Table Of ContentReducing quantum control forspin-spinentanglement distribution
F. Ciccarello1, M. Paternostro2, G. M. Palma3, and M. Zarcone1
1CNISM and Dipartimento di Fisica e Tecnologie Relative,
Universita` degli Studi di Palermo, Viale delleScienze, Edificio 18, I-90128 Palermo, Italy
2 School ofMathematics and Physics, Queen’s University, BelfastBT71NN, UnitedKingdom
3 NEST-INFM (CNR) , and Dipartimento di Scienze Fisiche ed Astronomiche,
Universita` degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy
Wepresentaprotocolthatsetsmaximumstationaryentanglementbetweenremotespinsthroughscattering
ofmobilemediatorswithoutinitialization,post-selectionorfeedbackofthemediators’state.Notime-resolved
tuningisneededand,counterintuitively,theprotocolgeneratestwo-qubitsingletstatesevenwhenclassicalme-
diatorsareused. Themechanismresponsibleforsucheffectisresilientagainstnon-optimalcouplingstrengths
0 anddephasingaffectingthespins. Theschemeusesitinerantparticlesandscatteringcentresandcanbeimple-
1 mentedinvarious settings. Whenquantum dots andphotons areused astrikingresult isfound: injectionof
0 classicalmediators,ratherthanquantumones,improvestheschemeefficiency.
2
n PACSnumbers:03.67.Bg,03.67.Hk,73.23.-b,42.50.Pq
a
J
Enforcingastateinaquantumsystem,ataskusuallyrequir-
0
3 ing quantumcontrol[1, 2], is key to the groundingof quan-
tumtechnology[3]. Typically,stateinitialisation,interaction-
] tuning, postselection and feedback are needed in order to
h
achieve a given state [4–14], especially when a system in-
p
volves remote parties requiring interaction-bridging media-
-
t tors[15–19, 21]. One wouldexpectthatquantumcoherence
n
a in the state of such mediatorsis neededto make two remote
u particlesinteract. Here, wediscussa protocolthatsets max-
q imumstationaryentanglementbetweenremotespinswithout
FIG.1:Generalset-upfortheproposedscheme.Twostaticspins,en-
[
initialization of mediators’ state, post-selection or feedback.
codedinthebi-dimensionalspace{|↑,↓i},separatedbyadistance
2 Notime-resolvedtuningisneededand,counterintuitively,the x ,interactwithastreamofflyingparticlese ’s,eachbeingaspin-
0 n
v protocolgeneratestwo-qubitsingletstates evenwhenclassi- 1/2preparedinaclassicalstatisticalmixture(1/2)(|↑ih↑|+|↓ih↓|).
2 cal mediators are used. The mechanism behind this process Whileingeneraleache ,aftermultiplescatteringbetween1and2,
n
3 is stable against non-optimal coupling strengths and robust canbereflectedback,wecollectatDonlytheparticlesthatsuccess-
8 againstdephasingaffectingthespins. Ourproposalusesfly- fullytrespasstheinteractionregionof0≤x≤x0.
0
ingparticles(conductionelectronsorphotons)andtwo scat-
.
