Index

A loophole-free tests, 224 Abelian Fourier transform, 320 non-locality, 72 Abelian groups, 310, 331 three variables, 69 Accessible information, 296 states, 105, 114, 240, 298 σ-additive measures, 80 Bernoulli sources, 46 Adiabatic theorem, 340 Binary entropy, 181 Affine maps, 89 Bi-photons, 226 σ-algebras, 12, 80 phase-momentum entanglement, 226 Algebras Birefringence, 227, 229, 230, 245 bicommutant, 84 Bit, 310 center, 85 string, 310 commutant, 83 scalar product, 310 commutative (Abelian), 83 Black box model, 317 dual, 89 Bloch equations, 259 factors, 85 Bloch sphere, 13, 87, 260, 285 matrix, 82 dynamics, 260 opposite, 92 Bloch vector, 260 unital, 81 Block positivity, 123 Algorithms Block-coding, 47 efficient, 311 Block-matrix representation, 22 Ancilla systems, 110, 187, 283 Boltzmann principle, 169 qubits, 314 Boolean functions, 312 Annihilation operators, 153, 225 Borel subsets, 12 Anti-unitary transformations, 171 Bose-Einstein condensation, 171 Aspect experiment, 70 Bose-Einstein statistics, 170 Asymptotic equipartition property (AEP), 51 Bosonic fields, 159 Atom-atom entanglement, 269 Bosons, 165, 225 Atom-field Hamiltonian, 264 Bound entanglement, 305 B92 protocol, 280, 286 B BB84 protocol, 243, 280 entanglement-based, 295 C Beam splitter, 225 Calderbank-Steane-Shor (CSS) code, 294 polarizing (PBS), 226 Canonical anti-commutation relations (CAR), quantum, 226 173 Bell Canonical commutation relations (CCR), 152 inequalities, 65, 220, 285 Canonical ensemble, 169 detection loophole, 224 Canonical quantization four variables, 71 CAR, 165

F. Benatti et al.: Index. Lect. Notes Phys. 808, 343–350 (2010) DOI 10.1007/978-3-642-11914-9 c Springer-Verlag Berlin Heidelberg 2010 344 Index

CAR relations, 172 Coherent attacks, 290 CCR, 165 Coherent control, 255 Cauchy-Schwartz inequality, 2 Coherent states, 155, 244 operations, 96 Bosons, 167 operators, 28 classical behavior, 155 Cavity QED, 269 exponential vectors, 166 CCR, 225 overcompleteness, 155 Channels Posissonian distribution, 155 capacity, 56, 184 squeezed, 157, 244 discrete, 182 Coherent superpositions, 253 Gaussian, 182 Communication channels, 54 memoryless, 54, 182 Completely positive maps, 22, 90, 110, 194 symmetric, 55 Completely positive trace-preserving (CPT) noisy, 54 maps, 117, 178 quantum, 194 Complexity classes, 312 Chebishev inequality, 48, 52 NP-complete problems, 312 Chemical potential, 170 NP problems, 312 Choi matrix, 121, 140 P problems, 312 CHSH inequality, 285 Compression rate Classical algorithms quantum, 188 deterministic, 318 Computational basis, 311 probabilistic, 325 Conditional entropy, 38, 303 Classical circuits, 312 Conditional probabilities, 38, 182 Classical-classical-quantum (ccq) states, 296 Contra-variant picture, 99 Classical computation Controlled-NOT operation (CNOT), 114 universal, 313 Control-qubit, 237 Classical computers, 309 Convergence in probability, 51 Classical information, 66 Convex hull, 78 bit, 66 Convex set, 78, 111, 178 Classical probability Correlation matrix, 171 commuting projections, 76 Coupling to classical sources, 166 Classical-quantum (cq) states, 296 Coupling to heat baths, 166 Clauser-Horn inequality, 222 Covariant picture, 102 Clauser-Horne-Shimony-Holt (CHSH) Co-prime integers, 329 inequality, 224, 285 Creation operators, 153, 225 Cloning strategies, 291 Cryptography, 278 CNOT ciphertext, 279 destructive, 237 cryptosystems, 278 gate, 107, 232, 235, 272 plaintext, 279 inverted, 315 quantum, 280 matrix, 107, 114 Cryptosystems probabilistic, 237 Vernam cipher, 279 transformation, 107 Codes, 47 D classical Data compression, 48, 181 linear, 293 quantum, 187 code-words, 47, 183 Decoders, 45 length, 55 Decoding rules, 58 CSS, 292 Decoherence, 135, 177, 195, 228, 255 dual, 294 Decoherence-free fixed length, 47, 181 states, 248 variable length, 47, 181 subspaces, 255 Coherences, 257 Decoherence-free propagation, 247 Index 345

