Direct Characterization of Quantum Dynamics

0.13 MB·English

by  M. Mohseni

Direct Characterization ofQuantum Dynamics
M. Mohseni1,2 and D. A. Lidar2,3
1Department ofPhysics, UniversityofToronto, 60St. GeorgeSt.,Toronto, Ontario, M5S1A7,Canada
2Department of Chemistry, Universityof Southern California, LosAngeles, CA 90089, USA
3Department of Electrical Engineering, and Department of Physics,
University of Southern California, Los Angeles, CA 90089, USA
The characterization of quantum dynamics isafundamental and central task inquantum mechanics. This
taskistypicallyaddressedbyquantumprocesstomography(QPT).Herewepresentanalternative“directchar-
acterizationofquantumdynamics”(DCQD)algorithm. IncontrasttoallknownQPTmethods,thisalgorithm
reliesonerror-detectiontechniquesanddoesnotrequireanyquantumstatetomography. Weillustratethat,by
construction, theDCQDalgorithmcanbeappliedtothetaskofobtainingpartialinformationaboutquantum
dynamics. Furthermore, wearguethat theDCQD algorithmisexperimentally implementable inavarietyof
7 prominentquantuminformationprocessingsystems,andshowhowitcanberealizedinphotonicsystemswith
0 presentdaytechnology.
0
2
PACSnumbers:03.65.Wj,03.67.-a,03.67.Pp

n
a
J Thecharacterizationandidentificationofquantumsystems characterizethe quantumdynamicsby preparingoneof four

9 areamongthecentralmodernchallengesofquantumphysics, possibletwo-qubitentangledstates,andaBell-statemeasure-
1 andplayanespeciallyfundamentalroleinquantuminforma- ment(BSM)attheoutput.ThisisillustratedinFig.1andTa-

tion science [1], and coherent control [2]. A task of gen- bleI. We also discussthe experimentalfeasibility ofDCQD
2
eraland crucialimportanceis the characterizationof the dy- inavarietyofquantuminformationprocessing(QIP)systems,
v
3 namics of a quantum system that has an unknown interac- andshowhowitcanberealizedwithlinearoptics.
3 tionwithitsembeddingenvironment. Knowledgeofthisdy- Inprinciple,thenumberofrequiredexperimentalconfigu-
0 namics is indispensable, e.g., for verifying the performance rationsintheAAPTschemecanbereducedbyutilizingnon-
1
of an information-processing device, and for the design of separable(global)measurementsfortherequiredstatetomog-
0
decoherence-mitigation methods. Such characterization of raphy,suchasmutuallyunbiasedbasis(MUB)measurements
6
0 quantum dynamics is possible via the method of standard [5],orageneralizedmeasurement[6]. Ingeneral,thesetypes
/ quantumprocesstomography(SQPT) [1, 3]. SQPT consists of measurements require many-body interactions which are
h
of preparing an ensemble of identical quantum systems in a not available experimentally. Here, we demonstrate that the
p
- memberofasetofquantumstates,followedbyareconstruc- DCQD algorithm requires only (n) single- and two-body
nt tionofthedynamicalprocessviaquantumstatetomography. operationsperexperimentalconfiOguration.
a We identify three main issues associated with the required We demonstrate the inherent applicability of the DCQD
u physicalresourcesinSQPT.(i)Thenumberofensemblemea- algorithm to the task of partial characterization of quantum
q
surementsgrowsexponentiallywiththenumberofdegreesof dynamicsin termsof coarse-grainedquantities. Specifically,
:
v freedom of the system. (ii) Often it is not possible to pre- we demonstratethatfora two-levelsystem undergoinga se-
Xi pare the complete set of required quantum input states. (iii) quenceofamplitudeandphasedampingprocesses,therelax-
Informationconcerningthedynamicalprocessisacquiredin- ation time T and dephasing time T can be simultaneously
r 1 2
a directly via quantum state tomography, which results in an determinedinoneensemblemeasurement.
inherentredundancyofphysicalresourcesassociatedwiththe QuantumDynamics.—Theevolutionofaquantumsystem
estimation of some superfluous parameters. To address (ii), (openorclosed)can,undernaturalassumptions,beexpressed
the method of ancilla-assisted process tomography (AAPT) intermsofacompletelypositivequantumdynamicalmap ,
E
was proposed[4]. However,the numberofmeasurementsis whichcanberepresentedas[1]
thesameinSQPTand(separable)AAPT[4].
