Furusawa A., van Loock P. Quantum Teleportation and Entanglement: A Hybrid Approach to Optical Quantum Information Processing

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Akira Furusawa and Peter van Loock

Quantum Teleportation and

Entanglement

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Akira Furusawa and Peter van Loock

Quantum Teleportation and Entanglement

A Hybrid Approach to Optical Quantum Information

Processing

WILEY-VCH Verlag GmbH & Co. KGaA

The Authors

Prof. Akira Furusawa

The University of Tokyo

Department of Applied Physics

Tokyo, Japan

akiraf@ap.t.u-tokyo.ac.jp

Dr. Peter van Loock

Physik Institut LS für Optik

Institut Theorie I

Universität Erlangen-Nürnberg

Max-Planck-Institut (MPL)

Erlangen, Germany

peter.vanloock@mpl.mpg.de

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Contents

Preface IX

Part One Introductions and Basics 1

1 Introduction to Quantum Information Processing 3

1.1 Why Quantum Information? 5

1.2 States and Observables 10

1.2.1 Qubit 12

1.2.2 Qumode 13

1.3 Unitaries 18

1.3.1 Qubit 19

1.3.2 Qumode 20

1.4 Non-unitaries 23

1.4.1 Channels 23

1.4.2 Measurements 27

1.4.2.1 POVM 27

1.4.2.2 Naimark Extension 29

1.5 Entanglement 31

1.5.1 Pure States 31

1.5.1.1 Qubits 33

1.5.1.2 Qumodes 33

1.5.2 Mixed States and Inseparability Criteria 34

1.5.3 Entanglement Witnesses and Measures 35

1.6 Quantum Teleportation 38

1.6.1 Discrete Variables 39

1.6.2 C ontinuous Variables 40

1.7 Quantum Communication 41

1.7.1 Ke y Distribution 42

1.7.2 R epeaters and Relays 45

1.7.3 Shannon Theory 47

1.8 Quantum Computation 49

1.8.1 Models 52

1.8.2 Universality 54

1.8.2.1 Qubits 54

Quantum Teleportation and Entanglement. Akira Furusawa, Peter van Loock

ISBN: 978-3-527-40930-3

VI Contents

1.8.2.2 Qumodes 57

1.9 Quantum Error Correction 62

1.9.1 Discretization 63

1.9.2 Stabilizer Codes 67

1.10 Experiment: Non-optical Implementations 70

2 Introduction to Optical Quantum Information Processing 79

2.1 Why Optics? 80

2.2 Quantum Optical States and Encodings 82

2.2.1 Field Quantization 83

2.2.1.1 Discrete Modes 84

2.2.1.2 Continuous Modes 87

2.2.2 Quadratures 88

2.2.3 Coherent States 89

2.2.4 Squeezed States 90

2.2.5 Phase-Space Representations 94

2.2.6 Photonic Qubits 98

2.2.7 Experiment: Polarization Qubits 99

2.2.8 Photonic Qumodes 100

2.2.8.1 Stabilizer States 101

2.2.8.2 Gaussian States 102

2.2.9 Experiment: Broadband Qumodes 107

2.3 Quantum Optical Unitaries 109

2.4 Gaussian Unitaries 113

2.5 Quantum Optical Non-unitaries 114

2.5.1 Channels 114

2.5.2 Measurements 116

2.6 Gaussian Non-unitaries 117

2.7 Linear Optics: Possibilities and Impossibilities 120

2.8 Optical Quantum Computation 121

Part Two Fundamental Resources and Protocols 125

3 Entanglement 127

3.1 Qubit Entanglement 128

3.1.1 Characterization and Witnesses 128

3.1.1.1 Two Parties 128

3.1.1.2 Three or More Parties 129

3.1.2 Cluster and Graph States 131

3.1.3 Experiment: Entangled Photonic Qubits 132

3.1.3.1 EPR/Bell State 132

3.1.3.2 GHZ State 135

3.1.3.3 Cluster States 138

3.2 Qumode Entanglement 142

3.2.1 Characterization and Witnesses 142

3.2.1.1 Two Parties 142

Contents VII

3.2.1.2 Three or More Parties 149

3.2.2 Cluster and Graph States 154

3.2.3 Experiment: Entangled Photonic Qumodes 160

3.2.3.1 Frequency-Domain EPR-Type State 160

3.2.3.2 Time-Domain EPR-Type State 161

3.2.3.3 GHZ-Type State 167

3.2.3.4 Cluster-Type States 172

4 Quantum Teleportation 179

4.1 Qubit Quantum Teleportation 180

4.1.1 Experiment: Qubit Quantum Teleportation 180

4.1.2 Experiment: Qubit Telecloning 183

4.1.3 Experiment: Qubit Entanglement Swapping 185

4.2 Qumode Quantum Teleportation 188

4.2.1 Experiment: Qumode Quantum Teleportation 189

4.2.2 Experiment: Qumode Telecloning 196

4.2.3 Experiment: Qumode Teleportation N e twork 203

4.2.4 Experiment: Qumode Entanglement Swapping 210

5 Quantum Error Correction 217

5.1 The Nine-Qubit Code 218

5.2 The Nine-Qumode Code 219

5.3 Experiment: Quantum Error Correction 220

5.3.1 Qubits 220

5.3.2 Qumodes 223

5.4 Entanglement Distillation 231

