 Ion‐trap quantum computer '95. (a) First step according to ...
Figure 23.3 (a) Ions stored in an array of microtraps. By addressing tw...
Figure 23.4 (a) Orbits in phase space of a single ion in a harmonic tra...
Figure 23.5 (a) Trajectory in phase space of the center‐of‐mass state o...
Figure 23.6 Laser setup and resulting optical lattice configuration in ...
Figure 23.7 Band structure of an optical lattice of the form
V
0
(
x
) =
Figure 23.8 (a) Simple three‐dimensional cubic lattice. (b) Sheets of a...
Figure 23.9 (a) Atomic fine and hyperfine structure of the most commonl...
Figure 23.10 Laser configuration for a state selective optical potentia...
Figure 23.11 (a) Interpretation of the BHM in an optical lattice as dis...
Figure 23.12 (a) Avoided crossings in the energy eigenvalues
E
for
n
= ...
Figure 23.13 We collide one atom in the internal state
|a
〉 (filled circ...
Figure 23.14 Illustrating how triangular configurations of atoms with n...
Figure 23.15 Creation of robust multiparticle entangled states in 1D be...
Figure 23.16 A spin 1/2 impurity
q
used as a switch: in one spin state ...
Figure 23.17 (a) The optical Feshbach setup couples the atomic state
...
Chapter 24
Figure 24.1 Stability diagram for a linear quadrupole configuration. In...
Figure 24.2 Different realizations of Paul traps. Each panel contains a...
Figure 24.3 Linear crystal with eight
Ca
ions imaged by a CCD sys...
Figure 24.4 Illustration of the model for the linear crystal, position ...
Figure 24.5 Eigenfrequencies of an
ion crystal. Black, radial modes...
Figure 24.6 Coherent dynamics on the
to
optical qubit transitio...
Figure 24.7
Ca
(a, b) and
Be
(c) level schemes, as examples...
Figure 24.8 (a) Level scheme for the Cirac–Zoller gate scheme: A red si...
Figure 24.9 The state evolution for the composite phase gate,
is vi...
Figure 24.10 Level scheme for the Sørensen–Mølmer gate operation. The b...
Figure 24.11 Calculation of the evolution of the probabilities for find...
Figure 24.12 Realization of the Sørensen–Mølmer gate operation with a p...
Figure 24.13 Geometric phase gate: (a) beam geometry: laser beams R1 an...
Figure 24.14 (a) Space–time diagram of the quantum teleportation algori...
Chapter 25
Figure 25.1 A quantum‐dot array for quantum computing according to (2)...
Figure 25.2 Two quantum dots in a scanning electron micrograph picture ...
Figure 25.3 Exchange coupling between two electrons is modified in the ...
Figure 25.4 A pair of coupled quantum dots containing one electron per ...
Figure 25.5 Exchange energy
(in meV) as a function of the magnetic ...
Figure 25.6 Exchange coupling
from HM (full line), Eq. 25.35, and f...
Figure 25.7 Exchange coupling
measured as a function of magnetic fi...
Figure 25.8 The SC flux qubit circuit studied in (42) in a scanning el...
Figure 25.9 (a) The simplest model of an SC qubit consists of a biased ...
Figure 25.10 An example of a circuit graph (a) and a tree of the same g...
Figure 25.11 (a) Schematic of the Delft circuit, Figure 25.8, where th...
Figure 25.12 Schematics of Josephson junctions produced by the shadow e...
Figure 25.13 (a) Double‐layer structure. Dashed blue lines represent th...
Figure 25.14 (a) Decoupling (red solid) and symmetric (blue dashed) cur...
Chapter 26
Figure 26.1 (a) Action of a beam splitter for spatial multiplexing: it ...
Figure 26.2 Scheme for (a) spatial and (b) temporal multiplexing for ph...
Figure 26.3 (a) Classical Galton board comprising four layers of pins a...
Figure 26.4 Comparison between the distributions of quantum (bars) and ...
Figure 26.5 (a) First and (b) second step of a quantum walk in time and...
Figure 26.6 Schematic of the implementation of a time‐multiplexed quant...
Figure 26.7 (a) Quantum walk on a two‐dimensional grid (the first step ...
Chapter 27
Figure 27.1 Graph representation of quantum dynamical control systems: ...
Figure 27.2 Branching diagrams showing all the irreducible simple subal...
Figure 27.3 Abstract optimization task: By following the gradient flow
Figure 27.4 Optimal control task: the quality function
is driven in...
Figure 27.5 (a) Block structure of operators in
(dark gray) and of ...
Chapter 28
Figure 28.1 Schematic representation of a liquid‐state NMR experiment. ...
Figure 28.2 (a) Schematic representation of a system consisting of
...
Figure 28.3 Schematic representation of BOC‐
‐
N‐
‐glycine‐fluori...
Figure 28.4 Experimental spectra (68) representing the result of the n...
Figure 28.5 Overview on the update schemes of gradient‐based optimal co...
