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quant-ph9705018.txt |@word limited:2 rev:8 results:1 computation:1 edu:1 causes:1 alamos:5 deterministic:1 γj:3 program:1 preserved:1 norm:2 supported:1 orthogonal:9 very:1 multi:4 seemingly:1 replication:2 condition:1 hamiltonian:1 knight:1 guo:3 noiseless:1 vector:3 science:2 small:3 eigenvalues:1 says:2 hillery:4 zero:1 further:2 linear:1 even:1 plays:2 diag:2 expressed:1 above:3 method:1 satisfy:2 here:3 desired:2 plenio:1 china:3 great:3 sender:1 leaves:1 allows:1 completely:1 suggests:2 products:1 schumacher:2 usually:1 summation:1 linearly:17 prove:5 remains:1 positive:19 negative:1 ψt:5 ψj:4 ψi:13 ψn:19 therefore:3 m2:3 marginally:1 more:1 formally:1 composite:2 becomes:1 correspondingly:1 normalized:1 ndimensional:1 equation:1 mn:3 following:8 proof:11 process:9 bennett:5 states:45 attain:3 information:3 buzek:4 holds:3 machine:7 how:1 significance:1 answer:2 gate:5 brassard:3 barnum:1 may:3 after:1 applications:3 data:2 constants:1 natural:1 joint:1 lemmas:1 yielding:2 so:6 fundamental:1 operate:1 numbers:2 satisfied:3 course:1 paper:1 through:2 eavesdropping:1 existence:1 fidelity:1 its:3 diagonalized:1 chosen:3 whether:6 lu:1 systems:1 mail:1 main:4 non:8 return:1 nonorthogonal:1 introduce:2 they:4 now:1 easily:1 finite:1 verifying:1 university:1 orthonormal:8 transmitted:1 each:1 found:1 2n:1 mean:1 schemes:1 programmed:1 measurement:5 quantum:33 our:1 operation:1 eq:12 maximizes:1 secretly:2 shown:1 matrix:13 space:4 since:2 crucial:2 increase:1 eprint:5 ηx:7 ref:2 independence:1 shows:1 foundation:1 theory:3 circumstance:1 succession:1 reason:1 complicated:1 jozsa:2 generate:1 forbids:2 programming:2 hence:3 onto:2 think:1 first:2 probability:4 u1:5 channels:1 adopted:1 another:2 impossible:1 array:4 open:2 checked:1 similarly:3 unknown:2 xij:3 system:6 construct:1 statement:2 programmable:1 wootters:3 gives:1 interests:2 accurately:1 broadcast:1 completes:1 fuchs:1 preparation:1 crepeau:1 result:4 fashion:1 ming:1 say:1 replicated:3 need:2 any:3 mermin:1 vedral:1 ideal:1 potential:1 φab:4 channel:1 though:1 probabilistic:2 considered:1 sometimes:1 braunstein:1 physics:1 fact:1 show:4 probe:4 chuang:2 find:1 based:1 popescu:1 state:11 should:1 lemma:9 only:6 continuity:2 do:2 cryptography:5 means:1 dependent:6 zurek:1 famous:1 report:1 evolution:11 arxiv:1 yield:3 arrays:1 london:1 set:3 constructing:1 exists:7 national:1 connections:1 cloning:33 alternative:1 concern:1 project:2 fails:1 definite:11 enough:3 simplest:1 caves:1 recently:1 reproduced:1 nature:2 qubits:1 however:3 hefei:1 efficiency:10 key:1 alters:1 last:1 receiver:1 expression:1 ci:1 cn:1 point:1 cc:2 corresponds:1 dimensional:2 described:2 raise:1 basis:2 copies:2 been:1 reduction:6 replicate:3 pj:1 ph:6 pn:1 abstract:1 ariano:1 subspace:1 remark:1 systematic:1 case:2 p0:5 p1:1 pacs:1 suppose:3 inaccurately:1 identical:1 unitary:21 refs:1 thus:4 surprisingly:1 duan:2 technology:1 different:1 perform:1 threatens:1 make:7 belong:1 same:5 imoto:1 quant:6 copying:3 φi:10 φj:12 φk:6 φn:12 development:1 independent:10 extended:2 hand:1 identity:1 recent:1 task:1 nevertheless:1 obviously:2 guang:1 just:1 when:1 yes:1 subjects:1 extends:1 lett:9 theorem:12 input:8 real:1 nielsen:2 koashi:1 posed:1 possible:3 department:1 projects:1 apart:1 success:3 semi:3 follows:1 reduces:1 yields:1 either:1 arbitrary:5 output:2 security:1 inter:1 essential:2 because:1 some:2 b1:2 b2:2 measurements:1 does:1 inaccurate:3 bn:2 constructive:1 yuen:2 hilbert:3 bz:1 noisy:1 eavesdropper:1 practical:2 mixed:1 communication:1 into:1 nsc:1 bruss:1 references:1 operator:8 matrixes:1 intervention:1 there:9 question:9 los:5 quadratic:1 inner:1 way:3 m1:3 was:1 archives:5 form:1 smolin:1 converse:2 but:7 cloned:15 copied:1 made:1 uu:1 cij:2 maximum:1 today:1 similar:2 mechanics:5 ab:2 proved:2 defined:1 certain:4 universal:1 measure:1 linearity:3 proven:1 general:2 peres:1 proves:1 exist:2 commuting:1 no:7 generally:1 other:2 role:2 np:2 zheng:1 superposition:1 separate:1 ustc:1 important:2 acknowledgment:1 longer:1 together:3 impossibility:1 time:2 failures:1
