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quant-ph9901002.txt |@word aguilera:2 monte:1 consider:2 results:2 leads:3 scalar:1 gb:16 whose:3 inverse:5 depend:1 exploited:1 according:2 stable:1 worth:1 kν:4 case:2 straightforward:1 far:1 implicitly:1 assuming:1 choice:2 exact:3 condition:11 solves:1 hamiltonian:6 ref:6 solution:9 hereafter:2 large:4 solved:1 convenience:1 small:2 eigenvalues:3 degeneracy:1 wronskian:1 occurrence:1 dealing:1 factorization:1 sign:1 approximation:1 second:5 insertion:1 further:2 odd:1 theory:3 trivial:4 section:2 giampiero:1 while:1 expressed:2 goes:1 new:3 method:5 satisfy:1 full:4 respectively:1 here:3 understanding:1 κx:1 ansatz:1 concluding:2 consideration:1 strong:1 equations:3 appears:1 logarithmic:1 obtained:2 scienze:1 involved:2 inspired:1 studied:6 trial:5 resulting:1 stressed:1 action:2 suitable:3 changes:1 involves:2 via:1 useful:1 operational:1 rψ:1 angelo:1 use:3 john:1 asymptotic:1 would:1 remains:3 contains:2 eρ:1 ψk:2 therefore:5 more:3 formally:2 becomes:2 operators:3 factor:1 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quant-ph9901003.txt |@word secondly:1 limited:1 magnetic:19 edn:1 results:6 leads:1 follow:2 expressions:4 3a0:14 depend:1 illustrates:1 containing:1 charge:1 th:1 case:4 rise:2 seemingly:1 wave:4 electron:5 figs:1 summarized:1 matrices:3 encounter:1 obvious:1 immediately:1 presented:2 skills:1 irreducible:2 solution:2 solved:1 vector:9 indices:1 1r:1 occurrence:2 findings:1 f21:1 sin5:2 sin3:7 sign:1 zero:1 further:2 even:1 index:1 what:1 arfken:1 appear:2 giving:1 3r:1 section:6 current:33 above:3 values:3 3a:3 increasing:1 method:4 wavefunctions:1 degree:1 multipole:23 component:10 here:1 atoms:1 desired:2 magnetism:1 cos4:1 cos3:2 cos2:6 represented:1 equations:3 appears:1 obtained:1 ref:1 ay1:1 study:1 studied:1 resulting:1 procedure:2 poisson:1 published:1 fourier:1 makes:1 involves:4 useful:1 although:1 substitution:6 sextet:1 neutrons:1 jrs:1 summation:2 total:9 plm:3 readily:1 derivatives:1 use:6 p31:1 superpositions:1 electrostatics:1 occupying:1 comparing:2 m1:3 therefore:3 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quant-ph9901004.txt |@word concept:6 consider:7 maximum:3 squared:7 magnetic:1 lack:1 brownian:1 results:4 four:3 commute:2 entropy:2 stochastically:1 causes:1 inverse:1 depend:1 calculate:2 pˆi:5 pˆj:2 tr:16 factoring:1 environment:7 pˆs:15 behave:1 indicate:3 those:4 under:2 stochastic:5 castin:1 applied:2 fax:1 orthogonal:6 far:1 frequencies:1 rise:1 springer:2 wave:1 should:6 summarized:2 resonant:1 affect:1 difference:3 matrices:2 condition:2 partitions:2 leads:5 δt:2 level:9 knight:1 proposals:1 deals:1 discrete:1 supplement:1 math:1 indices:1 inspection:1 mahler:4 measuring:1 smaller:1 consistent:2 isolate:1 direct:2 jump:62 imply:1 dissipation:1 further:2 dynamics:2 theory:1 even:2 index:2 what:2 ubner:2 plays:2 sub:2 rate:4 section:8 lead:1 brief:1 expressed:4 find:2 waiting:1 subspaces:1 new:3 ˆj:2 ˆk:1 ˆm:1 contrast:4 respectively:2 here:8 desired:1 plenio:1 compressed:1 groups:1 address:1 path:3 strong:1 equations:4 change:1 referring:2 obtained:2 exemplified:1 shift:1 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quant-ph9901005.txt |@word quasiperiodic:6 supported:1 ge:1 paper:1 through:1 aspect:1 edu:1 y1:3 eigenstates:4 σni:2 inverse:1 number:1 sheldon:1 decay:1 ln:1 character:1 should:1 jan:1 absence:1 valid:2 systems:1 circle:1 defining:1 equilibrium:2 yi:1 ei:1 yn:5 frequencies:6 de:2 wave:6 lyapunov:2 five:1 dx:17 φi:3 dt:1 densities:1 evolution:1 like:1 particle:6 goldstein:1 follows:2 university:1 arxiv:1 michael:1 solution:1 ai:2 vector:2 each:1 quasiperiodicity:3 found:1 side:2 configurations:1 square:1 functional:1 motions:1 energy:4 anchored:1 imply:1 growth:1 result:1 quantum:3 dynamics:4 will:3 satisfying:1 exponent:2 since:2 crucial:1 chaos:5 its:4 piscataway:1 kiessling:1 state:1 does:1 math:1 hence:1 after:1 barreto:2 theory:1 power:1 sole:1 weak:1 however:1 here:2 completely:1 no:2 address:1 alone:1 ratio:1 e0:1 bz:1 concentrated:1 almost:1 always:3 am:1 similarly:1 according:1 o:2 motion:10 carries:1 consequence:2 suggestions:1 otherwise:1 mechanics:2 presumably:1 rev:1 point:1 simple:1 variables:1 period:1 nsf:1 dimensional:2 right:3 decrease:1 another:1 impossible:2 electronic:1 mass:1 bound:1 iei:1 given:4 described:1 asymptotic:1 likely:1 prove:1 probabilities:1 there:2 system:2 a0:1 chaotic:1 way:2 ph:1 was:1 consistent:1 more:1 function:8 only:2 corresponding:1 grant:1 vanishing:1 distribution:2 continuous:1 odinger:2 but:1 bohmian:6 normalized:1 systematic:1 particular:2 convergence:1 helpful:1 line:5 general:3 than:1 case:2 ii:1 exception:1 determines:1 exponential:1 pacs:1 double:1 equation:3 work:1 iyi:1 depends:1 ωi:2 equivariance:3 while:1 rutgers:2 behavior:1 ωn:3 following:1 autonomous:2 property:1 inherits:1 claim:1 studied:1 associated:1 defined:2 flow:1 preserving:2 thus:3 spectrum:1 proven:1 evidence:1 states:2 fail:1 single:2 im:1 something:1 dms:1 any:7 ray:1 nj:1 january:1 perhaps:1 lim:1 means:1 when:1 feature:2 occur:1 field:2 potential:1 quant:1 department:1 mathematics:1 independent:1 dependence:1 superposition:4 numerical:1 upon:2 hand:2 most:2 important:1 moving:3 remarks:1 grateful:1 law:1 grow:1 recent:1 schr:2 dynamical:1 interval:1 well:2 order:2 bohm:4 part:1 so:1 time:4 position:1 normally:1 dimension:3 fact:4
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quant-ph9901006.txt |@word represent:1 gs:8 illustrates:1 induces:1 varying:1 edn:1 results:13 leads:6 eds:1 quadrature:11 ga:9 expressions:12 faster:1 causes:2 devoted:2 excludes:1 inverse:2 φgs:2 ws1:7 squeezing:38 th:2 under:8 bv1:2 amplitudes:3 2rj:1 advantage:1 induce:1 wave:1 figs:3 summarized:1 difference:1 matrices:2 condition:2 transmission:2 supports:6 solution:27 occurs:1 solved:5 indices:1 eigenvalues:1 optics:19 configurations:1 strongly:1 smaller:4 occurrence:5 periodicity:1 approximation:3 second:8 further:1 dynamics:5 nonlinear:21 sub:10 giving:1 sum:1 aˆk:3 expressed:2 cˆa:3 ˆb:2 above:17 3c:1 obvious:1 ˆj:6 increasing:5 waveguide:15 method:2 coefficient:1 evanescent:4 exchange:2 respectively:2 κa:41 understanding:1 behaviour:8 fluctuating:2 operating:2 κz:4 peˇrina:23 active:2 along:1 strong:4 equations:15 change:2 κs:45 obtained:15 listopadu:1 suitably:6 institute:1 involved:3 study:2 studied:3 simultaneous:1 characterizing:1 symmetrical:1 conveniently:1 suitable:2 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quant-ph9901007.txt |@word represent:1 gs:2 consider:1 remarkable:3 limited:1 caused:8 dynamic:1 results:13 leads:11 ρji:1 scalar:1 2n2:3 consists:1 gn:1 gm:5 faster:2 gk:1 gj:3 localized:6 devoted:2 preserves:1 ρsp:1 calculate:1 γs:7 technique:1 nakajima:2 environment:2 charge:1 squeezing:1 those:2 excluded:1 under:7 stochastic:1 include:2 silbey:5 summarizes:1 case:11 g5:1 g4:1 g3:5 g2:48 g1:16 very:1 fas:3 radius:1 conditions:4 g9:1 wave:2 electron:10 figs:5 resonant:1 difference:6 matrices:14 2γs:1 g8:1 annihilated:1 incoherent:1 presented:2 did:1 supports:1 applicable:2 t0:23 trf:3 aggregates:2 cas:1 vector:2 aas:2 small:6 optics:3 dimer:26 application:1 smaller:8 2k1:2 oscillation:1 renormalized:4 kato:1 sin2:1 pulse:23 sign:2 approximation:11 surrounding:3 zero:4 j22:1 j21:1 further:6 dynamics:46 estimated:2 expressions:2 linear:1 cˆn:1 index:2 cˆm:3 plays:1 stands:1 section:12 localization:2 expressed:4 ˆb:4 grover:1 ˆe:1 ˆf:1 elucidate:1 new:9 3a:3 ˆk:3 consisting:1 contrast:1 