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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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∗-algebra, 194 C∗-Algebra, 193, 194 H2(· | ·), 154 ⊕MIP[2], 144 ⊕MIP∗, 141 ⊕MIPNL, 141 ⊕MIP∗[2], 144, 145 k-vector, 207 1-2 OT, 13 practical, 171 1-2 oblivious transfer, 13 accessible information, 38 adversary, 6 adversary model, 164 algebra ∗-algebra, 194 C∗-algebra, 194 Banach, 194 center, 198 commutant, 198 GNS construction, 196 ideal, 195 invertible, 195 observable, 195
positive linear functional, 195 pure state, 195 semisimple, 195 simple, 195 spectrum, 195 state, 195 state space, 196 unital, 195 anonymous transmissions, 18 anti-commuting observables, 83, 205 Banach algebra, 194 basis, 187 Breidbart, 169 computational, 26 Hadamard, 26 measurement in, 30 MUB, 39, 81 mutually unbiased, 39 basis vectors, 187 BB84, 3, 163 BC, 10 Bell inequalities, 110 Bell inequality, 106 CHSH, 107 binary entropy, 36 bipartite state, 27 mutual information, 36 bipartite structure, 200 bit commitment, 9, 10 impossibility, 155 limitations, 151
bit string commitment, 11 impossibility, 156 possibility, 159 241
242 Index Bloch sphere, 33 Bloch vector, 33 bounded operator, 188 bounded storage classical, 6 composability, 20 due to noise, 163 quantum, 11, 43, 163 Boykin, P. O., 18 Breidbart basis, 169 center, 198 cheat-sensitive, 8, 11, 31 CHSH inequality, 107, 123 generalized, 125 classical ensemble, 27 classical information, 31
classical mutual information, 36 Clifford algebra, 83, 205 generators, 205 geometrical properties, 206 product, 206 uncertainty relation, 83 Clifford product, 206 coin tossing strong, 15 weak, 16 collision entropy, 37 commitment bit, 10
quantum bit string, 152 commutant, 198
compatible, 198 complementary, 198 complete, 105
complex Hadamard matrix, 40 composability, 20 bounded storage, 20 computational basis, 26 computational security, 7 conditional entropy, 36 cone, 189 conjugate transpose, 188 convex cone, 189 CP map, 30 CPTP map, 31 cq-state, 166 cryptography, 3 data hiding, 17 density matrix, 25 density operator, 25 depolarizing channel, 174 depolarizing noise, 174 dishonest, 6 distance from uniform, 165 trace, 32 variational, 35 distinguishability, 32 disturb, 31 dual, 190 dual function, 190 duality, 190 strong, 191 weak, 190 E91, 4 eigendecomposition, 188 eigenvalue, 187 eigenvector, 187 encoding, 26 ensemble, 26 classical, 27 entanglement, 8, 105 entropy binary, 36 collision, 37 conditional, 36 joint, 36 min-entropy, 37 R´enyi, 37 Shannon, 36 von Neumann, 37 EPR, 105 pair, 106
Paradox, 105 state, 106 feasible, 190 fidelity, 35 games non-local, 110 optimal strategies, 121 superposition, 148 XOR, 112
XOR, winning probability, 143 generalized Pauli matrices, 41 GNS construction, 196 Hadamard basis, 26 Hadamard transform, 29 Hermitian, 188 hidden variables, 106 Hilbert space, 187 bounded operators, 194 Holevo quantity, 38 honest, 5 honest-but-curious, 5, 32 ideal, 195 impossibility bit commitment, 155 bit string commitment, 156 information
accessible, 38 mutual, 36
information theoretic security, 6 inner product, 187
instrument, 31
interactive proof system, 139 invertible, 195
IP, 140
joint entropy, 36
Jordan Wigner transformation, 205 K-transform, 29 Kraus operator, 30 Lagrangian, 190 Latin square, 39 mutually orthogonal, 39 left ideal, 195 LOCKCOM, 159 locking, 8, 93 application, 159
mutually unbiased bases, 96 protocol, 94 uncertainty relations, 95 measurement, 29 delay, 8 in a basis, 30 observable, 29 operators, 29 post-measurement state, 29 POVM, 30 projective, 29 min-entropy, 37 MIP, 140 MIP[2], 140 mixed state, 26 mixture, 26 MOLS, 39 MUB, 39
