Transmission losses in optical qubits for controlled teleportation

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by  I. Medina

Transmissionlossesinopticalqubitsforcontrolledteleportation
I. Medina1 and F. L. Semia˜o1
1CentrodeCieˆnciasNaturaiseHumanas,UniversidadeFederaldoABC,09210-170,SantoAndre´,Sa˜oPaulo,Brazil
In this work, we investigate the controlled teleportation protocol using optical qubits within the single rail
logic. Theprotocolmakesuseofanentangledtripartitestatesharedbythecontrollerandtwofurtherparties
(users)whowillperformstandardteleportation.Thegoaloftheprotocolistoguaranteethattheteleportationis
successfulonlywiththepermissionofthecontroller. Opticalqubitsbasedoneithersuperpositionsofvacuum
andsingle-photonstatesorsuperpositionofcoherentstatesareemployedheretoencodeatripartitemaximal
slice state upon which the protocol is based. We compare the performances of these two encodings under
losseswhicharepresentwhenthequbitsareguidedthroughaopticalfibertotheusers. Finally,investigatethe
nonlocalityofthesharedtripartitestatetoseewhetherornotitimpactstheefficiencyoftheprotocol.
7
I. INTRODUCTION ing two of such encodings and studying their performance
1
0 in CT with realistic losses in the transmission of the qubits.
2 In the first case, qubit codification is considered to be based
Entanglementisapropertyofmultipartitequantumsystems
onsingle-modeorthogonalFockstates. Theparadigmaticex-
n which possess no complete classical analogue. Its existence
ample being the superposition of single photon and vacuum
a has been linked to fundamental aspects such as nonlocality
J states. This is known as single-rail quantum logic. Very im-
[1,2]orsteering[3]aswellaspracticalprotocolsinquantum
4 information [4, 5]. Quantum teleportation [4] is a protocol portant developments have been made for such qubits. For
2 instance,auniversalsetofnondeterministicgatescanbebuilt
where two parties, Alice (sender) and Bob (receiver), share
using linear optics and photon counting [14]. Improved pro-
a bipartite maximally entangled state (the quantum channel)
] cedures for performing gates with a significant reduction in
h which allows Alice to perfectly send the unknown state of a
p qubit to Bob by making only local measurements in her lab theresourceshavealsobeendiscussed[15]. Next,theuseof
- real-time feedback control was shown to further reduce the
and sending two bits of classical information to him. If the
t
n channel is not perfect, i.e., not a pure maximally entangled resourcesneededforscalablequantumcomputingwithsingle
a rail optical qubits [16]. A critical assessment of single rail
bipartitestate,thisprotocolallowsAlicetojustimperfectlyor
u quantum linear optical quantum computing can be found in
probabilisticsendthequbitstatetoBob[6,7].
q
[17]. Teleportationandviolationoflocalrealismforabipar-
[ With time, more sophisticated versions of the standard
titeextensionofthissingle-raillogicwasconsideredin[18].
quantumteleportationprotocolhavebeenproposed. Oneex-
1 Fromtheexperimentalside,thesestateshavebeengenerated
ampleisthesocalledcontrolledteleportation[8](CT)where
v withhighcontrolinspontaneousparametricdown-conversion
3 athirdpartycomesintotheplay. Charlie(controller)canlo- [19] or mixing single photon and coherent states in a beam
3 cally control the quality of the quantum channel shared by
splitter, followed by a conditioned measurement [20]. As a
8 AliceandBobthusaffectingthequalityofthestandardtele-
matter of fact, even the teleportation protocol as proposed in
6 portation they intend to perform. To achieve this control,
[18]hasalreadybeenexperimentallyrealized[21].
0
CharliecansharewithAliceandBobamaximallyentangled
.
1 three-qubitGreenberger-Horne-Zeilinger(GHZ)state[9],and Inthesecondcase,thecodificationrelyontheuseofnon-
0 by making or making not a local projective measurement on orthogonalsingle-modecoherentstatesofoppositephases. It
7
his qubit, Charlie can influence the fidelity of teleportation is a generalized type of single-rail qubit in the sense that it
1
[8]. When Charlie do not perform the measurement, Alice tendstothestandardcaseforlargecoherentstateamplitudes.
:
v and Bob can only achieve teleportation fidelities up to 2/3. In this case, the two opposite-phase coherent states used in
Xi Inpracticalterms,thismeansnoquantumteleportationsince the encoding approach orthogonality. On the other hand, for
suchfidelitiescanbeachievedwithouttheuseofanyquantum small amplitudes, the non-orthogonality usually brings very
r
a channel [10]. In other words, Alice and Bob need the per- interesting effects when energy relaxation acts on the qubits
mission of Charlie (his performing a local measurement and [22]. Theuseofcoherentstatesinquantuminformationisin
informing the outcome) in order for them to achieve fideli- factaveryactiveresearchtopicgivingthatthegeneration,ma-
ties higher than the classical bound 2/3. In [11], this prob- nipulationandcharacterizationofcoherentstatesisverywell
lem is studied for a general family of pure three-qubit states developedinquantumoptics[23]. Theuseofcoherentstates
andin[12]aclassofstatescalledmaximalslicestates(MS) for quantum teleportation and quantum logic is carefully de-
areshowntobeparticularlyusefulforCT.Whatissurprising tailed in [24–27]. Entangled coherent states, such the ones
here is that such states are not, in general, maximally entan- usedinthiswork,haveinfactbeenexperimentallygenerated
gled triparte states such as the GHZ state. Both the standard incircuitquantumelectromagneticsetupswherefullquantum
andcontrolledteleportationprotocolsfindapplicationsinthe state tomography via quantum nondemolition measurements
fieldofsecurequantumcommunication[13]. ofthejointphotonnumberparityoperatorhavebeensucess-
Inthecontextofquantumoptics,differentphotonicqubits fulyimplemented[28]. Ingeneral,oneneedssomenonlinear-
have been considered for practical implementation quantum ity in the Hamiltonian to dynamically superpose or entangle
information protocols. In this work, we will be consider- coherentstates. Proposalsarefoundin[29,30].
