Quantum partial teleportation as optimal cloning at a distance

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by  Radim Filip

Quantum partial teleportation as optimal cloning at a distance
Radim Filip
Department of Optics, Palack´y University,
17. listopadu 50, 772 07 Olomouc,
Czech Republic
(Dated: February 1, 2008)
We propose a feasible scheme of conditional quantum partial teleportation of a qubit as optimal
asymmetric cloning at a distance. In this scheme, Alice preserves one imperfect clone whereas
otherclone isteleported toBob. Fidelities oftheclonescan besimply controlled byan asymmetry
in Bell-state measurement. The optimality means that tightest inequality for the fidelities in the
asymmetriccloningissaturated. Furtherwedesignaconditionalteleportationassymmetricoptimal
N →N+1cloningfromN Alice’sreplicasonsingledistantclone. Weshortlydiscussedtwofeasible
experimentalimplementations,firstoneforteleportationofpolarizationstateofaphotonandsecond
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one, for teleportation of a time-bin qubit.
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0 PACSnumbers:
2

n
a I. INTRODUCTION tation are limited by the fidelities in optimal quantum
J cloning. Previously, universal symmetric optimal quan-
8 tum cloners were theoretically proposed to locally dis-
One frommain tasksof quantuminformationprocess-
tributeanunknownpurequantumstateofaqubittothe
ing is how to optimally distribute an unknown quantum
1 copies[5,6,7,8,9]andwerealsorealizedexperimentally
state Ψ of a qubit to another distant qubit. A perfect
v | i [10]. To locally duplicate an unknown quantum state of
9 quantum teleportation [1] where Alice completely trans-
a qubitwith unbalancedfidelities, the asymmetric quan-
3 mits unknown qubit state Ψ to distant Bob’s qubit, is
| i tum clonersweretheoreticallydiscussed[11]. The asym-
0 a particular example of this task. As a resource, they
1 share a pair of qubits in a maximally entangled state metric 1 → 2 optimal cloning produces two copies from
0 a single replica of an unknown state and obtained state-
which can be distributed aprioriandthen, in advantage,
4 they perform only localoperations and classicalcommu- independent fidelities FS and FS′ of the copies saturate
0 cloning inequality [11]
nication (LOCC) in an actual time of the state trans-
/
h mission. After this complete quantum teleportation, Al-
p ice has no information about input qubit state. A con- (1−FS)(1−FS′)≥(1/2−(1−FS)−(1−FS′))2. (1)
-
t ditional version of qubit teleportation of a polarization
n This inequality sets the tightest no-cloning bound on fi-
stateofphotonwasexperimentallydemonstratedusinga
a delities of 1 2 cloning device that duplicates an un-
u simple Bell-stateanalyzerbasedonbalancedbeamsplit- known qubit→state to another qubit with isotropic noise.
q ter [2]. Recently,alsoconditionallong-distancequantum
Thus if the equality occurs in (1) then for given fidelity
: teleportation of a time-bin qubit in telecommunication
v FS one cannotobtained a better fidelity FS′. Previously
fibers has been realized [3].
i experimentally performed symmetric quantum cloning
X
In this paper, we extend the conditional teleportation with identical fidelities FS,S′ = 5/6 arises as a partic-
ar scheme to partial optimal teleportation of single replica ular case. An enhancement of the fidelities FS′ =5/6 of
ofunknownqubitstate. Thepartialteleportationmeans a single additional copy can be obtained only if we have
that Alice preserves an imperfect copy ρS of input state N > 1 identical replicas of the input state and imple-
andBobobtainstheotherimperfectcopyρS′. Inthepar- mentsymmetricN N+1cloning[6,8]. Thenasingle
tial teleportation the fidelities of copies FS = ΨρS Ψ , additional copy of q→uantum state can be produced with
h | | i
FS′ = ΨρS′ Ψ canbecontroledbyanasymmetryinthe fidelity
h | | i
Bellstatemeasurement. The optimalityinthiscasesays
that for a given fidelity of Alice’s copy with initial state, (N +1)2+N
Bob cannot in principle obtain a higher fidelity of his FN N+1 = , (2)
→ (N +1)(N +2)
copy. This extension of teleportation can be straightfor-
wardly implement into the recent conditional teleporta- which approaches unity as the number N of replicas in-
tion experiments [2], using the Bell-state analyzer beam creases. From the point of view of quantum cloning,
splitter or fiber coupler with a variable reflectivity. Fur- the schemes proposed below can be also reviewed as a
ther,weproposeaschemeforteleportationfromN iden- conditional implementation of optimal universal quan-
tical replicas of a qubit state to a single distant copy. tum cloning at a distance. These proposal can be also
The partialtransmissionofa quantum state is dissim- view as a new tele-cloning procedurein comparisonwith
ilar to classical information processing because perfect Ref. [12].