quant-ph9705019.txt |@word emch:1 sci:2 consider:2 rev:3 global:1 desirable:1 leads:3 roy:1 go:1 follow:1 preserves:3 simon:6 number:2 paris:1 concerned:1 equivalence:3 environment:1 charge:2 critical:1 ny:3 preserved:1 under:7 γc:3 γb:5 differing:1 norm:2 belonging:1 provides:1 advantage:1 orthogonal:14 springer:2 geometrical:1 corollary:1 noncyclic:1 emphasized:1 difference:3 entire:1 symplectic:3 hamiltonian:2 cosine:3 leave:1 discrete:4 vector:7 manifestly:1 pramana:1 antiunitary:2 optics:2 der:1 mathematique:1 rri:1 sin2:1 direct:1 literature:1 sign:1 fold:1 rate:1 arnold:1 gn:1 further:1 even:2 what:2 abc:5 appear:2 trivial:1 section:7 current:1 indian:1 above:4 new:1 journal:2 filled:1 satisfy:4 led:1 chose:1 deduces:1 component:4 residual:1 here:7 understanding:1 behaviour:2 reported:1 tangent:3 excess:5 concluding:1 path:2 along:1 strong:1 appears:1 change:1 great:1 institute:1 involved:2 study:5 vectors:5 allows:2 resulting:2 bhandari:2 pick:2 action:2 studies:1 quotient:1 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quant-ph9705020.txt |@word monte:4 customary:1 remarkable:1 both:1 quadrature:1 whose:1 inverse:1 tr:19 finally:2 lavoro:1 those:1 alized:1 analogous:1 very:3 springer:2 sono:1 summarized:1 northwestern:1 presented:1 confirmed:1 optics:2 funzioni:2 reα:2 noise:1 carlo:4 nel:1 stands:1 sul:1 analogously:1 ˆa:2 ˆd:1 ˆe:3 obvious:1 positivit:1 method:1 pioneering:1 edited:1 respectively:2 groups:1 equations:13 change:1 obtained:3 convert:1 delle:1 allows:1 molto:1 products:2 usually:2 symmetrical:1 studi:1 makes:1 glu:1 via:1 useful:2 convolution:1 marginal:1 namely:4 highly:1 names:1 orderings:1 planck:8 positiveness:2 use:1 sacchi:5 usa:1 positive:5 negative:5 definition:2 properties:1 densit:1 historically:1 cahill:4 operators:9 odinger:1 berlin:2 appendix:2 eλ:1 particular:4 known:3 cases:2 iii:1 easiest:1 high:1 keeping:1 equation:2 work:1 oscillator:3 cat:1 remain:1 values:4 concepts:2 following:10 averages:1 example:6 proof:2 states:3 numbers:1 fortunato:1 provide:1 s2:16 holds:1 recovers:1 write:1 how:1 pure:1 field:6 parameter:1 fs:13 efficient:1 singular:1 transforming:1 derived:1 consequence:1 plane:4 multiplication:5 representations:4 derives:1 plank:2 branches:1 green:1 representation:4 playing:1 wigner:40 over:1 symbols:1 thermal:1 paper:1 through:3 volta:1 aλ:1 derive:2 la:2 lo:1 solving:1 expectations:1 boson:2 actually:1 forms:9 pavia:2 schmidt:1 amenable:1 esempio:1 they:2 easily:6 inversion:2 2s:4 always:1 university:1 composition:1 mode:1 each:1 universit:1 macchiavello:1 characterized:1 weyl:2 ed:1 ea:17 displacement:1 related:1 connect:2 measure:1 quantum:11 our:1 damped:1 eq:14 shown:1 tombesi:1 matrix:8 space:3 nuovo:1 since:1 rev:4 rew:1 denotes:2 may:1 acting:1 denoted:1 formula:5 ref:4 formule:2 shows:1 theory:2 gardiner:1 fisica:1 quite:1 york:1 practically:1 applications:2 bassi:1 motion:1 signature:1 hence:3 dipartimento:1 origin:1 probability:2 laguerre:1 number:2 applicazioni:1 precisely:1 given:1 ordering:3 eqs:4 relations:3 glauber:4 master:6 statement:1 recovered:1 complesso:1 optical:1 positively:1 zachary:1 cimento:1 see:7 annihilation:1 1o:1 providing:1 generalization:1 recover:2 any:4 viewed:1 academic:1 studying:2 definite:2 ideal:1 generalized:18 δnm:1 formulas:3 trace:9 normal:3 diffusion:1 phase:3 class:1 considered:1 order:2 exp:6 mechanical:2 left:3 physics:3 review:1 dover:1 show:1 relation:4 d2:12 only:3 factor:1 various:1 piano:1 di:7 express:1 expectation:2 da:1 joint:1 new:1 dv:1 during:1 symmetrically:1 evolution:1 arxiv:1 simplification:1 view:1 detection:1 alessandro:1 computer:1 result:2 techniques:1 αw:1 positivity:4 αa:13 state:7 αe:1 between:1 stato:1 αm:3 αn:3 publishers:1 terms:1 nature:1 however:1 le:1 ws:24 bose:1 come:1 queste:1 equazioni:2 questo:1 quantization:1 estimation:1 expression:1 con:2 among:1 colons:1 point:1 simple:3 harmonic:2 infinitely:1 proved:3 aˆ:2 three:2 damping:1 interest:1 basic:1 ph:1 