2
This work is organized as follows. In Sec. II, we quickly the permission? In this case, we want to evaluate the non
reviewtheCTprotocolandpresentourinvestigationproblem conditionedteleportationfidelityF . Itisevaluatedwiththe
nc
which is the performance of different optical qubits in con- correspondingdensityoperatorfortheMSstateinEq.(2),af-
trolledteleportationunderdissipation. Sec.IIIisdedicatedto ter the controller has been traced out. This corresponds to a
our results. In particular, we compare and contrast the per- scenariowherenomeasurementswereperformedbythecon-
formanceofsingle-railencodingusingorthogonalFockstates troller. Inthiscase,onefinds[12]
withthatusingcoherentstates.Wealsolookintothetripartite
2+d
nonlocality of the state shared by the parties to see whether F = . (4)
or not nonlocality and efficiency are related. In Sec. IV, we nc 3
sumarizeourfindingsandpresentourfinalremarks. Finally,
From this result, it is clear that for d = 0(θ = 0), when the
intheAppendix,webrieflyreviewamethodtoevaluatetele-
MS state in Eq.(1) reduces to the GHZ state, Alice and Bob
portationfidelities.
fails completely in the attempt of cheating [8]. In this case,
alltheyobtainistheclassicalfidelity2/3. Asdincreases,the
power of the controller starts to diminish. In the limit case
II. BASICELEMENTS
d = 1(θ = π/2), theteleportationissuccessfullyperformed
without permission of the controller. Again, this is a simple
A. Controlledteleportationprotocol consequence of form of the MS state in Eq.(1). When d =
1, Alice and Bob are left in a Bell state which is completely
In the context of CT, we will assume that the controller uncorrelatedwiththequbitinCharliepossession.
(Charlie) possesses qubit 1, while the sender (Alice) and the In [12], 1 − F is called control power. Here, we will
nc
receiver (Bob) possess qubits 2 and 3, respectively. In the renormalizeitinawaythatitexplicitlytakesintoaccountthe
ideal case (no dissipation), these three qubits are in the pure unsuitabilityoffidelitiesbelow2/3forquantumteleportation.
MSstatedefinedas[12,31] ThisnormalizedcontrolpowerC varyingbetweenzeroand
p
onereads
1
|MS(cid:105)123 = √2(|0,0,0(cid:105)+c|1,1,1(cid:105)+d|0,1,1(cid:105))123, (1) C =(cid:26)1−3(cid:0)Fnc− 32(cid:1), if Fnc >2/3, (5)
p 1, if F ≤2/3
nc
wherecanddarerealssubjectedtoc2 =1−d2. Actually,in
thesimulations,wewillbeusingc=cosθandd=sinθwith Notice that Fnc = 1 implies Cp = 0. On the other hand,
θ ∈ [0,π/2]. AliceandBobwillusetheirpartofthisstateto fidelities equal or below the classical bound implies Cp =
performthestandardteleportation[4]oftheunknownstateof 1. These two limits correspond to complete success or fail
a forth qubit in the possession of Alice. To see how Charlie of Alice and Bob when they proceed without permission of
can affect the performance of the teleportation protocol, we Charlie.
rewriteEq.(1)as The CT protocol is intrinsically a twofold problem. First,
when no permission is given, C must be as close to one as
p
1 possible. Second, when the permission is given, the condi-
|MS(cid:105) = [(1+d)|0(cid:105)+c|1(cid:105)] ⊗|Φ+(cid:105)
123 2 1 23 tioned fidelity Fc must be higher than 2/3 and as close as
1 possibletoone. Thesetwofactsledustodefineaefficiencyη
+ [(1−d)|0(cid:105)−c|1(cid:105)] ⊗|Φ−(cid:105) , (2)
2 1 23 fortheCTprotocolas
√ (cid:26)
where|Φ±(cid:105)23 =1/ 2(|0,0(cid:105)±|1,1(cid:105))23aremaximallyentan- η = Cp[1+3(Fc−1)], if Fc >2/3, (6)
gledBellstates. Now,itiseasytoseethatifCharlieperforms 0, if Fc ≤2/3
aprojectivemeasurementontheorthonomalbasis[12]
NoticethatηvariesbetweenzeroandoneandwheneverF ≤
c
1 2/3orC =0theefficiencyisnull. Thismeansthat,inorder
|ξ±(cid:105) = [(1±d)|0(cid:105)±c|1(cid:105)] , (3) p
1 (cid:112)(1±d)2+c2 1 for the control to be effective, it is of no use to have a high
control power C when the conditioned teleportation fidelity
p
Alice and Bob end up with one of two Bell states. Upon re- Fcisundertheclassicalbound. Whenpermissionisgiven,no
ceivingaclassicalmessagefromCharlieinforminghismea- quantumteleportationwillbeachievedinthisscenario.