9 teringcentres(substitutionalimpuritiesinnanowiresorquan-
0 tumdotsinwaveguides). statethee ’sarepreparedin. Westartexaminingthegeneral
n
9
In order to best present these ideas, we use a setup- symmetriesenjoyedbyHamiltonian(1). Thesearisefromthe
0
independent language that lets us stress the flexibility of commutationrules[Hˆ ,Sˆ2] = [Hˆ ,Sˆ ] = 0, whereSˆ =
: n n n z,n n
v our mechanism. We consider two static spin-1/2 parti- σˆ +Sˆ isthetotalspinoperatorofthee −1−2systemand
i n 12 n
X cles, labelled 1 and 2, separated by a distance x0 in a one- Sˆ = Sˆ +Sˆ . Foreach left-incominge , theoverallspin
12 1 2 n
dimensionalstructure, as in Fig. 1. A low-densitystream of
r space is eight-dimensional. Moreover, the only free-energy
a mobile spin-1/2particles, e ’s, propagatesalongx: each of
n term in Eq. (1) is the kinetic one so that, for an e injected
n
them undergoes scattering by particles 1 and 2 whenever at with wavevectork , the system’s energyis E =k2/(2m)
theirrespectivesites. Inournotation,each(staticormobile) n kn n
witheightassociate(degenerate)stationarystates,eachcorre-
spin is encoded in the bi-dimensional space {|↑,↓i}. The
spondingtoadifferentspinstate. Knowledgeoftheirformal-
interaction Hamiltonian ruling the bilocal coupling between
lowsustodeterminetheevolutionofthesystem’sinitialspin-
eachmobilespine andthestaticonesis
n stateuponeachscatteringevent.Itisstraightforwardtoprove
Hˆ =pˆ2/(2m)+Jσˆ ·[Sˆ δ(x)+Sˆ δ(x−x )], (1) (cfr. AppendixA) that if knx0=qπ (with q a positive inte-
n n n 1 2 0 ger), a situation which we call “resonance condition” (RC),
wherepˆn(m)isthemomentumoperator(mass)ofen,J isthe dynamicstakesplaceasifparticles1and2occupythesame
interactionstrengthandσˆ ,Sˆ andSˆ arethespinoperators site. We have the formulaδ (x)=δ (x−x ) (the sub-
n 1 2 RC RC 0
of e , 1 and 2, respectively(throughoutthis paper we adopt scriptsremindusoftheRCcondition).Thiseffecthasaclear
n
units such that ~=1). Our first task is to demonstrate that, interpretationrelatedtothephase-factorse±iknx0 acquiredby
under proper geometric conditionson this generalsetup and e ’swavefunctionuponmultiplereflectionsatthesitesofpar-
n
bysimplyrequiringtheconservationofthenumberofmobile ticles 1 and 2 (cfr. Fig. 1). Under RC e±iknx0 = 1 and no
particles crossing the scattering region, maximum stationary relative phase-shiftoccursbetween reflected and transmitted
entanglement[3]between1and2canbeset,regardlessofthe componentsof the wavefunction, as if the scattering centers
2
D countsasmanyclicksasthenumberofparticlesthathave
beeninjected)invirtueofthepreviousanalysisweconclude
that, after a sufficiently large number of mediators’ injec-
tionsparticles 1 and 2 necessarily are in a singlet state. Our
proposal de facto embodies a Bell-like projective measure-
ment [3] performed over the remote spins. This occurs re-
gardlessof the spin-state of each e , which can be prepared
n
evenin the classicalstatistical mixtureρ =1/2(|↓i h↓|+
en en
|↑i h↑|)=I /2,thereforedemonstratingthatmaximumen-
FIG. 2: Performance of the scheme. (a) State fidelity F(n) = tanegnlementceannbedistributedbetweentworemotespinswith
12hΨ−|ρ(1n2)|Ψ−i12betweenthetargetstate|Ψ−i12 =2−1/2(|↑↓i− virtuallynoquantumcontrolneededonthestateofeachme-
|↓↑i) andtheremote-spins’stateρ(n) obtainedafternmediators diator. Wenowshowthatthescheme’sefficiencydependson
12 12
havebeeninjectedandallcountedatD. (b)Associatedprobability parametersthatcanbeengineeredoff-line.
ofsuccessP(n). WeusetheHamiltonianmodelinEq.(2)andcall
Westartbyclarifyinghowmanymediatorsoneneedstoin-
v=k/mthemediatorvelocity(assumingallthemediatorstohave
jectandcountatDbeforeprojectiononto|Ψ−i isachieved
thesamewavevectork). Thecurvesareinsensitivetothespinstate 12
withsignificantfidelity.Thisdependsonthecouplingstrength
ofeache ,whichcanthereforebeevenunknown.