ariano:8 corresponding:1 easy:1 commutes:1 coherent:2 last:1 fokker:10 present:1 exception:1 pacs:1 value:1 evaluating:2 property:1 procedure:1 shumovsky:1 ii:1 il:1 italy:1 distributions:2 eigenvalue:1 complex:4 quant:1 advances:1 physical:1 couple:1 higher:1 used:2 hand:2 kim:1 identity:3 task:1 schr:1 well:2 analysis:1 defining:1 solids:1 position:2 usual:4 just:1 being:1 λa:1 using:7 moroni:1 dimλ:1 eigenstates:1 rapid:1 kluwer:1 density:8 photons:1 guaranteed:1 evanston:1 quadratures:2 conjugated:1 lett:1 versa:3 role:1 har:1 eigenstate:1 momentum:2 iφ:1 around:1 represents:2 rules:1 evaluate:2 matrice:1 corresponds:1 possibly:2 vacuum:1 press:1 immediately:2 mod:1 follows:2 qualitative:1 indication:1 integrals:1 become:1 because:1 often:1 classical:2 creation:1 some:1 examples:1 choosing:1 table:2 oscillations:1 tutorial:1 lead:2 measurements:2 integration:1 per:2 connecting:3 expressing:1 steps:1 symbol:1 2a:4 differentiable:1 slight:1 performed:1 equivalent:3 continuing:1 eiφ:1 super:6 hilbert:1 bz:1 che:1 od:1 o:26 simulation:1 mechanics:1 statements:1 into:3 functional:1 odd:1 iπa:1 references:1 difficult:1 operator:12 amsterdam:1 methods:1 degli:1 there:1 way:1 was:1 function:27 form:4 true:1 considerable:1 vice:3 maximum:1 converted:1 spectroscopy:1 distribution:2 more:4 obtains:1 aa:1 ordered:5 ad:1 abstract:1 am:2 denote:2 engineering:1 proves:2 right:4 simulations:2 identities:3 check:1 dα:1 functions:18 compared:1 whereas:2 generally:2 coefficient:2 other:3 electrical:1 department:2 defines:2 conclusion:1 differential:7 elements:1 preceding:1 nonclassical:1 polynomials:1 astronomy:1 quasiprobability:1 monic:1 eαa:3 requires:1 original:2
quant-ph9705021.txt |@word represent:2 customary:1 consider:2 switched:1 results:1 four:1 quadrature:11 amplification:1 tr:3 quasi:1 compensate:1 squeezing:17 g3:2 g2:2 g1:3 very:3 engineered:1 choice:3 mixing:1 experimentally:4 splitter:8 hamiltonian:10 large:1 admissible:1 settle:1 small:1 optics:3 setup:1 compensated:1 approximation:1 second:1 squeezer:3 implemented:3 port:2 displacement:5 will:15 appear:1 apparatus:1 ˆx:2 ˆy:3 xφ:1 expressed:1 cell:6 consistently:1 experiment:1 above:1 capital:1 new:1 learned:1 method:2 respectively:2 understanding:1 inverse:1 represented:1 path:1 along:2 strong:3 change:3 obtained:1 leaves:2 changed:2 yˆ:7 evading:1 accomplished:1 action:3 conveniently:1 changes:1 glu:1 york:1 uniquely:1 formulated:1 useful:1 convolution:1 bs:3 plenum:1 jˆz:2 univ:1 use:3 sacchi:1 working:3 photocurrent:1 finally:2 theoretical:2 more:2 becomes:1 operators:7 apparent:1 controlling:1 foundations:1 iii:1 easiest:1 must:1 dominicis:1 involve:1 valued:1 indirectly:1 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quant-ph9705028.txt |@word ibid:1 represent:3 cosines:1 consider:1 switched:1 rev:34 focus:1 quadrature:1 selection:1 consists:2 displaying:2 whose:1 detects:1 terms:3 paris:1 iii:2 technique:2 tr:4 acting:1 supported:1 reductions:2 preserved:1 th:1 under:4 stochastic:1 characterised:1 norm:2 alternating:1 applied:1 probe:6 amplitudes:2 advantage:1 wji:1 very:2 frequencies:2 choice:2 maali:2 resonant:2 affect:1 difference:1 matrices:2 condition:1 europhys:1 smithey:1 incoherent:1 vibrational:2 level:2 knight:2 complexvalued:1 convenience:1 vector:2 indices:3 shows:1 optics:1 upper:1 work:1 occurrence:1 cessa:1 appearance:2 subtracting:1 direct:5 pulse:3 discriminated:1 displays:1 further:2 dynamics:8 linear:1 even:3 established:1 nonlinear:3 sub:5 section:1 brief:1 dunn:1 reconstruction:3 new:2 ground:1 method:11 contrast:1 realizing:1 full:7 degree:2 respectively:6 here:5 atoms:3 behaviour:1 desired:5 cos2:1 relaxation:1 represented:2 consideration:2 strong:6 equations:1 appears:1 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quant-ph9705030.txt |@word partial:1 consider:2 caused:2 leads:1 schatten:1 generalize:1 controversial:1 whose:1 depend:2 uders:1 equivalence:1 tr:20 editors:2 tn:5 under:1 norm:1 applied:1 random:2 every:2 lecture:2 condition:2 nonselective:2 school:1 hamiltonian:5 