Conditional implementation of asymmetrical universal quantum cloning machine

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by  Radim Filip

Conditional implementation of asymmetrical universal quantum cloning machine
Radim Filip
Department of Optics, Palack´y University,
17. listopadu 50, 772 07 Olomouc,
Czech Republic
(Dated: February 1, 2008)
Weproposetwofeasibleexperimentalimplementationsofanoptimalasymmetric1→2quantum
cloningofapolarization stateofphoton. Bothimplementations arebased onapartial andoptimal
reverse of recent conditional symmetrical quantum cloning experiments. The reversion procedure
is performed only by a local measurement of one from the clones and ancilla followed by a local
operation on the other clone. The local measurement consists only of a single unbalanced beam
splitter followed in one output by a single photon detector and the asymmetry of fidelities in the
cloning is controlled bya reflectivity of thebeam splitter.
4
0 PACSnumbers: 03.67.-a
0
2
Copying the quantum states is apparently dissimilar To build quantum cloners, quantum networks using
n
a to classical information processing since it is impossible CNOT gates for both optimal SUQCM and AUQCM
J to precisely duplicate an unknown quantum state as a were proposed [10]. However, a strength of the state-

8 consequence of a linearity of quantum mechanics [1]. To of-the-art nonlinear interaction at a single photon level
cloneanunknownquantumstateatleastapproximately, is unfortunately too weak to produce a deterministic

