859

Index a atomic kaleidoscope 680 Abelian symmetry 629–630 atomic spin operator 704 absolutely maximally entangled states atomic trajectory 679 317 atom–light interaction 802–805 absorption 691, 810–811 atoms in a cavity 661 accuracy 805, 814 attenuated laser pulses 360 active transformations 48 Aunt Martha 369 additivity 224 authentication 357 adiabatic approximation 477, 478 axioms of probability 405 adiabatic computation 104–105 adiabatic theorem 105 alphabet 403 b Anderson localization 599 base norm 226 angular momentum 635 BB84 protocol 278, 355, 369 annihilation operator 196 BBPSSW protocol 242 anti-Jaynes–Cummings Hamiltonian beam splitters 199, 201, 371, 409–410, 488 681 approximate quantum cloning Bell asymmetric cloning 69–70 basis 241 experimental quantum cloning inequality 149 70–71 no-cloning theorem 56–57 measurement 438, 718 phase-covariant cloning 63–65 Bell inequalities probabilistic cloning 70 communication complexity reduction state-dependent cloning 57–63 285–286 universal cloning 65–69 Bell states 242, 336, 424 area law 311 analysis 377 asymmetric cloning 69–70 basis 376 asymptotic activation effect 273 measurement 336–337 asymptotic continuity 223 Berry phase 477 asymptotic superactivation 277, 284 binary symmetric channel (BSC) 9 atom-cavity microscope 680 binding entanglement channel atom chip 684 281–282 atomic ensemble 702, 724, 731 binomial coefficient 64 atomic frequency comb 725–727 bipartite 195

Quantum Information: From Foundations to Quantum Technology Applications. Edited by Dagmar Bruß and Gerd Leuchs. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA. 860 Index

bipartite entanglement distillation c defined 267 Calderbank–Shor–Stean code (CSS) LOCC operations 265–267 358 bipartite system 233 Caley’s hyperdeterminant 300 biseparable 308 canonical commutation relations 45 bit-flip error 116 canonical transformations 717 Bloch ball 42 Cartan decomposition 653 Bloch band 497 Cauchy–Schwarz inequality 203 Bloch equation 634 cavity 678, 698 Bloch–Messiah reduction 846 lifetime 678 Bloch sphere 417 QED 572 Bloch vector 42 CD-decomposition 464 centraliser 610, 634 block code 10 channel 75 Boolean function 404, 432 binary symmetric 9 bosonic system 613, 614 classical 83–84 bosons, Gaussian state 314 covariant 88 bound entanglement dephasing 113 activation effects 284–285 depolarizing 87, 112 applications 275–277 discrete 4 asymptotic activation 277–278 entanglement breaking 87 asymptotic irreversibility 272 expansion 77 asymptotic regime 277 forgetful 86 Bell inequalities and communication Gaussian 89 complexity reduction 285–286 Heisenberg picture 77 binding entanglement channel ideal 86–87 281–282 with memory 83–84 bipartite 238 memoryless 85 continuous variables 287 noiseless 4 enhanced probabilistic quantum noisy 4 teleportation 276–277 noisy time evolution 77–78 feedback to classical theory product 113 286–287 restriction 77 information phenomenon 281 Schrödinger picture 77 multipartite 248 unitary time evolution 77 quantum cryptography 278–280 channel capacity remote quantum information discrete memoryless 8 concentration 285 discrete noiseless 5 states 206 quantum 114 bound information 281 channel noise 358, 691 bound states 600 characteristic function 46 branching diagram 610, 613 charge qubit 644, 656 breeding 246 Choi–Jamiolkowski isomorphism Brownian motion 593 78–80, 277 Bures distance 219 Cirac–Zoller 95, 490 bus mode 490 classical–classical state 180 by-product operator 452 classical Gaussian noise 50 Index 861 classical regime 811 conditional variance 348 Clifford gates 428 connection 693 Clifford group 453, 459–461 constrained optimisation 616 cluster state 249, 449, 838 continuous variables 195 three-qubit 313 entanglement 339, 340 two-dimensional 314 quantum teleportation 343–347 cluster states 849 systems 384–386 CNOT gate 653, 656, 676 control engineering 607 multiply controlled 653 controllability 607, 628 C-numerical range 614 pure state 628 code strong 628 block 10 symmetry condition 612 CSS 122 controllability criteria 608 dual 14 controlled NOT 407, 429 Hamming 14–15 operation 242 quantum 114 controlled reversible inhomogeneous rate 10 broadening 724 repetition 10 controlled Sign gate (CSIGN) 445 stabilizer 123 controlled unitary gate cU 658 coherent attack 360 controlled-Z-gate 313 coherent evolution 593 control system 608 coherent pulses 372 bilinear 608, 613, 620 coherent quantum control 493 infinite-dimensional 626 coherent spin state 705, 712 linear 608 coherent states 45, 47, 783–786 control theory 651 coherent superposition 677 convex roof construction 224 collective attack 360 Cooper pair boxes 684 collective controls 633 correlation matrix 198, 199 coloring 27–28 correlation operator 241, 249 commutant 610 coset 651, 653 commutation relations leader 15 angular momentum 704 coupling topology 653–655 commutator 196 covariance matrix 136, 315 compact Lie algebra 614 criterion of separability 138 completely positive map 75–78 creation operator 207 Gaussian 716 cross norm 227 trace preserving 76 criterion of separability 135 unital 77 cryogenic environment 673 complete positivity 44 cryptography 33 composability 357 CSS code 122 compositeness 26–27 composition 44 concurrence 154 d conditional entropy 8, 223 decoherence 411, 597, 677, 716 conditional probability distribution free coding 377 182 DEJMPS protocol 243 862 Index