from Latin squares, 39 from Pauli matrices, 41 mutual information, 36
classical, 36
mutually unbiased basis, 39 from Latin squares, 39 from Pauli matrices, 41 no-cloning, 7, 27
noise
depolarizing, 174 noisy channel, 6
noisy quantum storage model, 163 noisy storage, 163
non-local games, 110 non-uniformity, 165 oblivious transfer, 163
244 Index 1-2, 13 practical, 171 Rabin, 13 observable, 29, 195 algebra, 195 anti-commuting, 83 compatible, 198 complementary, 198 operator norm, 188 orthogonal projector, 189 OT, 163 1-2, 13 Rabin, 13 outer product, 207 Pauli matrices, 29 generalized, 41 strings of, 41 perfect secrecy, 6 physical reality, 105 PI-STAR, 47, 116 algebraic framework, 63 AND, 57 gap, 116 lower bound, 54 no quantum memory, 54 perfect prediction, 63 quantum memory, 63 three bases, 70 to succeed at, 48 two bases, 66 XOR, 60 player, 5 positive definite, 189
positive linear functional, 195 positive semidefinite, 189 post-measurement information, 43, 47 algebraic framework, 63 AND, 57 gap, 116 lower bound, 54 no quantum memory, 54 perfect prediction, 63 quantum memory, 63 three bases, 70 two bases, 66 XOR, 60 post-measurement state, 29 POVM, 30 primal, 190 privacy amplification, 154, 155 private input, 5
private shared input, 5 private shared randomness, 5 projective measurement, 29 projector, 189 orthogonal, 189 public information, 5 pure state, 25 purification, 28 QBSC, 152 impossibility, 156 possibility, 159 QIP, 140 QIP(2), 141, 145 QKD, 3, 9
quantum bit string commitment, 152 quantum channel, 31
trace preserving, 30 unital, 31
quantum key distribution, 3, 9 quantum seals, 19
quantum signatures, 19 quantum strategy, 121 qubit, 25
RAC, 132
random access code, 132 unbalanced, 133
randomized oblivious transfer, 166 rank, 188
reduced state, 28 right ideal, 195 ROT, 166
SDP, 190 dual, 190 dual function, 190 duality, 190 example, 35, 57, 59, 61 feasible, 190 Lagrangian, 190 primal, 190 Slater’s conditions, 191 standard form, 190 strong duality, 191 weak duality, 190 secret classical, 17 quantum, 17 secret sharing, 17
secure function evaluation, 12 secure multi-party computation, 16 semidefinite program (SDP), 190 semidefinite programming, 187 semisimple, 195 decomposition, 199 SFE, 12 Shannon entropy, 36 simple, 195 SMP, 16 spectrum, 195 STAR, 47, 116 AND, 50 Boolean functions, 50 to succeed at, 48 XOR, 51 state algebra, 195 bipartite, 27 discrimination, 32
discrimination with post-measurement information, 43 EPR pair, 106 invariant, 202 mixed, 26 post-measurement, 29 pure, 25, 195 reduced, 28
von Neumann entropy, 37 Werner, 110
state discrimination, 47 AND, 50
Boolean functions, 50
with post-measurement informa-tion, 47
XOR, 51 state space, 196
string commitment, 151 string of Pauli matrices, 41 strong duality, 191 superoperator, 31 superposition game, 148 superselection rules, 7, 10 tensor product, 188 trace, 188 trace distance, 32 trace norm, 33 trace out, 28 traceless, 18 transpose, 188 trusted initializer, 6 Tsirelson bound, 108, 123 construction, 115, 122 vectors, 115 two-party protocols, 153 two-sided ideal, 195
two-universal hash function, 154 uncertainty, 75
uncertainty relation, 8, 75 Clifford algebra, 83 for all MUBs, 82
for anti-commuting observables, 83 for Latin square MUBs, 80
for MUBs in square dimensions, 79
for two bases, 78 Heisenberg, 75
246 Index
Maassen and Uffink, 78 meta, 89
unconditional security, 6
unforgeable subway tokens, 3, 19 unital algebra, 195 quantum channel, 31 unitary, 188 unitary evolution, 29 URAC, 133 variational distance, 35 verifier, 110
von Neumann entropy, 37 weak duality, 190