tra:9 notes:2 proposals:1 t1:25 solved:1 ueda:1 equation:1 consistent:1 measurable:1 rigorous:1 further:1 linear:1 even:1 established:1 what:1 plays:1 apparatus:21 annals:1 above:4 bm:1 new:6 method:1 ihτ:7 satisfy:3 simultaneously:1 desired:1 others:1 represented:1 change:11 obtained:7 changed:1 prior:8 simultaneous:1 changes:8 york:2 uniquely:1 formulated:4 inference:1 projection:25 family:1 marginal:4 plenum:2 outcomes:9 apply:1 total:1 univ:1 conventional:1 prove:1 positive:5 hudson:1 nagoya:2 next:1 therefore:2 theoretical:1 more:1 duality:1 composite:6 operators:7 odinger:1 berlin:2 normalized:2 foundations:4 states:2 springer:2 valued:4 join:1 science:2 originally:1 following:4 ρx:2 example:1 proof:1 process:3 sample:1 numbers:1 allowed:1 counting:2 information:7 instant:1 s2:6 s1:8 maps:1 ensemble:2 simplicity:1 product:2 may:2 after:9 derived:2 physik:1 dynamical:2 light:1 counter:1 element:1 davies:1 fundamental:4 operate:1 discussion:1 typical:1 wigner:1 developed:1 pointed:2 htot:1 experiments:1 london:1 statistical:5 during:1 japan:3 satisfies:4 its:2 before:2 derive:1 how:3 affine:3 imoto:1 systems:1 decades:1 introduce:1 they:1 now:1 discuss:1 association:2 eigenvalue:1 university:1 characterized:1 mean:1 formulate:1 map:2 t2:22 principles:1 related:2 measurement:64 quantum:33 out:5 shown:1 tombesi:1 space:14 since:3 rev:1 denotes:1 specifications:1 reader:1 formula:6 occurs:1 math:5 shows:2 philosophy:2 theory:2 free:4 standard:3 completely:1 base:2 bayes:15 publ:1 interaction:13 derivation:5 determining:2 times:1 nondemolition:1 hence:7 initial:1 probability:34 variables:1 number:1 directly:1 precisely:1 open:1 illustrate:1 given:12 introduction:1 system:37 postulate:23 speaking:1 gives:1 ih2:8 continuous:3 representing:3 specify:1 roles:1 namely:1 appealing:1 classes:1 were:1 outcome:22 dynamically:1 turned:1 argument:2 say:1 any:11 depending:1 note:1 sciences:1 potential:1 measurability:1 spectral:1 ih1:5 trace:5 shall:2 measures:4 most:4 successive:3 device:1 measured:7 class:4 inconsistency:1 order:2 principle:19 physics:1 notion:2 fact:2 show:2 probe:16 inconsistent:1 based:1 just:5 kyoto:1 aψ:2 should:2 only:3 local:9 amplifying:1 over:1 mixture:1 means:1 joint:14 cannot:1 dx:3 compute:1 evolution:4 arxiv:1 paper:2 view:2 apparatuses:1 set:1 observer:1 physically:2 see:1 belavkin:1 subsequent:2 convex:1 initially:1 state:57 difficulties:2 between:2 interpretational:1 statistically:1 rims:1 importance:1 however:2 extent:1 problem:2 observable:23 realizable:1 prescription:1 hint:1 according:6 otherwise:1 among:1 cn:2 point:2 proved:1 ogawa:1 mathematically:1 pr:29 sensible:1 described:5 three:1 been:2 reduction:17 describes:1 basic:1 ph:1 ann:1 define:1 corresponding:1 commutes:1 am:1 photon:3 precision:1 spectrum:2 last:1 general:2 irrelevant:1 case:2 subsets:1 discrete:2 will:7 suppose:6 applies:1 squeezed:1 nonrelativistic:1 identical:1 detailed:1 unitary:4 von:1 thus:9 ih:1 reads:1 dual:1 interacts:3 journal:1 distributions:6 same:2 noninteracting:3 quant:1 finite:1 composition:2 φn:5 extended:1 upon:3 hand:1 purpose:1 changing:1 schr:1 locality:1 roughly:1 academic:1 without:7 model:8 know:1 proposed:1 obtain:1 press:1 ozawa:12 letting:1 density:6 characterization:2 evidences:1 instrument:1 easy:1 theorem:7 real:1 depends:1 rules:3 possible:3 entangled:3 confined:1 necessary:1 follows:5 communications:1 yields:1 arbitrary:3 page:5 mathematical:2 neumann:1 sequence:1 conditional:4 some:2 deduce:1 specified:1 duration:1 cp:3 lead:2 measurements:11 does:7 assuming:3 measuring:16 object:13 s12:4 equivalent:2 informatics:1 includes:1 hilbert:5 stage:1 pike:1 carried:1 o:1 mutually:1 range:1 h2:6 h1:9 functional:1 references:1 determine:2 operator:12 statistics:1 there:4 borel:1 way:1 was:1 naturally:1 function:3 but:2 idealized:1 posterior:7 properly:1 ha:5 considerable:1 applying:1 whether:1 devices:1 distribution:37 called:3 mechanics:6 constant:1 defined:4 abstract:1 measure:3 linearity:1 denote:1 engineering:1 formulation:3 commuting:2 physical:1 no:3 when:2 application:1 other:1 papers:1 conclusion:1 conclude:1 hirota:1 nonclassical:1 representable:1 prepared:3 a1:10 important:1 bounded:1 ξm:2 ξn:3 coupling:1 assume:1 interval:1 rule:2 time:27 observables:8