Decoy states, 302 EPR Dense coding, 67, 108, 240 Bohm’s formulation, 221 Density matrices, 13, 77, 111, 177 elements of reality, 221 reduced, 18, 112, 178 EPR pairs, 242 Dependencies, 94 Error correction, 281, 282, 293 Deutsch circuit, 318 parity check matrix, 293 Diffusion transform gate, 334 quantum, 294 Dilation, 89 syndrome, 293 Distillable key rate, 306 Error probability, 55 Distinguishability, 284 Errors Bell states, 233 quantum, 195 classical states, 176 locally independent, 196 Holevo χ-quantity, 210 sequentially independent, 196 orthogonal states, 177 Events, 34, 80 Double commutant theorem, 84 independent, 34 Dual maps, 89, 121 Extremal points, 78, 111, 178 Duality, 121 F Factoring algorithms, 309 E polynomial time, 310 Eavesdropping strategies, 284, 288 Factoring problem, 279 Effective Hamiltonian, 218, 272 Fano inequality, 41 Efficiently solvable problems, 311 generalized, 43 Einstein-Podolski-Rosen argument (EPR), Fast Fourier transform (FFT), 321 207, 221 Fermi-Dirac statistics, 174 Ekert protocol, 243, 280, 285 Fermions, 165, 172 Electromagnetically induced transparency thermal states, 174 (EIT), 234 Fidelity, 29, 188, 246, 289, 297 Electromagnetic field, 164 Flip operator, 124 Hamiltonian density, 216 Floquet picture, 267 Embedding theorem, 97 Fock number states, 157 Encoders, 45 Fock space, 161 Entangled photons, 215 Fock states, 263 energy-time, 226 Fourier sampling, 332 phase-momentum, 219 polarization, 229 G time-bin, 228 Garbage removal, 314 Entanglement, 89, 109, 256 Gaussian states, 171 bipartite, 109 entangled, 171 bound, 131 GHZ states, 127 cost, 132 polarization entangled, 231 detection, 129 Gibbs inequality, 39 distillation, 131, 292 Glauber-Sudarshan representation, 157 EPR pairs, 207 Gleason theorem, 81 multipartite, 127 Gran-canonical ensemble, 170 witness, 124, 305 Greatest common divisor, 329 Entanglement cost, 304 Group Entanglement of formation, 133, 304 Abelian, 319 Entanglement measures, 133, 307 character, 320 concurrence, 133 Entropy, 34 H concavity, 35 Hadamard differential, 36 gate, 107, 242, 315 Environment, 20, 110, 177, 195 matrix, 107, 114 346 Index

product, 23 K transformation, 107, 281, 315 Kerr crystals, 234 Hamiltonian, 20 Kossakowski matrix, 135 Heisenberg canonical quantization, 152 Kraus operators, 22, 136, 198 Heisenberg picture, 90, 135, 153 Kraus representation, 22, 89, 110 Heisenberg principle, 98, 100, 101 Heisenberg uncertainty relations, 156 L Heralded single photon sources, 219 Lattice states, 145 Hidden subgroup problem, 331 LHV theories Hidden variables, 65 realistic, 222 Hilbert-Schmidt scalar product, 28 Linear operators, 4 norm, 28 adjoint, 5, 152 Hilbert spaces, 1, 80 eigenvalues, 8, 153 algebraic tensor products, 6 eigenvectors, 8, 153 antisymmetric tensor products, 7, 173 matrices, 4 bra-ket notation, 2 modulus, 9 canonical basis, 3 norm, 4, 81 dual, 92 partial isometries, 94 norm, 2 positive, 9, 77, 153 orthogonal expansions, 4 projections, 4, 75 orthogonal vectors, 3 self-adjoint, 5 orthonormal bases, 3 spectrum, 8, 103, 153 scalar product, 1, 81 square root, 9 symmetric tensor products, 8, 162 support projection, 27 tensor products, 6, 311 unitary, 5 Holes, 174 Linear optics, 235 Holevo χ Liouville-von Neumann equation, 134 quantity, 202 Local hidden variables (LHV) theories, 221 bound, 202 Locality, 72 Holevo capacity Local realism, 72, 220 additivity, 211 LOCC operations superadditivity, 212 dilution, 132 Holevo-Schumacher-Westmoreland (HSW) distillation, 131 theorem, 210 Logical gates, 312 Homodyne detection, 244, 248 AND, 312 I CNOT, 315 Individual attacks, 288 FANOUT, 314 Information, 34 NAND, 312 Information of formation, 304 NOT, 312 Information rate, 49 OR, 312 Information sources, 45 quantum π/ memoryless, 180 8, 316 quantum, 187 one-qubit, 316 Intrinsic information, 303 two-qubit gates, 316 Isotropic states, 125 Toffoli, 313 universal, 313 J Log-likelihood, 50 Jaynes-Cummings interaction, 266 Jaynes-Cummings model, 262 M Jensen inequality, 28 Mach-Zender interferometers, 226 Joint entropy, 37 Franson scheme, 226 Joint probabilities, 37 Macroscopic scale, 255 Jordan-Wigner transformation, 173 Marginal distributions, 18, 38 Index 347