n
J. In fact, for a given state ρ 6=|Ψ−i hΨ−|, each e has
12 12 n
a non-null probability to be reflected back without reaching
wereatthesamesite. Thisintroducesadditionalsymmetryto D. Such a reflection probabilitygrowswith J, while we al-
the effectiveinteractionbetween 1 and 2 mediated byen, as readyknowitiszeroifρ12=|Ψ−i12hΨ−|,regardlessofJ. A
witnessed by the new conservationlaw [Hˆ ,Sˆ2 ] = 0. This largevalueofJ thusmakesusconfidentthatonlyafewtrans-
n 12
arises simply by noticing that under RC k =k =qπ/x mittedmediatorsneedtobecountedatDbeforetheeffective
n RC 0
andEq.(1)takestheeffectiveform Bell-projection is achieved. Indeed, in this case, we will be
confident that the transmission of all e ’s is associated with
n
Hˆ =q2π2/(2mx2)+J/2(Sˆ2−σˆ2−Sˆ2 )δ (x), (2) particles 1 and 2 being in a state very close to |Ψ−i . We
RC,n 0 n n 12 RC 12
remark that, although the number of required mediators de-
while Sˆ2 commutes with each term in Eq. (2). We use the pendsonthecouplingstrength,theconvergencetothesinglet
12
statefidelityF(n)= hΨ−|ρ(n)|Ψ−i . InFig.2(a),wecon- isasymptoticallyachievedforanyvalueofthisparameter.No
12 12 12 fine setting of J is requiredby our scheme. Furthermore, in
sidertheinitialproductstateρ =|↑,↓i h↑,↓|andplotF(n)
12 12 linewiththefeaturesofscattering-basedprotocols[18],only
againstnandtheratioJ/v(v=k/m,assumingallthemedi-
averyweakrequirementonthecontrolofinteractiontimesis
atorstohavethesamewavevectork).
inorder:thetimeelapsedbetweentheinjectionoftwosucces-
The above features are enough to explain the claimed in-
sivemediatorsshouldexceedthecharacteristictimeT taken
sensitivity of entanglement generation to the mediators’ in- s
byeachscatteringprocess. Undereasily-matchedconditions
ternal state. Let us consider an initial spin state of the form
and for quasi-monochromatic Gaussian wavepackets of the
|χi |Ψ−i , where |χi is an arbitrary spin state of e
en 12 en n mediators [23] we have T ∼ 1/(v ∆k) with v the ve-
and |Ψ−i = 2−1/2(|↑↓i − |↓↑i ) is the maximally en- s k0 k0
tangledsin12gletstateof1an12d2. |χi 12|Ψ−i isaneigenstate locityassociatedwiththecarrierwavevectork0 and∆k each
en 12 wavepacket-width in k-space. This tells us that the charac-
ofSˆ2 withzeroeigenvalue.Ontheotherhand,|χi |Ψ−i teristic time taken by each scattering event depends only on
12 en 12
isalsoaneigenstateofSˆ2 witheigenvalue3/4,whichbrings kineticparametersofthemediatorwavepacketandnotonthe
n
usdirectlytotheveryspecialeffectiveHamiltonianHˆ = spin-spincouplingmechanism. The mediators’wavepackets
RC,n
q2π2/(2mx2), where the spin degrees of freedom of parti- can be taken so as to make Ts ≪ Td, the latter being the
0
characteristictimeofcoherentdynamicsofagivensetup(i.e.
clese , 1 and2are absent. Despite itssimplicity,thisresult
n
the time-scale afterwhich decoherenceaffectingspins1 and
bringsabouttwocrucialconsequences.First,eache istrans-
n
2yieldssignificanteffects).
mitted through the interaction region with 100% probability
whenever 1 and 2 are in the singlet state. That is, by plac- While the robustnessofthe protocolpresentedhereto de-
ingaGeiger-likeparticlecounterD attheright-handsideof phasingaffectingthestaticspinsisdiscussedlateron,inFig.2
the setup (as in Fig. 1), the numberof flying mediatorstres- westudytheefficiencyofthescheme. Weconsidertheinitial
passing 1 and 2 is conserved. Second, each initial spin state product state ρ = |↑,↓i h↑,↓ | and use the fidelity F(n)
12 12
|χi |Ψ−i isleftunchangedbythescatteringdynamics(re- to measure how close to |Ψ−i is the 1-2 state after that n
en 12 12
mindthat|χi isarbitrary). Thispropertyiseasilyextended successivemediatorshavebeeninjectedandhaveallbeende-
en
toarbitrarymixedspinstatesofeache . Thesingletstateof tected at D. We call P(n) the correspondingsuccess proba-
n
spins 1 and 2 is the only initial state to enjoy such features, bility. In agreementwith our explanations, F(n) grows with
as can be easily proved by considering simple properties of n and progressively approaches F(n) = 1, which marks the
additionofangularmomenta.Also,noticethatourarguments generation of a singlet state. On the other hand, Fig. 2(b)
arevalidforanycouplingstrengthJ. showshowtheprobabilityP(n)thatallthemediatorsarecol-
In retrospect, if in an experiment the number of particles lected at D convergesto 1/2. The curves in these plots are
passing through the interaction region is conserved (that is, insensitivetothedegreeofpurityofthestateofeache (cfr.