quant-ph9705031.txt |@word saturating:1 magnetic:1 desirable:1 four:15 entropy:2 transversally:1 feeding:1 aug:1 under:7 recycle:1 digit:1 experimentally:1 cause:7 solution:1 enhance:1 estimates:2 elegant:1 second:8 implemented:7 machines:1 even:20 established:1 errors:116 selected:1 decoding:1 above:5 resilient:1 new:7 pioneering:2 1997b:3 1997a:3 never:1 here:4 reported:1 protection:1 represented:2 obtained:7 study:2 approaches:2 procedure:28 complicated:2 fourier:8 9i:1 golden:1 highly:1 turchette:2 sloane:5 achievable:1 total:3 linearly:2 would:35 codewords:12 century:1 calt:1 leaks:2 call:2 therefore:5 type:6 until:3 tolerated:3 successful:2 expose:1 correspondingly:1 must:30 work:9 my:2 example:11 bennett:3 want:12 keep:1 recovery:48 turn:1 provide:1 verify:7 feature:2 machine:9 how:23 nontrivial:2 env:6 gate:61 faulty:5 description:2 ancilla:74 after:5 demanding:2 wrong:2 gottesman:15 minimizes:1 law:30 parallel:3 types:1 attempt:2 third:7 complexity:6 maintain:1 algorithms:3 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quant-ph9705036.txt |@word emerging:1 consider:1 show:2 formula:3 random:1 closeness:1 results:5 distortion:3 illustrated:1 course:1 paper:1 through:1 computation:1 mentioned:1 fidelity:2 its:1 aσ:2 ensembles:2 density:6 concerned:1 conclude:1 character:1 state:1 sˆ:5 source:1 only:4 other:3 tool:1 theorem:2 input:2 adequate:1 nonzero:1 into:2 ingredient:1 real:1 σn:1 above:1 cryptography:1 was:4 very:2 minimal:1 simplest:1 nonorthogonal:1 choice:1 newman:1 define:1 negative:1 words:2 shown:1 sσ:1 using:1 sρ:11 now:10 matrices:5 reiterating:1 ρaµ:1 evolution:2 transmission:3 fields:1 arxiv:1 2x:2 noiseless:1 solved:1 inequality:13 transmitted:2 contrast:1 reasonable:1 arbitrary:1 output:2 trr:1 ρk:2 csiszar:1 yerevan:2 strengthened:7 set:2 smolin:1 methods:1 classical:6 physically:1 ψi:8 some:6 kodai:1 relative:4 cs:2 second:4 result:2 context:1 proper:1 ohya:2 sem:1 alternative:1 shannon:3 superposition:2 hilbert:3 specific:1 phases:1 capacity:1 tensor:1 space:3 rep:2 normal:1 rev:2 brief:1 ak:2 symmetric:1 acting:1 vector:1 identification:1 alikhanian:1 between:2 conceptual:1 new:2 evaluation:1 interchanging:2 commun:1 foundation:1 theory:14 degree:1 qubits:3 moscow:1 bn:2 processing:8 here:4 understanding:1 completely:2 pauli:1 armenia:1 found:1 represented:3 bk:2 appears:1 von:1 noisy:4 σk:1 definition:4 about:1 toward:1 derivation:3 barnum:1 most:1 institute:1 spectrums:2 gantmacher:1 prove:2 step:1 upper:1 products:1 δij:1 schumacher:4 introduced:3 cˆ2:9 bloch:1 first:7 ck:2 ci:1 there:1 useful:1 cn:1 follows:2 simple:4 communication:1 variables:1 coding:1 bound:1 system:1 dimensional:2 hermitian:1 strengthening:1 decrease:2 operator:6 respect:2 xˆ1:1 unit:1 math:3 use:1 described:2 compression:1 basis:1 distinct:1 positive:2 contains:1 due:2 been:5 properties:2 ψj:2 needed:1 linear:1 describes:1 calculations:1 basic:1 taken:3 ph:4 scheme:1 type:1 knowledge:1 a2:2 more:2 main:1 preserving:1 form:3 orthonormal:1 operators:3 converse:1 proving:1 but:3 spectrum:1 korner:1 pj:2 final:1 fixed:1 aµ:2 authors:2 known:1 encoded:1 lindblad:6 bennett:1 than:2 entropy:7 must:4 case:6 kraus:1 account:2 pi:5 bσ:1 fuchs:1 bits:3 equation:2 zero:1 value:2 ρn:2 recall:1 ρr:5 suppose:1 were:1 reviewed:1 problem:1 property:1 c1:1 example:3 b1:2 mechanics:2 references:1 theorems:1 proved:4 nauka:1 defined:4 well:1 abstract:1 inaccessible:1 ann:1 allahverdyan:2 nielsen:3 general:8 states:6 quantum:27 strengthen:1 exist:2 our:2 need:1 purification:1 conversely:1 any:3 physical:2 information:21 mutual:10 formalism:1 because:1 us:1 min:3 rather:1 sˆt:1 get:5 eigenvalue:1 tr:1 write:1 field:1 nontrivial:1 potential:1 quant:4 take:1 see:1 eprint:3 φi:1 ordinary:2 many:1 ensemble:1 channel:7 independent:1 brothers:1 trace:1 sˆ1:10 may:2 cρ2:1 cρ1:1 after:2 problems:1 φr:7 possible:3 a1:4 applications:1 a3:2 important:3 log:2 vectors:1 entanglement:5 representations:1 minimization:2 contain:1 data:9 types:1 matrix:4 max:1 effective:1 characterizes:1 assume:2 sˆ2:10 st:1 concrete:1 representation:2 cˆ1:4 physics:1 notion:1 quantity:4