cloning of an unknown quantum state is impossible [4]. The paper is organized as follows. In the Sec. II,
Thus the fidelities of the copies after the partial telepor- we design the partial conditional teleportation scheme
2
ALICE ρ (R) = (1 F (R))Ψ Ψ +F (R)Ψ Ψ
anti−clone BOB distant clone I − I | ih | I | ⊥ih ⊥|
clone (5)
R S’
I S with the following fidelities:
ψ
1
F (R) = (1 2R)2+(1 R)2 ,
S (cid:0) (cid:1)
Ψ 2P(R) − −
1 (1 R)2
FS′(R) = (cid:0)R2+(1 R)2(cid:1), FI(R)= − ,
2P(R) − 2P(R)
FIG.1: Schemeofconditionalpartialteleportationasoptimal
asymmetric 1→2 cloning. (6)
where P(R) = 1 3R + 3R2. It can be proved that
and prove that it represents optimal asymmetric 1 → 2 the fidelities FS an−d FS′ saturate the inequality (1) and
cloning at a distance. We also discuss local implemen- therefore the distribution of input state between clone
tation of U-NOT gate, LOCC reversibility of partial S and distant clone S is optimal. The symmetric dis-
′
teleportation and the sequential partial teleportation. tribution can be obtained for the reflectivity R = 1/3.
Further, in Sec. III the partial conditional symmetric In this case we also obtain optimal U-NOT gate with fi-
N N+1teleportationisdescribedanditisprovedthat delity F =2/3 if we take the anticlone I as output
→ UNOT
itproducesN+1copieswithoptimalfidelities. Simulta- of the U-NOT. This U-NOT optimally approximates a
neously,thisschemelocallyrealizesoptimalU-NOTgate transformation Ψ Ψ [13], only by mixing the in-
forN multiple replicasofinputstate. InthelastSec.IV put state with |thei →ran|do⊥mi mixed state on unbalanced
experimental implementations of these schemes for the beam splitter with R=1/3.
polarization and time-bin qubits are shortly discussed. Now we show that we can probabilistically transform
any asymmetric cloner with R,T =0 to complete condi-
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tional teleportation with unit fidelity only by local mea-
II. OPTIMAL 1→2 ASYMMETRIC CLONING surements on Alice’s clone and ancilla, classical commu-
AT A DISTANCE
nication with Bob and state filtration on Bob’s qubit.
Assuming input state Ψ = αV +β H , the state
S S S
Aschematicsetupforpartialconditionalteleportation afterprojection(4)can|beiexpan|dedi inthe| foillowingway:
of a qubit is depicted in Fig. 1. In fact, this is a feasible
modificationofthe previousexperimentonteleportation α(1 2R)VVH SIS′ β(1 2R)HHV SIS′
ofapolarizationstateofphoton[2]. Itisbasedoncondi- − | i − − | i −
tional and partial Bell-state measurement which can be α(1−R)|VHViSIS′ +αR|HVViSIS′ +
simplyimplementedbyanunbalancedbeamsplitterwith β(1 R)HVH SIS′ βRVHH SIS′. (7)
− | i − | i
a variable reflectivity R, 0 R 1/2. After mixing two
≤ ≤ Generalizing an idea of the state restoration from
photons S,I on beam splitter we restrict our teleporta-
Ref.([14]), wecanmeasurepolarizationinbasis V , H
tiononlytosuchcaseswhenbothphotonsleavethebeam
| i | i
on Alice’s clone and ancilla and the results send to Bob.