n
3
waveguideofferingtheadvantagesofreliablephotonictrans-
port. Two embedded multi-level quantum dots (QDs) em-
bodytwoeffectivepseudospin-1/2staticparticles. Thespec-
trum of each dot consists of a ground doublet and one ex-
cited state. The waveguide accommodates two frequency-
degenerate,orthogonally-polarisedmodesofradiation. Each
mediatore and its spin states are respectivelyembodiedby
n
a photon ph and its (orthogonal) polarization states, which
n
we abstractly indicated as |↓,↑i [cfr. Fig. 3(a)]. For a
phn
large enough detuning between each pseudospin transition-
FIG. 3: Flexibility of the scheme to alternative coupling models. frequencyandthephotonicmediators,theexcitedstateisonly
(a) Level-configuration of the effective static pseudospin-1/2 used
virtuallypopulatedandthetransitionbetweenthepseudospin
inacavity-quantumelectrodynamicsimplementationofourscheme.
states is achieved via two-photon Raman processes with as-
(b) Percentage difference between the state fidelity F(n) achieved
upon use of models HˆRC,n = q2π2/(2mx20)+J/2(Sˆ2n − σˆ2n − sociatedcoherentscatteringofaphotonbetweenstates|↓iphn
Sˆ212)δRC(x) and HˆRC,n = qπvph/x0+J[σˆ−,n(0)Sˆ+,12 +h.c.], and|↑iphn. Weconsideralinearphotonicdispersionrelation
againsttherescaledspin-spininteractionstrengthandforafewsig- E =v k foreachpropagatingphotonso thatthe Hamil-
kn ph n
nificantvaluesofn.Resultsquantitativelyveryclosetothoseshown tonian describing the dynamics of QD’s and each photonic
herearefoundforthepercentagedifferencesinsuccessprobability mediatorunderRCtakestheeffectiveform[15]
P(n).
Hˆ =qπv/x +J σˆ (0)Sˆ +h.c. , (3)
RC,n 0 −,n +,12
h i
Appendix A). In particular, as we stated above, these same
where σˆ (x) = σˆ† (x) are effective pseudospin density
featuresholdforaninjectedstreamofparticleseachprepared +,n −,n
photon operators at position x (cfr. Appendix A) whereas
in the unpolarized state ρ = I /2. This classical statis-
tical mixture gives us theelneast poesnsible informationon each Sˆ±,12 =Sˆ±,1+Sˆ±,2 and S+,i =S−†,i is the ladder operator
mediator’sspin state. Notice that the rate of convergenceof of QD i = 1,2. With due changes, the approach followed
bothF(n) andP(n) increaseswithJ/v(visthemediatorve- in order to demonstrate our scheme holds under the model
locity), in agreement with our predictions: the stronger the in Eq. (3). The results correspondingto this case are indeed
spin-spincoupling,thesmallertherequiredn.Forinstance,at verysimilartothosereportedinFig.2. Infact,bothF(n)and
J/v ≃ 1.6,F(5)>95%withP(5)>50%,whileatJ/v&7.5 P(n) correspondingto model(3) deviate byless than 6%, at
evena single mediatorbeingsent andcollected is enoughto small values of J, from the analogous quantities in the case
achievethesevalues. of Eq. (2). The difference drops to zero in the range of pa-
rameters that guarantee high efficiency of the effective Bell-
Aninterestingarenaforasolid-stateimplementationofthe
projection.Fig.3(b)showsthepercentagedifferencebetween
protocoldescribedsofarisservedbyaone-dimensional(1D)
such quantities, which explicitly shows the closeness of the
CdTenanowire. Astreamofconductionelectronswouldem-