quant-ph9705037.txt |@word 2k0:1 code:34 consider:4 just:1 show:2 over:3 results:1 macwilliams:2 gf:5 enumerators:3 proposition:6 positions:2 still:2 d2:8 its:3 d1:6 aj:1 inequalities:1 based:3 generator:8 ln:2 krawtchouk:3 γq:1 technique:1 tr:3 written:2 ωa2:1 should:1 better:2 lemma:2 only:2 version:1 transformed:1 fig:2 details:1 4k0:9 arguments:2 correcting:1 case:5 them:1 dk:5 quantph:3 get:10 straightforwardly:1 b2:9 know:1 words:1 new:2 using:4 shadow:1 now:2 matrices:4 2t:1 inf:2 follows:3 arxiv:1 list:1 make:1 yield:1 2d:1 2e:1 iff:1 spaces:1 arbitrary:4 sloane:1 2n:17 shor:2 2k:6 upper:3 ik1:1 set:1 ei:2 2k1:5 classical:9 logq:3 national:1 designs:1 some:3 mceliece:2 up:2 jr:1 polynomial:2 see:8 elements:2 b0:1 b1:7 lnk:2 combining:1 quantum:24 laboratory:1 discr:1 ep:4 analog:2 techniques:1 shown:3 errors:1 3k1:4 dimensions:1 rate:1 nondegenerate:9 ieee:2 since:2 analogously:1 3t:1 vol:2 3i:3 smallest:1 tends:4 print:7 obvious:1 unrestricted:1 order:1 conditions:1 consisting:2 math:1 satisfy:1 log2:1 theory:7 complementary:4 respectively:1 were:2 bj:1 bi:9 expansion:1 2h4:1 wt:2 complicated:1 here:2 less:3 estimate:1 standard:1 ωa1:1 welch:2 simpler:1 obtained:1 definition:1 re2:1 many:2 taking:1 codewords:1 programming:2 equal:3 o:2 called:1 correction:1 length:4 nonbinary:3 place:1 symplectic:2 alamos:1 hence:1 complimentary:1 generalizations:1 pp:4 ji:1 xj:1 22n:1 singleton:2 contains:1 useful:1 projection:1 into:2 asymptotically:1 coding:4 bound:30 h4:5 assertion:2 delsarte:2 ik0:1 references:1 coordinates:1 operator:2 respect:2 linearly:1 second:1 a3:1 use:5 rows:2 additive:7 asymptotic:7 pt:3 los:1 previous:1 there:1 due:1 next:1 their:1 however:1 approach:1 inner:2 pe:1 taken:1 pk:3 ph:5 pi:1 was:4 universal:1 via:2 ir:2 corresponding:1 form:7 weight:1 becomes:1 operators:1 hq:1 calderbank:1 k0:10 ik:4 aaltonen:1 levenshtein:1 somewhat:1 particular:1 known:4 cases:1 cardinality:1 than:8 present:1 alexei:1 distance:14 account:2 applying:1 resulting:1 type:3 work:1 bounds:13 value:1 will:4 minimum:14 suppose:1 stabilizer:10 error:3 trans:2 root:1 example:5 proof:1 f0:3 block:1 lanl:7 associated:1 dual:1 figure:7 laflamme:2 ai:9 stronger:2 abstract:1 describe:1 knill:1 mixed:5 proven:1 general:1 ii:2 say:5 ij:1 exist:1 reports:1 degenerate:5 grater:1 seem:1 element:1 identities:1 check:1 greater:3 ni:2 nj:1 codes:30 completeness:1 zero:3 author:1 lengths:3 fixing:1 latter:1 orthogonal:1 nd:2 subcode:1 any:1 note:6 k1:34 valid:2 quant:5 best:4 rumsey:2 so:4 defines:1 ashikhmin:1 apply:2 difficult:1 meets:1 product:2 used:1 infinity:1 binary:7 may:1 according:1 polynomials:3 a1:5 a0:1 independent:1 a2:3 remarks:1 fi:7 structure:1 matrix:20 philips:1 no:1 observation:1 linear:4 well:1 4q:1 4n:1 rains:4 weights:2 4k:2 hamming:4 res:1 cleve:1 representation:1 first:1 dimension:4 fact:1 quantity:1
quant-ph9705038.txt |@word ideality:1 concept:1 partial:2 consider:5 represent:1 rev:8 global:13 results:10 leads:9 ceased:1 scalar:9 generalize:1 computation:2 expressions:1 denotes:3 reasoning:2 whose:1 supposedly:1 c0:25 const:1 calculate:1 untilded:1 fellowship:1 technique:1 tr:7 supported:2 parametrization:1 finally:4 behave:1 device:1 present:2 under:3 tr1:1 tr2:2 resources:1 division:1 probe:7 already:2 straightforward:1 orthogonal:7 far:1 could:4 a18:1 rise:1 keeping:1 seemingly:1 choice:1 wishes:1 every:1 difference:3 matrices:2 condition:16 four:7 interact:1 terhal:1 presented:1 transmission:1 turns:2 solution:4 damped:1 leave:1 vector:14 indices:2 entropy:1 shows:4 noted:1 dieks:1 upper:3 work:2 villa:1 variables:2 maximization:2 direct:2 hillery:9 approximation:1 second:6 design:1 ρa0:1 equal:2 further:2 defines:1 implemented:1 nothing:1 what:3 appear:2 rate:1 3s:3 section:7 selected:2 