dense coding capacity 160 entangled beams 829 density matrix 427 entangled state 712 density operator 647 entangled two-mode quantum states thermal 647 839 dephasing channel 113 pair 303, 334, 338 depolarization 241 paradox 318 depolarizing channel 112 electron spin 563, 643 depumping rate 706 electro-optic modulator 598 designer atom 820 element distinctness 104 detection 589, 590 encoded operator 119 mode 853 energy constraint 41–42 operator 615, 634 ensemble quantum 643 Deutsch–Josza algorithm 94–96, 645, computing 648 648, 653, 661 ensemble states 661 Deutsch’s algorithm 93 entangled mixed states 131 dipole-blockade 508 entangled pure states 130, 294 dipole–dipole interaction 507 entangled states 392, 396, 712, 794 discrete log 101 of atomic ensembles 712 discrete-time quantum walk 593 entanglement 195, 231, 334–336 discretization 588 activation 188–190 displacement 811 based QKD 375 operator 46 combing 305 distillability 252–253 cost 155, 219 distillable entanglement 155, 220, 236 distillation 210 Rains’ bound 226 of distillation 236 distillation 232 entropy 303 distributed-phase-reference 373 fidelity 113 divide-and-conquer 22 genuine 299 Doppler cooling 802 measure 305 dual code 14 monotonicity 305 dual-rail encoding 440 multi-partite 293 Duan separability criterion 712 pumping 243 dynamical trapping 523–524 purification 232 dynamic disorder 599 spectrum 295 dynamic group 633 structure, cloning 68–69 dynamic system algebra 610, 614 swapping 693, 719 witness 141, 203, 309 entanglement measures 305 e additivity 224 eavesdropping 354, 360 additivity on pure states 223 effective Rabi frequency 816 asymptotic continuity 223 eigenvalue 418, 419 axioms for 221 symplectic 315 convex roof construction 224 eigenvector 418 cross norm measure 227 Einstein–Podolski–Rosen (EPR) 195, distillable entanglement 220 675 entanglement cost 219 Index 863

entanglement monotones 222 extension field 12 entanglement of formation 224 extrinsic constraints 620 entropy of entanglement 220, 223 extremality of distillation and cost 223 f logarithm of the negativity 225 Fabry–Perot 678 norm based monotones 226 factoring 97 ordering of 223 factorization 32 pure states 616 faint pulse QKD regularization 223, 224 fiber based 371–373 relative entropy of entanglement free space 373–375 225 Faraday mirror 372, 373 robustness of entanglement 226 fast Fourier transform (FFT) 98 squashed entanglement 228 feedback loop 597, 598 uniqueness on pure states 223 feed-forward 389–390 entanglement of formation 153, 224 fermionic system 610 additivity 224 fermions strong super additivity 224 Gaussian state 314 fidelity 240, 241, 339, 676 entropy entanglement 113 conditional 8 field conservation 662 extension 12 entanglement 181 prime 12 of entanglement 153, 220, 234, 303 field-ionization 673 entropy of entanglement 220, 223 filtering 239 function 9 operation 299 joint 7 five-qubit quantum computer 646 relative 306 flux qubit 644 Shannon 5–6 focal Hamiltonian 311 von Neumann 303 free atom limit 811 error free-space QKD 373–375 basis 115 frequency combs 837 bit-flip 116 full controllability 644, 645 correction 377–378 symmetry condition 612 model 253–254 operator 112 sign-flip 116 g syndrome 14 gain 345, 346 threshold 259 Galilei group 635 weight 115 gapped phase 311 Eulerian path 20 gate exchange coupling 554–556 complexity 654 expectation value 615, 616 model 319 quantum computer 644 Gaussian measurements 51–52 experimental control 644 Gaussian operations 199–200 experimental quantum cloning 70–71 Gaussian quantum cloning 51 864 Index