twomodels. TheresultsinFig.3areobtainedforastreamof
bodyparticlese ’sandsubstitutionalMnatomscouldactas
n
unpolarizedphotonicmediators,eachinρ =I /2. This
scatteringcenters1and2.IntheCdTehost,Mnatomsarenot phn phn
bringsustothenextquestion:howdoesthestateofeachme-
ionized due to valence-number matching, which suppresses
diatoraffecttheperformanceoftheschemeforthe modelin
any mediator-static spin electrostatic interaction and leaves
Eq.(3)[wealreadyknowthattheefficiencyobtainedthrough
onlyanexchange-typespin-spincoupling. Theconditionsof
Eq.(2)isinsensitivetothemediators’spinstate]? Toassess
extremely low control required by our scheme fit very well
this, we consider the case that each mediating photon is in-
with the experimental capabilities in current spintronics set-
jectedinthestateρ =[(1−r)|↓i h↓|+(1+r)|↑i h↑|]/2,
tingssuchastheonesketchedhere.Asnopreparationorpost- phn phn phn
selection of the mediator’s spin-state is required, there is no where r ∈ [0,1] determines the purity of the polarization
need forexperimentallychallengingspin-filteringoperations state. In Fig. 4 we set J/v = 1.5 and plot F(n) and P(n)
performed over mobile electrons (the efficiency of currently against n and r. Surprisingly, the efficiency of the scheme
availablespinfiltersistothebestofourknowledgestillquite decreaseswithr: thedistributionofmaximumentanglement
low [24]). On the other hand, the fabrication of Mn-doped is optimized by sending fully unpolarized photons. Despite
nanostructuresiscurrentlyexperiencingsomeimpressiveim- itscounter-intuitivenature,thisresultiseasilyunderstoodby
provements, includingthe implantationof a single Mn atom noticingthat,infullanalogywiththemodelinEq.(2),ifthe
intoaquantumdot[25].AlthoughaMnatominaCdTecom- pseudospinsareinasingletstate,thespin-spinterminEq.(3)
poundhasquantumspinnumbers = 5/2,whilethedescrip- vanishesregardlessofthemediators’state, implyingthatthe
tionofourproposalinvolvedspin-1/2particles,theworking injected photonswill all be collected at D. However,unlike
principleofourschemeremainsvalid,withdueadjustments, thecaseofanexchangeinteraction,asinEq.(1),thisisnotthe
inthecaseofspin-sscatteringparticles(cfr. AppendixA). onlycasewheresuchbehaviouroccurs.Indeed,foranincom-
ingphotoninρ =|↑i h↑|(r=1)withthepseudospinsin
A second interesting scenario for testing our predictions phn phn
is provided by a cavity-quantum electrodynamics (QED) ρ =|↑,↑i h↑,↑|, itisHˆ = v k (foranyk ). How-
12 12 RC,n ph n n
setup. Our proposalconsidersa 1D semiconductorphotonic ever, when r6=1 this is no longer true since the photon has
4
Programme 2007-2008. MP is supported by EPSRC
(EP/G004579/1).
Note added. After completion of the manuscript we be-
cameawareofarelatedproposalputforwardbyK.Yuasain
arXiv:0908.4377v1[quant-ph].
APPENDIXA:Technicaltools
FIG.4:Effectsofknowledgeofmediators’state.(a)Fidelityand(b)
Throughout these Appendices we omit the subscript n in
successprobabilityoftheproposedscheme,whenthemodelvalidin
the scattering mediator since the features discussed here do
thecavity-quantumelectrodynamicsimplementationisused,against
n and the parameter r entering the state of the injected mediators. notdependonthislabel.