surementoutput,theycanperformtheteleportationwithunit WenowproceedtoapplythesequantifiersFc,Cp,andηin
fidelity. Theteleportationfidelityundertheseconditionswill the analyze of the performance of different optical qubits in
be called conditioned fidelity F . The local measurement by CTwithMSstatesanddamping.
c
CharlieandtheclassicalcommunicationoftheoutputtoAlice
andBobiswhatismeantbycontrollerpermission. Itisinter-
esting to see that the MS state allows unit F for any choice B. Theproblem
c
ofd(orθ). Inotherwords,thereisnoneedofagenuinemax-
imallyentangledtripartitestatesuchasaGHZstate. The physical situation we want to address in this work is
WhatifAliceandBobtrytocheatCharlie? Inotherwords, depictedinFig.1. ThekeyfigureintheCTprotocolisChar-
how does the teleportation work when Charlie does not give lie,thecontroller. Itisthennaturaltothinkthatitishimwho
3
in the fibers [33]. After individually studying these figures
of merit, we will then directly compare the encodings using
the efficiency as defined in Eq.(6). Finally, we will investi-
gatewhetherthereisanymeaningfulrelationbetweentripar-
titenonlocalityandtheefficiencyoftheprotocol.
A. Vacuumandsingle-photon
The first step is to rewrite the MS state in Eq.(1) in
FIG.1. (Coloronline)CharliepreparestheMSstateinEq.(1),send
terms of the physical qubits of the VSP encoding in Eq.(8).
oneopticalqubittoAliceandanothertoBob,boththroughoptical
Then, Eq.(7) is solved with the initial condition ρ(0) =
fiberswithdissipationrateΓ.Duringthispropagation,thequbitthat
stayswithCharlieisassumedtoapproximatelynotdissipateenergy |MS(cid:105)123(cid:104)MS|. In the case of the non-conditioned density
norsufferdecoherence(closedsystem). operatorforAliceandBob,qubit1(controllerqubit)istraced
outresultingin
1
willpreparethetripartiteMSstateanddistributeonequbitto ρ (t)= [(1+r4)|¯0,¯0(cid:105) (cid:104)¯0,¯0|+τ4|¯1,¯1(cid:105) (cid:104)¯1,¯1|
AliceandanothertoBob.Inhislab,Charliekeepalong-lived nc 2 23 23
stationary matter qubit based, for instance, on electronic de- +τ2sinθ(|¯0,¯0(cid:105) (cid:104)¯1,¯1|+|¯1,¯1(cid:105) (cid:104)¯0,¯0|)
23 23
gree of freedom of trapped ions or states of artificial atoms +r2τ2(|¯0,¯1(cid:105) (cid:104)¯0,¯1|+|¯1,¯0(cid:105) (cid:104)¯1,¯0|)], (10)
23 23
in circuit quantum electrodynamics. However, for Alice and
√
Bob, Charlie must employ flying or optical qubits that will where τ = e−Γt/2 and r = 1−τ2 is a normalized time
be sent to their laboratories. Typically, this is made with the varying from zero to one. Physically, r represents the pres-
help of optical fibers which inevitably involve losses (ampli- enceofthelossyfiber. Thebiggerthenormalizedtimer,the
tude damping). Time evolution under amplitude damping is strongertheeffectofthelosses.
describedbythemasterequation[32] FortheconditionedfidelityF ,insteadoftracingoutqubit
c
1, the evolved density operator is projected onto the basis in
∂ρ
=Jρ+Lρ, (7) Eq.(3). Thetwopossibleoutcomesare
∂t
whereJρ=Γ(cid:80)3 a ρa†andLρ=−(Γ/2)(cid:80)3 (a†a ρ+ ρ±(t)= 1[(1+r4)|¯0,¯0(cid:105) (cid:104)¯0,¯0|+τ4|¯1,¯1(cid:105) (cid:104)¯1,¯1|
i=2 i i i=2 i i c 2 23 23
ρa†a )witha (a†)theannihilation(creation)operatoracting
i i i i ±τ2(|¯0,¯0(cid:105) (cid:104)¯1,¯1|+|¯1,¯1(cid:105) (cid:104)¯0,¯0|)
on mode i, and Γ is the dissipation rate of the optical fibers. 23 23
Noticethat,inourproblem,onlythemodesaddressedtoAlice +r2τ2(|¯0,¯1(cid:105)23(cid:104)¯0,¯1|+|¯1,¯0(cid:105)23(cid:104)¯1,¯0|)], (11)
andBob(i=2,3)aresubjectedtodampinginthefibers.
whichshowupwiththeprobabilities
Asmentionedbefore,wewillbeworkingwithtwodistinct
physicalqubitsinsingleraillogic. Theyare 1±sinθ
P± = . (12)
1. Vacuum and Single-Photon States (VSP): in this case, 2
thephysicalqubitsaretheFockstatescorrespondingto
Now,theteleportationfidelities
zerooronephotoninthemode,i.e.,
F =P+F(ρ+)+P−F(ρ−), (13)
{|0(cid:105),|1(cid:105)}−→{|¯0(cid:105),|¯1(cid:105)} (8) c c c
F =F(ρ ) (14)
cn nc
The bars are used in the VSP to avoid confusion be-
can be evaluated using the method discussed in Appendix.
tweenphysicalandlogicalqubits.