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quant-ph9901012.txt |@word equalities:1 results:1 mℓ:2 edu:4 query:3 gj:1 depend:1 iii:1 edward:1 th:2 under:2 multilinear:1 case:3 orthogonal:2 unstructured:1 lefthand:1 condition:1 presented:1 michael:1 fewer:2 large:2 upper:1 cooperative:1 established:1 what:1 fℓ:1 sum:2 above:1 neu:1 degree:3 here:1 search:2 ctp:1 institute:2 items:3 simultaneous:1 fc02:1 makes:1 divide:1 wim:1 orderings:1 prove:1 distinct:1 give:2 classically:4 therefore:2 ψf:6 theoretical:1 more:2 sort:1 becomes:2 operators:1 particular:1 known:1 foundations:2 cases:1 hold:2 producing:1 must:2 high:1 valued:1 ma:3 science:3 work:4 mj:2 grover:2 values:2 following:2 example:2 proof:1 f0:7 f1:1 f2:1 pℓ:4 beals:1 states:1 minimum:1 numbers:1 information:3 buhrman:1 holds:8 how:3 94er40818:1 symposium:3 mar:1 after:1 fa:3 fd:1 representations:1 fj:4 summing:1 complexity:1 so:4 farhi:2 order:2 over:2 paper:4 y1:3 before:1 derive:2 interesting:1 distinguishing:2 main:1 yh:3 dam:2 xg:3 half:1 sorted:1 now:5 easily:1 term:1 equality:1 january:1 university:1 alternately:1 orthonormal:1 sufficiently:1 inequality:1 each:6 2n:15 side:1 2k:5 goldstone:2 domain:2 independent:1 energy:1 hard:1 measurement:1 quantum:20 shown:1 space:3 furthermore:1 since:3 formula:1 math:1 shows:2 foundation:1 theory:2 vazirani:1 pairs:1 starts:1 computers:1 hence:1 xi:1 xn:1 probability:5 succeeds:2 number:2 done:1 fast:1 precisely:1 size:1 given:4 checked:1 introduction:1 least:1 needed:1 siam:1 bits:1 final:1 gives:1 part:1 x2:1 x1:4 than:3 coefficients:3 distinguishes:1 distinguished:11 generality:1 boston:1 any:4 viewed:1 answering:1 indexes:1 note:3 take:1 algorithms:2 mit:4 price:1 most:2 oracle:11 axy:3 proceedings:3 mechanical:1 physics:1 fact:1 particularly:1 show:1 supported:1 state:3 lemma:2 only:4 announce:1 bernstein:1 do:1 his:1 subspaces:1 means:2 de:2 cannot:1 runs:1 arxiv:1 queries:16 set:6 achieved:1 violated:1 see:1 computer:2 result:1 best:4 exploited:1 mechanically:3 consequence:1 written:1 between:1 succeed:1 distinguish:1 suppressed:1 both:1 last:1 many:2 equal:2 northeastern:1 39th:2 polynomial:3 sam:1 basis:1 been:1 defined:2 pd:1 pj:2 ph:5 define:2 corresponding:4 subspace:1 observe:1 p2:1 p1:1 value:2 suppose:3 applies:1 unitary:2 almost:1 vi:1 vk:2 thus:1 ii:1 technology:2 binary:1 grant:1 make:1 complex:1 quant:5 meets:1 jeffrey:1 functions:26 v1:1 qjr:2 lower:2 center:1 database:1 ccr:1 without:1 solve:1 sets:2 model:1 dimension:1 identify:2 yet:1 application:1 showing:1 posed:1 possible:5 interrogation:1 know:2 using:1 loss:1 identifying:2 follows:1 yields:2 always:1 gutmann:1 because:1 classical:4 successfully:2 some:1 examples:3 28th:1 measurements:2 does:2 run:1 journal:1 agreement:1 nowhere:1 hilbert:2 comparison:1 constraint:2 o:1 wolf:1 projectors:2 subset:1 cambridge:2 into:1 bound:10 nsf:1 getting:1 national:1 operator:1 sipser:2 van:2 additional:1 there:3 question:1 long:1 was:2 function:3 form:6 but:1 algorithm:10 whether:1 bounds:1 required:2 associated:2 computing:2 ai:1 certain:1 initial:1 ax:3 commuting:1 mosca:1 nk:2 no:2 whereas:1 when:1 tight:2 d2n:1 department:3 mathematics:2 problems:1 polynomials:1 a1:1 a0:3 massachusetts:2 calls:1 factors:1 cleve:1 n1:2 n2:1 requires:1
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quant-ph9901013.txt |@word concept:2 consider:4 forbidden:1 commute:1 follow:1 relationships:1 alamos:1 under:1 worth:2 very:2 difference:3 hamiltonian:2 notes:1 settle:1 small:2 lapse:1 prevent:1 approximation:1 even:1 what:1 appear:1 uniform:1 experiment:1 above:1 new:1 margolus:2 simultaneously:1 exchange:2 along:1 change:2 great:1 institute:1 suggests:1 simultaneous:1 stressed:2 action:4 via:1 useful:2 total:1 use:1 would:3 give:1 therefore:3 taken:1 theoretical:1 more:1 introducing:1 particular:2 known:3 foundations:1 account:1 equation:1 theories:1 obstacle:1 concepts:2 making:1 example:1 history:1 stream:1 stated:4 neglected:2 states:1 slit:1 huge:1 get:1 how:2 parameter:1 abstraction:1 sommerfeld:2 information:1 may:1 after:1 law:1 remember:1 neglect:3 complexity:1 so:2 allow:1 order:1 september:1 pointed:1 interpreted:1 course:1 paper:1 during:4 its:3 interesting:1 better:1 happened:1 introduce:1 landau:2 hamilton:1 they:1 hands:1 oxford:1 dirac:2 always:1 university:2 asher:1 each:1 found:1 energy:10 principles:1 measurement:12 quantum:16 our:1 eq:1 out:1 missed:1 space:1 rev:2 eprint:1 issue:1 theory:4 quite:1 completely:1 york:1 definition:2 language:2 times:1 motion:2 think:1 already:1 number:1 least:1 aharonov:1 given:3 exists:1 los:1 bologna:1 system:11 construct:1 unruh:1 speaking:1 relationship:4 took:1 regards:1 authors:1 translation:3 than:1 explodes:1 dynamical:3 kind:1 target:1 princeton:4 double:1 approximations:1 see:2 toward:1 peierls:2 measure:1 say:1 seen:1 any:1 self:1 note:3 take:1 forces:1 heisenberg:4 lectures:1 though:2 object:1 plenty:1 phase:1 observation:1 consequences:1 dordrecht:1 average:1 later:1 steady:1 physica:1 points:1 principle:4 physics:8 particularly:1 reznik:1 find:1 ground:1 causality:1 ment:1 state:5 should:1 jan:1 only:2 do:2 notation:1 means:3 zurek:2 bear:1 cannot:2 words:1 reconstruct:2 dt:1 evolution:2 arxiv:1 twice:1 investigate:1 common:1 isolation:1 wrote:1 reference:3 physically:1 depends:2 result:1 said:1 interacting:1 nature:5 written:1 closed:4 between:6 email:1 publishers:1 comparable:1 complementary:1 however:4 observable:2 instant:1 many:1 uncertainty:1 according:1 quantization:1 connection:1 logic:1 point:1 instants:2 simply:1 moreover:2 speak:1 described:2 kinematic:1 addition:1 trajectory:1 much:1 linguistic:1 ph:2 indeterminacy:6 mind:1 wheeler:2 precision:1 oppenheim:1 external:1 pacs:1 while:1 formulation:1 equal:1 itself:1 claimed:1 different:1 italy:1 anymore:1 same:2 clearly:1 quant:2 flows:1 hand:1 recent:1 nevertheless:2 well:2 bohr:2 academic:2 without:2 english:2 position:1 popescu:1 less:1 obtain:1 actions:2 questions:2 macroscopic:1 kluwer:1 had:2 other:2 momentum:2 real:1 possible:4 know:2 press:4 necessary:1 follows:1 δq:1 satisfactory:1 because:3 methods:1 classical:3 exposed:1 some:4 examples:1 cnr:2 avoid:1 measurements:2 does:2 refer:1 measuring:1 bo:1 clarendon:1 step:1 bz:1 about:6 luigi:1 levitin:2 into:1 bound:1 nothing:1 references:1 operator:2 your:1 often:3 deal:1 strictly:1 there:3 question:3 strict:1 way:4 was:1 questionable:1 plunge:1 phenomena:1 but:4 he:2 made:1 variations:1 considered:2 problem:2 called:1 mechanics:9 sometimes:1 constant:2 int:1 certain:2 abstract:1 describe:1 peres:2 rovelli:2 formalism:1 archive:1 physical:2 compared:1 no:3 whereas:1 when:3 valid:2 role:1 foschini:3 you:2 theor:1 miscellaneous:1 important:1 observables:4 variable:1 structure:1 longer:1 interval:4 impossibility:1 time:31 understand:1 gobetti:1