For r = 0 we prepare initial statistical mixtures with no quantum
coherence,whichneverthelesscorrespondtotheoptimizedeffective
Bell-projectiondescribedinthetext. 1.Stationarystates
Here, we give a brief account of the method followed in
now some probability to be sent in |↓i h↓ |. In this case,
phn order to determine the stationary states correspondingto the
the previousargumentdoes nothold and some “impedance”
generalization of the interaction models in the body of the
tothemediator’stransitarisesevenwhenρ =|↑,↑i h↑,↑|:
12 12 paper to remote static particles having spin quantum num-
thehigherr,thestrongertheimpediment. Asthelatteristhe
ber s ≥ 1/2. We call m =↓,↑ and m =−s,..s the quan-
basicresourceharnessedbyourschemeinordertodiscrimi- e i
tum numbers of σ and S respectively (i = 1,2). A spin
nate thesingletstate, ouranalysisexplainsand clarifieswhy z iz
state |m ,m ,m i is labeledby µ={m,m,m}. We con-
theprotocolisoptimizedwhenr=0. e 1 2 e 1 2
sider the three sections into which the x axis in Fig. 1 has
Wehaveshownthepossibilityofguidingthestateofadis-
been divided. The system’s stationary states can be writ-
tributedsystem ofremotespinsviaa control-relaxedscatter- tenas|Ψµ′i= dxΨµ′(x)|µi|xiwiththewavefunctions
ing. Besidepropergeometricalarrangementofthesetup,im- µ µ
plying a properly set inter-spin distance, the mere conserva- Ψµµ′,I(x)=δµ,µP′eikxR+rµµ′e−ikx,Ψµµ′,II(x)=Aµµ′eikx+Bµµ′e−ikx
tion of the numberof scattered mediatorsis requiredfor the and Ψµ′ (x) = tµ′eikx, each valid in section I, II and III
successoftheprotocol. Preparationandpost-selectionofthe µ,III µ
respectively. Here, coefficients A’s, B’s, r’s and t’s are
mediators’ internal state are not demanded by our scheme,
computed by imposing proper boundary conditions [27] on
which sets steady-state entanglement insensitive, by defini- Ψµ′(x) and, depending on the interaction model, its deriva-
tion, to timing imperfection. Againstany expectations, even µ
tiveatx=0andx=x . ItshouldberemarkedthatΨµ′(x)as
classical mediators can be used to establish maximum en- 0 µ
wellasA,B,randtimplicitlydependonkx andJ/v.
tanglement. We havedemonstratedthatthe scheme is stable 0
againstnon-optimalcouplingstrengthswhileitexhibitsanin-
trinsic robustness against collective dephasing-likenoise. In
fact, our effective Bell projection progressively extracts the 2.Ontheconsequencesofresonanceconditions
singlet-state component from the initial state of the remote
spins,anditiswell-knownthatcollectivescramblingmecha- The previous section allows us to prove that under reso-
nismsaffectingthephase-relationwithinthestate ofthedis- nance condition (RC) δ (x) = δ (x−x ). In the basis
RC RC 0
tributed spins is quenched by the symmetries of the singlet {|Ψµ′i} the matrix representation of the operator δ(x−x′)
state[26].Thequantitativeaspectsofthisstudy,togetherwith M(x′)iseasilyfoundtobeM(x′)µ′′,µ′=hΨµ′′|δ(x−x′)|Ψµ′i=
those relativeto the case ofindividualnoise-mechanismsaf- Ψ¯µ′′(x′)Ψµ′(x′)(z¯isthecomplexconjugateofz).Match-
fecting the spins, are presented in Appendix B. Within the µ µ µ
iPngofΨµ′(x)atx=x impliesthatAµ′+Bµ′e−2ikx0=tµ′,
context of quantum-state guidance, our proposal stems as a µ 0 µ µ µ
milestonedemonstratingthatfulladvantagecanbetakenfrom which for kx0=qπ becomes Ψµµ′(0)=e−iqπΨµµ′(x0). This
suitablesymmetriesinordertosignificantlyreducequantum givesM(0)=M(x0)andthusδRC(x)=δRC(x−x0).
control.
3.Descriptionofthespinstateafterascatteringevent
Acknowledgments
Here we assess the form taken by the spin state of the
We thank A. Acin, G. Fishman and J. M. Lourtioz for remote spins after scattering of a mediator takes place.
