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Publié par | mijec |
Nombre de lectures | 23 |
Langue | English |
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nepOQuantumRandomWalks
∗
S.Attal
1
,F.Petruccione
2
,C.Sabot
1
andI.Sinayskiy
2
1
Universite´deLyon
Universite´deLyon1,C.N.R.S.
InstitutCamilleJordan
21avClaudeBernard
69622Villeubannecedex,France
2
QuantumResearchGroup
SchoolofPhysicsandNationalInstituteforTheoreticalPhysics
UniversityofKwaZulu-Natal
Durban4001,SouthAfrica
Abstract
Anewmodelofquantumrandomwalksisintroduced,onlattices
aswellasonfinitegraphs.Thesequantumrandomwalkstakeinto
accountthebehaviorofopenquantumsystems.Theyaretheexact
quantumanalogueofclassicalMarkovchains.Weexplorethe“quan-
tumtrajectory”pointofviewonthesequantumrandomwalks,thatis,
weshowthatmeasuringthepositionoftheparticleaftereachtime-
stepgivesrisetoaclassicalMarkovchain,onthelatticetimesthe
statespaceoftheparticle.Thisquantumtrajectoryisasimulation
ofthemasterequationofthequantumrandomwalk.Thephysical
pertinenceofsuchquantumrandomwalksandthewaytheycanbe
concretelyrealizedisdiscussed.Connectionsanddifferenceswiththe
alreadywell-knownquantumrandomwalks,suchastheHadamard
randomwalk,areestablished.Weexploreseveralexamplesandcom-
putetheirlimitbehavior.WeshowthatthetypicalbehaviorofOpen
QuantumRandomWalksseemstobeverydifferentfromHadamard-
typequantumrandomwalks.Indeed,whilebeingveryquantumin
theirbehavior,OpenQuantumRandomWalkstendtobecomemore
andmoreclassicalastimegoes.
∗
WorksupportedbyANRproject“HAM-MARK”,N
◦
ANR-09-BLAN-0098-01,by
SouthAfricanResearchChairInitiativeoftheDepartmentofScienceandTechnologyand
NationalResearchFoundation
1
Contents
1Introduction3
2GeneralSetup5
3OpenQuantumRandomWalks9
4Examples11
44..12AAnnEExxaammpplleeoonna
Z
G.r.ap.h...........................................1121
5InhomogeneousCase13
6RecoveringClassicalMarkovChains14
7QuantumTrajectories15
8PhysicalImplementation17
9ComingBacktoExamples21
9.1TheExampleon
Z
........................21
9.2TheExampleonaGraph....................22
10UnitaryQuantumRandomWalks24
11Exampleson
Z
27
11.1AWalkWithOnlyOneSteptotheLeft............27
1111..23AExaMmorpeleQsuwaitnhtuSmevEerxaalmGpaleus.si.an.s.................................2298
11.4AnExampleinDimension5...................31
11.5Exampleson
Z
2
..........................32
12ExamplesonGraphs35
12.1O.Q.R.W.onthe2-Graph....................35
1122..32AExnciEtxataimonpleTroannsapo4r-tGr.ap.h.........................................3376
13Appendix:ProofsoftheLemmas38
2
1Introduction
Nowadaysquantumrandomwalks,suchastheHadamardquantumrandom
walk,arequiteasuccessfuldomainofresearch,withimportantapplications
inQuantumInformationTheory(see[Kem]forasurvey).Thesequantum
randomwalksareparticulardiscrete-timequantumdynamicsonastatespace
oftheform
H⊗
C
Z
d
.Thespace
C
Z
d
standsforastatespacelabelledbya
net
Z
d
,whilethespace
H
standsforthedegreesoffreedomgivenoneach
pointofthenet.Thequantumevolutionconcernspurestatesofthesystem
whichareoftheform
X
|
Ψ
i
=
|
ϕ
i
i⊗|
i
i
.
dZ∈iAfteronestepofthedynamics,thisstateistransformedintoanotherpure
state,
X
|
Ψ
0
i
=
|
ϕ
i
0
i⊗|
i
i
.
dZ∈iEachofthesetwostatesgivesrisetoaprobabilitydistributionon
Z
d
,the
donewewouldobtainbymeasuringthepositionon
C
Z
:
Prob(
{
i
}
)=
k
ϕ
i
k
2
.
Thepointisthattheprobabilitydistributionassociatedto
|
Ψ
0
i
cannotbe
deducedfromthedistributionassociatedto
|
Ψ
i
by“classicalrules”,thatis,
thereisnoclassicalprobabilisticmodel(suchasaMarkovtransitionkernel,
orelse)whichgivesthedistributionof
|
Ψ
0
i
intermsoftheoneof
|
Ψ
i
.One
needstoknowthewholestate
|
Ψ
i
inordertocomputethedistributionof
0.iΨ|Thesequantumrandomwalks,thatweshallcallUnitaryQuantumRan-
domWalks(forareasonwhichwillappearclearinSection10)havebeen
successfulfortheygiverisetostrangebehaviorsoftheprobabilitydistribu-
tionastimegoestoinfinity.Inparticularonecanprovethattheysat
√
isfya
rathersurprisingCentralLimitTheoremwhosespeedis
n
,insteadof
n
as
usually,andthelimitdistributionisnotGaussian,butmorelikefunctions
oftheform(see[Kon])
√1
−
a
2
(1
−
λx
)
√,→7xπ
(1
−
x
2
)
a
2
−
x
2
where
a
isaconstant.