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quant-ph9901014.txt |@word represent:1 monte:1 concept:2 customary:2 systematic:1 rev:15 amplifier:2 results:3 eds:5 quadrature:2 issues:1 consider:1 certainly:1 inverse:1 depend:1 technique:1 vs:1 program:1 squeezing:5 those:2 th:1 under:1 stochastic:1 report:1 figs:6 experimentally:5 condition:4 northwestern:2 level:1 transmission:1 ref:5 large:1 small:4 shows:1 optics:3 limitations:1 work:1 verifiable:1 vasilyev:1 hillery:1 sign:1 second:1 plenum:2 radon:1 odd:2 degraded:4 even:6 established:1 nonlinear:2 homodyning:1 experiment:4 reconstruction:12 new:2 method:4 contrast:1 evolving:1 full:1 lee:2 respectively:3 tomographically:3 grey:4 here:2 reported:1 simon:1 nonunit:2 alone:1 along:4 change:1 knowledge:3 obtained:9 remark:1 spite:1 larger:1 allows:1 shumovsky:1 example:2 usually:1 studi:1 fourier:1 glu:1 involves:2 york:2 via:1 convolution:2 hermite:1 cylinder:1 marked:1 total:2 unit:1 prem:1 confidence:1 sacchi:1 usa:1 probabilities:7 positive:4 hong:1 classically:1 sample:3 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quant-ph9901015.txt |@word consider:1 magnetic:1 leads:1 roy:1 re:1 shanghai:2 orthonormalized:1 worth:1 case:1 1ph:2 wave:4 englert:2 fan:4 lynch:2 emphasized:1 difference:7 annihilated:1 riemann:10 immediately:1 turns:1 vector:13 science:3 manifestly:2 shows:1 optics:6 force:1 sin2:1 jump:2 further:1 port:1 spectral:1 established:1 abf:1 expressed:1 experiment:2 new:8 momentum:1 contrast:1 degree:1 respectively:1 sono:1 here:2 behaviour:2 china:5 meet:1 ratio:5 sine:1 observing:1 appears:1 box:1 involved:1 vectors:1 allows:1 completely:1 usually:1 conveniently:2 makes:1 composed:1 examining:1 technique:1 polonica:1 establish:1 unit:1 prove:1 a51:1 comparing:1 bialynicki:2 a54:1 associated:1 theoretical:1 more:1 introducing:1 formally:1 pike:1 becomes:1 operators:3 falloff:1 zaidi:1 normalized:2 particular:1 iv:1 iii:2 must:1 states:20 quantized:7 obliquely:2 recall:1 work:3 oscillator:2 resembles:1 following:2 making:1 root:1 example:1 beijing:1 compare:1 process:1 preserving:1 high:1 indicates:2 want:1 sense:2 axis:4 information:1 write:1 trieste:1 acta:1 parameter:1 circular:1 product:1 description:3 may:2 after:1 different:1 plane:1 fk:17 parallel:4 natural:1 resemble:1 light:1 correspond:1 so:1 representation:6 orientations:2 oblique:1 years:1 through:1 roots:1 anhui:1 decay:1 d35:1 glogower:1 exhibits:1 writing:1 yi:3 them:1 noh:2 polarizations:3 material:1 introduce:5 p10:2 they:3 oxford:2 now:3 yu:1 represents:1 dirac:5 name:1 always:1 nfm:6 orthonormal:3 mode:1 each:2 x10:3 entirely:3 side:1 square:2 a52:1 energy:2 idea:2 interferometer:1 ey:4 ex:4 measurement:1 quantum:7 laboratory:1 eq:6 special:1 out:2 principles:1 shown:1 3rd:1 rs:2 rep:1 since:3 rev:9 denotes:1 attracted:1 acting:2 induction:1 hcl:1 predicted:1 foundation:1 wiley:1 localization:1 reason:1 a114:1 definition:1 feasible:1 introduction:1 yue:1 probability:3 number:2 carry:1 passes:1 given:1 their:2 ordering:1 eqs:3 construct:2 needed:1 rates:1 brevity:1 john:1 shifted:1 relationship:3 optical:2 shifter:1 x1:3 modes:2 relative:4 kind:2 polarized:6 supposed:1 fˆ:1 a49:1 exhibit:2 ann:1 seen:1 seek:1 any:2 angle:1 polarization:30 absorbed:1 coincide:1 modulus:1 electric:3 normal:1 quote:1 most:1 eight:1 letter:1 nothing:2 intrinsic:1 constructed:1 observation:2 considered:1 physica:1 proceedings:1 part:1 principle:1 exp:5 mechanical:1 left:6 physics:3 laws:1 silberstein:11 supported:1 relation:2 substituting:1 d2:7 d3:5 postulates:1 based:4 claude:1 state:16 operational:1 only:1 bases:1 ambiguity:1 factor:1 local:2 do:2 means:1 beams:2 express:1 expectation:2 beam:3 cannot:2 words:1 international:2 a86:1 fields:1 arxiv:1 summary:1 qe:1 common:1 london:1 set:2 reference:2 sep:1 physically:1 national:1 klauder:2 detection:1 see:4 sec:3 result:2 eliminating:1 quantizing:1 phases:1 satisfying:1 mechanically:1 written:1 various:1 between:11 limit:1 comparable:2 terms:5 nature:3 attention:1 physicist:1 hefei:1 however:1 key:1 analyzed:1 both:1 cos:4 σn:1 equal:1 involve:1 regarded:1 reside:1 aa:1 cn:2 tong:1 point:1 leipzig:1 homodyne:2 simply:1 spans:1 moreover:2 conference:1 strong:1 described:4 basis:1 equations:1 been:1 much:1 ph:1 cohen:1 define:1 partly:2 subjecting:1 photon:24 coherent:1 determination:1 input:1 present:1 applied:2 p1:2 engineering:1 value:3 will:1 iξ2:1 applies:1 eik:1 property:1 opt:1 unitary:1 vol:1 thus:2 according:1 ii:1 sudden:1 technology:1 hong:3 italy:1 perform:1 make:4 same:2 clearly:1 complex:11 quant:2 gets:1 optic:3 effect:1 hand:1 susskind:1 recent:1 eigenvectors:2 schr:1 center:1 poland:1 without:1 components:1 usual:2 ck:1 proposed:1 freyberger:2 being:3 world:1 note:1 density:2 photons:1 conclude:1 seems:1 lett:4 iπ:1 eigenstate:1 match:1 resolved:1 iθ:10 eigenvalue:2 iα:1 iδ:1 vacuum:1 press:1 using:2 follows:2 signal:1 yields:2 arbitrary:1 university:2 therein:1 right:7 classical:5 absolutely:1 creation:2 some:1 born:1 b2:1 ccast:1 eiα:2 describing:5 eiδ:2 electromagnetic:6 integration:2 provides:1 proc:1 strada:1 contrasting:1 bf:4 bb:1 jumps:2 perpendicular:6 powerlaw:1 although:3 properties:1 comparison:1 discussed:2 o:1 2i:2 mutually:1 wolf:1 introduced:1 soc:1 jiao:1 pergamon:1 into:7 within:2 phase:15 hermitian:1 because:2 references:2 determine:2 operator:25 additional:1 tannoudji:1 cosine:2 mandel:3 was:1 function:2 complete:2 form:3 but:3 displacement:1 costiera:1 kf:2 arranged:1 whether:1 up:2 us:1 crystal:2 un:2 operationally:3 ωn:2 distribution:1 semiclass:1 similar:4 agree:1 eigenvector:6 mechanics:3 ab:1 ordered:1 af:4 defined:3 certain:1 describe:2 general:1 single:6 pˆ:1 dθ:1 our:4 qiu:1 reflecting:1 formalism:2 sarkar:1 physical:3 functions:1 completeness:1 no:1 when:3 handed:13 field:7 conceptions:1 other:4 spatial:1 department:2 picture:1 superposition:2 time:1 odinger:1 a1:3 commutator:1 meaning:1 a2:3 birula:2 dimensions:1 implies:2 governed:1 eiθ:14 quantity:2
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quant-ph9901016.txt |@word consider:6 chinese:1 crucial:1 magnetic:3 lack:1 results:1 integration:4 leads:3 edu:1 follow:1 displaying:1 inverse:1 calculate:3 aug:1 shanghai:1 violation:1 charge:3 th:7 divergent:2 applied:2 worse:1 advantage:1 returning:1 than:4 implicitly:2 induce:1 wave:5 electron:18 summarized:1 li:2 emphasized:1 difference:5 bringing:1 encounter:1 treating:1 four:1 hamiltonian:4 ref:2 applicable:1 large:1 small:5 prd:1 smaller:3 renormalized:2 warns:1 coefficient:2 literature:2 approximation:2 second:5 value:9 chen:4 subordinate:1 substitute:1 ˆp:1 trick:1 appear:1 contributions:2 3p:3 3j:1 experiment:3 huber:1 new:2 contributes:1 method:5 body:3 here:3 china:4 ratio:1 eikr:1 knowledge:1 zwanziger:1 great:1 140m:1 splitting:2 shift:6 study:1 leaving:1 loop:2 suitable:1 composed:1 via:2 followed:1 extra:2 manage:1 nanjing:1 brought:1 coinciding:1 decrease:3 total:2 unit:1 wang:3 use:3 would:3 covariant:4 positive:1 contains:1 negative:2 next:2 process:1 nucl:2 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quant-ph9901017.txt |@word consider:2 caused:2 magnetic:4 separates:1 alternatives:1 results:1 leads:2 go:1 hole:11 whose:1 ac03:1 segments:2 magnetostatic:2 charge:8 does:3 those:1 under:2 connecting:2 must:1 case:3 random:1 mainstream:1 electron:1 difference:3 four:1 exits:1 presented:1 did:4 evb:1 t2:2 solution:2 correct:2 large:2 discrete:12 vector:2 science:1 noted:1 inserting:1 says:1 ten:1 renormalized:2 ted:1 approximation:1 second:4 linear:1 even:3 what:4 chicago:1 current:1 19th:1 sequential:4 above:2 flowing:1 international:1 counters:3 body:3 niche:1 justification:1 simultaneously:3 sinks:2 respectively:1 rug:1 tangent:1 sink:2 separation:1 active:1 path:2 along:7 equations:9 change:13 obtained:1 arc:6 shift:1 spacial:1 studied:1 allows:3 leaving:1 usually:1 kauffman:3 changes:4 criticism:1 adjustable:1 boolean:1 apply:1 glance:1 should:4 establish:1 use:4 would:5 examined:1 few:2 call:3 relativistic:2 taken:1 type:1 historically:2 more:3 yes:1 started:1 conducting:1 warn:1 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quant-ph9901019.txt |@word secondly:1 concept:1 consider:5 switched:1 rev:5 hats:1 particular:1 dynamic:7 results:4 leads:3 