Gaussian states 47, 200–202 hashing 245, 252, 256 of bosons 314 inequality 224 of fermions 314 Heisenberg algebra 624 Gaussian transformations 387–389 Heisenberg–Langevin equations 707 generator matrix 13 Heisenberg limit 780, 782, 791 genuine entanglement 845 Heisenberg picture 704 genuine multipartite entanglement Hermitian conjugate 413 162 hidden subgroup problem 101, 653 geodesic 651, 655 abelian 653 Riemannian 653 hidden translation problem 102 sub-Riemannian 653 higher dimensions, cloning 68 geometric 654 high-finesse cavity 683 control 644, 651, 661 highly charged ions 822 control theory 653 Hilbert space 57, 343 measure of entanglement 306 Holevo bound 159 phase 475, 476 Holevo’s theorem 316 GHZ-class 164 holonomic quantum computation GHZ game 318 application to 479–480 GHZ state 235 geometric phase 475–479 GKS-Lindblad generator 622 holonomy 478 glass fiber 371 homodyne detection 389–390 GMon 624 homodyne detector 344, 346 Gottesman/Chuang trick 438–439 homodyne method 677 gradient echo memory 724 Hubbard model 501 gradient flow 617 derivation 500 on Lie groups 619 Mott insulator 495 on Lie subgroups 619 offsite interaction 501 graph isomorphism 33, 102 onsite interaction 502 graph state 249, 449 superfluid 495 Greenberger–Horne–Zeilinger (GHZ) tunneling 502 820 tunneling matrix elements 501 group orbit 614 Huffman coding 7 Grover’s algorithm 103 hyperdeterminant 300 Grover’s search 102 hyperpolarisation 648

h i Hadamard gate 92, 416, 425 incomparable states 218 Hahn–Banach theorem 141 individual attack 360 Hamiltonian 419, 421 individual controls 632 local 311 infinite-dimensional 195 path 24 inhomogeneous walk 598–600 Hamming code 14–15 inseparability criteria 202–209 Hamming distance 10 inseparable Hamming weight 10 fully 308 Hanbury Brown and Twiss 681 inseparable 202, 203 Index 865 instance 19 k-SAT 29 intensity correlation 681 k-separable 308 interaction phase 506 interference 414–416 interferometer 780–783 l internal state preparation 807–808 Lagrange multiplier 62 intrinsic constraints 620 Lamb–Dicke 803 intrinsic information 281 approximation 803 invariants 296 factor 532 ion limit 487, 811 crystal 526–529 Landau symbols 21 delta kick 494 Langevin stochastic forces 707 internal states 487 laser beam 829 motional states 486 laser-cooling 679 Paul trap 521, 522 laser diodes 373 phase gate 490 laser spectroscopy (LS) 799 quantum control gate 492 Lie algebra 653, 656, 661 qubit 546 compact 614 single qubit gate 489 definition 610 state dependent interaction 491 exceptional 610 traps 485, 519, 661 orthogonal 610, 613 two-qubit gate 490 rank condition 608 Ising interaction 645, 653 semi-simple 614 simple 610 (unitary) symplectic 610, 613 j Lie semialgebra 622 Jaynes–Cummings Hamiltonian Lie semigroup 622 704 Lie wedge 623 Jaynes–Cummings model 631, 661 light–atoms interface 701 controllability 632 light force 683 Jaynes principle 49 light pulses 372 joint entropy 7 light source 680 joint probability distribution 177 Lindblad-generator 622 Jones polynomial 657, 658 linear optics 440 Josephson device 644, 656, 661 linear spin chain 653 linear transformations 199 local broadcasting 184–186 k local C-numerical range 616 kinematic phases 506 local coherence 187–188 KLM scheme local operations and classical (Knill–Laflamme–Milburn) communication (LOCC) 131, 437 190, 216, 217, 234, 297 knot theory 657 asymptotic transformations 219 Königsberg bridges 20 operation 265–267 Kraus operator 298 single copy transformations 218 Kraus operators 112 stochastic 298 866 Index