Thepurposeofthisarticleistointroduceanewtypeofquantumrandom
walks,thatwesuggesttocall
OpenQuantumRandomWalks
(O.Q.R.W.).
3
Thesequantumrandomwalksalsodealwithaquantumdynamicsonastate
space
H⊗
C
Z
d
,buttheyconsidertheevolutionofdensitymatrices
Xρ
=
ρ
i
⊗|
i
ih
i
|
.
dZ∈iMoreorless,theprincipleisthesameasabove,andthedynamicsleadsto
anewdensitymatrix
X
ρ
0
=
ρ
i
0
⊗|
i
ih
i
|
.
dZ∈iToeachofthemisassociatedtheprobabilitydistributionobtainedwhen
measuringtheposition
Prob(
{
i
}
)=Tr(
ρ
i
)
,i
∈
Z
d
.
ThisnewtypeofquantumrandomwalksisverydifferentfromtheUnitary
QuantumRandomWalks.Itseemsthatthereisnoinclusionwhatsoever,
thoughweproveinSection10averystronglinkbetweenthetwowalks,in
thewaytheycanbephysicallyimplemented.
Actually,thelimitbehaviorofOpenQuantumRandomWalksshowsup
adissipativecharacter,ittendstoconvergetoaclassicalbehavior,thatis,
itseemstogiverisetoclassicalCentralLimitTheorems:onecanseethe
distributionconvergingtoGaussianlimits,ortomixturesofGaussianlimits.
Thepointtobestressedisthegeneralityofoursetup.Itallowsto
consideraverywideclassofquantumrandomwalksonnetsaswellason
graphs.Oursetupistheexactquantumgeneralizationoftheconstructionof
aclassicalMarkovchainonanet,oronagraph.Bytheway,weshallshow
thatOpenQuantumRandomWalkscontainalltheclassicalMarkovchains
asparticularcases.
Ourconvictionisthatthistypeofquantumrandomwalksgivesrisetoa
vastfieldofexplorationforthebehaviorofopenquantumsystems.Itmaybe
asrichastheoneofclassicalMarkovchainsanditshallgiverisetothesame
typeofquestions:existenceofinvariantstates,ergodicbehavior,Central
LimitTheorems,LargeDeviationPrinciple,recurrenceandtransience,etc.
Manyoftheexamplesthatwehaveexploredleadustothinkthatthese
quantumrandomrandomwalksmayapplyinmanyrealisticphysicalsitua-
tions.Theirdissipativebehaviormakesthemphysicallymorerealistic,while
keepingaveryquantumbehavior.Forexample,someoftheexamplesthat
weshallexploreinthisarticlemakeusthinkofpossibleapplications,such
asheatconductionandquantumFourier’slawforaonedimensionalmodel
4
(suchasthequantumversionofthe“SimpleExclusionProcess”,see[Bod])
andrealisticmodelforexcitationtransportonachainofquantumsystems.
Notethatthemainphysicalimplicationsofthisarticlehavealreadybeen
announcedandsummarizedinaletter[APSS].
Also,ithastobesaidthattheideaofconsideringmatricesofcompletely
positivemapssuchthatthelines(orcolumns,dependingonthepointof
view)formaso-calledquantumoperation,appearedearlierin[Gud].This
approachispresentedasa“quantumMarkovchain”.Theseobjectspresent
clearlyseveralcommonpointsintheirstructurewithourOpenQuantum
RandomWalks,buttheyarenotstudiedasgivingrisetoquantumrandom
walks.Exceptattheendofthearticlewhereanincorrectparallelwith
UnitaryQuantumRandomWalkisclaimed.
2GeneralSetup
WenowintroducethegeneralmathematicalandphysicalsetupoftheOpen
QuantumRandomWalks.Forsakeofcompletenesswerecallinthissection
severaltechnicallemmaswhichensurethatourdefinitionsareconsistent.
TheproofsoftheselemmasarepostponedtoSection13.
Wearegivenaset
V
ofvertices,whichmightbefiniteorcountable
infinite.Weconsideralltheorientededges
{
(
i,j
);
i,j
∈V}
.Wewish
togiveaquantumanalogueofarandomwalkontheassociatedgraph(or
lattice).
Weconsiderthespace
K
=
C
V
,thatis,thestatespaceofaquantum
systemwithasmanydegreesoffreedomasthenumberofvertices;when
V
isinfinitecountableweput
K
tobeanyseparableHilbertspacewithan
orthonormalbasisindexedby
V
.Wefixanorthonormalbasisof
K
whichwe
shalldenoteby(
|
i
i
)
i
∈V
.
Let
H
beaseparableHilbertspace;itstandsforthespaceofdegreesof
freedom(or
chirality
astheycallitinQuantumInformationTheory)given
ateachpointof
V
.Considerthespace
H⊗K
.
Foreachedge(
i,j
)wearegivenaboundedoperator
B
ji
on
H
.This
operatorstandsfortheeffectofpassingfrom
j
to
i
.Weassumethat,for
each
j
X
∗B
ji
B
ji
=
I,
(1)
iwheretheaboveserie