roy:1 edu:1 follow:1 reasoning:6 depend:1 calculate:2 pˆi:3 eˆ:3 suspending:1 charge:2 present:2 under:1 lord:1 p2:11 very:2 desirable:2 rise:1 account:2 focus:1 sons:1 every:1 osi:1 hamiltonian:7 positions:1 henri:1 t2:2 xˆi:1 ox:1 try:1 discrete:1 math:1 gσµ:1 limitations:2 force:3 haar:1 miller:1 consistent:3 measurable:1 rate:4 ter:1 chem:1 canonical:6 even:1 clock:44 gρµ:2 xρ:1 uniform:1 experiment:3 goes:1 new:4 satisfy:2 here:2 okinawa:2 slow:1 free:1 address:1 represented:2 equations:3 change:1 box:2 clarendon:1 shift:3 amount:2 simultaneous:1 resulting:1 characterizing:1 poisson:1 stressed:1 changes:1 cancellation:1 involves:1 composed:1 named:1 vid:1 elaborated:1 wij:3 brought:1 total:1 salecker:2 inc:1 confidence:1 covariant:1 positive:3 negative:2 next:1 approach:2 therefore:10 more:2 flat:1 operators:4 limitation:2 curvature:1 correspondingly:1 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quant-ph9901020.txt |@word represent:1 partial:1 consider:9 rev:6 magnetic:1 results:9 leads:2 roy:1 ice:1 follow:1 gm:8 row:1 whose:3 calculate:3 concerned:1 rayleigh:2 plasmons:5 presents:1 tm:16 under:1 mendon:1 analogous:1 te:3 divergent:1 amplitudes:3 g2:1 advantage:1 g0:7 very:5 frequencies:5 modes:9 keeping:1 wave:9 sidebands:13 resonant:3 tridiagonal:1 exact:6 condition:7 entire:1 presented:1 evaluation:1 solution:7 correct:1 large:1 vector:4 small:8 optics:3 configurations:1 smaller:2 eberlein:3 oscillation:2 sin2:1 mg:1 approximation:15 inconsistent:1 rate:24 enhancement:3 expressions:1 linear:1 spectral:3 index:2 what:2 abc:3 brackets:1 giving:1 trivial:1 section:11 bc:6 while:1 d3:4 reminiscent:1 experiment:1 δq0:23 establishes:2 expresses:1 contributes:1 boost:1 method:2 3d:2 satisfy:2 agreement:3 evanescent:4 relates:1 divergence:2 component:11 here:2 understanding:1 behaviour:3 cos2:1 situation:1 directions:4 along:22 strong:1 equations:5 appears:1 change:1 obtained:5 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quant-ph9901021.txt |@word remarkable:1 priori:1 roy:3 abelian:1 computation:2 consists:2 query:1 remarkably:1 issues:1 preserves:4 amplification:1 clarification:2 acting:2 finally:1 string:3 straightforward:2 orthogonal:12 choice:2 unstructured:3 every:1 matrices:1 rotating:2 school:1 level:1 solution:1 singularly:1 vector:5 small:1 enhance:1 noted:1 specially:1 deutsch:1 dealing:1 sign:8 second:1 linear:2 what:5 adds:1 richard:1 giving:1 sum:2 uniform:1 access:3 version:2 above:3 new:1 journal:3 full:2 component:3 here:7 ranges:1 simon:1 directions:1 along:1 standard:2 appears:5 interpret:1 calculate:1 box:6 search:3 increments:1 involved:2 leaves:1 vectors:1 action:3 suitable:1 uniquely:2 via:1 followed:2 extra:1 prefer:1 marked:2 replace:1 apply:4 total:1 unit:3 readily:2 formalised:1 use:3 takes:1 working:1 e1:4 pedagogical:1 hilbert:1 classically:1 m1:1 m2:1 more:7 clearly:1 becomes:1 operators:2 apparent:1 aware:1 particular:2 known:5 foundations:2 effort:1 must:1 high:2 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quant-ph9901022.txt |@word consider:2 remarkable:1 rev:2 leads:4 scalar:2 go:1 transiting:1 preserves:1 inverse:1 catastrophe:4 golshani:1 pˆi:2 pˆj:3 answered:1 finally:1 advantage:1 orthogonal:2 conditions:1 wave:1 witten:1 jacob:1 experimentally:2 condition:5 four:2 school:1 hamiltonian:4 louis:1 solution:6 solved:1 dealt:1 gauge:2 challenge:1 work:1 force:1 sign:1 zero:2 insertion:1 pass:1 canonical:5 ˆr:6 g22:1 appear:1 section:8 d3:3 ˆa:1 above:2 new:4 consisting:1 satisfy:2 iran:1 respectively:1 here:6 separation:1 path:1 annales:1 calculate:1 box:1 study:1 vectors:5 ferrara:2 periodic:1 usually:2 suitable:1 der:1 appropriately:1 orthonormality:1 julia:2 replace:1 aˆν:5 summation:1 unit:1 covariant:2 negative:1 linear:1 therefore:3 themselves:1 theoretical:4 more:1 successful:3 becomes:2 operators:6 conducting:1 validity:1 casey:1 normalized:1 cases:1 must:1 states:5 local:2 equation:5 fµν:1 oscillator:1 remain:1 fock:2 obvious:1 following:7 example:1 numbers:2 cosmological:5 stated:1 process:1 carroll:1 involve:1 assumptions:1 something:1 want:1 absolute:3 verify:2 discussion:2 write:1 how:1 answer:2 coleman:1 tried:3 may:5 after:1 derived:1 confirmed:2 choice:2 undesired:1 summing:1 cosmology:2 so:2 enter:1 banks:1 order:1 g11:1 commutator:1 interpretation:4 over:3 course:4 paper:1 aˆµ:5 facilitates:1 still:2 its:1 before:1 lf:1 nieuwenhuizen:2 la:1 energies:1 choose:2 supersymmetry:1 dam:1 polarizations:1 introduce:3 mention:4 scipp:1 now:3 easily:2 2v:6 university:3 presented:1 mode:1 reasonable:3 each:1 found:2 side:2 em:5 energy:13 ef:1 principles:1 related:3 weinberg:1 expect:1 quantum:12 our:6 aˆ0:1 special:1 shown:3 pˆ0:1 furthermore:1 since:1 lagrangian:4 particles:4 δrs:1 shows:1 theory:12 fondation:1 free:6 california:1 aˆr:1 hc2:3 impose:1 introduction:1 think:1 first:3 number:1 restrict:1 relations:4 impossible:1 given:1 reference:1 indicate:1 ordering:7 unknown:1 least:2 physically:2 calculations:2 final:1 damaging:1 gives:1 akad:1 authors:1 tehran:1 probably:1 roles:1 abbassi:1 second:1 annihilation:2 toward:2 perturbative:1 bridge:1 generalization:4 seem:1 dragon:1 any:1 shojai:1 εs:1 εr:2 note:2 sciences:1 finding:1 destroy:3 broglie:1 unless:1 normal:7 phase:1 commutation:11 kk:1 calculated:1 unreasonable:2 physica:1 feynman:6 points:3 pˆν:5 ned:1 fact:2 quantity:2 particularly:1 show:3 anyway:1 relation:18 solutions:1 find:5 based:3 commutators:2 explain:1 should:5 jan:1 only:2 going:3 g00:1 interested:1 dk:1 means:5 de:2 famous:1 cannot:2 despite:1 report:1 aν:2 fields:15 arxiv:1 remove:1 set:1 exists:1 imposing:3 frame:1 relative:1 see:3 result:5 alternative:1 quantizing:1 gravitation:1 tensor:1 supergravity:1 state:3 difficulties:1 between:6 invariant:2 boundary:2 approach:4 ˆb3:1 recently:1 ˆb2:1 ikx:6 however:1 hutp:1 harvard:1 essentially:1 many:1 taking:1 equal:2 quantization:1 hω:3 casimir:3 contribution:2 expression:1 ck:1 point:1 simple:1 corresponds:1 microcausality:3 lagrange:1 ct:2 described:1 aˆ:1 scherk:2 been:4 hawking:2 ph:2 systems:1 field:12 corresponding:1 photon:3 astron:1 c2:10 worked:1 those:1 case:2 pacs:1 expanded:1 value:4 will:2 while:1 suppose:1 eik:2 property:1 procedure:1 ik:10 limit:1 vanish:1 turner:1 worry:2 different:3 verified:1 radiation:5 same:2 clearly:1 parts:1 quant:2 plates:1 independent:2 zumino:1 used:2 infinity:5 effect:4 hand:2 recent:1 fourier:1 without:5 solve:2 components:1 position:3 dimension:1 usual:5 proposed:2 buchmuller:1 generally:1 lorentz:3 questions:1 momenta:1 letting:1 polarization:4 ˆb1:1 preprint:2 density:5 photons:2 seems:1 valid:1 lett:4 wet:1 excited:2 momentum:5 real:2 bij:1 rules:2 harmonic:1 cruz:1 know:1 vacuum:15 press:1 using:4 accepted:1 g33:1 like:1 particle:4 specific:1 deltafunction:1 either:1 integrals:1 arbitrary:1 works:2 replacement:1 mathematical:1 right:2 often:1 discrete:1 creation:1 some:1 proper:2 affects:1 annu:1 shall:1 integration:1 does:2 linearly:1 proc:1 equivalent:1 expansion:1 disappeared:1 step:2 santa:1 hilbert:2 comparison:1 about:2 constraint:2 discussed:1 dependence:1 o:2 tˆ0:1 important:4 h2:1 maxwell:1 integral:2 predictions:3 because:1 references:1 operator:5 qed:6 van:4 removing:1 there:5 question:2 nucl:3 way:2 propagator:2 function:5 complete:1 but:4 cremmer:2 repeated:1 aµ:12 bagger:1 delta:2 adler:1 girardello:2 up:1 us:2 considered:6 fermi:2 problem:6 nµ:1 mechanics:1 constant:7 worthwhile:1 abstract:1 am:1 doesn:3 tˆ:3 ar:6 pˆ:7 check:1 functions:2 completeness:1 no:5 when:2 vanishes:3 virtual:1 ng:1 application:1 other:2 role:2 department:1 massless:3 worthwile:1 conclusion:1 calculating:1 hδij:1 ξr:4 time:3 independency:1 a1:2 a0:1 a3:2 a2:2 physics:1 space:9 electromagnetic:8 n1:6 n2:5 n3:5