discussions. We acknowledge support from PRIN 2006 Let ρ = ρ ρ be the initial spin state. Its decompo-
e 12
“Quantum noise in mesoscopic systems”, EUROTECH sition in the spin basis {|µi} introduced above is ρ =
and the British Council/MIUR British-Italian Partnership µ,µ′cµ,µ′|µihµ′|. The corresponding scattering process is
P
5
fully described by the state ρs = µ,µ′cµ,µ′|ΨµihΨµ′| (no- 5.Cavity-quantum-electrodynamicsmodel
tice that, unlike ρ, ρs refers to botPh the spatial and spin de-
grees of freedom). The transmitted part of ρs is ρt(x,x′)= InthisSectionweprovideafewkeydetailsonthederiva-
µ,µ′cµ,µ′ νtµνeikx|µi ν′t¯µν′′e−ikx′hν′|, where tµν’s are tionoftheeffectiveHamiltonianEq.(3).Foragenericphoton
tPransmissionPamplitudes asPsociated with scattering between wavevector, we have Hˆ = Hˆ + Vˆ, where Hˆ is the free-
0 0
spinstates|µiand|νi. Thereflectedpartcanbedetermined, Hamiltonian of the waveguide and Vˆ is the atom-photonin-
likewise, in terms of analogous reflection amplitudes rνµ’s. teraction.Hˆ0reads
Once trace of ρ over e is performed, the final state of the
s
raermefoutnecstpioinnssodfeptheendtrsanosnmoipsseiroantoarnsdTˆrmmeefl′ee’sctainodnaRˆmmmpe′el’istu,dwehsitc’hs Hˆ0 =−i X X Z dxvβcˆ†β,γ(x)∂xcˆβ,γ(x), (A-3)
β=R,Lγ=↑,↓
andr’s,respectively(cfr. followingSection).Theiranalytical
expressionsaretoocumbersometobepresentedhere. where v = −v = v and cˆ† (x) [cˆ† (x)] is the bosonic
R L R,γ L,γ
operatorcreatingaright(left)propagatingphotonofpolariza-
tionγ=↑,↓atpositionx. Vˆ hastheform
4.Effectofmediatorcounts
Vˆ= dxσ (x)[Sˆ δ(x)+Sˆ δ(x−x )] +h.c. , (A-4)
Z + 1− 2− 0
In order to give a quantitative account of how recording
clicks at the Geiger-like detector D (shown in Fig. 1) af- where σˆ (x) = σˆ†(x) = cˆ†(x)cˆ (x), along with σˆ (x) =
fectstheremote-spinstateweadoptthelanguageofquantum + − ↑ ↓ z
maps [3]. Assume that a mediating particle e has been pre- [cˆ†↑(x)cˆ↑(x)−cˆ†↓(x)cˆ↓(x)]/2,arepseudo-spindensityopera-
paredinthestate|m′ =↓,↑iandsenttotheinteractionregion, tors such that σˆ = dxσˆ(x) is a spin-1/2 vector operator.
e
while spins 1 and 2 have been preparedin ρ . Without the Fromthis,thedescripRtioninourworkiseasilyestablished.
12
Geiger-likedetector,thestateof1and2afterscatteringis
ρ′ = (Rˆm′eρ Rˆm′e†+Tˆm′eρ Tˆm′e†), (A-1) APPENDIXB:Resilienceagainstphase-scrambling
12 me 12 me me 12 me
X
me=↑,↓
In the case of the cavity-quantumelectrodynamicsimple-
whereRˆm′e’s(Tˆm′e’s)aretheoperators(introducedinthepre- mentationproposedinourpaper,oneshouldconsidertheef-
me me fectsof spuriousenergy-conservinginteractionsbetween the
viousSection)describinghowareflected(transmitted)media-
spin degree of freedom of each static particle and a back-
torin|m =↓,↑iaffectsthestaticspins’state.Inthecasethate
e ground of phononic modes belonging to the substrate onto
isinjectedinastateoftheformρ ==r|↓i h↓|+(1−r)|↑i h↑|,
e e e which each particle resides. The effects of such couplings
recordingaclickatDchangestheinitialstateρ into
12 canbeeffectivelymodelledasrandomphase-kicksovereach
spin’sstateoccurringwithacharacteristictimeT . Formally,
d
E˜(1)(ρ )=r Tˆ↓ ρ Tˆ↓†+(1−r) Tˆ↑ ρ Tˆ↑†. they can be appropriately accounted for by standard quan-
12 me 12 me me 12 me
mXe=↑,↓ mXe=↑,↓ tum maps [3]. As stated in the body of the paper, a po-
(A-2) tential setup for testing our scheme is a GaN nanowire with
As, in general, e may also be reflected back, state (A-2) embeddedGaInN QDs. In this host, typical photonic wave-
is not normalized. Thus, the right-hand side has to be di- lengths and group velocities are λ ∼400 nm and v ∼ c/2,
ph
vided by the probabilityP(1)(ρ12) = Tr[E˜(1)(ρ12)] that e is respectively. Under the conditions assumed in our study,
transmitted. Therefore, the complete map transforming ρ12 each mediator takes a time Ts ∼1/(vk0∆k) to be scattered
into the static spins’ state ρ(1) after the injection and col- off of spins 1 and 2 (see Ref. [23] for details). By taking
12
lection at D of a single mediator reads ρ(1) = E(1)(ρ ) = x0 = π/k0 and ∆k/k0 ∼ 10−2, so as to fulfill the RC, we
12 12 obtain T ∼ 10−14s, which is at least one order of magni-
E˜(1)(ρ )/P(1)(ρ ). Thestateρ(n)afternofsuchinjection- s
12 12 12 tudesmallerthanthetypicaldecoherencetimesinthissetups.