Theresultis
2. Coherent States: coherent states |α(cid:105) are special
1
superpositions of F√ock states defined as |α(cid:105) = Fnc = 6(3+2sinθ|r2−1|+|1−2r2+2r4|), (15)
een−c|αo|n2d/2in(cid:80)g,∞nw=e0(αwnil/l bne!)e|mn(cid:105)plwoyitihngαccoohmerpelnetx.staItnesthoisf Fc = 61(3+2|r2−1|+|1−2r2+2r4|). (16)
sameamplitudebutoppositephases[24–27],i.e.,
Please, notice that the conditioned fidelity F defined in
c
{|0(cid:105),|1(cid:105)}−→{|α(cid:105),|−α(cid:105)}. (9) Eq.(13)isactuallyanaveragefidelityweightedbytheproba-
bilitiesofobtainingeachoftheconditioneddensityoperators
uponCharliemeasurement.FortheparticularcaseoftheVSP
III. RESULTS qubits,theseprobabilitiesaregivenbyEq.(12). Forthequbits
basedoncoherentstatestobediscussednext,theseprobabil-
For each of the photonic qubits considered in this work, itieswillnaturallychange.
andtheinitialMSstateinEq.(1),wenowstudyhowthecon- In Fig. 2, we present the effect of losses in the fiber for
ditionedfidelitiesandthecontrolpowerareaffectedbylosses the control power evaluated with Eqs. (5) and (15) and the
4
conditioned fidelity (16) for different values of θ, i.e., some thebasisinEq.(3),andthetwopossibleoutcomesarenow
choices of c and d in the MS state Eq.(1). First, it is worth ρ±(t)=M (α)2[|γ,γ(cid:105) (cid:104)γ,γ|+|−γ,−γ(cid:105) (cid:104)−γ,−γ|
noticingthatforθ =0thecontrolpowerremainsoneregard- c ± 23 23
less of the losses in the fiber. This happens because for for ±e−4r2|α|2|γ,γ(cid:105)23(cid:104)−γ,−γ|
θ = 0 the initial MS state is actually a maximally entangled ±e−4r2|α|2|−γ,−γ(cid:105) (cid:104)γ,γ|], (18)
23
GHZstatewhosereducedstate(tracingoutthecontroller)is
(cid:112)
a statistical mixture with no entanglement. Since the local where M±(α) = 1/ 2±2exp[−4|α|2]. The probabilities
dynamics of losses acting on this non-entangled state is un- forρ±c (t)arenow
abletocreateentanglement,AliceandBobwillalwaysfailto 1±sinθ
P ± = (1±e−4|α|2). (19)
perform quantum teleportation. One the other hand, as θ is α 2(1+sinθe−4|α|2)
progressively increased in the range [0,π/2], the initial state
These probabilities will be used to evaluate the conditioned
with Alice and Bob tends continually to a maximally entan-
fidelityF = P +F(ρ+)+P −F(ρ−),asexplainedbefore.
gled Bell state causing Cp to decrease. In this context, the c α c α c
In order to once again use the methods discussed in the Ap-
lossesinthefiberstendtofavorthecontroller(increasingC )
p
pendix, weneedtoredefineapropermagicbasisintermsof
as they destroy the entanglement in the reduced state of Al-
coherentstates. Itnowreads
iceandBob. Thisiswheretheintroductionoftheefficiency
quantifierinEq.(6)willplayanimportantrole. Thepresence |mγ(cid:105)= √1 (|γ+,γ+(cid:105)+|γ−,γ−(cid:105)),
oflosseswillalsoaffecttheconditionedfidelityobtainedwith 1 2
thepermissionofthecontroller. Consequently,theincrement i
|mγ(cid:105)= √ (|γ+,γ+(cid:105)−|γ−,γ−(cid:105)),
ofCp duetothelossesarenotnecessarilygoodforCT.This 2 2
twofoldaspectoftheprotocolrevealedbytheanalysisofthe
i
efficiencywillbediscussedlateronthispaper. |mγ(cid:105)= √ (|γ+,γ−(cid:105)+|γ−,γ+(cid:105)),
3 2
1
|mγ(cid:105)= √ (|γ+,γ−(cid:105)−|γ−,γ+(cid:105)),
1.0 1.0 4 2
0.8 0.9
where |γ±(cid:105) = (2 ± 2e−2|γ|2)−1/2(|γ(cid:105) ± |−γ(cid:105)) are even
CP0.6 Fc0.8 and odd coherent states which are naturally orthonormal
0.4 0.7 (cid:104)γ+|γ−(cid:105) = 0. The expressions for F and C are now a bit
c p
0.2
0.6 cumbersome, so we will only perform a graphic analysis of
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
thesequantities.