Figure 28.6 The QFT in linear coupling topologies
: (a) gate complex...
Figure 28.7 The
NOT gate on complete coupling topologies
: (a) ne...
Figure 28.8 Time‐optimal pulse sequences for synthesizing the propagato...
Figure 28.9 (a) Times required for simulating the trilinear coupling Ha...
Figure 28.10 Standard knots and links that relate to the braid group wi...
Figure 28.11 NMR pulse sequences implementing the set of controlled uni...
Figure 28.12 Experimental results (110) with real and imaginary parts ...
Chapter 29
Figure 29.1 A cavity QED setup using circular Rydberg atoms, prepared i...
Figure 29.2 Experimental quantum Rabi oscillation. The probability for ...
Figure 29.3 A single resonant atom prepares a coherent superposition of...
Figure 29.4 Experimental setup of an optical cavity QED experiment: col...
Figure 29.5 Photon statistics of a deterministic single‐photon source (...
Figure 29.6 Time‐resolved quantum‐beat experiment with two single‐photo...
Chapter 30
Figure 30.1 Purification loop: Connection of
L
elementary pairs and rep...
Figure 30.2 Nested purification with an array of elementary EPR pairs. ...
Chapter 31
Figure 31.1 A level structure of the model atoms with total angular mom...
Figure 31.2 Geometry of the experimental setup. A weak quantum light be...
Figure 31.3 Experimental setup. (a) Two cesium samples in glass cells a...
Chapter 32
Figure 32.1 Atomic energy level structure (top) and projection of the B...
Figure 32.2 (a) The absorption profile of an atomic frequency comb. The...
Figure 32.3 (a) Two‐level RASE. A
‐pulse inverts the population (i) ...
Chapter 33
Figure 33.1 Initial‐state probability
P
(
t
) as a function of time
t
in u...
Figure 33.2 Spontaneous decay of an initially excited qubit embedded in...
Figure 33.3 Spontaneous decay of an initially excited qubit embedded in...
Chapter 34
Figure 34.1 Diagrammatic representation of the excitations contributing...
Figure 34.2 Diagrammatic representation of the excitations contributing...
Figure 34.3 Diagrammatic representation of basic processes: representat...
Figure 34.4 A forbidden diagram: The diagram describes a process in whi...
Figure 34.5 (a)‐(c) Diagrammatic representation of the spontaneous phot...
Figure 34.6 A schematic setup: Two photons with initial states
and
Figure 34.7 Diagrams describing the scattering of two photons by a sing...
Figure 34.8 Two atoms coupled to a multimode radiation field: Photonic ...
Figure 34.9 Time dependence of the probability of exciting both atoms: ...
Chapter 35
Figure 35.1 Schematic diagram of a Mach–Zehnder interferometer.
Figure 35.2 Phase diagram representing a light field.
Figure 35.3 An interferometer with a coherent and a vacuum input state....
Figure 35.4 Phase resolution of interferometer with bright coherent inp...
Figure 35.5 Comparison of the phase resolution for a bright‐ and a vacu...
Figure 35.6 Phase resolution in quantum interferometry for different in...
Chapter 36
Figure 36.1 Linear ion trap setup with two ions. Radial confinement to ...
Figure 36.2 Simple quantum logic spectroscopy sequence. Shown are two l...
Figure 36.3 Quantum logic spectroscopy of the
transition in
. (a...
Figure 36.4 Quantum logic‐enabled internal state preparation scheme. Sh...
Figure 36.5 Quantum logic spectroscopy sequence used for optical clock ...
Figure 36.6 Absorption cross section for a trapped atom in the weak‐bin...
Figure 36.7 Illustration of photon recoil spectroscopy. (a) Starting fr...
Figure 36.8 Photon recoil spectroscopy of the
line. (a) Partial...
Figure 36.9 Quantum logic‐enabled nondestructive molecular state detect...
Figure 36.10 Quantum logic spectroscopy of an optical molecular transit...
Figure 36.11 Typical energy level shifts of the
state caused by...
Figure 36.12 Interferometric sequence for measuring small forces using ...
Chapter 37
Figure 37.1 Measuring the pointing direction of a laser beam with a qua...
Figure 37.2 (a) Light emitted by the process of spontaneous parametric ...
Figure 37.3 (a) Image without amplification, (b) amplified image in a p...
Figure 37.4 Two‐photon imaging of an unknown object: Light going throug...
Chapter 38
Figure 38.1 Parametric downconversion of a femtosecond comb.
Figure 38.2 Quadrature noise in the different supermodes (normalized to...
Figure 38.3 Experimental setup.
Figure 38.4 Sketch of a frequency‐multiplexed homodyne measurement setup.
Figure 38.5 Example of experimental result: difference between the expe...
Figure 38.6 Mean noise levels and uncertainties (dB) for each of the or...
Figure 38.7 Amplitude spectra of the successive principal modes.
Figure 38.8 Different four‐mode cluster states, and the corresponding a...
Figure 38.9 Nullifier squeezing values of various cluster states in dB ...
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