splitter separately. Then we may effectively describe the
The measurement can be experimentally implemented
unbalanced beam splitter by the following transforma-
using polarization beam splitter followed on both out-
tion:
puts by single photon detectors which is, in fact, an
ΨΨ (T R)ΨΨ , asymmetric version of Bell state measurement in tele-
SI SI
| i → − | i
portation experiment [15]. If we select only such results
ΨΨ T ΨΨ RΨ Ψ , (3)
SI SI SI
| ⊥i → | ⊥i − | ⊥ i when the orthogonal polarizations V S H I (H S V I)
| i | i | i | i
which corresponds to the projection are detected then the Bob’s state changes to new one,
proportionaltoα(1 R)V S′ βRH S′ (αRV S′+β(1
− | i − | i | i −
Π−SI(R)=((1−2R)1S ⊗1I +2R|Ψ−iSIhΨ−|) (4) R)|HiS′). Thesestatescanbeconditionallytransformto
initial state Ψ S by the local filtering RV S′ V (1
oinnginthpautt epnotlaanrigzlaetdionstastteate|Ψo−fiItSw′o =phot√o12n(s|.VHAisIsSu′m−- Rha)n|Hd,iSB′ohHb |c|a(Rni|hHeilpS′AhHlic|+e(t1o−cRon)d|VitiiSo′nhaVll||y).riOesnthotrh|ee−iontithiea−rl
a|HndVBiIoSb′), =we√c12a(n|ΨpΨro⊥vieItSh′a−t|ΨAl⊥icΨeicIaSn′)cisonshdaitrieodnablylyApliecre- sintabteasoisn Vher, cHlonoe.n qBuobbithSa′s atnodpAerlfiocremthmeesaasmureemmeenat-
| i | i
foltooonwrBmainospgbta.alrotPtecieaaorllffotstertolmaettpaienlosgsroytpafsrttcoielomjonencoe|tΨsifvaSSeni,m|SuΨen′−akasinnuIoSdrwe′amnwneeqtniu-otcbbΠliotta−SnsiIetn(aRItthe)e⊗|Ψf1oiSlS-′ (t(sα|ouVr(se1itmIa−|tHee2niRtpSr)o′|o)nVptioqhSruetb+nioitβnaa(Is1l.t−atIotfeRαt)oh(|f1eHt−dihSeeRt)eA)wc|tlVhiecidiecShsc−tlioasntβeee(q1iuiss−a|lcH2otoRniIvi)|ne|VHrittiieiSaSd′l
stateafterconditionalstateprojection(1 2R)V V
S
− | i h |−
ρS,S′(R) = FS,S′(R)Ψ Ψ +(1 FS,S′(R))Ψ Ψ , (1 R)H S H ((1 2R)H S H +(1 R)V S V ).
| ih | − | ⊥ih ⊥| − | i h | − | i h | − | i h |
3
ALICE BOB1 BOB2 anti−clone clone
anti−clone clone anti−clone distant clone distant clone I
ALICE BOB
S
I1 R1 S1 I2 R2 S2 S3 ψRN N clone distant clone
ψ R2 S2clone S’
Ψ Ψ N copies ψ R1 S1
ψ
Ψ
FIG. 2: Scheme of sequential conditional partial teleporta-
tion. FIG. 3: Scheme of conditional partial teleportation as N →
N+1 symmetric optimal cloning.
Thuswecanatleastconditionallyproveinfeasibleexper-
imentthatasymmetriccloningprocedureisconditionally must adjust the reflectivities according to
LOCC reversible.