local oscillator 707 molecule 815 local unitary (LU) 295 monogamy of entanglement 164, 302 logarithmic negativity 157, 225, 272 motional eigenfrequencies 528 logical Heisenberg picture 456–457 motional eigenmodes 520 logical qubit 119 motional sideband 488 logic gates 414 multimode quantum light 837 Lorentz group 635 multipartite 210 lossy channel 50, 360 multipartite bound information 286 multi-partite entanglement 845 multipartite entanglement distillation m 268–269 Mach–Zehnder interferometer multipartite fully-separable states 411–412 162 macroscopic superposition 512 multipartite k-separable states 162 magnetic resonance 643 multipartite pure state entanglement majorization 218, 620 235 majorization criterion of separability multipartite quantum secret sharing 134 851–852 Markov condition 624 multiparty entanglement purification master equation 620 249 matrix multiparty quantum channels 287 generator 13 multiphoton path representation parity check 13 761–766 product state 311 multiplexed homodyne detection 842, realignment criterion of separability 843 135 multiplexing, spatial 588, 590 maximally entangled set 298 maximally entangled states 217 multiplexing, temporal 589, 590 Maxwell–Bloch equations 705 multiply-controlled NOT gate 653, measurement 421–422 655 measurement-based quantum multiscale entanglement computation (MBQC) 319, renormalization ansatz (MERA) 450, 466, 837, 852–853 312 Mermin GHZ game 318 mutual information 8, 316 Mermin–Klyshko inequalities 285 metrology 320 micromaser 674 n millennium problems 31 near-field image 833 minimal reversible entanglement negative partial transpose (NPT) 237 generating set (MREGS) 304 negative Wigner functions 849 minimum distance 11 negativity 157, 225 quantum 119–120 network 587 minimum spanning tree 34 complexity 643, 653, 655, 661 mixed state 427 Nielsen’s theorem 298 entanglement 235–236 N→M purification protocol 245 mixing 42 NMR quantum computer 646, 648 Index 867 no-broadcasting theorem 72 one-time pad 354 no-cloning theorem 56–57, 116, 335, one-way function 33 340, 370 one-way pattern 452 noiseless coding theorem 6–7 open system 683 noisy apparatus 252–257 optical amplification 831 noisy coding theorem 9 optical clocks 805–806 noisy operation 253–254 optical density 706 non Abelian quantum Fourier transform optical dipole 815 102 optical experiments 678 nonclassical states 819–821 optical Feshbach resonance 514 nondestructive state detection optical frequency combs 838 815–819 optical lattice 496 nondeterministic polynomial 25 blue detuning 500 non-Gaussian operations 52–53, 849, controlled coherent collision 505 853 geometry 498 non-Gaussian transformations impurity 512 390–391 loading 502 nonlinear-sign-shift gate (NSS) 445 defect suppression 503 non-local games 318 irreversible scheme 503 non-locality 318, 675 maximally entangled state 511 nonseparable state 335 red detuning 500 nonunity gain teleportation 347 single qubit gate 504 normal form 296 site offset 498 normal modes 801 spontaneous emission 500 decomposition 49, 315 state dependence 498 NP 24 state dependent interaction 507 NP-complete 31 state selective movement 499 NP-hard 37 optical potential 496 n-positive map 79 optical qubits 439–440 nuclear magnetic resonance (NMR) 71 optimal control 655 nuclear spin 573–574 theory 643 nullifiers 850 optimization 34–37 number states 45 oscillator algebra 634 numerical optimal control 624, 651 dynamo platform 651 GRAPE algorithm 651 p Krotov algorithm 651 Pancharatnam phase 476 platform DYNAMO 624 para-hydrogen 661 numerical range 614 parallel computing 661 NV centers 661 parallel transport condition 476 parametric fluorescence 681 parity check matrix 13 o partial transpose 205, 208 observable 615 partial transposition 132, 206, 208, all expectation values 634 225–226, 237 occupation-number qubit 439 criterion of separability 133 868 Index

