1 Probability and information .................................. 1
1.1 Introduction ............................................ 1
1.2 Conditional probabilities ............................... 2
1.3 Entropy and information ................................. 7
1.4 Communications theory .................................. 17
Suggestions for further reading ............................. 26
Exercises ................................................... 27
2 Elements of quantum theory .................................. 31
2.1 Basic principles ....................................... 31
2.2 Mixed states ........................................... 37
2.3 Unitary operators ...................................... 43
2.4 Qubits ................................................. 45
2.5 Entangled states ....................................... 49
Suggestions for further reading ............................. 54
Exercises ................................................... 55
3 Quantum cryptography ........................................ 59
3.1 Information security ................................... 59
3.2 Quantum communications ................................. 66
3.3 Optical polarization ................................... 70
3.4 Quantum key distribution ............................... 76
Suggestions for further reading ............................. 83
Exercises ................................................... 84
4 Generalized measurements .................................... 89
4.1 Ideal von Neumann measurements ......................... 89
4.2 Non-ideal measurements ................................. 92
4.3 Probability operator measures .......................... 93
4.4 Optimized measurements ................................. 98
4.5 Operations ............................................ 106
Suggestions for further reading ............................ 111
Exercises .................................................. 112
5 Entanglement ............................................... 115
5.1 Non-locality .......................................... 115
5.2 Indirect measurements ................................. 121
5.3 Ebits and shared entanglement ......................... 125
5.4 Quantum dense coding .................................. 127
5.5 Teleportation ......................................... 129
Suggestions for further reading ............................ 136
Exercises .................................................. 136
6 Quantum information processing ............................. 141
6.1 Digital electronics ................................... 141
6.2 Quantum gates ......................................... 144
6.3 Quantum circuits ...................................... 148
6.4 Quantum error correction .............................. 153
6.5 Cluster states ........................................ 158
Suggestions for further reading ............................ 160
Exercises .................................................. 161
7 Quantum computation ........................................ 165
7.1 Elements of computer science .......................... 165
7.2 Principles of quantum computation ..................... 169
7.3 The quantum Fourier transform ......................... 175
7.4 Shor's factoring algorithm ............................ 181
7.5 Grover's search algorithm ............................. 185
7.6 Physical requirements ................................. 189
Suggestions for further reading ............................ 191
Exercises .................................................. 192
8 Quantum information theory ................................. 197
8.1 The von Neumann entropy ............................... 197
8.2 Composite systems ..................................... 202
8.3 Quantitative state comparison ......................... 205
8.4 Measures of entanglement .............................. 211
8.5 Quantum communications theory ......................... 214
Suggestions for further reading ............................ 227
Exercises .................................................. 227
A The equivalence of information and entropy .............. 231
В Lagrange multipliers .................................... 235
С Stirling's approximation ................................ 239
D The Schmidt decomposition ............................... 241
E Number theory for cryptography .......................... 243
E.l Division properties ................................ 243
E.2 Least common multiple and greatest common
divisor ............................................ 243
E.3 Prime numbers ...................................... 244
E.4 Relatively prime integers and Euler's
(φ-function ........................................ 245
E.5 Congruences ........................................ 245
E.6 Primitive root modulo p ............................ 245
E.7 Diffie-Hellman cryptosystem ........................ 246
E.8 RSA cryptosystem ................................... 246
F Quantum copying ......................................... 249
G Quantized field modes ................................... 253
H Position and momentum eigenstates ....................... 257
I Necessary conditions for a minimum-error POM ............ 261
J Complete positivity ..................................... 263
К Hardy's theorem ......................................... 269
L Universal gates ......................................... 271
M Nine- and five-qubit quantum codewords .................. 275
N Computational complexity ................................ 277
0 The Bernstein-Vazirani algorithm ........................ 279
P Discrete Fourier transforms ............................. 283
Q An entropy inequality ................................... 285
R Quantum relative entropy ................................ 287
S The Araki-Lieb inequality ............................... 289
T Fidelity for mixed states ............................... 291
U Entanglement of formation for two qubits ................ 295
Index ......................................................... 297
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