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Cryptography in a quantum world
Wehner, S.D.C.
Publication date 2008
Link to publication
Citation for published version (APA):
Wehner, S. D. C. (2008). Cryptography in a quantum world.
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Contents
Acknowledgments xv
I
Introduction
1
1 Quantum cryptography 3
1.1 Introduction . . . 3
1.2 Setting the state . . . 5
1.2.1 Terminology . . . 5
1.2.2 Assumptions . . . 6
1.2.3 Quantum properties . . . 7
1.3 Primitives . . . 9
1.3.1 Bit commitment . . . 9
1.3.2 Secure function evaluation . . . 11
1.3.3 Secret sharing . . . 17
1.3.4 Anonymous transmissions . . . 18
1.3.5 Other protocols . . . 19
1.4 Challenges . . . 19
1.5 Conclusion . . . 20
II
Information in quantum states
23
2 Introduction 25 2.1 Quantum mechanics . . . 25 2.1.1 Quantum states . . . 25 2.1.2 Multipartite systems . . . 27 2.1.3 Quantum operations . . . 29 2.2 Distinguishability . . . 322.3 Information measures . . . 36
2.3.1 Classical . . . 36
2.3.2 Quantum . . . 37
2.4 Mutually unbiased bases . . . 39
2.4.1 Latin squares . . . 39
2.4.2 Generalized Pauli matrices . . . 41
2.5 Conclusion . . . 42
3 State discrimination with post-measurement information 43 3.1 Introduction . . . 43
3.1.1 Outline . . . 45
3.1.2 Related work . . . 46
3.2 Preliminaries . . . 47
3.2.1 Notation and tools . . . 47
3.2.2 Definitions . . . 47
3.2.3 A trivial bound: guessing the basis . . . 48
3.3 No post-measurement information . . . 49
3.3.1 Two simple examples . . . 49
3.3.2 An upper bound for all Boolean functions . . . 50
3.3.3 AND function . . . 50
3.3.4 XOR function . . . 51
3.4 Using post-measurement information . . . 54
3.4.1 A lower bound for balanced functions . . . 54
3.4.2 Optimal bounds for the AND and XOR function . . . 57
3.5 Using post-measurement information and quantum memory . . . 63
3.5.1 An algebraic framework for perfect prediction . . . 63
3.5.2 Using two bases . . . 66
3.5.3 Using three bases . . . 70
3.6 Conclusion . . . 72
4 Uncertainty relations 75 4.1 Introduction . . . 75
4.2 Limitations of mutually unbiased bases . . . 78
4.2.1 MUBs in square dimensions . . . 79
4.2.2 MUBs based on Latin squares . . . 80
4.2.3 Using a full set of MUBs . . . 80
4.3 Good uncertainty relations . . . 83
4.3.1 Preliminaries . . . 84
4.3.2 A meta-uncertainty relation . . . 89
4.3.3 Entropic uncertainty relations . . . 89
5 Locking classical information 93
5.1 Introduction . . . 93
5.1.1 A locking protocol . . . 94
5.1.2 Locking and uncertainty relations . . . 95
5.2 Locking using mutually unbiased bases . . . 96
5.2.1 An example . . . 96
5.2.2 MUBs from generalized Pauli matrices . . . 99
5.2.3 MUBs from Latin squares . . . 101
5.3 Conclusion . . . 101
III
Entanglement
103
6 Introduction 105 6.1 Introduction . . . 105 6.1.1 Bell’s inequality . . . 106 6.1.2 Tsirelson’s bound . . . 1086.2 Setting the stage . . . 109
6.2.1 Entangled states . . . 109
6.2.2 Other Bell inequalities . . . 110
6.2.3 Non-local games . . . 110
6.3 Observations . . . 113
6.3.1 Simple structural observations . . . 113
6.3.2 Vectorizing measurements . . . 115
6.4 The use of post-measurement information . . . 116
6.5 Conclusion . . . 119
7 Finding optimal quantum strategies 121 7.1 Introduction . . . 121
7.2 A simple example: Tsirelson’s bound . . . 123
7.3 The generalized CHSH inequality . . . 125
7.4 General approach and its applications . . . 128
7.4.1 General approach . . . 128
7.4.2 Applications . . . 129
7.5 Conclusion . . . 130
8 Bounding entanglement in NL-games 131 8.1 Introduction . . . 131
8.2 Preliminaries . . . 132
8.2.1 Random access codes . . . 132
8.2.2 Non-local games and state discrimination . . . 134
8.3 A lower bound . . . 134
8.5 Conclusion . . . 138
9 Interactive Proof Systems 139 9.1 Introduction . . . 139
9.1.1 Classical interactive proof systems . . . 139
9.1.2 Quantum multi-prover interactive proof systems . . . 140
9.2 Proof systems and non-local games . . . 142
9.2.1 Non-local games . . . 142
9.2.2 Multiple classical provers . . . 143
9.2.3 A single quantum prover . . . 145
9.3 Simulating two classical provers with one quantum prover . . . 145
9.4 Conclusion . . . 148
IV
Consequences for Crytography
149
10 Limitations 151 10.1 Introduction . . . 151 10.2 Preliminaries . . . 152 10.2.1 Definitions . . . 152 10.2.2 Model . . . 153 10.2.3 Tools . . . 15410.3 Impossibility of quantum string commitments . . . 156
10.4 Possibility . . . 159
10.5 Conclusion . . . 161
11 Possibilities: Exploiting storage errors 163 11.1 Introduction . . . 163
11.1.1 Related work . . . 165
11.2 Preliminaries . . . 165
11.2.1 Definitions . . . 165
11.3 Protocol and analysis . . . 170
11.3.1 Protocol . . . 170
11.3.2 Analysis . . . 170
11.4 Practical oblivious transfer . . . 171
11.5 Example: depolarizing noise . . . 174
11.5.1 Optimal cheating strategy . . . 175
11.5.2 Noise tradeoff . . . 183
11.6 Conclusion . . . 185
A Linear algebra and semidefinite programming 187
A.1 Linear algebra prerequisites . . . 187
A.2 Definitions . . . 189
A.3 Semidefinite programming . . . 190
A.4 Applications . . . 191
B C∗-Algebra 193 B.1 Introduction . . . 193
B.2 Some terminology . . . 194
B.3 Observables, states and representations . . . 195
B.3.1 Observables and states . . . 195
B.3.2 Representations . . . 196
B.4 Commuting operators . . . 198
B.4.1 Decompositions . . . 199
B.4.2 Bipartite structure . . . 200
B.4.3 Invariant observables and states . . . 202
B.5 Conclusion . . . 203
C Clifford Algebra 205 C.1 Introduction . . . 205
C.2 Geometrical interpretation . . . 206
C.2.1 Inner and outer product . . . 206
C.2.2 Reflections . . . 207 C.2.3 Rotations . . . 208 C.3 Application . . . 212 C.4 Conclusion . . . 215 Bibliography 217 Index 241 Symbols 249 Samenvatting 251 Summary 255