d2 1
Here we develop an algorithm for direct characterization −
ofquantumdynamics(DCQD),whichdoesnotrequirequan- E(ρ)= X χmnEmρEn†. (1)
m,n=0
tum state tomography. The primary system is initially en-
tangled with an ancillary system, before being subjected to Hereρisthesysteminitialstate,andthe E areasetof(er-
m
{ }
theunknowndynamics. Completeinformationaboutthedy- ror) operator basis elements in the Hilbert-Schmidtspace of
namicsis thenobtainedby performinga certain setof error- linearoperatorsactingonthesystem,satisfyingTr(Ei†Ej) =
detecting measurements. We demonstrate that for character- dδ . The χ are thematrixelementsofthesuperopera-
ij mn
{ }
izing a non-trace preserving quantum dynamical map on n torχ,whichencodesalltheinformationaboutthedynamics,
qubitsthe numberof requiredexperimentalconfigurationsis relative to the basis set E [1]. For an n qubit system,
m
{ }
reducedfrom24n, forSQPT andseparableAAPT,to 22n in thenumberofindependentmatrixelementsinχis24n fora
DCQD. E.g., for a single qubit, we show that one can fully non-trace-preservingmap,or24n 22nforatrace-preserving
−
2
ements χ and χ , respectively). The required input state
A 03 12
Bell state BSM is a non-maximally entangled state φC = α00 + β 11 ,
preparation | i | i | i
B with α = β = 0,whosesolestabilizerisZAZB. Now,by
| | 6 | | 6
separatingrealandimaginaryparts,theprobabilitiesofnobit
FIG.1:Schematicofcharacterizationalgorithmforsinglequbitcase, flip error and bit flip error events become: Tr[P+1 (ρ)] =
E
consistingofBellstatepreparations,applyingtheunknownquantum χ00 + χ33 + 2Re(χ03) ZA and Tr[P 1 (ρ)] = χ11 +
map, ,andBell-statemeasurement(BSM). χ + 2Im(χ ) ZA , whherei ZA −Tr(EρZA) = 0 (be-
E 22 12 h i h i ≡ 6
cause α = β = 0), with ρ = φ φ . We already
C C
| | 6 | | 6 | ih |
knowthe χ fromthepopulationmeasurementdescribed
mm
{ }
map.ThematrixχispositiveHermitian,andTrχ 1. Thus above, so we can determine Re(χ ) and Im(χ ). Af-
03 12
≤
χ can be thought of as a density matrix in Hilbert-Schmidt ter measuring ZAZB the system is in either of the states
space, whencewe oftenrefer,below, toits diagonalandoff- ρ = P (ρ)P /Tr[P (ρ)]. Now we measure the
1 1 1 1
diagonal elements as “quantum dynamical population” and ex±pectation±vEalue o±f a (com±mEuting) normalizer operator U
“coherence”,respectively. (suchas XAXB) [7]. Thuswe obtainTr[Uρ ] = [(χ
+1 00
−
Characterization of Quantum Dynamical Population.— χ ) U +2iIm(χ ) ZAU ]/Tr[P (ρ)]andTr[Uρ ]=
33 03 +1 1
(Heenrseemwbeled)emmoeanssutrraetmeehnotwfotor adestienrgmleinequ{bχitm. mL}eitnuas sminagxlie- [(ZχA11h,−iUχ2,2a)nhUdiZ−A2UhiRea(rχei1a2l)lhnZoAn-UzeiEr]/oTarn[dP−al1rEea(ρdy)],kwn−ohwerne.
h i h i h i
mally entangle two qubits A (primary) and B (ancillary) as Therefore, we can obtain the four independent real parame-
ψ = (0A0B + 1A1B )/√2, and then subjectonly qubit ters needed to calculate the coherence components χ03 and
A| itoam|ap . iFrom| hereion,forsimplicity,wedenote χ12,bysimultaneouslymeasuring,e.g.,ZAZB andXAXB.