partition 308 pseudo-pure state 647, 648 passive transformations 48 pseudo-spin system 644, 656, 661 Pauli group 122 pulse shaping 842 Pauli matrices 59, 60, 420 pure product states 130 percolation 594 pure states 43, 427, 661 period finding 97 entanglement 233 phase conjugation 50 purification loop 695 phase-covariant cloning 63–65 purification step 240 phase distribution 677 phase gate 428 phase space 45 q phase-space variables 197 QND interaction 706 photon 698 quadrature covariance matrix 842 antibunching 680 quadrature operators 706 bunching 680 of atomic ensemble 705 echo 724, 725 quadratures 345 recoil spectroscopy 811–814 quantized motion 801 statistics 680 quantized radiation field 761 photonic band-gap 684 quantum photon-number resolving detection bits 175 589–592 channel capacity 114 photon-number splitting attack 360 channels 43, 370, 691 physical qubit 119 circuit 428–433, 561, 578 π-pulse 675 code 114 π/2 pulse 675 coherence 175 plug&play-system 372 computation 259–260 Pockels cells 369 control 643, 651, 661 polarizing beam splitter (PBS) 378 Cramer-Rao bound 792, 853 polynomial reduction 29–30 cryptography 278–280 positive maps 144 Darwinism 186 positive operator valued measure defined 175 (POVM) 66 dense coding 158, 376–377, positive partial transposition (PPT) 391–393 237 discord 180–184 positive Shannon entropy 181 dot 553, 564, 698 primality 27 error correction 259 prime field 12 gate operation 521, 526 Prim’s algorithm 35 harmonic oscillator 196 privacy amplification 359 information theory 265 probabilistic cloning 70 interface 672 probabilistic gates 444 laser pointer 828–830 probability amplitudes 410–414 logic gates 377 product channel 113 logic spectroscopy 806 product state 294 memories 683 proof checking 33 messengers 683 proof existence 33 metrology 853–854 Index 869

microlithography 828 quantum error correction code (QECC) mutual information 182 114, 358 neural network 661 quantum Fisher information 780, noise 707, 827 792 non demolition 676, 706 quantum Fourier sampling 96 operation 43 quantum Fourier transform 98, 653, phase gate 676 654 regime 811 quantum gate repeaters 691, 713 local 645 secret sharing 317 universal 645 sensing 793, 795 quantum imaging Shannon theory 316 laser pointer 828–830 simulator 595 spatial quantum noise 830–832 systems theory 607 two-photon 832–833 teleportation 333 quantum key distribution (QKD) 357, walk 104, 593–595 393–396 quantum algorithm 521, 544, faint pulse 371–375 817 quantum map divisible 621 for binary search 91 Markovian 620, 622 DQC1 by NMR 659 time-dependent Markovian 622 for graph problems 91 time-independent Markovian 622 quantum-beat 682 quantum memory 712, 728 quantum-classical boundary retrieval 715 quantum-classical state 180 storage protocol 713 quantum cloning 720 quantumness 2 into atomic memory 720 quantum protocol coherent states 720 multi-party 317 quantum communication 232, 257, quantum teleportation 375, 718 691 of light onto atoms 719 error correction 377–378 quasifree states 47 quantum dense coding 376–377 quaternion 634 quantum compiler qubits 65–68, 804 CISC 661 logical 119 RISC 661 physical 119 quantum correlations alternative characterizations 186–190 r beyond entanglement 175 Rabi flopping 803 general desiderata 190–191 Rabi frequency 680, 803 interpreting 184–186 Rabi pulse 675 quantifying 180–184 Radon–Nikodym theorem 82–83 vs. quantumness 176–180 Rains’ bound 226 quantum cryptography 70, 71 Raman process 680 entanglement based QKD 375 random walk 593 faint pulse QKD 371–375 range criterion of separability 148 870 Index