5.5 Experiment: Entanglement Distillation 233

5.5.1 Qubits 233

5.5.2 Qumodes 236

Part Three Measurement-Based and Hybrid Approaches 243

6 Quantum Teleportation of Gates 245

6.1 Teleporting Qubit Ga tes 246

6.1.1 KLM 246

6.1.2 Experiment: Qubit Gates 248

6.2 Teleporting Qumode Gates 252

6.2.1 Experiment: Gaussian Qumode Gates 252

6.2.1.1 Universal Squeezer 252

6.2.1.2 Quantum Non-demolition (QND) Sum Gate 259

6.2.2 Universal Qumode Gates 266

7 Cluster-Based Quantum Information Processing 271

7.1 Qubits 272

7.1.1 Elementary Qubit Teleportations 272

7.1.2 Experiment: Qubit Cluster Computation 273

7.2 Qumodes 280

VIII Contents

7.2.1 Elementary Qumode Teleportations 281

7.2.2 Gaussian Computation 282

7.2.3 Experiment: Gaussian Qumode Cluster Computation 283

7.2.3.1 Quadratic Phase Gate 283

7.2.3.2 Fourier and Squeezing Gates 291

8 Hybrid Quantum Information Processing 299

8.1 How to Create Non-Gaussian States, Cat States 300

8.2 Experiment: Creation of Non-Gaussian States, Cat States 303

8.3 Hybrid Entanglement 310

8.4 Hybrid Quantum Teleportation 312

8.4.1 Experiment: Broadband Qumode Teleportation

of a Non-Gaussian Wavepacket 313

8.4.2 Experiment: Broadband Qumode Teleportation

of a Polarization Qubit 314

8.5 Hybrid Quantum Computing 315

8.5.1 Hybrid Hamiltonians 315

8.5.2 Encoding Qubits into Qumodes 317

8.5.3 GKP 318

References 323

Index 333

Preface

The ﬁeld of quantum information processing has reached a remarkable maturi-

ty in recent years with regard to experimental demonstrations. In particular, to-

wards an extension of optical communications from the classical into the quantum

realm, many proof-of-principle experiments were performed including the genera-

tion and distribution of photonic entangled states over free-space or ﬁber channels.

As an application, unconditionally secure quantum key distribution systems have

emerged and even developed into a commercially available technology.

Light systems, apart from their obvious usefulness for communication, have now

as well turned out to be a serious contender for approaches to quantum compu-

tation. A breakthrough in this context was the theoretical discovery of so-called

measurement-based models: quantum algorithms no longer depend on sequences

of reversible quantum gates, each enacted through well controlled interactions be-

tween, for instance, two or more qubits; instead, sequences of measurements on

parts of an entangled resource state prepared prior to the computation will do the

trick. In other words, quantum entanglement, already known to be a universal re-

source for quantum communication in conjunction with quantum teleportation,

represents a universal resource for quantum computation too – and again the ex-

ploitation of the entangled resource relies upon quantum teleportation which, in

its ultimate form, achieves arbitrary quantum state manipulation s.

The aim of this book is to give a fairly general introduction to two com-

plementary approaches to quantum information processing: those based upon

discrete-variable “qubit” systems and those utilizing quantum oscill ator systems

(“qumodes”) most naturally represented by continuous quantum variables such

as amplitude and phase. In quantum optics, the corresponding photonic sys-

tems would consist of just a few photons or they would correspond to ﬁelds with

extremely high mean photon numbers, respectively. The qubit may then be repre-

sented by the polarization of a single photon, while a qumode state is encoded into

an inﬁnite-dimensional phase space. Entangled states can be deﬁned, formulated,

and experimentally realized in either dimension, including their use for quantum

teleportation. Since either approach encounters somewhat different complications

when it comes to more sophisticated quantum information protocols, a recent

trend in optical quantum information is to combine the two approaches and to

exploit at the same time discrete and continuous degrees of freedom in a so-called

Quantum Teleportation and Entanglement. Akira Furusawa, Peter van Loock

ISBN: 978-3-527-40930-3