r r
InFig.3,wepresenttheplotsofthecontrolpowerandcon-
FIG.2. (Coloronline)Controlpower(left)andconditionedfidelity ditionedfidelityfordifferentvaluesofθeα. Whentheinitial
(right) as a function of normalized time and different values of θ statecorrespondstoatypeofGHZ(θ =0),thecontrolpower,
fortheVSPencoding. Leftpanel:θ =0(solid),θ =π/6(dashed),
besides being independent of r, becomes now also indepen-
θ=π/4(dot-dashed),andθ=π/3(dotted).Rightpanel:F turned
c dentofα. Onceagainthishappensbecausethereducedstate
outtobeindependentonθ [seeEq.16]andthedashedstraightline
of Alice and Bob possess no entanglement for θ = 0. This
indicatesdeclassicalfidelity2/3.
isnotobservedforF whichpresentsastrongdependenceon
c
αevenwhenθ = 0. Inparticular,wewanttodrawattention
tothedifferentcrossingsthathappeninC andF whenr is
p c
varied. Thismeansthatdependingonthenormalizedtimeor
B. Coherentstate howmuchthelossesactedonthestate,itismoreappropriate
tochooselargeorsmallcoherentstateamplitudesαtoachieve
Wenowproceedtothesecondkindofphotonicqubitcon- morecontrolorhigherconditionedfidelities.Thishappensfor
sideredinthiswork. Now, thefirststepistorewritetheMS two reasons. First, superposition of coherent states are more
stateinEq.(1)intermsoftheencodinginEq.(9).Itisalsonec- affectedbythelosseswhenthetheiramplitudesarelarge[34].
essary to renormalize the state since the coherent states used However, the same large amplitude coherent states produces
intheencodingarenotorthogonal.Next,Eq.(7)issolvedwith better conditioned fidelities because the states in Eq.(9) tend
theinitialconditionρ(0)=|MS(cid:105) (cid:104)MS|. Aftertracingout tobecomeorthogonalinthislimit. Itisthiscompromisebe-
123
qubit1(controllerqubit),wegetthenonconditioneddensity tweendissipationandorthogonalitywhichproducesthecross-
operator ingsobservedinFig.3forintermediatevaluesofr. Thisbe-
haviorofthecoherentstatesencodingmakesitaninteresting
ρ (t)=N(α)2[|γ,γ(cid:105) (cid:104)γ,γ|+|−γ,−γ(cid:105) (cid:104)−γ,−γ| candidateforimplementationofvariedquantuminformation
nc 23 23
tasks[22,41]. Asafinalremark,atr = 0,thecontrolpower
+e−4r2|α|2sinθ|γ,γ(cid:105) (cid:104)−γ,−γ|+
23 decreasedwithθ because,inthelimitθ → π/2,thereduced
+e−4r2|α|2sinθ|−γ,−γ(cid:105) (cid:104)γ,γ|], (17) stateisaquasi-Bellstate
23
(cid:112) |Φ±(cid:105)=(2±2e−4|α|2)−1/2(|αα(cid:105)±|−α;−α(cid:105)) (20)
where γ = ατ and N(α) = 1/ 2+2sinθexp[−4|α|2]. α
For the conditioned case, we again consider projection upon whichallowsalmostperfectteleportationwithlargeα[22].
5
Θ=0HGHZL Θ=0HGHZL Fig.4, it is the VSP encoding with θ = 0 (GHZ state) that
2.0 1.0
presentsthebestefficiency.Asθisvaried,however,crossings
1.5 0.9 mayhappeninvolvingtheVSPandthecoherentstatesencod-
Cp1.0 FC0.8 ing. Forinstance, forθ = π/3itisquiteclearthatforweak
0.5 0.7 ormoderatelossesr ≤ 0.3thecoherentstatesdoabetterjob
than the VSP states. Once again, the use of small amplitude
0.0 0.6
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 orlargeamplitudecoherentstateswilldependontheprecise
r r
valueofronehas,i.e.,thequality/lengthoftheopticalfibers.
Θ=Π(cid:144)6 Θ=Π(cid:144)6
1.0 1.0 Finally, it worthwhile to notice that the efficiency is not, in
0.9 0.9 general, a monotonically decreasing function of the normal-
0.8 ized time what might be very useful for practical purposes.
Cp FC0.8
0.7 Thishappensbecausetheefficiencytakesintoaccount,onan
0.6 0.7 equalfooting,thetwofoldaspectsoftheCTprotocol: control
0.5 0.6 powerandconditionedfidelity.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
Θ=Π(cid:144)4 Θ=Π(cid:144)4 Θ=0HGHZL Θ=Π(cid:144)6
Cp00000001........34567890 FC00001.....67890 Η000001......024680æææææææææææææææææææææ Η000000......012345æææææææææææææææææææææ
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r r r
Θ=Π(cid:144)3 Θ=Π(cid:144)3 Θ=Π(cid:144)4 Θ=Π(cid:144)3
Cp00001.....24680 FC00001.....67890 Η000000000.........001122334050505050æææææææææææææææææææææ Η0000000.......00112230505050æææææææææææææææææææææ
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r r r
FIG.3. (Coloronline)Controlpower(left)andconditionedfidelity FIG. 4. (Color online) Efficiency of the CT protocol as a function
(right)asafunctionofnormalizedtimeanddifferentvaluesofθand of normalized time for the VSP and coherent states encoding and
αforthecoherentstatesencoding.Thedashedstraightlineindicates differentvaluesofθ. LinewithcirclesisusedtotheVSPcase. For
theclassicalfidelity2/3.Theusedamplitudesofthecoherentstates thecoherentstates,wehaveα = 0.20(solid),α = 0.50(dashed),
areα = 0.20(solid),α = 0.50(dashed),α = 1.25(dot-dashed), α=1.25(dot-dashed),α=2.50(dotted).