E E ⊗I
byE.Inthiscasetheerrorbasis{Em}3m=0becomestheiden- In orderto characterizethe remainingcoherenceelements
tityoperatorandthePaulioperators: I,X,Y,Z . Thestate of χ we make an appropriate change of basis in the prepa-
{ }
ψ isa+1eigenstateofthecommutingoperatorsZAZBand ration of the two-qubit system. For characterizing χ and
| i 01
XAXB, i.e., it is stabilized underthe action of these “stabi- χ , we can performa Hadamardtransformationon the first
23
lizeroperators”,anditisreferredtoasa“stabilizerstate”[1]. qubit, as HA φ = α+ 0 + β 1 , where =
C
| i | i| i |−i| i |±i
Any non-trivialoperatoracting on the state of the first qubit (0 1 )/√2. We then measure the stabilizer operator
anticommutes with at least one of ZAZB and XAXB, and X|AiZ±B,|aind a normalizer such as ZAXB. For characteriz-
thereforebymeasuringtheseoperatorswecandetectanarbi- ingχ andχ ,wepreparethesysteminthestabilizerstate
02 31
traryerroronthefirstqubit,i.e.,byfindingtheeigenvaluesof SAHA φ =α +i 0 +β i 1 ,andmeasurethestabi-
C
| i | i| i |− i| i
eitheroneorbothoperatorstobe 1. Measuringtheobserv- lizeroperatorYAZBandanormalizersuchasZAXB,where
−
ablesZAZB andXAXB isequivalenttoaBSM,andcanbe Sisthesingle-qubitphasegate,and i =(0 i1 )/√2.
representedbythefourprojectionoperatorsP0 = φ+ φ+ , Therefore, we can completely chara|c±teriize th|eiq±uan|tuim dy-
| ih |
P1 = ψ+ ψ+ , P2 = ψ− ψ− , P3 = φ− φ− , where namicalcoherencewiththreeBSM’soverall.Wenotethatthe
| ih | | ih | | ih |
φ± = (00 11 )/√2, ψ± = (10 01 )/√2, form bases 0 , 1 , and i are mutually unbiased,
| i | i±| i | i | i±| i {| i | i} {|±i} {|± i}
the Bell basis of the two-qubit system. The probabilities of i.e.,theinnerproductsofeachpairofelementsinthesebases
obtaining the no error outcome I, bit flip error XA, phase havethesamemagnitude[5].
flip error ZA, and both phase flip and bit flip errors YA,
A summaryoftheschemeforthecase ofa singlequbitis
on the first qubit, become p = Tr[P (ρ)] = χ , for
m m mm presented in Fig. 1 and Table. I. This table implies that the
E
m = 0,1,2,3respectively. Therefore,wecandeterminethe
requiredresourcesinDCQDareasfollows:(a)preparationof
quantum dynamicalpopulation, χ 3 , in a single en-
{ mm}m=0 amaximallyentangledstate(forpopulationcharacterization),
semble measurement (e.g., by simultaneously measuring the
(b)preparationofthreeother(nonmaximally)entangledstates
operatorsZAZB andXAXB)onmultiplecopiesofthestate
(forcoherencecharacterization),and(c)afixedBell-statean-
ψ .
alyzer. We remark that a generalized DCQD algorithm for
| i
CharacterizationofQuantumDynamicalCoherence.—In
order to preserve the coherence (χ ) in the quantum dy-
m=n
6
namicalprocess,weperformasetofmeasurementssuchthat
we always obtain partialinformationaboutthe nature of the TABLEI:Onepossiblesetofinputstatesandmeasurementsfordi-
errors. The simplest case is a measurement of the projec- rectcharacterization ofquantum dynamics (χij)forasinglequbit,
where α = β =0,and 0 , 1 , , i areeigenstates
tiψo+n oψpe+rat+orsψP+1ψ= c|oφr+reishpφo+n|d+ing|φto−ithhφe−ei|gaennvdalPu−es1+=1 oftheP|au|l6io|pe|ra6torsZ,X{,|aind|Yi}. {|±i} {|± i}
− −
| ih | | ih |
and 1ofmeasuringthestabilizerZAZB. Theoutcomesof inputstate Measurement output
−
thismeasurementrepresenttheprobabilitiesofnobitfliper- Stabilizer Normalizer
ror and bit flip error on qubit A, without telling us anything (0 0 + 1 1 )/√2 ZAZB,XAXB N/A χ00,χ11,χ22,χ33
| i| i | i| i
abouta phaseflip error. Therefore,wepreserveonlytheco- α|0i|0i+β|1i|1i ZAZB XAXB χ03,χ12
herence between operators I and ZA, and also between the α|+i|0i+β|−i|1i XAZB ZAXB χ01,χ23
XA and YA (which are represented by the off-diagonal el- α|+ii|0i+β|−ii|1i YAZB ZAXB χ02,χ13
3
MMeeaassuurreemmeenntt
PPrreeppaarraattiioonn
TABLEII:Comparisonoftherequiredphysicalresourcesforchar-
qacutbeirtisz.inNginaannodnN-trmacreesppreecsteivrevliyngdeCnPotqeutahnetunmumdbyenraomfirceaqlumireadpionnpunt HHHH1111 QQQ111 QQQQ2222HHHH2222 PPPBBBSSS 111DDD NNNooonnnllliiinnneeeaaarrr CCCrrryyyssstttaaalll