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quant-ph9901023.txt |@word gr:1 similarity:1 rev:6 lack:1 results:1 illustrated:2 gc:2 broader:1 kato:1 causes:1 particles:1 inverse:1 illustrates:2 spirit:1 contours:7 under:1 nodes:1 presentation:1 traditionally:1 encircle:2 global:1 orthogonal:1 rise:3 wave:16 affect:1 difference:2 continued:2 generalised:1 four:1 riemann:1 hamiltonian:6 solution:1 large:2 solid:2 eigenvalues:4 degeneracy:10 upper:1 encloses:1 occurrence:1 findings:2 sign:2 fold:2 biorthogonal:1 pass:2 implemented:1 selfadjointness:1 linear:1 even:1 will:2 nonlinear:1 appear:2 section:1 rotter:1 experiment:1 above:5 judicious:1 crossed:1 journal:1 contrast:1 ek0:1 respectively:1 loose:1 behaviour:3 address:1 path:4 along:2 change:4 intersect:1 obtained:2 incoming:1 involved:1 study:1 losses:1 resulting:1 suggestion:1 resonance:1 suitable:3 changes:3 sire:1 hulthen:1 africa:1 plenum:1 private:1 viz:2 should:2 linearly:1 experimentation:1 use:1 would:3 next:1 ψk:1 nucl:1 therefore:3 bender:1 taken:1 type:2 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quant-ph9901024.txt |@word replacing:1 limited:1 code:27 partial:8 consider:3 keep:2 global:2 results:10 chaos:1 four:1 roy:2 commute:1 entropy:1 computation:42 projective:1 go:1 follow:1 gi:1 soc:2 whose:1 supposedly:2 leung:1 alamos:8 technique:2 acting:1 send:1 environment:1 finally:6 protocols:3 set:1 briegel:3 presentation:1 resources:2 ingredient:2 applied:1 supported:1 outside:1 far:1 plausibility:1 choice:6 tradeoffs:2 every:8 anonymous:1 affect:3 associated:1 matrices:1 securely:2 arrange:1 leads:1 level:1 tra:1 evaluation:1 notes:1 solution:2 fewer:3 ri:3 noiseless:1 math:1 science:5 small:3 chuang:1 shows:2 challenge:1 related:1 lloyd:1 smaller:1 specially:2 ten:1 foundation:2 abortion:2 consistent:2 grover:1 direct:2 hillery:1 outcomes:2 zero:3 design:2 slotine:1 qeccs:4 pass:2 further:1 leakage:1 goes:3 even:6 aim:1 what:1 abc:1 section:1 selected:1 shares:5 access:4 19th:1 version:2 above:14 shared:7 public:4 contrast:1 3d:6 satisfy:1 edited:2 margolus:1 full:4 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quant-ph9901025.txt |@word proposition:6 code:18 partial:1 consider:8 remarkable:2 interference:1 global:2 desirable:1 shor:3 existing:1 four:3 roy:1 concept:2 computation:3 dist:2 relationships:1 excludes:2 dimensionality:1 leung:1 alamos:2 technique:1 containing:1 environment:2 finally:1 tk:1 under:1 alberta:1 kwong:1 applied:2 supported:2 straightforward:1 returning:1 very:1 aigner:1 corollary:5 worst:1 hewlett:1 rm:2 difference:1 matrices:1 securely:2 paz:1 leads:2 disturb:1 t6:1 turns:2 fewer:3 ri:1 vector:1 erasures:5 small:1 work:1 says:3 29th:1 zd:5 hillery:3 second:3 noise:1 orthogonal:1 implemented:1 zq:1 theory:5 even:1 errors:6 richard:1 t2n:1 giving:1 deeply:1 shares:40 decoding:2 henceforth:1 errorcorrecting:4 above:3 designing:1 bundling:1 shared:1 unrestricted:1 method:3 contrast:1 full:1 degree:1 ρab:1 applies:1 component:1 teleportation:1 jointly:3 explore:2 focused:1 hkl:1 alone:1 z0:5 appears:1 deduced:2 obtained:5 shift:1 vectors:2 contradiction:1 resulting:4 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quant-ph9901026.txt |@word consider:3 similarity:1 rev:2 dynamic:2 results:3 leads:1 roy:4 commute:2 excludes:1 violate:3 concerned:1 equivalence:1 tr:3 does:2 rangwala:6 worth:1 applied:1 broadly:1 orthogonal:7 misunderstanding:1 wave:40 englert:1 every:2 emphasized:1 difference:1 exact:1 condition:3 splitter:2 europhys:1 did:1 turns:1 t2:5 vector:1 unsharp:2 trt:1 direct:1 past:1 imply:1 perspective:1 rigorous:1 greenberger:1 click:1 linear:1 spectral:1 section:11 brief:1 expressed:2 version:2 above:2 new:2 simultaneously:3 respectively:3 here:14 understanding:2 behaviour:1 groups:1 others:1 represented:5 path:21 tunneling:11 institute:1 vectors:1 through:3 p2:2 amount:2 simultaneous:4 oversimplified:1 invalidate:1 projection:14 merely:1 explained:1 plenum:1 replace:1 apply:1 counts:1 unit:1 ψr:13 positive:1 mirrors:1 ψt:9 hilbert:16 few:1 m1:2 m2:1 taken:2 entails:1 more:5 formally:1 duality:2 operators:6 visibility:1 excellent:1 known:1 must:5 none:3 challenge:1 values:1 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quant-ph9901027.txt |@word measurements:1 partial:4 consider:2 pointing:3 rev:5 forbidden:1 results:2 scb:4 scalar:1 computation:1 consists:2 expressions:1 φcb:1 contributions:1 supposes:1 causes:1 certainly:1 compose:2 depend:4 remarkable:1 concerned:2 tr:1 finally:1 critical:1 th:5 ti:3 circumstances:1 norm:4 straightforward:1 sitting:1 far:1 rise:1 choice:1 denoted:1 corollary:3 every:5 orientations:1 dim:1 tra:4 positions:3 trc:1 trb:2 solved:1 vector:13 phase:1 math:1 eigenvalues:4 antiunitary:3 der:2 smaller:1 dealing:2 direct:4 sign:1 translating:1 zero:1 equal:2 further:4 dynamics:1 requiring:1 haag:1 telecommunication:1 even:1 established:1 what:3 abc:3 apparatus:2 sum:3 bc:9 expressed:1 access:1 ˆa:1 ˆb:3 version:1 above:5 new:6 satisfy:1 simultaneously:1 exchange:3 deduces:1 respectively:2 realizations:1 teleportation:9 here:2 modular:3 situation:1 represented:2 equations:3 change:2 convert:1 sender:3 vectors:13 excursion:1 prior:1 resulting:1 durch:1 action:6 bob:20 proof:3 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quant-ph9901028.txt |@word consider:3 results:4 chaos:4 consists:1 follow:1 expressions:1 causes:1 looking:1 depend:1 clarification:1 tr:3 corrections:1 critical:1 under:3 stochastic:2 must:1 case:15 straightforward:3 very:7 rise:1 wave:3 every:5 difference:1 exact:3 arnold:1 hamiltonian:10 notes:1 large:2 discrete:1 force:1 mantica:1 approximation:2 rate:3 displays:4 dynamics:10 substitute:1 haake:1 even:5 what:1 nonlinear:1 iterating:1 near:1 a231:1 above:3 new:3 aℓm:4 coefficient:4 edited:1 kicked:7 degree:3 whose:1 respectively:1 diagonal:1 behaviour:1 appears:1 change:1 obtained:6 study:1 accomplished:1 periodic:3 action:7 resonance:1 suitable:1 makes:1 involves:1 useful:1 projection:2 namely:1 explained:1 unitary:3 readily:1 use:1 takes:1 would:2 probabilities:3 comparing:1 therefore:2 a56:1 associated:1 more:5 sort:1 introducing:1 becomes:1 operators:1 limitation:1 berlin:2 basically:1 particular:4 cases:2 given:1 discontinuously:1 evolves:2 springer:3 erratum:1 equation:2 ρn:3 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quant-ph9901029.txt |@word code:4 consider:1 ron:1 wreath:2 consists:1 gm:7 contributions:1 graph:7 equivalence:4 finally:2 swap:3 g2:1 g1:5 orthogonal:5 p0:3 decide:2 difference:2 høyer:3 dim:3 ro:1 presented:2 solution:1 solved:1 vector:4 smaller:1 second:1 odd:1 canonical:1 even:1 index:1 what:1 appear:1 richard:1 complexities:1 sum:3 international:1 implementable:3 full:1 respectively:1 here:1 groups:2 november:1 change:1 remark:3 larger:2 allows:1 resulting:1 suggestion:1 alamos:4 fourier:6 via:1 projection:1 apply:1 use:1 working:1 contains:1 vol:2 therefore:2 theoretical:1 more:1 rigid:1 particular:1 known:2 must:1 logarithm:1 states:5 isomorphism:9 science:3 work:3 ch1:2 stabilizer:1 ms:1 example:2 proof:2 do:2 december:1 high:1 information:3 s2:6 discussion:1 write:1 nontrivial:1 answer:1 symposium:1 product:5 may:7 produce:1 efficiently:4 rudolf:1 element:2 algorithms:2 sn:7 september:1 over:4 paper:3 including:1 omitting:1 ieee:1 its:1 group:9 thank:1 personal:1 