α=2.50(dotted).
C. Efficiency D. Tripartitenon-locality
From the previous discussions, it is clear that the control Therelationbetweennonlocality, inthesenseofviolation
powerCpandtheconditionedfidelitiesFc,usuallyemployed of Bell inequalities, and teleportation has a long history in
intheanalysisofCT[12],arenotenoughtograspallaspects quantum information [10]. In the bipartite scenario of stan-
oftheproblemwhendissipationispresentandnonorthogonal dard teleportation, it can be shown that violation of a Bell
basis states are used in the encoding. In particular, the plots inequality by the state shared by Alice and Bob implies that
in Fig.3 clearly show that, for small r (low dissipation), an the state is useful for quantum teleportation (provides fideli-
increaseinαcanhelpFc, whatisdesirableinCT.However, tiesabovetheclassicallimit2/3)[6]. However,noviolation
thisspoilsCp,whatisverymuchundesirable. Thissituation does not imply that the state is useless for quantum telepor-
becomes even more evolved for moderate dissipation where tation [10]. When looking at more general scenarios, it can
crossingsdohappen. Itisforthisreasonthatwenowproceed beshownthatanybipartitestatewhichisusefulforteleporta-
toanalyzetheefficiencyηasdefinedinEq.(6). Thisquantity tion is also a nonlocal resource [35]. This means that a Bell
will allow us to finally compare the performance of the two inequality is always violated deterministically when a suffi-
differentopticalqubits. cientlylargenumberofcopiesofthestateisprovided.Forthe
In Fig.4, we present the plots of efficiency for both phys- tripartite scenario of the CT protocol, there is yet no general
ical encodings: VSP in Eq.(8) and coherent state in Eq.(9). results linking legitime tripartite nonlocality and the perfor-
Ourintentnowistoreallycomparetheperformanceofthese mance of the protocol. In order to shed some light on this
twokindofphotonicqubitsintheCTprotocolunderthesce- problem,wewillnowstudyviolationoftripartitenonlocality
nariodepictedinFig.1. Accordingtoourresultsdisplayedin anditspossible(orlackof)relationtotheefficiencyoftheCT
6
protocol.
5.75 5.75
OnepossibilityistheuseoftheBell-Svetlichnyinequality 5.50 5.50
whichwasthefirsttoolproposedtotestgenuinemultipartite max55..0205 max55..0205
nonlocality[37]. SupposeasetodichotomicobservablesA2 Sv4.75 Sv4.75
(A(cid:48)),B (B(cid:48)),C (C(cid:48)),whereA isanobservableactingon È4.50 È4.50
the2state3spac3eofq1ubi1t2(Alice’sl2aboratory),C(cid:48) isanobserv- 4.25 4.25
1 4.00 4.00
ableactingonthestatespaceofqubit1(Charlielaboratory), 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0
r Η
etc. At each run of the experiment, each party chooses one
ofthetwoobservablesatrandomtomeasure. Fordichotomic FIG. 5. (Color online) Left: Maximized Bell-Svetlichny function
observableswithspectrum±1,theBell-Svetlichnyinequality |Sv|maxfortheVSPencodingEq.(8)asafunctionofthenormalized
canbewrittenas[36] timeandfordifferentvaluesofθ.Right:MaximizedBell-Svetlichny
|S | against efficiency η as the normalized time varies in the
v max
|S |=|E(A,B,C)+E(A,B,C(cid:48))+E(A,B(cid:48),C) interval[0,1]. OnlytheregionwheretheBell-Svetlichnyinequality
v
isviolatedthatisshown. Weusedθ =0(solid),θ =π/6(dashed),
+E(A(cid:48),B,C)−E(A(cid:48),B(cid:48),C(cid:48))−E(A(cid:48),B(cid:48),C)
θ = π/4(dot-dashed)eθ = π/3(dotted). Thedottedstraightline
−E(A(cid:48),B,C(cid:48))−E(A,B(cid:48),C(cid:48))|≤4, (21) indicatemaximalviolationoftheBell-Svetlichnyinequalityallowed
byquantummechanics.
whereE(A,B,C) ≡ Tr{ρO}istheexpectationvalueofthe
three-body operator O = ABC and ρ is the tripartite state
whose nonlocal features one is interested in. We call S the In the right panel of Fig.5, we present parametric plots
v
Bell-Svetlichny function. The GHZ state is√known to maxi- where each point corresponds to (η(r),|Sv(r)|max). Again,
mally violate this inequality with |S | = 4 2 [37, 38]. It is only the nonlocal region is displayed. From this plot, it is
v
importanttohaveinmindthat,ifonestatedoesnotviolatea possible to see |S | and η are not trivially related. For
v max
giveninequality,itdoesnotimmediatelyimplythatitwillnot the GHZ (θ = 0), the better the efficiency the stronger the
violateotherinequalitybuiltwithadifferentsetofdichotomic nonlocality and vice versa. However, for finite θ, there can
observables[39]. be situations where efficiency grow is accompanied |S |
v max
For the VSP encoded MS state evolving according to depletionoreventheotherwayround.