analogously:2 expressed:1 fortschritte:1 above:11 capital:1 new:3 order:6 public:3 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quant-ph9705039.txt |@word sci:1 consider:1 sousa:1 brownian:6 results:1 leads:2 eds:1 scalar:1 c2:4 remarkably:1 compact:1 concerned:2 tn:2 155b:1 charge:3 preserved:1 under:1 tf:1 stochastic:4 differing:1 tc:1 p2:4 very:2 zachos:1 choice:2 electron:3 difference:1 deformations:4 illustrated:1 level:4 july:1 solution:4 large:1 small:1 eigenvalues:1 oscillation:4 zk:2 consistent:1 direct:1 past:1 integrability:3 dordrecht:1 dynamics:1 canonical:2 spectral:4 what:1 nonlinear:7 appear:3 xσ:4 sum:2 d3:2 current:1 establishes:1 international:1 increasing:1 satisfy:3 multipole:1 spacing:3 mev:1 free:3 sinh:7 quite:1 equations:1 interpret:1 suq:2 obtained:2 institute:1 involved:1 products:1 fisika:1 published:1 action:1 fourier:1 studies:2 matos:1 useful:1 although:1 nearest:1 vii:1 plenum:1 replace:1 total:1 tools:1 planck:1 seligman:1 use:2 vogel:1 would:2 commun:1 macfarlane:1 h0:1 next:1 classically:1 therefore:4 hirsch:2 qci:1 type:8 more:2 composite:1 becomes:2 operators:19 k0:1 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quant-ph9705041.txt |@word represent:1 code:4 consider:4 results:3 illustrated:1 entropy:3 computation:10 go:1 query:35 compares:1 gk:7 gi:7 equiprobable:2 whose:1 simon:1 terms:2 violate:3 combinatorial:1 factoring:2 th:4 include:1 alternating:1 division:1 digit:4 charles:1 g3:2 string:13 g1:2 orthogonal:1 far:1 pcol:3 rise:3 sons:2 worst:1 dim:1 presented:2 transmission:1 developing:1 large:1 item:7 small:9 generalize:1 barbara:1 limitations:1 smaller:1 10n:1 zk:4 direct:3 further:1 implemented:1 investment:1 peredachi:1 linear:2 army:1 what:3 adds:1 appear:3 section:4 version:2 new:2 order:1 method:1 full:1 discrete:1 here:6 understanding:1 following:2 explore:1 separation:1 represented:4 depend:1 strong:2 search:11 property:1 provably:1 institute:1 items:2 larger:1 leaving:1 p2:1 amount:3 resulting:1 postprocessing:4 permutation:1 implement:1 fourier:1 makes:3 uniquely:2 useful:1 extra:1 merely:1 marked:8 unitary:2 analyzing:1 amherst:1 summation:1 apply:1 total:9 select:1 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quant-ph9705042.txt |@word sergei:1 sci:1 consider:1 rev:2 splitting:2 illustrated:1 generalize:1 computation:1 consists:1 expressions:1 zinn:1 sch:1 whose:2 depend:5 tt:3 equivalence:1 γv:1 nonlocal:3 γt:2 tn:1 charge:1 icx:2 γn:10 under:3 stochastic:33 analogous:1 must:2 xti:1 applied:1 fax:1 provides:1 advantage:2 orthogonal:2 far:1 than:1 springer:2 choice:4 barnich:1 affect:2 difference:1 exact:2 condition:7 treating:1 xt1:1 hamiltonian:3 turns:1 permits:1 applicable:1 leave:1 convenience:1 phase:4 gauge:4 petersburg:1 configurations:2 dcn:2 force:3 grateful:1 arnimallee:1 go:1 zero:1 l2t:2 dynamics:2 substitute:1 canonical:9 will:2 what:2 nonlinear:1 bined:1 itˆ:4 sum:4 supermultiplet:3 steiner:1 version:2 axel:1 above:4 contrast:2 coefficient:2 condensed:1 mem:1 respectively:2 desired:2 represented:1 path:29 δzk:1 equations:8 appears:1 δzn:1 obtained:1 generalizing:2 colored:1 shift:1 drs:1 studied:2 resulting:1 action:40 makes:2 uniquely:1 via:2 useful:1 extra:1 ipn:1 replace:3 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quant-ph9705049.txt |@word concept:2 semantic:2 consider:5 crucial:1 equalities:1 results:1 leads:2 commute:1 follow:1 whose:1 depend:5 calculate:2 irrelevant:1 equivalence:3 kδ:1 case:4 expressing:3 wave:4 every:3 difference:3 matrices:3 condition:1 hamiltonian:2 solution:1 discrete:1 vector:2 consistent:1 direct:1 conjunctions:11 imply:1 qualitatively:1 canonical:1 what:1 axes:1 expressed:1 above:1 expresses:1 simultaneously:1 relates:1 here:2 understanding:1 met:1 objects:6 clns:1 cos2:2 indeterminism:3 represented:4 appears:1 contradiction:1 products:1 permutation:2 action:1 makes:2 york:1 term:5 logical:14 translational:4 apply:2 planck:2 use:1 superpositions:1 usa:1 prove:1 remains:1 beginning:1 ψp:1 next:1 call:1 therefore:3 type:8 theoretical:2 more:1 