We shortly discuss a sequence of two partial telepor- 1
R = , (8)
n
tationsinwhichAlicecanconditionallysymmetriclydis- n+2
tribute an unknown quantum state among three Bobs if
where n = 1,...,N and then Alice obtains N optimal
they share singlets Ψ , as is depicted in Fig. 2. Since
this scheme is a se|qu−enice of partial teleportations we clones of input state in modes S1,...,SN, single anti-
clone in mode I and on the other hand, Bob has at a
need to generally evaluate the fidelities of clones after
distanceasinglecloneinmodeS . Toprovethis,wecal-
every step of the procedure. To find R1,...,RM 1 for ′
a symmetric distribution of M clones we have to−solve culate the probabilities that state Ψ can be detected
in the particular output modes S a|nd⊥iI
a set of quadratic equations for fidelities with condition ′
F1,...,FM, which can be performed numerically. Ana-
N N
lytically,wecanpresentthesimplestexampleforM =3, 1 1
inwhichwesetR1 =3/8,R2 =1/3toobtainsymmetric p⊥S′ = P(N) Y(1−2Rk)2, p⊥I = P(N) Y(1−Rk)2.
distribution. At result, we obtain the same fidelity of all k=1 k=1
(9)
three clones F = 29/38 0.763. It is slightly worse in
≈ and in modes Sn
comparisonwith the fidelity F =7/9 0.777 of optimal
≈
1 3 cloning [6, 8]. Thus we cannot generally use a
n 1 N
sfoerq→mueantcioenotfoamsyamnmyeutsreircsoinptoimptailmcalolnwearys.tIotdisisatpripbaurteenitnly- p⊥Sn = P(1N)Rn2 Y− (1−Rk)2 Y (1−2Rk)2. (10)
k=1 k=n+1
dissimilar with classical-like universal cloning when we
with a given probability swap an unknown state to one For R given by (8), the total probability P(N) of suc-
k
from the M users and to the others we send completely cess can be determined from normalization condition
randomized state. In this case, the fidelity of cloning N p + p + p = 1 and is equal to P(N) =
FM = 12(cid:0)1+ M1 (cid:1) is always less than optimal universal 4P/(n(=N1 +⊥Sn1)(N⊥S+′ 2)).⊥IConsequently, the fidelity of n-th
cclloonniinngg Fca1n→Mbe=im2pM3leM+m1enbtuedt tbhyisaclsaesqsiuceanl-cleikeof11→ M2 scilmonpelyispFrnov=e1t−hapt⊥SnalalnpdhiontsoenrtsiningrtehfleecmtiovditeisesS(18,).w..e,cSaNn
→
classical-like cloners. have the same fidelity equal to (2). The distant Bob’s
clone in the mode S has a fidelity FS′ = 1 p and
′ n+1 − ⊥S′
using (8) we can subsequently prove that the clone has
fidelity equal to (2). Apart from N +1 clones the setup
III. OPTIMAL N →N+1 CLONING AT A
DISTANCE produces also single anti-clone in the mode I. Using (9)
forfidelitybetweentheanti-cloneandstate Ψ ,wecan
simply calculate that the final anti-clone ha|s t⊥hie fidelity
The setup for symmetric teleportation from N iden-
tical replicas of input state Ψ S on single distant N +1
copy is depicted in Fig. 3. | Iti is an extension of FI = . (11)
N +2
the previous setup by additional unbalanced beam
splitters BS2 BSN placed in mode I which have As the number of replicas increases we obtain a better
−
specific reflectivities R2,...,RN. Thus we imple- and still optimal approximation of the U-NOT gate for
ment the following sequence of projective measurements N replicas. For demonstration of local U-NOT with N
Π−SN,I(RN)...Π−S2,I(R2)Π−S1,I(R1)⊗1S′ onastateofto- replicas we need no source of entanglement and only N
talsystem Ψ SN... Ψ S2 Ψ S1 Ψ IS′ andoptimizethe unbalanced beam splitters having reflectivity according
reflectivitie|s Ri in su|chia|waiy t|o a−cihieve symmetric dis- to Eq. (8) with single port in completely random polar-
n
tribution of state Ψ in N +1 copies. To obtain it we ization state is required.