forms:1 hidden:10 main:1 non:2 usa:1 they:1 easily:1 stated:1 january:1 university:1 theory:1 each:2 2m:5 2n:3 polynomially:1 subset:1 idea:1 knill:4 quantum:9 laboratory:2 shown:1 space:6 furthermore:1 since:3 research:2 print:1 gov:2 suffices:1 occurs:1 semidirect:1 foundation:1 linear:1 definition:1 could:1 mathon:1 explored:1 onto:2 first:2 utilizes:2 probability:3 number:1 directly:1 given:2 los:4 introduction:1 least:1 returns:1 sixteenth:1 hoyer:1 relationship:1 exactly:1 b230:1 than:1 preparation:1 projections:1 generality:1 danish:1 argument:1 disjoint:1 clearly:1 efficient:1 carlo:1 able:2 note:2 finding:1 cosets:1 germane:1 connected:1 automorphism:5 nothing:1 acknowledgements:1 class:1 unequal:1 vertices:3 graphs:8 proceedings:1 fact:1 thomas:1 show:2 trivial:1 find:2 tapp:1 knowledge:1 implementation:1 nonisomorphic:1 written:2 jan:1 only:1 generators:2 abelian:4 normalizations:1 dk:2 quantph:1 denmark:1 arxiv:1 involutive:5 view:2 reveals:1 set:2 decides:1 exists:1 national:3 aarhus:2 operates:1 observed:1 see:4 computer:4 result:4 discussions:1 tensor:4 erez:1 state:3 difficulties:1 gian:1 between:2 email:2 available:1 notice:2 recently:1 complement:1 however:2 originates:1 both:4 many:1 ch:3 cm:4 point:1 simply:1 polynomial:4 pp:2 described:1 basis:4 union:1 been:3 mark:3 reduction:1 basic:1 ph:1 empty:1 define:3 corresponding:1 observable:20 sufficient:1 xxx:1 subspace:1 noncommutative:3 c2:3 c1:7 present:2 case:2 hughes:1 p1:5 exponential:1 suppose:2 transactions:1 unitary:1 almost:3 thus:3 itself:1 triple:1 march:1 different:1 s2n:1 same:1 any:4 quant:1 emphasize:1 products:2 several:2 difficult:1 http:1 v1:1 v2:1 nis:1 subgroups:4 well:2 without:1 solve:2 dimension:1 subgroup:7 yes:1 conclusions:1 except:1 easy:2 theorem:2 transformation:1 identical:2 aspects:1 showing:1 sums:1 possible:1 spanned:1 know:2 using:1 21m:1 like:1 loss:1 helpful:1 follows:1 roth:1 yields:1 martin:1 rota:1 terminology:1 classical:2 some:1 peter:3 swaps:2 centre:1 refer:1 processing:1 although:1 found:1 refers:1 hilbert:6 about:1 would:2 certainty:1 equally:1 computational:1 letters:1 communication:1 h0:2 h1:9 isomorphic:6 references:2 determine:1 beth:1 support:1 there:2 question:2 transform:1 thirdly:1 resulted:1 was:2 function:1 form:4 differences:1 alain:1 coset:3 algorithm:4 whether:1 us:2 below:1 demonstrate:1 problem:12 proved:1 defined:1 abstract:1 our:2 archive:1 codes:1 nm:1 no:1 whereas:1 when:2 other:1 papers:1 department:1 conclusion:1 elements:1 generated:1 important:1 vectors:4 implies:1 calls:1 assume:2 lanl:2 equivalently:1 time:4 oracle:1 requires:1
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quant-ph9901030.txt |@word limited:1 consider:1 maximum:2 krainov:1 forbidden:4 suggests:1 results:21 leads:2 iim:6 edu:1 research:2 contributions:2 certainly:1 inverse:2 schiff:2 kpeak:5 technique:2 baym:1 tp:1 containing:1 monotonically:1 spain:1 under:7 te:3 ingredient:1 case:23 amplitudes:1 advantage:3 very:4 frequencies:1 translations:1 seemingly:1 wave:2 refractive:2 exact:9 condition:7 arrange:1 ve2:1 presented:9 transmission:20 louis:1 solution:2 large:3 stehle:1 unrelated:2 small:1 shows:1 mo:1 upper:3 work:2 illustrating:1 sinh2:1 sin2:2 cosh:9 approximation:10 second:1 cost:2 manner:1 further:4 canonical:1 even:5 established:1 what:2 appear:2 iterating:1 dei:1 giving:1 trivial:1 section:4 coef:1 henceforth:1 version:2 above:9 lifshitz:2 replaced:1 increasing:2 acknowledgments:1 seeks:1 satisfy:8 placing:1 here:2 understanding:1 grav:1 selective:2 sink:2 sinh:9 path:1 extrema:8 equations:16 implicit:1 change:1 obtained:1 search:1 incoming:1 yablonovitch:1 changed:1 example:2 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quant-ph9901031.txt |@word concept:1 gs:2 consider:14 remarkable:1 rev:1 dynamic:1 focus:2 leads:3 scalar:4 commute:1 edu:1 expressions:1 localized:1 dunod:1 inverse:1 depend:2 by7:1 paris:1 concerned:1 finally:1 those:2 stable:1 include:1 case:6 advantage:1 very:3 desirable:2 results:1 electron:2 sons:1 every:3 matrices:4 condition:5 preuss:1 reexamination:1 hamiltonian:8 turns:3 dig:1 solution:1 try:1 discrete:3 math:1 dealt:7 small:8 eigenvalues:1 equation:15 concluded:1 oscillation:11 second:4 estimated:4 even:3 established:3 appear:2 clock:11 sun:1 section:7 expressed:1 experiment:2 above:3 new:16 consisting:2 satisfy:2 led:2 chose:1 degree:1 respectively:4 never:2 here:4 okinawa:2 address:1 represented:3 strong:1 equations:1 appears:6 change:1 conviction:1 shift:2 contradiction:1 p2:6 usually:1 suitable:2 spacetime:1 der:1 named:1 formulated:1 projection:1 replace:1 sincerely:1 analyzing:1 apply:3 unit:2 use:4 would:3 covariant:1 positive:5 negative:1 next:1 therefore:11 m2:6 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quant-ph9901032.txt |@word consider:6 results:6 correo:1 broader:1 expressions:1 whose:1 depend:1 iii:2 aria:1 presents:1 tm:1 those:3 under:2 tb:1 lake:1 sent:1 p2:2 very:15 mn1:1 modes:2 p0:1 wave:2 maali:1 resonant:7 condition:4 school:1 mc:1 level:4 transmission:15 positions:2 tanh:1 solution:4 try:1 coincidences:1 small:10 inspection:1 setup:1 ps:1 concluded:1 upper:4 andrews:1 smaller:3 ten:6 dealing:1 sin2:1 measurable:1 approximation:3 zero:6 odd:5 ues:1 even:14 will:8 appear:3 morse:1 section:1 evaporative:1 agrees:1 experiment:1 above:2 sup:1 crossed:1 increasing:1 erieure:1 acknowledgments:1 contrast:2 coefficient:6 edited:1 κn:52 predicting:1 exchange:4 respectively:1 equals:1 here:4 atoms:29 cylindrical:4 eikz:2 separation:2 november:1 logarithmic:4 obtained:3 instituto:1 eigensolution:1 shift:1 studied:2 experience:1 eberly:1 usually:1 resonance:9 conveniently:1 narrow:1 useful:1 although:2 discontinuous:1 unitarity:1 analyzing:1 total:5 derivatives:3 confidence:1 takes:1 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quant-ph9901033.txt |@word represent:1 consider:2 rev:5 focus:1 proposition:2 computation:1 go:1 reasoning:2 whose:1 simon:1 depend:1 calculate:1 re:2 equivalence:2 those:2 under:1 division:1 amplitudes:1 advantage:1 orthogonal:25 difference:1 condition:4 generalised:6 dim:1 hamiltonian:2 t2:9 solution:3 t1:8 gur:2 bangor:2 valle:2 gauge:1 says:1 grateful:1 consistent:1 sin2:1 sign:2 zero:1 street:1 further:1 carmo:1 imposing:1 linear:2 what:4 expressed:4 above:4 new:1 net:1 satisfy:12 margolus:1 full:1 here:1 understanding:1 cos2:2 pati:6 path:6 along:4 obtained:2 remark:1 study:8 studied:2 amount:1 horesh:3 via:2 useful:1 projection:1 salamo:2 scaled:1 unit:2 vivid:1 would:4 prove:6 distinct:1 next:2 ψj:10 ψk:1 live:1 ψi:13 therefore:2 taken:1 theoretical:2 more:1 becomes:2 particular:1 known:2 projective:4 cases:2 hold:1 kumar:1 ma:1 recall:1 equation:13 work:1 remain:1 following:2 example:1 proof:2 impose:1 give:1 calculation:1 india:1 states:54 sphere:2 minimum:1 numbers:3 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quant-ph9901034.txt |@word limited:1 results:1 gm:4 research:2 gj:1 gi:13 knill:2 b265:1 g3:1 g2:10 g1:13 orthogonal:3 høyer:2 gov:4 exponential:2 presented:3 solution:1 small:4 direct:1 fold:1 second:1 perspective:1 further:1 even:1 appear:1 above:2 international:1 implementable:2 equals:1 here:1 groups:4 november:1 observing:1 vectors:2 allows:1 accomplished:1 periodic:4 resulting:5 postprocessing:1 alamos:6 fourier:1 projection:2 apply:2 takes:1 usa:2 prove:2 distinct:2 ψi:8 call:1 therefore:1 theoretical:1 more:1 formally:1 outputs:1 normalized:1 known:2 must:1 states:8 science:3 work:2 values:1 stabilizer:1 ms:2 following:1 example:1 proof:2 