statesandthenumberofnoncommutativemeasurementsforeachin- HHH000QQQ000 QQQ QQQuuuaaarrrttteeerrr WWWaaavvveeeppplllaaattteeesss
putstate,andNexp ≡NinNmisthenumberofrequiredexperimental UUUVVV 555000///555000 DDD222 HHH HHH555aaa000lllfff/// 555WWW000 aaaBBBvvveeeeeeaaapppmmmlllaaa tttSSSeeepppsssllliiitttttteeerrr
configurations. 333DDD PPPooolllaaarrriiizzziiinnnggg BBBeeeaaammm SSSpppllliiitttttteeerrrsss
SScQhPeTme dim2(nH) N4nin N4nm N16enxp PPPBBBSSS DDD444 DDDMMMeeeiiitttrrreeerrrooocccrrrtttooosssrrrsss
non-separableAAPT 22n 1 4n+1 4n+1
DCQD 22n 4n 1 4n FIG. 2: A schematic layout of a linear-optical realization of the
DCQD algorithm. A pair of entangled photons, (H H +
A B
eiϕ V V )/√2, is generated via parametric down-co|nversioni in
A B
| i
anonlinearcrystalfromaUVpumplaser,where H and V rep-
qudits,withdbeingapowerofaprime,ispossible,andwill resenthorizontalandverticalpolarization[12]. Th|eqiuarter|andi half
bethesubjectofafuturepublication[8]. waveplatesareusedtocreatenon-maximallyentangled states[13],
AnadditionalfeatureofDCQDisthatalltherequireden- andchangethepreparationandmeasurementbases. Two-photonin-
terferometryoccursatthe50/50beamsplitters. Thisallowsfordis-
semble measurements, for measuring the expectation values
criminatingtwoofthefourBellstatesdeterministically,byutilizing
of the stabilizer and normalizer operators, can also be per-
polarizingbeamsplittersandphotodetectorsasanalyzers.
formedinatemporalsequenceonthesamepairofqubitswith
onlyoneBell-state generation. Thisis becauseat the endof
eachmeasurement,theoutputstateisinfactinoneofthefour
criminatingamongBellstatesispossible[12].Thenumberof
possible Bell states, which can be utilized as an input stabi-
ensemblepreparationsinthisspecificsetupisthuseffectively
lizerstate.
increased by a factor of two over an implementation of the
For characterizing a quantum dynamical map on n qubits DCQDalgorithmusinganidealBellstateanalyzer.However,
we need to perform a measurement corresponding to a ten- even in this optical realization, the number of experimental
sor product of the required measurements for single qubits. configurationsisstillreducedbyafactorof2n(forcharacter-
Animportantexampleisa QIPunitwithnqubitswhichhas izing a quantumdynamicalprocess on n qubits) over SQPT
a 2n-dimensionalHilbert space ( ). DCQD requiresa total and separable AAPT schemes. Alternatively, a small-scale
H
of 4n experimentalconfigurationsfor a complete characteri- anddeterministiclinear-opticalimplementationofDCQDcan
zation of the dynamics. This is a quadratic advantage over in principle be realized by a multirail representation of the
SQPT andseparableAAPT, whichrequirea totalof16n ex- qubits[14].
perimental configurations. In general, the required state to- To increase the efficiency of an all-optical Bell-state ana-
mographyin AAPT could also be realized by non-separable lyzer,wehavethreedifferentstrategies: (i)introducingsome
(global)quantummeasurements.Thesemeasurementscanbe nonlinearity,suchasanopticalswitch[15],(ii)utilizingpost-
performedeitherinthesameHilbertspace,with4n+1mea- selectedmeasurements,suchasanopticalCNOT [16],which
surements,e.g.,usingaMUBmeasurement[5],orinalarger hasbeendemonstratedexperimentallyforfilteringBellstates
Hilbertspace, with a singlegeneralizedmeasurement[6]. A withafidelityofabout79%[17],(iii)employinghyperentan-
comparisonoftherequiredphysicalresourcesispresentedin glement between pairs of photons for a complete and deter-
TableII. AdetailedresourcecostanalysiscomparingDCQD ministiclinear-opticalBell-statediscrimination[18,19]. The
tootherQPTmethodswillbereportedinafuturepublication lattermethodisbasedonthefactthatthepairsofpolarization-
[9]. entangledphotonsgeneratedbyparametricdown-conversion
PhysicalRealization.—Forqubitsystems,theresourcesre- have intrinsic correlations in time-energy and momentum.