Eq.(7), the three-body operator O we will be using to eval- For the coherent states, it is necessary to adapt the three-
uate(21)is body operator O to an hybrid scenario [41]. We need now
dichotomicoperatorsactingonthewholestatespaceofeach
OVSP =Ω1(χ)Ω2(λ)Ω3(µ), (22) field mode and not just on a subspace of two states as in the
VSP.Inthiswork,wewillemploy
where each dichotomic operator (eigenvalues ±1) is defined
asΩj(ζ) = Rj(ζ)σzjRj†(ζ),withσzj theusualPauli-z op- Ocoh =Ω1(χ)Π2(λ)Π3(µ), (24)
erator acting on the state space of qubit j (j = 1,2,3), and
Rj(ζ) a rotation operator acting on the same space. In the whereΠj(β)isthedisplacedparityoperator[42]
Pauli-zbasisitreads
∞
(cid:88)
(cid:32) cos|ζ| ζ sin|ζ|(cid:33) Πj(β)=Dj(β) (|2n(cid:105)j(cid:104)2n|−|2n+1(cid:105)j(cid:104)2n+1|)Dj†(β),
Rj(ζ)= −ζ∗ sin|ζ| |ζc|os|ζ| , (23) n=0
|ζ| (25)
whereζ =−ωe−iδ/2with0≤ω ≤πand0≤δ ≤2π. Itis withD (β)thedisplacementorGlauberoperator[43]acting
j
worthnoticingthatbychoosingζ complex,ourmaximization on modes j (j = 2,3), β a complex number, and |n(cid:105) Fock
j
is more apt to detect stronger violations [41]. Finally, in the statesofmodej. Forqubit1,possessedbyCharlie,wekeep
notationusedinEq.(21),onehasA =Ω (λ),A(cid:48) =Ω (λ(cid:48)), using the same dichotomic operator as in the VSP case giv-
2 2 2 2
B =Ω (µ),andsoforth. ingthathestillperformstheprojectiononthediscretesetin
3 3
In Fig.5, we present |S | maximized over the twelve pa- Eq.(3).
v
rametersthatcomprisethetwomeasurementseachpartycan Again,theBell-Svetlichnyfunctionwilldependontwelve
choose. In the left panel, we see how the maximal Bell- variablesbecausetheargumentofeachoperatorinEq.(24)is
Svetlichny function evolves under dissipation. For the ideal a complex number. In Fig.6, we present its maximization as
fiber r = 0, the maximally entangled GHZ (θ = 0) indeed afunctionofnormalizedtime(leftpanels)andtheparametric
maximally violates the Bell-Svetlichny inequality. Also at plots(η(r),|S (r)| )(rightpanels).Wevarytheamplitude
v max
r = 0, as θ increases in the range [0,π/2], the violation be- α of the coherent states in Eq.(9) as well as the parameter θ
comesprogressivelysmaller. Thiscanbeunderstoodbylook- whichdefinestheMSstateinEq.(1). Onecanseethat,except
ingatthetripartiteentanglementofthepureMSstateinEq.(1) forθ = 0, onlythestateswithlargeamplitudeviolateofthe
as quantified by the tangle Υ [40]. For this state, Υ = cosθ Bell-Svetlichnyinequality. Also,therearecaseswhereweak
[31], i.e., it is a monotonically decreasing function of θ as violations provide better efficiencies than strong violations
it varies from 0 to π/2. The observed decay with r is the [see, for instance, the case θ = π/4]. The Bell-Svetlichny
expected effect of the losses which, in general, depletes the function,liketheefficiency,controlpowerandconditionedfi-
quantumfeatures. delities, also presents crossings. Finally, from Figs.(4), (5),
7
Θ=0HGHZL Θ=0HGHZL fibers used by the controller to distribute the optical qubits
5.75 5.75
5.50 5.50 to the other parties. We introduced an efficiency quantifier
max55..0205 max55..0205 which takes into account the main features of the protocol.
Sv4.75 Sv4.75 WefoundthatthebestperformanceisachievedwiththeVSP
È4.50 È4.50 encoding and the GHZ state, which is a proper limit of the
4.25 4.25
4.00 4.00 MS states considered in this work. For general MS states,
0.00 0.05 0.10 0.15 0.20 0.6 0.7 0.8 0.9 1.0
however,thelossesaffecttheefficienciesfortheVSPandthe
r Η
Θ=Π(cid:144)6 Θ=Π(cid:144)6 coherentstatesinanontrivialway. Dependingonstrengthof
5.2 5.2 thelosses,eithertheVSPorthecoherentstatesmightbethe
5.0 5.0
bestchoice. Wealsoinvestigatedapossiblerelationbetween
max44..68 max44..68 tripartite nonlocality, under the scope of the Bell-Svetlichny
ÈSv4.4 ÈSv4.4 inequality, and the efficiency of the protocol. We found that
4.2 4.2 thereisnosimpleormonotonicrelationbetweenthesephysi-
4.0 4.0
0.00 0.05 0.10 0.15 0.20 0.475 0.485 0.495 calquantities.