| i
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IV. EXPERIMENTAL IMPLEMENTATIONS into account only detection events when both detectors
register only single photon in different time-bins we can
An experimentalrealizationof partialteleportationas describe an action of the variable coupler on basis states
optimalasymmetric1 2cloningofapolarizationstate 1,0 and 0,1 by the same relations as in Eqs. (3). A
of a photon depicted in→Fig. 1 is straightforwardmodifi- |calcuilation|of piartial teleportation of time-bin qubit can
cation of a well-knownprevious experiment demonstrat- be done with the help of previous analysis. Let us con-
ing the total teleportation of polarization state [2]. We sider thought unitary operation US′ which converts the
only have to be able to control the reflectivity of the state (12) to the state √12(|1,0iI|0,1iS′ −|0,1iI|1,0iS′).
beam splitter in the Bell-state measurement. Using sin- This operation consists of mutual flip of basis states
glepairofphotonsinstate Ψ andmoreinputphotons 1,0 S′ 0,1 S′ and phase shift 1,0 S′ 1,0 S′,
prepared as the identical re|pl−icias which can be directly |0,1iS′ ↔ |0,1iS′. Then we obtain| anialog→ical−t|elepior-
| i → | i
extracted from pump beam by strong attenuation Ψ tation scheme with shared Ψ− -like state as has been
| i | i
we can implement teleportation as N N +1 optimal discussedabove. Aftersuccessfulteleportationweimple-
→
cloning. Inthisway,wealsoexperimentallydemonstrate ment second thought unitary operation US†′ on time-bin
the usefulness of the multiple copies to locally realize U- qubit S′ and due to US†′US′ = 1 we obtain in fact the
NOT gate with a higher fidelity. same result as with the state (12) shared between Al-
These schemes can be also implemented in the exper- ice and Bob. Thus after this teleportation we have the
iments on long-distance teleportation of time-bin qubit same fidelities F (R) and F (R) with input state Ψ ,
S I S
[3]. A time-bin qubit is a quantum superposition of a howevertheBobtime-binqubitisinthestatehavin|gtihe
wphhoetroenbiansiasdstiaffteere1n,t0tiSmceo-brrinessp|ΨonidSs=toαfi|r1s,t0tiiSm+e-βb|i0n,1ainSd, FTSh′e(rRef)owreitBhotbrahnassfotrompeedrfsotramteu|Ψni′tia=ryαp|h0y,1siicSa′l−oβp|e1r,a0tiiSo′n.
| i
|0,1iS to the second one. To teleport time-bin qubit Al- US′ on time-bin qubit S′ to obtain the demanded state
ice and Bob use shared time-bin entangled state having fidelity FS′(R) with state Ψ S.
| i
1 Inthispaperweproposetwoextendedconditionaltele-
|Φ+i= √2(|1,0iI|1,0iS′ +|0,1iI|0,1iS′), (12) portationschemes as asymmetric 1 2 and N N+1
→ →
cloning at a distance which can be straightforwardlyim-
which can be produced from type I nonlinear down- plemented in the recent quantum teleportation experi-
conversion,wherethepumppulseissplittedtotwosepa- ments. Further, we discuss an experiment on the condi-
rate ones by unbalanced Michelson interferometer. If we tionalLOCCreversibilityofthepartialteleportationand
restrict only to cases when two photons are emitted ei- aconditionalrealizationofoptimalU-NOToperationon
therbyfirstpumpingpulseorsecondonewehaveexactly the multiple copies.
state (12). The Bell-state measurement was performed
by mixing of two time-bin qubits in balanced fiber cou-
pler followed by two single photon detectors and if both Acknowledgments The work was supported by the
the detectors register photons in different time-bins the project LN00A015, Post-doc grant 202/03/D239 of
teleportation (up to an unitary operation on Bob side) Czech grant agency and CEZ: J14/98 of the Ministry
has been successfully performed [3]. of Education of Czech Republic. I would like to thank
Toimplementourideaofpartialteleportationtotime- to Markus Aspelmeyer, Cˇaslav Brukner, Nicolas Cerf,
binqubitweneedonlyanopticalfibercouplerwithvari- Jarom´irFiura´ˇsek,PetrMarekandL.MiˇstaJr.,forstim-
able coupling for the Bell-state projection. If we take ulating and fruitful discussions.
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