quired in order to implement the DCQD algorithm are Bell Theseadditionaldegreesoffreedomcanbeexploitedinorder
statepreparationandmeasurement,andsinglequbitrotations. todistinguishbetweenthesubsetsofBell-stateswhichother-
These tasks play a central role in quantum information sci- wisecannotbediscriminatedbyastandardHong-Ou-Mandel
ence, since they are prerequisites for quantum teleportation, interferometer. Notethatemployingthetime-energycorrela-
quantumdensecodingandquantumkeydistribution[1]. Be- tionsin the contextofimplementingtheDCQD algorithmis
causeoftheirimportance,thesetaskshavebeenstudiedexten- feasibleifthequantumdynamicalmapactsonlyonthepolar-
sively andsuccessfully implementedin a varietyof different izationdegreesoffreedom.
quantum systems, e.g., nuclear magnetic resonance (NMR) PartialCharacterizationofDynamics.—DCQDcanbeap-
[10]andtrappedions[11]. Thus,theDCQDalgorithmisal- plied, by construction, to the task of partial characterization
readywithinexperimentalfeasibilityofessentiallyallsystems ofquantumdynamics,wherewecannotaffordordonotneed
whichhavebeenusedtodemonstrateQIPprinciplestodate. a fullcharacterizationof the system, or when we have some
Here,weproposeaspecificlinear-opticalimplementation, a priori knowledge about the dynamics. In particular, we
realizable with present day technology – see Fig. 2. Using can substantially reducethe numberof measurements, when
only linear-optical elements, at most 50% efficiency in dis- weareinterestedinestimatingthecoherenceelementsofthe
4
superoperator for only specific subsets of the operator ba- hasbeenshownthatDCQDcanbeusedforrealizationofgen-
sis and/or subsystems of interest. E.g., we need to perform eralized quantum dense coding [8]. Moreover, DCQD can
a single ensemble measurement if we are required to iden- be efficiently appliedto (single-and two-qubit)Hamiltonian
tify onlythe coherenceelementsχ and χ of a particular identificationtasks[9]. Finally,thegeneraltechniquesdevel-
03 12
qubit. Here, we present an example of such a task. Specif- opedherecouldbefurtherutilizedforclosed-loopcontrolof
ically, we demonstratethatthe DCQD algorithmenablesthe openquantumsystems.
simultaneousdeterminationofcoarse-grainedphysicalquan-
ThisworkwassupportedbyNSERC(toM.M.),NSFCCF-
tities,suchasthelongitudinalrelaxationtimeT andthetrans-
1 0523675, ARO W911NF-05-1-0440,and the Sloan Founda-
verserelaxation(ordephasing)timeT . Assumethatwepre-
2 tion (to D.A.L.). We thank R. Adamson, J. Emerson, D.F.V.
pareatwo-qubitsysteminthenon-maximallyentangledstate
James, K. Khodjasteh, D.W. Leung, A.T. Rezakhani, A.M.
φ = α00 +β 11 . Then we subject qubitA to an am-
C Steinberg,andM.Zimanforusefuldiscussions.
| i | i | i
plitudedampingprocessfordurationt ,followedbyaphase
1
damping process, for duration t . The elements of the final
2
densitymatrixthenread 0ρ 0 =1 exp( t1)(1 α2)
h | f| i − −T1 −| |
and 0ρ 1 =exp( t′ )α β,where t′ = t2 + t1. Inor-
h | f| i −2T2′ ∗ T2′ T2 T1 [1] M. A. Nielsen and I. L. Chuang, Quantum Computation
dertodetermineT ,wemeasuretheeigenvaluesofthestabi-
1 andQuantumInformation(CambridgeUniversityPress,Cam-
lizeroperatorZAZB.Weobtaineither+1or 1,correspond-
bridge,UK,2001).
−
ing to the projective measurementP+1, or P 1. The proba- [2] H.Rabitzetal.,Science288,824(2000).