computers:1 stated:2 december:1 high:3 insignificantly:1 information:4 how:1 pure:1 symposium:1 product:4 may:5 after:1 membership:1 arhus:1 efficiently:8 register:1 element:1 emanuel:1 complexity:1 algorithms:1 course:1 paper:1 mentioned:1 existence:1 group:7 better:1 choose:1 hidden:10 main:3 non:3 they:1 now:2 yu:1 january:1 university:1 identified:1 orthonormal:1 each:2 2m:6 identifies:1 subset:3 ei:17 computable:1 measurement:3 quantum:16 our:2 shown:1 space:2 furthermore:1 since:2 denotes:1 print:2 ro:1 induction:1 foundation:1 linear:1 e0:1 determining:1 omit:1 times:1 ordering:1 exponentially:5 hence:2 onto:2 first:3 probability:6 xm:2 number:4 directly:1 open:2 given:2 introduction:1 least:1 1m:1 final:1 sixteenth:1 hoyer:1 exactly:2 discover:1 b230:1 x1:2 than:1 determines:1 cyclic:1 distinguishes:3 determined:1 tion:1 outcome:2 danish:1 kitaev:1 say:4 manner:1 seen:1 any:3 efficient:1 indexed:1 so:1 centre:1 beth:1 class:1 proceedings:1 left:2 fact:1 thomas:1 show:2 random:1 implying:1 d2:1 find:1 jan:1 lemma:6 only:4 abelian:5 dk:2 quantph:1 words:1 denmark:1 arxiv:1 view:1 exists:4 national:3 aarhus:1 computer:4 result:3 best:1 tensor:1 state:5 discovers:1 correctly:2 geometric:1 email:3 available:2 notice:1 complement:1 however:1 answers:1 unnormalized:1 picking:1 alters:1 observable:1 improved:1 among:3 key:1 polynomial:5 throughout:1 basis:1 due:1 been:1 mark:2 basic:1 pk:7 ph:2 abstract:1 define:3 corresponding:3 xxx:2 subspace:1 noncommutative:2 generalization:1 present:1 case:2 subsets:1 suppose:1 error:1 unitary:1 almost:2 sequentially:1 different:2 inquire:1 amount:1 quant:2 finite:7 http:2 i2:1 hand:2 nis:1 subgroups:3 lower:1 well:1 dimension:2 subgroup:13 obtain:2 identify:4 previous:1 candidate:1 cosets:3 theorem:3 aspects:1 dihedral:1 possible:5 spanned:1 department:1 know:3 necessary:1 follows:2 either:1 martin:1 works:1 classical:2 some:1 b1:2 b2:1 peter:2 nonzero:1 measurements:2 does:1 bm:2 arbitrary:3 continuing:1 bi:6 step:1 although:2 hilbert:2 o:2 remains:1 range:1 important:1 raised:1 bound:2 references:1 operator:3 theoretically:2 log:9 prepare:1 strictly:3 there:5 question:3 transform:2 los:6 inner:2 was:1 function:8 form:2 regard:1 but:1 hk:3 repeatedly:1 true:2 utilizing:1 cic:1 applying:1 algorithm:13 whether:3 distinguishability:1 us:1 below:3 problem:3 informational:1 called:1 constant:1 choosing:1 povm:2 ak:1 am:2 denote:1 exist:2 laboratory:2 archive:2 nm:2 application:1 other:2 test:2 conclusion:1 elements:4 superposition:1 a1:2 generated:1 a2:1 implies:3 calls:6 assume:1 lanl:6 equivalently:1 2im:2 time:6 oracle:7 requires:2
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quant-ph9901035.txt |@word consider:8 forbidden:1 informally:1 results:3 four:3 generalize:1 montr:4 broader:1 consists:2 follow:1 whose:5 depend:1 vs:1 send:2 γj:1 th:3 under:1 alberta:1 stochastic:2 include:1 resources:1 random:6 infinite:1 string:2 continues:1 focus:1 interpreting:1 summarized:1 difference:1 exact:3 condition:1 entire:1 solves:1 turns:1 defeats:1 solution:1 correct:1 large:2 phase:3 round:1 ville:1 smaller:1 zi:2 ten:1 deutsch:3 sin2:5 approximation:2 second:5 cost:2 odl:1 8n:1 odd:1 even:1 index:1 what:2 richard:1 t2n:1 celebrated:1 negates:1 uniform:1 version:1 above:5 designing:1 new:1 satisfy:1 full:1 respectively:10 component:3 gilles:1 behaviour:8 test:1 cos2:11 consideration:1 along:1 sending:2 shift:1 larger:1 allows:1 amount:3 usually:1 bob:41 proof:7 tasks:1 regardless:2 ix:1 explained:1 outcomes:10 apply:2 clockwise:1 unit:3 permitted:3 would:5 prove:2 positive:1 next:3 scenarios:5 call:2 therefore:3 centre:1 calgary:3 more:8 separated:4 formally:1 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quant-ph9901036.txt |@word sch:2 chinese:1 ge:1 generalize:2 gm:3 research:1 consider:2 ihep:1 inverse:8 calculate:1 concerned:1 technique:1 finally:1 anharmonic:1 kν:1 case:2 far:1 wave:11 electron:1 affect:1 exact:7 did:1 turns:1 solution:4 eigenvalues:1 dealing:1 b21:1 sign:3 approximation:1 zero:1 recalculate:1 chem:2 even:1 interatomic:1 section:5 expressed:1 above:4 lifshitz:1 new:2 ground:10 method:6 satisfy:1 27a:3 27b:2 27c:2 simultaneously:2 respectively:6 here:2 atoms:2 china:4 ansatz:7 address:1 equations:6 dα1:1 box:3 remark:1 institute:2 study:6 vogt:1 products:1 plotted:4 univ:1 unit:1 readily:3 use:2 takes:1 negative:5 vol:2 therefore:1 m2:3 type:1 odinger:21 normalized:5 bates:1 cases:3 bernstein:1 ma:6 science:2 equation:26 work:1 london:1 values:14 following:9 making:2 arrive:3 example:1 beijing:3 compare:3 figure:4 give:1 calculation:3 organized:1 high:2 imϕ:1 numbers:1 axis:4 discussion:1 write:1 hou:1 parameter:8 lam:1 singular:1 after:3 wrong:3 choice:2 constants:2 short:1 natural:2 dynamical:1 succeeded:1 leroy:2 joachain:1 so:1 fundamental:1 order:3 indeed:1 course:2 paper:3 through:1 mentioned:2 satisfies:3 ln:2 fix:1 actually:2 systems:1 derivative:1 main:2 wen:1 them:1 they:8 oxford:3 now:4 discuss:3 eigenvalue:2 hereafter:2 advent:1 2g:1 2a:1 2b:2 radial:10 higher:1 constraints:2 another:3 hadronic:1 ed:1 energy:7 hard:1 idea:1 used:1 quantum:4 laboratory:1 eq:30 special:1 out:2 3rd:1 furthermore:1 since:1 rev:1 denotes:5 semiconductor:1 rm:8 cα1:2 correct:1 foundation:1 theory:1 8b:2 8c:2 8a:4 standard:1 reason:1 york:2 besides:2 dgm:1 e0:1 interaction:6 r0:2 r2:2 sextic:2 parashar:1 retain:2 first:5 origin:1 major:1 already:1 features:1 rydberg:1 dong:6 2a1:1 impossible:2 electronic:1 angular:4 intermolecular:1 given:5 likely:1 similarly:1 introduction:1 eqs:3 tow:1 relations:1 their:7 calculations:2 realization:1 molecular:2 behaved:2 2ad:2 2ab:1 2ac:1 ryzhik:1 asymmetric:1 academy:1 bα1:1 17b:2 17a:1 attempt:1 wide:1 ab11:1 unfortunately:1 cimento:2 depends:1 submitted:3 matter:2 determined:1 were:1 ann:1 say:1 wannier:1 mie:1 instance:1 potential:33 take:2 forces:1 xi:1 coincide:1 play:1 rℓ:4 wells:1 gradshteyn:1 octic:2 calculated:2 later:1 principle:2 exp:6 physics:6 fact:2 supported:1 spaces:1 relation:2 solutions:10 d2:8 find:6 d1:2 parameters:13 explain:1 written:3 jan:1 analytic:1 only:6 molecules:1 developed:1 employed:1 2bd:1 2bc:3 2ba:1 do:1 distance:1 dg:1 during:1 table:1 dr:15 prime:1 intuitive:1 fields:3 arxiv:1 2b1:2 acknowledgments:1 qi:6 shi:6 fixed:1 discussing:1 national:1 see:2 result:2 muonic:1 techniques:1 eigenfunctions:2 dipole:2 state:15 closed:1 between:3 progress:1 however:2 unnormalized:5 both:6 c21:1 last:3 equal:1 hψ:1 figures:3 yℓm:1 cn:1 16d:2 16e:2 each:1 16a:3 16b:1 16c:2 dimensional:9 simply:1 cr:6 throughout:1 considering:2 longman:1 three:15 been:2 interest:1 ph:1 choose:2 obtained:4 arises:1 corresponding:5 stems:1 c2:3 c1:7 applied:1 pacs:1 value:3 will:7 a21:1 almost:1 according:3 il:2 technology:1 different:5 grant:1 make:3 same:4 quant:1 widely:1 k2c:1 several:1 rigby:1 difficult:3 extended:1 hand:2 ion:1 arrived:1 potentials:5 frequently:1 lower:2 schr:19 center:1 well:3 academic:2 sets:4 obtain:12 mistaken:1 mistakes:2 atomic:1 simsek:4 world:1 hai:6 sciences:1 smith:1 except:1 simple:1 lett:3 easy:1 role:1 excited:5 momentum:4 generalized:1 eigenfunction:6 read:3 advances:1 possible:1 know:2 press:4 using:2 accepted:1 advanced:1 follows:3 continue:1 integrals:1 arbitrary:1 become:1 essential:1 because:1 methods:1 people:2 some:4 growth:1 b1:5 b2:5 respect:3 dimensions:28 lead:1 2c1:1 does:2 rℓ1:4 power:8 2c:1 br:8 properties:1 simpler:2 constrains:1 carried:1 constraint:2 discussed:2 o:3 important:1 submited:1 computational:1 pergamon:1 integral:1 references:1 often:1 low:1 way:2 series:1 was:1 function:6 form:1 neutral:1 atom:1 evident:1 problem:1 similar:1 mechanics:4 ac:1 ab:2 constant:3 ad:2 classic:1 abstract:1 denote:2 ar:6 exist:1 aα1:1 functions:8 compared:2 eψ:2 no:1 eα1:1 zhong:6 when:3 vanishes:1 field:1 other:4 republic:2 nr:1 conclusion:2 node:1 draw:1 calculating:2 problems:1 occurred:1 a1:9 a2:3 variable:3 implies:1 requires:1 n0:4 4b:2 nuovo:2 4a:4