r Η It is important to remark that the teleportation fidelities in
Θ=Π(cid:144)4 Θ=Π(cid:144)4
4.8 4.8 thisworkareevaluatedassumingthatBobisabletoperforma
4.7 4.7 setofunitaryrotationsonhisqubit,asdemandedbythestar-
4.6 4.6
max44..45 max44..45 dardquantumteleportationprotocol[4]. Theopticalelements
Sv4.3 Sv4.3 demandedtoperformsuchrotationsarewellknownsincethe
È4.2 È4.2 veryfirstproposalsfortheuseofcoherentstatesinquantum
4.1 4.1
4.0 4.0 information tasks [24–26]. On Alice side, the Bell measure-
0.00 0.04 0.08 0.12 0.16 0.29 0.30 0.31 0.32 0.33 0.34
ment she performs is typically probabilistic when using co-
r Η
Θ=Π(cid:144)3 Θ=Π(cid:144)3 herentstates. Thisisaconsequenceofthenonorthogonality
4.4 4.4 ofthebasisstates.Bellmeasurementswithcoherentstatesare
4.3 4.3 carefullydiscussedin[44,45].Simpleschemesmaygivesuc-
max4.2 max4.2 cessprobabilitiesapproaching50%asdiscussedin[44]. The
ÈSv4.1 ÈSv4.1 probabilisticnatureoftheBellmeasurementsdoesnotaffect
our calculations giving that only the successful events have
4.0 4.0
0.00 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20 beenconsidered. Detailsintheappendix.
r Η
Wehopeourworkmaybeusefultoabetterunderstanding
FIG. 6. (Color online) Left: Maximized Bell-Svetlichny function ofthecontrolledteleportationprotocolandmotivateitsfuture
|Sv|max for the coherent state encoding Eq.(9) as a function of experimentalimplementationwithopticalqubits.
the normalized time and for different amplitudes α. Right: Maxi-
mized Bell-Svetlichny |S | against efficiency η as the normal-
v max
izedtimevariesintheinterval[0,1].OnlytheregionwheretheBell-
Svetlichnyinequalityisviolatedthatisshown. Weusedα = 0.50
ACKNOWLEDGMENTS
(dashed), α = 1.25 (dot-dashed), α = 2.50 (dotted). The dotted
straightlineonthetoppanelsindicatemaximalviolationoftheBell-
Svetlichnyinequalityallowedbyquantummechanics. IMacknowledgessupportbytheCoordenaodeAperfeioa-
mentodePessoaldeNvelSuperior(CAPES).FLSacknowl-
edgespartialsupportfromCNPq(grantnr. 307774/2014-7).
and (6), one can see that the nonlocality is more affected by
losses than the performance of CT protocol. In other words,
evenafter|S |startsassumingvaluescompatiblewithaclas-
v
sical local and realist theory, the CT protocol still produces APPENDIX
results compatible only with quantum theory. In this sense,
forthephysicalsettingdescribedinFig.1,theCTprotocolis 1. TeleportationFidelity
morerobustagainstdissipationthanthenonlocalityalaBell-
Svetlichny.
LetussupposethatAliceandBobshareageneralbipartite
statedescribedbythedensityoperatorρ,andtheywanttouse
this state as a channel for teleportation [4]. Assuming that
IV. FINALREMARKS Alicesucceedsinprojectingthelocalstateofhertwoqubits
inaBellstate,andthatBobcanperformarbitraryrotationson
his own qubit, it can be shown that the maximal fidelity for
Inthiswork,wepresentedacarefulstudyofthecontrolled
thestandardteleportationprotocolcanbewrittenas[46]
teleportationprotocolwithtwoofthemostusedopticalqubits
in the single rail logic: superpositions of vacuum and single
photon (VSP) states and superposition of coherent states of 2f +1
F(ρ)= , (26)
oppositephases. Theanalysistookintoaccountlossesinthe 3
8
withf beingthesocalledfullyentangledfractiondefinedas referredtoasthemagicbasis
[47]
1
|m (cid:105)=|Φ+(cid:105)= √ (|00(cid:105)+|11(cid:105)),
1
2
f =max(cid:104)φ|ρ|φ(cid:105), (27) |m (cid:105)=i|Φ−(cid:105)= √i (|00(cid:105)−|11(cid:105)),
|φ(cid:105) 2
2
i
|m (cid:105)=i|Ψ+(cid:105)= √ (|01(cid:105)+|10(cid:105)),
3
2
wherethemaximizationisoverallmaximallyentangledstates
1
|φ(cid:105), i.e., all states that can be obtained from a singlet using |m (cid:105)=|Ψ−(cid:105)= √ (|01(cid:105)−|10(cid:105)).
4
localunitarytransformations. 2
The fully entangled fraction can be found using a simple Thefullyentangledfractionf isthenevaluatedsimplyasthe
methodwhosestepswenowbrieflydescribe[48]. First,the biggest eigenvalue of the real part of ρ when written in the
density operator ρ must be written in a special basis usually magicbasis[48].
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