−
bilities of either of these outcomes, e.g., Tr(P 1ρf) are re- [3] J.F.Poyatos,J.I.Cirac,andP.Zoller,Phys.Rev.Lett.78,390
latedtoT throughtherelation 1 = 1 ln(1 −2Tr(P−1ρf)). (1997);I.L.ChuangandM.A.Nielsen,J.Mod.Opt.44,2455
1 T1 −t1 −1 Tr(ZAρ) (1997); A. M. Childs, I. L. Chuang, and D. W. Leung, Phys.
In order to obtain informationaboutT , we meas−ure the ex-
2 Rev. A 64, 012314 (2001); M. W. Mitchell et al., Phys. Rev.
pectation value of any normalizer of the input state, such as
Lett.91,120402(2003);M.Howard,etal.,NewJ.Phys.8,33
XAXB, yielding t′ = 2lnTr(XAXBρf). Since the op- (2006).
T2′ − Tr(XAXBρ) [4] D. W. Leung, PhD thesis (Stanford University, 2000); G. M.
eratorsZAZB and XAXB commute, we can measure them
D’ArianoandP.LoPresti,Phys.Rev.Lett.86,4195(2001);J.
simultaneously. Therefore we can find both T and T in a
1 2 B.Altepeteretal.,Phys.Rev.Lett.90,193601(2003).
Bell-statemeasurement. [5] W.K.WoottersandB.D.Fields,Ann.Phys.191363(1989);
Outlook.—OnecancombinetheDCQDalgorithmwiththe J. Lawrence, Cˇ. Brukner, and A. Zeilinger, Phys, Rev. A 65,
method of maximum-likelihood estimation [20], in order to 032320(2002).
minimize the statistical errors in each experimentalconfigu- [6] G.M.D’Ariano,Phys.Lett.A300,1(2002).
[7] A normalizer operator isa unitary operator that preserves the
ration. Moreover, a new scheme for continuous characteri-
stabilizersubspace.
zation of quantum dynamics can be introduced, by utilizing
[8] M.MohseniandD.A.Lidar,quant-ph/0601034.
weakmeasurementsfortherequiredquantumerrordetections
[9] M.Mohseni,A.T.Rezakhani,andD.A.Lidar,inpreparation.
inDCQD[21]. [10] M. Mehring, J. Mende, and W. Scherer, Phys. Rev. Lett. 90,
For quantum systems with controllable two-body interac- 153001(2003).
tions(e.g.,trapped-ionandNMRsystems),DCQDcouldhave [11] D.B.Leibfriedetal.,Nature422,412(2003).
near-termexperimentalapplicationsforcompleteverification [12] K.Mattleetal.,Phys.Rev.Lett.76,4656(1996).
[13] A.G.Whiteetal.,Phys.Rev.Lett.83,003103(1999).
ofsmallQIPunits. Forexample,DCQD,reducesthenumber
[14] M. Mohseni, et al., Phys. Rev. Lett. 91, 187903 (2003); M.
ofrequiredexperimentalconfigurationsforsystemsof3or4
Mohseni, J.A.Bergou, andA.M.Steinberg,Phys.Rev.Lett.
physicalqubitsfrom5 103 and6.5 104 (inSQPT)to64
93,200403(2004).
× ×
and256,respectively.Asimilarscale-upcanonlybeachieved [15] K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev.
by utilizing non-separable AAPT methods. Complete char- Lett.89,037904(2002).
acterization of such dynamics would be essential for verifi- [16] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46
cation of quantum key distribution procedures, teleportation (2001);J.L.O’Brienetal.,Nature426,264(2003).
[17] Z.Zhazoetal.,Phys.Rev.Lett.94,030501(2005);N.K.Lang-
units,quantumrepeaters,andmoregenerally,inanysituation
fordetal.,Phys.Rev.Lett.95,210504(2005).
inquantumphysicswhereafewqubitshaveacommonlocal
[18] P.G.KwiatandH.Weinfurter,Phys.Rev.A58,R2623(1998).
bath and interact with each other. Furthermore, as demon-
[19] C.Schuck,etal.,Phys.Rev.Lett.96,190501(2006).
strated here, DCQD is inherentlysuited to extractpartialin- [20] R.Kosut,I.A.Walmsley,andH.Rabitz,quant-ph/0411093.
formationaboutquantumdynamics. Severalother examples [21] C. Ahn, A. C. Doherty, and A. J. Landahl, Phys. Rev. A 65,
ofsuchapplicationshavebeendemonstrated. Specifically,it 042301(2002).