collectionstepsisgivenbythen-timeapplicationofthemap
Therefore,insuchregimephasekickstakeplaceinthebuffer
asρ(1n2)=E(n)(ρ12)=E[E[··[E(ρ12)]]]withassociatedproba- time-windowTbbetweentwosuccessivescatteringevents.To
bilityP(n)=Tr[E˜(n)(ρ12)],whereE˜(n)(ρ12)]isobviouslyde- testthe resilienceofourprotocolagainstTb/Td we haveper-
finedasthen-timeapplicationofthemapinEq.(A-2)above. formeda Monte Carlo numericalsimulationof the proposed
Asdiscussedinthebodyofthepaper,whenρ12=|Ψ−12i Ψ−12 scheme.Foragivennumberofstepsn,theapplicationofmap
eisalwayscollectedatD and,inaddition,theremote(cid:10)spins(cid:12)’ E has been interspersed n times with that describing phase
(cid:12)
state is not affected. Thus, under RC E(n)(|Ψ−i Ψ− ) = kicks on 1 and 2, as coming from dephasing baths attached
12 12
|Ψ−i Ψ− and P(n)(|Ψ−i Ψ− )=1 for any n,(cid:10)r an(cid:12)d J. to each static spin. We explicitly allow for the possibility of
12 12 12 12 (cid:12)
Inthe(cid:10)caseo(cid:12)ftheHamiltonian(cid:10)give(cid:12)ninEq.(2),thisistheonly correlations between the baths in the problem. The dephas-
(cid:12) (cid:12)
spinstateenjoyingsuchproperty,whichcharacterizesthesin- ing map depends on the dimensionless parameter [3] T/T
b d
gletastheonlyfixedpointofmapE. ForEq.(3),ontheother and the degree of noise correlation [28] µ. This quantity is
hand,itistheonlyfixedpointprovidedthatr 6=0. the probability that 1 and 2 undergo correlated phase-kicks
6
FIG.5: Behavioragainstdephasingnoiseaffectingtheremotestaticspins1and2whenmodel(3)isconsideredandforvarioussettingsof
parametersnandJ/v. Panels(a),(b)and(c): statefidelityF(n)= hΨ−|ρ(n)|Ψ−i forstreamsofunpolarizedmediators(i.e. forr = 0)
12 12 12
againstT/T andµ.Panels(d),(e)and(f):P(n)againstT/T andµ.
b d b d
[28] so that we have µ = 0 (µ = 1) for fully-uncorrelated actly1forµ = 1,regardlessoftheratioT /T . Asthebaths
b d
(fully-correlated)baths. In Fig. 5 we study the resilience of consideredbecomeonlypartiallycorrelated,thisstrikingand
thestatefidelityagainstT /T andµ. Remarkably,forµ=1 clearrobustnessisonlypartiallyspoiled:statefidelityremains
b d
state fidelity is insensitive to dephasing noise. In fact, both veryhighevenforaratherpessimisticT /T andquitealarge
b d
thestatesenteringtheeffectivedynamicalmapresponsiblefor numbernofstepsconsidered. Weconcludebystressingthat
theprogressiveprojectionontothemaximallyentangledsin- the spoiling effects of dephasing can be counterbalancedby
glet state of the remote spins are known to be unaffectedby the choice ofa couplingstrengthlargeenoughto achievean
perfectly correlated dephasing baths [26] (they are basically effectiveprojectionontothe singletstate ofthe remotespins
decoherence-freestatesforthisclassofnoise).Thisisclearly (withhighfidelity)withonlyasingle-mediatorinjection.
reflected in Figs. 5(a), (b) and (c), where state fidelity is ex-
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