Markov sources, 46 Operator-sum representation, 21 stationary, 46 Oracles, 317 Master equation, 135 Lindblad form, 135 P Matrix convexity, 11 Parallelogram law, 3 Matrix monotone functions, 10 Parameter estimation, 282, 286 Maximally entangled states, 105, 116, 292 Parametric down conversion (PDC), 215 Measurable subsets, 12, 80 conjugated photons, 217 Measurement, 282 type II, 218 Measurement processes, 100 quantum, 15, 110, 311 Partial isometries accessible information, 200 final space, 94 joint, 104 initial space, 94 partial, 311 Partial trace, 112, 178, 239 POVM, 15, 179, 286, 312 Partial transposition, 124 von Neumann, 102 Particles, 174 Mesoscopic scale, 255 Partition sums, 169 Momentum representation, 156 Pauli matrices, 14, 86, 173, 195, 264, 281 Monotonicity Pauli operators, 197 of fidelity, 30 PDC Multiplication theorem, 97 bi-photon, 216 Mutual entropy idler photon, 216, 218 conditional, 303 pump photon, 218 Mutual information, 38, 184, 207, 289 signal photon, 216, 218 Mutually unbiased orthonormal basis, 287 type I, 218 type II N Hamiltonian, 246 No-classical coding, 100 Peres criterion, 124 No-cloning, 67, 237, 238, 280 Peres-Horodecki criterion, 127 theorem, 99 Phase, 13 No-faster than light transmission, 237 Phase-entanglement, 228 Noisy channels Phase-transitions, 171 coding, 54, 181 Phonons, 165, 272 error correction, 54, 182 Photon number splitting (PNS) attacks, 301 Non-classical states, 157 Photons, 68, 86 Non-linear optics, 234 Polarizers, 68 Non-local effects, 227 Polarizing beam splitter (PBS), 235 Normal ordering, 153 Pooling inequality, 40 NPT states, 124 Position representation, 156 Number operator, 153 Positive maps, 110 decomposable, 128 O Positive operator-valued measures (POVM), 12 Observables, 14 Positivity under partial transposition (PPT), expectation values, 16, 222 124, 128 local, 111 Post-selection, 228 self-adjoint operators, 14 POVM Operation elements, 21 Kraus representation, 92, 110, 198 Operations projection-valued measures, 12, 312 classical, 88 PPT entangled states, 129 coding, 100 PPT states, 124 decoding, 100 Preparation, 282 LOCC, 131 Privacy amplification, 282, 286 quantum, 90, 110, 117, 178 Probability density, 36 348 Index