rank Schmidt tensor 300 coefficients or entanglement rare-earth ion 728–730 spectrum 295 reachable set 620, 645 decomposition 130, 196, 233, 295 closed system 614 measure 303 open systems 623 rank 303 realtime measurements 678 Schrödinger cat 676 receiver 4 Schrödinger-cat states 821 recoil kicks 683 secure function evaluation 318 recurrence protocol 240, 250 secure links 374 reduced density matrix 427 secure state distribution 258–259 reduction 29–30 self-homodyning 707 criteria 238 semi-simple Lie algebra 614 from factoring to period finding 97 separable 201, 202 regularization 223, 224 class 308 relative C-numerical range 616 k-separable 308 relative entropy states 131, 216 of entanglement 225 Shannon entropy 5–6, 177, 181 regularized 224 Shannon’s theorem 432 subadditivity 225 Shor’s algorithm 98 relative entropy of entanglement 156 shot noise 827 remote quantum information sidebands 707 concentration 285 transitions 804 repeater 691 sifting 356 rephased amplified spontaneous sign-flip error 116 emission 727–728 Simon’s algorithm 96 resources 696 simple Lie algebra 610 state 319 simulability 613 retarded time 705 single-atom transistor 512 reversible classical computation 92 single photon processes in cavity 767 reversible computation 405, 431 single-photon source 682 Riemannian exponential map 618 solid-state 553 Riemannian manifold 617 spatially multimode beams 838 Riemannian metric 617 spectrum estimation 295 Riemannian symmetric space 653 spin robustness of entanglement 226, 302 definition 635 rotating wave approximation (RWA) orbit 558 743, 761, 803 polarization 662 rotational state 816 qubit 563 RSA 33 system 613 Rydberg atoms 672 spin-1/2 675 spontaneous emission 674, 680 squashed entanglement 228 s squeezed states 196, 200, 386, 393, satisfiability 29 711, 786–792 scalability 648, 649, 654, 661 of atomic ensemble 711 Index 871 squeezing 196 synchronously pumped optical stability 805 parametric oscillators (SPOPOs) diagram 523–524 838 stabilizer code 123 system algebra 634 stabilizer formalism 455 systems theory stabilizer group 122 closed quantum systems 609 standard array 15 open quantum systems 620 standard form 209 Stark effect 673 Stark eigenstate 507 t state-dependent cloning 57–63 tangent space 619 state preparation 819 3-tangle 307 states 42 technical noise 707 space 42 tensor 296 transformation 234 network state 311 static disorder 599 product 406–407 Stinespring dilation theorem 80 rank 300 Stinespring representation 80 thermal state 648, 661 threshold 254 ancilla form 81 time complexity 643, 654, 655, 661 Kraus form 82 time-optimal 645, 651, 653–655 minimal 80 time reversal 206 stochastic local operations and classical Toffoli gate 429, 432, 656 communication (SLOCC) 298 topological order 311 stochastic matrices 404, 405 topological quantum computing 657 Stokes vector 704 trace distance 219 strip-line resonators 684 transmitter 4 strong-binding 811 traveling salesman problem 36 strong controllability 628 tripartite 202 strong coupling 678 T-transform 623 strong phase reference pulse 363 T-V diagram 348 strong subadditivity 316 two-colorable graph 250 subgroup orbit 614, 619, 620 2D quantum walks 600 super additivity 114 two-mode squeezing 717 superconducting 672 two-particle scattering 600 qubits 575–582 two photon processes in cavity 767 supermodes 840, 847 two-photon quantum 832–833 superradiance 683 2π pulse 675 super-resolution techniques 834 two qubit gate 505 switchable noise 623 symmetry 610 breaking of 630 u condition 612 unbalanced interferometer 371–372 of a dynamic system 634 uncertainty relation 198 quadratic 612 unconditional security 359 symplectic transformations 199 unitary matrix 423, 430 872 Index

unitary operation 44 von Neumann entropy 178, 201, 209, unitary orbit 645 662 unitary transformations 199 unity gain teleportation 347 universal cloning 65–69 w entanglement structure 68–69 Wannier function 497 higher dimensions 68 W-class 164 qubits 65–68 weak-binding regime 811 universal control 634 Werner state 241 universal gate set 91, 428 Weyl operator 46 universal quantum gates 635 Wigner function 46, 197, 200, 385, universal quantum simulators 508 386, 677, 840 Heisenberg Hamiltonian 510 Williamson eigenvalues 846 Ising interaction 509 Williamson normal form 315 one-qubit term 509 words quantum phase transition 512 ε-typical 6 two-qubit term 509 worst case 21

v y vacuum-Rabi frequency 680 yield 236, 253 vacuum-Rabi oscillation 675 vacuum-Rabi period 678 vacuum-Rabi splitting 680 z Vernam cipher 354 Zeeman level 680