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quant-ph9901037.txt |@word concept:1 consider:1 chinese:1 desirable:1 ge:1 ihep:1 whose:1 simon:1 anharmonic:4 very:1 4ec:1 wave:13 affect:1 exact:8 solution:6 pramana:1 dealing:1 sign:1 approximation:1 linear:1 section:2 selected:1 above:2 new:2 acknowledgments:1 coefficient:1 simultaneously:1 respectively:1 here:2 desired:1 inverse:1 china:4 ansatz:8 address:1 equations:2 obtained:2 institute:2 study:6 studied:1 khare:1 fourier:1 studies:2 plotted:1 univ:1 unit:1 readily:2 eψ:1 positive:1 whittaker:1 predictions:1 next:1 few:1 m2:1 today:1 more:3 introducing:1 becomes:1 odinger:28 normalized:3 particular:1 known:1 foundations:1 behalf:1 states:1 ma:4 βr:3 science:2 equation:33 work:2 modified:6 london:2 values:7 following:5 arrive:5 example:4 beijing:3 compare:1 figure:1 give:2 organized:1 topic:1 imϕ:1 numbers:1 want:2 axis:2 means:1 coleman:1 parameter:7 may:2 after:2 containing:1 summarize:1 amin:1 natural:2 dynamical:1 succeeded:1 so:1 fundamental:1 order:3 16c2:1 paper:3 through:5 mentioned:1 satisfies:1 before:1 sciences:1 ln:1 fix:1 actually:1 choose:1 derivative:1 main:1 deformation:1 13c:2 13b:2 13a:2 combination:1 13d:2 introduce:1 framework:1 oxford:1 now:2 structural:2 eigenvalue:1 dirac:1 hereafter:2 university:1 2a:3 reasonable:1 radial:12 higher:1 constraints:1 adjointness:1 energy:4 principles:1 related:1 used:1 obtains:1 quantum:5 laboratory:1 transitions:1 eq:14 out:2 furthermore:1 rev:1 denotes:3 attracted:1 rm:4 newly:1 foundation:1 theory:3 8e:2 formation:1 york:1 besides:1 e2:1 r2:3 sextic:1 turn:1 parashar:1 retain:1 first:4 origin:2 features:1 symmetry:2 vacuo:1 number:3 dong:4 electronic:1 angular:3 given:5 introduction:1 unknown:1 eqs:1 observations:1 scheme:1 behaved:1 academy:1 authors:1 attempt:1 wide:1 unfortunately:1 submitted:1 matter:1 college:1 function:12 false:1 ann:1 modern:2 academic:1 self:1 instance:1 high:2 potential:22 analogy:5 lectures:2 play:1 who:1 algebraic:1 αr:2 phase:1 octic:10 calculated:1 principle:1 exp:3 physics:6 fact:1 son:1 supported:1 discovered:1 spaces:8 relation:1 solutions:5 carefully:1 substituting:1 d4:1 d2:9 find:1 ground:5 knowledge:1 parameters:7 written:3 jan:1 only:2 employed:1 isham:1 distance:1 familiar:1 watson:1 words:1 dr:12 fields:1 arxiv:1 mathematicians:1 qi:4 shi:4 fixed:1 set:1 national:1 depends:1 sec:1 result:1 pm0:3 satisfying:1 eigenfunctions:7 state:5 closed:3 between:1 progress:1 rm0:5 recently:2 importance:1 attention:1 however:4 chatterjee:1 unnormalized:2 many:1 according:1 against:1 hψ:1 cn:1 nonrelativistic:1 appeared:1 16a:1 16b:1 dimensional:20 respect:1 throughout:1 considering:1 due:1 been:3 much:3 interest:1 ph:2 fig:1 box:3 corresponding:2 understand:1 denote:1 case:2 pacs:1 value:2 choices:1 will:8 property:1 refs:1 them:1 ii:1 8e2:2 vanish:1 technology:1 different:1 13e:2 grant:1 make:2 same:1 quant:2 proved:1 several:1 difficult:2 literature:1 infinity:1 singapore:1 hand:1 arrived:1 purpose:2 potentials:4 frequently:1 normalization:1 euro:1 relies:1 lower:1 task:1 schr:28 center:1 well:3 analysis:2 concrete:1 solids:1 solve:1 r00:2 obtain:6 atomic:1 world:1 hai:4 application:1 seems:1 except:1 simple:7 lett:2 easy:1 role:1 transformation:4 momentum:3 aspects:1 read:2 ame:1 possible:1 know:3 press:4 using:1 accepted:1 advanced:1 follows:2 arbitrary:7 stationary:1 mathematical:2 methods:2 deal:1 people:2 some:4 understood:1 cr:3 dimensions:4 lead:1 2b:1 clarendon:1 br:3 hilbert:1 simpler:2 constrains:1 central:3 carried:1 o:3 consequently:2 cambridge:3 into:2 three:1 references:1 twodimensional:1 imperial:1 physicists:1 way:1 was:1 polaron:1 form:2 phenomena:1 reed:1 filed:1 esposito:2 applying:3 born:1 whether:1 erice:1 demonstrate:1 define:1 mechanics:2 ac:1 constant:2 worthwhile:1 abstract:1 general:2 ar:3 our:2 functions:2 compared:1 no:1 zhong:4 when:1 field:1 other:4 republic:2 er:2 calculating:1 problems:1 important:1 variable:1 implies:1 n0:4 rep:1
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quant-ph9901038.txt |@word secondly:1 unavoidable:1 similarity:1 rev:6 varying:1 desirable:1 results:1 four:1 gf:27 scalar:1 generalize:1 mℓ:1 consists:1 spaced:1 contributions:3 depend:1 calculate:1 producing:3 concerned:1 technique:2 passage:1 supported:1 γi:6 th:4 eprint:1 include:2 case:9 confirmation:1 amplitudes:2 outside:1 very:4 frequencies:2 exception:1 choice:1 maali:2 every:1 figs:4 resonant:28 experimentally:1 associated:2 continued:1 randomness:1 induced:1 hamiltonian:8 solution:2 large:7 discrete:2 vector:5 small:10 eigenvalues:1 dip:14 equation:6 zn:2 averaged:2 cooperative:3 oscillation:1 estimates:2 direct:2 matrices:1 located:2 second:11 dynamics:1 spectral:2 will:1 nonlinear:4 trapping:1 sum:1 collet:1 expressed:1 below:1 experiment:3 destroyed:1 judicious:1 new:2 ground:4 ever:2 method:5 coefficient:1 edited:1 full:1 component:3 κa:1 here:4 atoms:4 desired:1 selective:1 path:2 along:1 strong:5 equations:6 differential:1 obtained:1 turchette:1 noticeable:1 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quant-ph9901039.txt |@word proposition:1 consider:2 chain:1 pth:1 results:2 commute:3 ρt0:1 denotes:1 devoted:2 whose:1 calculate:4 equivalence:1 tr:9 tm:1 under:1 applied:1 straightforward:1 mpi:1 print:6 corollary:3 every:3 hamiltonian:8 turns:2 list:1 solution:3 t0:64 t1:61 vector:4 ρh:3 gauge:1 ga:1 htv:3 imply:1 rigorous:2 further:1 linear:4 index:1 sum:1 brief:1 uniform:1 sequential:1 above:4 dtγ:4 new:5 acknowledgments:1 notions:1 respectively:11 ρv:2 here:9 governing:1 address:1 directions:1 path:3 along:31 equations:12 appears:1 change:2 morphisms:14 remark:1 suitably:1 institute:1 shift:1 study:1 vectors:4 permutation:1 suitable:1 via:11 equality:1 ls:1 summation:1 apply:1 unit:1 conventional:4 feb:1 prove:1 positive:2 contains:1 bundles:5 vol:2 ψi:10 taken:1 msc:1 theoretical:1 possible:1 more:1 bulgarian:1 operators:9 quant:6 odinger:8 curvature:15 particular:2 known:2 iii:43 easiest:2 must:1 evolves:1 account:1 rη:9 ma:1 recall:1 equation:22 work:4 reproduce:1 remain:2 values:5 ρt:1 concepts:1 following:3 something:1 give:1 december:3 organized:1 purposes:1 states:14 numbers:4 want:1 conversely:1 end:2 turn:1 provide:1 pae:1 discussion:1 meanwhile:1 sec2:1 criterion:1 fourth:3 map:4 product:4 description:20 may:2 use:1 derived:7 hence:3 representations:3 law:1 parallel:1 short:1 natural:2 dynamical:3 chauss:1 lines:1 so:5 lγ:12 fundamental:2 representation:2 superscript:1 restriction:1 fγ:17 interpretation:1 september:2 pointed:1 mainly:1 satisfied:1 produced:1 ended:1 paper:3 through:1 statistical:2 mentioned:1 satisfies:1 its:3 sciences:1 chosen:1 absence:1 systems:1 mail:1 lx:4 emphasize:1 them:4 pure:6 propose:1 introduce:1 framework:1 they:2 oxford:2 now:7 developments:1 term:1 dirac:3 january:1 presented:1 appropriate:1 iff:2 academic:1 editor:1 found:3 entirely:1 characterized:1 mean:5 preceded:1 differential:2 ei:1 evolve:1 ee:1 series:4 energy:1 revised:2 ea:1 principles:1 investigated:3 expect:1 combining:1 quantum:42 our:2 special:2 out:1 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