Probability distributions, 34, 80 Quantum information sources discrete, 34 memoryless, 189 expectations, 34, 68 Quantum joint entropy, 185 Probability space, 66 Quantum key distribution (QKD), 243, 280 Projective measurement, 239 Quantum measurement process, 98 Quantum metrology, 219, 247 Q Quantum mutual information, 185 Quadrature operators, 157, 248 discarding systems, 186 Quantum algorithms, 309 quantum operations, 187 adiabatic, 340 Quantum noise, 136, 255 Deutsch-Josza, 310 Quantum novelties, 104, 237 Grover, 310 Quantum operations unstructured search, 334 n-fold controlled, 316 Shor, 309 Quantum phase gate Simon, 325 controlled, 233, 273 Quantum channels, 66, 179, 281 Kerr controlled, 233 amplitude damping, 199 Quantum probability, 66 bit-flip, 198 Quantum probability spaces, 81 classical capacity, 208 Quantum registers, 255 depolarizing, 25, 199 scalability, 255 Holevo capacity, 210 Quantum relative entropy, 27, 117 memoryless, 194 joint convexity, 117 noisy, 179 monotonicity, 117 Pauli, 24 positivity, 117 product state capacity, 210 Uhlmann monotonicity, 27, 205 Quantum circuits, 312 Quantum states adiabatic quantum algorithms, 340 entangled, 111 initialization, 312 expectation functionals, 81 measurement process, 312 expectation values, 16, 178 unitary dynamics, 312 mixed, 14, 77, 111, 177 Quantum coin toss, 86 pure, 13, 77, 178 Quantum complex amplitude, 152 separable, 111 Quantum computation, 20, 67 Quantum swapping, 243 Quantum computers Quantum systems quantum circuit models, 309 composite, 16, 111, 185, 255 quantum Turing machines, 309 open, 110, 177, 255 universal, 309 Quantum walks, 338 Quantum conditional entropy, 185 Qubit, 13, 254, 282, 309 Quantum copier, 98 double-well potentials, 254 Quantum cryptography, 66 energy eigenstates, 254 entanglement-based protocols, 280 logical, 254 prepare-and-measure protocols, 280 photonic, 234 Quantum errors, 244 polarization states, 75, 176, 254 Quantum fields Qudits, 244 local structure, 163 Query complexity, 317 smeared operators, 163 Ququats, 244 Quantum Fourier transform (QFT), 322 Qutrits, 244 Quantum gates, 20 Quantum harmonic oscillator, 152 R Quantum impossibilities, 98, 237 Rabi oscillations, 259 Quantum information, 66 vacuum, 267 qubit, 66 Random variables (RV), 34 Quantum information processing, 255 binary, 75 Index 349

conditionally independent, 41 State transformations, 20, 110 discrete, 37 convex set, 22 IID, 180 extremal points, 22 weak law of large numbers, 52 Statistical operators, 14 IID sequences, 52 Stimulated absorption, 264 Independent Stimulated emission, 264 Identically Distributed (IID), 46 Stinespring theorem, 92 independent, 39 Stochastic equivalence principle, 89 sequences, 51 Stochastic processes, 66 vectors, 43 Stochastic variables, 111 Randomness, 37 Subgroup Reduced density matrices, 239 cosets, 331 Reduced dynamics, 110 Super-selection rules, 165 Relative entropy, 38 Symplectic matrix, 171 Relative entropy of entanglement, 307 Relative modular operator, 28 T Reliable transmission, 45 Teleportation, 67, 105, 233, 241 Rivest-Shamir-Adleman (RSA) protocol, 279 Tensor products of Hilbert spaces, 78 Rotating-wave approximation (RWA), 258 Tensor products of operators, 79 RSA protocol, 309 Thermal states, 169 Thermodynamic limit, 170 S Time development, 19 Schatten decomposition, 26 Time-ordering, 168 Schmidt coefficients, 112 Totally symmetric state, 115 Schmidt decomposition, 15, 112 Trace, 10, 178, 239 Schrödinger picture, 20, 99, 135 cyclicity, 10, 204 Schumacher’s quantum coding theorem, 189 partial, 19 Schur lemma, 320 Trace map, 128 Schwartz positivity, 96 Trace-norm, 27 Secret key establishment, 282 Trace-preserving maps, 22 Secret key rate, 303 Transmission channels, 45 Secure keys, 292 Transmission rates, 55 Shannon entropy, 37, 115, 180, 288 quantum, 184 Subadditivity, 40 reliable, 55 Shannon’s first coding theorem, 51, 180 Transposition, 18, 90, 110 Shannon’s second coding theorem, 59, 184 Trapped cold atoms, 272 Shor’s algorithm, 233 Trapped ions, 158, 272 Sifting, 282 Typical sequences, 190 Simon criterion, 171 Typical subspaces, 189 Simplex, 78 theorem, 190 Six state protocol, 287 Source coding, 45 U Space of states, 78, 89 Uncertainty relations, 157 Spectral decomposition, 14 Unconditionally secure protocols, 297 Spectral measures, 13 Unitary propagator, 20 Spectral theorem, 13 Universal quantum computation Spin-Boson Hamiltonian, 254 reversible, 314 Squeezed states, 157 Universal quantum gates, 314 Stark shift, 259 State-dependent phase-shift, 240 V State-flip, 240 Vacuum, 225 State purification, 113 Vacuum state, 153 State space, 111 squeezed, 157 350 Index

Von Neumann entropy, 26, 37, 115, 170, 185 subadditivity, 118 axiomatic characterization, 26 triangle inequality, 118 concavity, 211 conditioning, 186 W continuity, 27 Werner states, 130 convexity, 118 Weyl operators, 154 mixing property, 26 Weyl relations, 154 strong subadditivity, 29, 118, 186, 203 Woronowicz theorem, 128