introducing:4 formally:1 initial:3 theorem:5 floyd:1 particular:1 discreteness:4 cases:1 must:2 none:1 rotational:2 equation:4 assumptions:3 values:28 cal:2 following:3 under:3 arrive:1 example:5 proof:2 give:1 permutations:5 tag:1 numbers:1 sense:1 axis:1 needs:1 acts:1 negation:1 θmax:2 answer:5 instead:2 parameter:3 displacements:1 product:1 description:2 may:4 after:1 containing:1 wrong:1 plane:4 choice:2 parallel:1 constants:1 sz:2 natural:1 maybe:1 su:1 greek:1 so:1 fundamental:2 order:1 se:1 typical:1 exclusive:5 interpretation:27 cos4:1 indeed:1 interpreted:1 existence:4 its:3 group:2 symmetry:10 chosen:1 interesting:1 affine:3 coordinates:3 absence:1 λq0:7 systems:2 defining:3 non:6 combination:1 introduce:2 newman:1 false:1 now:5 nor:1 possess:2 february:1 finite:1 university:1 unpredictable:1 truth:28 each:4 found:1 2n:1 quantum:24 laboratory:1 operation:1 beyond:1 shown:2 matrix:6 space:11 since:6 interference:5 denoted:2 formula:3 correct:10 2sin2:1 given:7 possibility:1 mapping:2 clarify:1 reason:1 yuri:1 e1:5 cornerstone:1 e2:5 definition:2 pairs:2 r1:3 length:1 leading:1 onto:1 first:3 origin:1 already:1 features:1 probability:5 number:3 exact:1 least:1 another:5 impossible:1 open:1 properties:2 rewritten:2 their:2 introduction:1 unknown:2 system:7 construct:1 needed:2 assumption:5 statement:13 final:1 gives:3 exactly:1 part:3 semantics:3 connections:1 second:1 inversely:1 letters:1 symmetries:3 need:1 border:1 any:6 angle:1 zero:2 latter:2 analogy:1 coincide:1 continuous:1 though:2 formulae:2 pair:1 measured:1 deviations:1 observation:1 orlov:1 considered:7 clear:2 tautology:1 points:3 principle:1 mechanical:2 fact:1 show:2 spaces:1 find:1 q0:4 should:9 aϑ:1 only:11 probabilities:6 do:5 means:4 dependent:1 simplest:1 cannot:1 aforementioned:1 new:1 aθ:2 conjunction:4 ds:1 dq:1 continuum:1 arxiv:1 λp0:20 qi:1 fixed:1 requirement:1 set:6 constructing:1 exists:3 violated:1 observed:1 depends:2 result:2 insoluble:1 yes:4 best:1 hilbert:1 satisfying:1 nuclear:1 label:1 symmetric:1 state:1 enumerated:1 between:6 invariant:4 neither:1 conditions:5 terms:4 extend:1 however:2 answers:1 distribution:2 both:1 interpretations:23 uncertainty:1 according:5 quantization:2 etc:1 manifestations:1 logic:3 whole:2 arbitrarily:1 among:2 point:1 simple:1 others:1 dimensional:1 simply:1 constant:3 described:1 transformations:2 basis:1 dqdq:1 three:3 combined:2 ph:1 define:3 arises:1 corresponding:2 sufficient:1 coherence:9 consecutive:1 spin:4 p2:4 p0:6 p1:2 uncertain:3 value:3 will:13 while:2 suppose:1 applies:1 situation:3 real:3 thus:3 them:3 equal:1 ii:2 itself:2 different:4 make:1 same:6 complex:1 quant:1 physical:3 events:1 ordinal:1 independent:3 erence:2 numerical:2 effect:3 hand:2 invariance:1 normalization:1 eigenvectors:2 studies:1 analysis:1 obviously:1 concrete:1 without:2 coordinate:3 possesses:1 questions:3 using:1 λk:1 λi:3 ithaca:1 λn:1 λl:3 density:1 cornell:1 priori:1 transformed:1 location:2 propositions:1 momentum:3 identical:8 around:1 rules:1 possible:1 integer:1 moreover:1 formal:1 meaning:3 necessary:1 particle:8 follows:1 δq:1 either:4 arbitrary:2 λqi:1 because:1 classical:7 some:5 searching:1 examples:1 choosing:2 table:2 interf:2 describing:1 per:1 does:5 connecting:1 provides:1 obey:1 analysed:2 step:4 although:1 permitted:1 about:9 would:1 origins:1 dependence:1 culate:1 essential:1 mutually:5 sin4:1 own:2 statements:19 presence:1 into:1 proportional:1 additional:2 assumed:2 there:10 question:2 was:1 function:5 arise:1 δkl:2 differences:1 analyse:1 but:4 construction:1 true:13 whether:2 inside:1 probabilistic:1 up:1 us:3 placed:2 below:1 problem:3 mechanics:5 ordered:2 defined:8 certain:14 abstract:1 general:1 single:2 exist:3 our:14 formalism:4 distributions:1 defines:3 functions:5 no:4 when:10 other:5 perpendicular:1 problems:1 a1:3 commutator:1 generated:1 θmin:2 disjunction:2 structure:2 delineate:1 independently:1 algorithm:5 assume:2 interval:2 describable:1 having:1 observables:1 indirectly:1 once:1
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