Open Quantum Random Walks

Camille Jordan

Description

Niveau: Supérieur, Doctorat, Bac+8
Open Quantum Random Walks? S. Attal 1 , F. Petruccione 2 , C. Sabot 1 and I. Sinayskiy 2 1 Universite de Lyon Universite de Lyon 1, C.N.R.S. Institut Camille Jordan 21 av Claude Bernard 69622 Villeubanne cedex, France 2 Quantum Research Group School of Physics and National Institute for Theoretical Physics University of KwaZulu-Natal Durban 4001, South Africa Abstract A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogue of classical Markov chains. We explore the “quan- tum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time- step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Connections and differences with the already well-known quantum random walks, such as the Hadamard random walk, are established. We explore several examples and com- pute their limit behavior. We show that the typical behavior of Open Quantum Random Walks seems to be very different from Hadamard- type quantum random walks.

• hadamard quantum


• random walks


• limit behavior


• behavior makes


• time- step gives rise


• ??i? ?


• quantum random


• markov chains

Sujets

• Jordan
• Markov chain

Informations

• Publié par: mijec
• Nombre de lectures: 23
• Langue: English

Extrait

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