1 universal quantum cloning machines (UQCM) were de- and efficient CNOT operation only by a direct interac-
v
veloped [2]. The UQCM is a device that universally and tion between photons. For this reason,the deterministic
6
3 optimally produces a copy ρS′ from an unknown quan- SUQCMandAUQCMstillhavenotbeenexperimentally
tum state Ψ of the original. Specifically, an optimal implemented in quantum optics. Netherless, stimulated
0 S
| i
1 symmetrical 1 2 UQCM (SUQCM) for qubits cre- or spontaneous parametric down-conversions were used
→
0 atesacopyρS′ withamaximalstate-independentfidelity to realize a conditional implementation of the SUQCM
4 FS′ =S ΨρS′ Ψ S = 5/6. Simultaneously, a pure state for a polarization state of photon [11, 12, 13]. However,
0 of originhal|cha|ngies to mixed state ρ exhibiting maxi- to the best of our knowledge, no feasible experimental
S
/
h mally the same fidelity FS =S ΨρS Ψ S = 5/6 as the setup for an optimal 1 2 AUQCM has been presented
h | | i →
p clone. To optimally control the fidelities FS and FS′, yet. On the other hand, an experimental realization of
- a concept of asymmetrical UQCM (AUQCM) has been optimal asymmetrical cloning machine has already been
t
n theoretically developed [4, 5, 6]. The optimal 1 2 proposed for coherent states [15].
a AUQCM produces the copies having state-indepen→dent
u In this paper, we propose optimal 1 2 AUQCM
fidelities controlledby a parameterR in sucha waythat →
q which is a simple and experimentally feasible extension
: foragivenfidelityofthe copyFS′(R),the fidelity FS(R) of the recent experiments on the conditional symmetri-
v of the originalis maximal. More specifically, assuming a
cal cloning of the polarization state of a photon. Our
i
X qubit in an unknown state Ψ then the original S and method is basedon a partial optimal reverseof SUQCM
| i
clone S leaving 1 2 AUQCM can be represented by
r ′ → byaspecificcontrollablejointmeasurementononeofthe
a the following density matrices [4, 5, 6]
copies and an auxiliary photon leaving the cloning pro-
cess. By this partial reverse the quantum information
ρS,S′ =FS,S′ Ψ Ψ +(1 FS,S′)Ψ Ψ (1)
| ih | − | ⊥ih ⊥| between the disturbed original and copy can be redis-
where the fidelities 5/6 FS 1 and 1/2 FS′ 5/6 tributed posteriori only using the local operations and
≤ ≤ ≤ ≤
satisfy the cloning relation classical communication. It can be experimentally ac-
complishedaddingonlyasingleunbalancedbeamsplitter
(1 FS)(1 FS′) (1/2 (1 FS) (1 FS′))2, (2) followedbyasingle-photondetectorintherecentcloning
− − ≥ − − − −
experiments [11, 12, 13].
where the equality corresponds to an optimal AUQCM,
inthesensethatforaalargerF cannotbe obtainedfor An experimental realization of the conditional 1 2
S
→
given FS′. The Eq. (2) is the tightest no-cloning bound SUQCM [11, 12] was based on a nonlinear parametric
for the fidelities of the 1 2 cloner which copies an un- down-conversion stimulated in a signal beam by a sin-
→
knownqubitstatetotheanotherwithanisotropicnoise. gle photon prepared in an unknown polarization state
Therecentexperimentalefforttobuilddifferentquantum Ψ = aV +bH , where V and H denote the ver-
S S S
| i | i | i
clonersismainlystimulatedbytheiruseasindividualat- tical and horizontal polarizations. The experimental ar-
tacks in quantum communication and cryptography [8]. rangementisdepictedinFig.1. Theinput singlephoton
More information about this practical application of the extractedfromalaserpulseispreparedinthe state Ψ
S
| i
asymmetrical universal cloning as optimal attack for a inapreparationdeviceusingλ/2andλ/4waveplates. A
cryptographic protocol can be found in Ref. [9]. moreintensivepartofthelaserpulseisfrequencydoubled
2
State
preparation
λ/2λ/4 S’
I Asymmetrizer S’
Asymmetrizer
I
R BBO
L BS1 2ω PtyBpBeO II BS2 S’S BS3 D1 L BS4 2ω Ptype II D1 BS3 R
BS1 D2
D3 S’
clone λ/2λ/4 S’ BS2 S
anti−clone PBS D4 λ/2 λ/4
State analyzer State
preparation clone anti−clone
FIG. 1: Setup for conditional AUQCM based on stimulated
FIG. 2: Setup of conditional AUQCM based on spontaneous
parametric down-conversion: L – laser, BS1-BS3 beam split-
parametric down-conversion: L – laser, BS1-BS3 beam split-
ters, PBS – polarization beam splitter, 2ω – frequency dou-
ters, PBS – polarization beam splitter, 2ω – frequency dou-
bler, BBB – nonlinear BBO type II crystal, λ/2,λ/4 – wave
bler, BBB – nonlinear BBO type II crystal, λ/2,λ/4 – wave
plates, D1– single photon detector.
plates, D1-D2 – single-photon detectors.
and used to pump a BBO non-linear crystal. An action
1
of a non-degenerate type II parametric down-conversion 0,1 S 0,1 S′, (4)
√2| i | i
process in the crystal can be described by the Hamilto-
nian HI = i¯hχ(a†Hb†V −a†Vb†H)+ h.c., where χ is pro- where S and S′ are the signal modes. In the next pro-
portionalto a nonlinearsusceptibility ofthe crystal,and
cedure only such cases when a single photon is present
h.c. denotes the hermitian conjugation. Here the anni-
inthe mode S areconsidered. Returningto the previous
hilation operators a,b are assumed to be acting on the
notation 1,0 = H and 0,1 = V , the SUQCM
i i i i
selected signal mode S and idler mode I, respectively. | i | i | i | i
transformation
A short-time approximation of the evolution operator
Uma=tioenxpis(−coirHreIctt/¯hfo)r≈th1is−kiinh¯tdHoIfitshueseexdp.erTimhiesnatsp,psrionxcie- |ΨiS →r23|ΨΨΨ⊥iSS′I − √13|Ψ+iSS′|ΨIi, (5)
the gain g = χt is usually very small (g 1). Within
| | ≪
tHhe short1-t,i0me aapnpdroVximation0,,1polaorifzathtieoninbpaustispshtoattoens where |Ψ+i = √12(|ΨΨ⊥iSS′ +|Ψ⊥ΨiSS′) and |Ψ⊥i =
| iS ≡ | iS | iS ≡ | iS a∗ H b∗ V is the orthogonal state to Ψ , is actually
evolve according to the following rules | i− | i | i
performed. This SUQCM is optimal and transforms an
U 1,0 0,0 1,0 0,0 +g(√22,0 0,1 unknown state Ψ S of the original to the disturbed one
S I S I S I | i
| i | i ≈ | i | i | i | i and a copy, with the fidelities F =F =5/6. Both the
1,1 1,0 , S S′
−| iS| iI output states of photons S and S′ have to be measured
U 0,1 0,0 0,1 0,0 g(√20,2 1,0 using the state analyzer composed from the λ/2-wave
S I S I S I
| i | i ≈ | i | i − | i | i
1,1 0,1 . (3) plate and λ/4-waveplate, the polarizationbeam splitter
S I
−| i | i PBS and a pair of single photon detectors D3,D4, as
Here,theproducedstates 2,0 and 0,2 representanef- depicted in Fig. 1. Note, in the cloning experiment [12],
| i | i
fectofastimulatedemissionwhichisusedtopreparethe the fidelities were approximately FS,FS′ 0.81 which
≈
clone, and the state 1,1 corresponds to an unavoidable are really close to the optimal value of 5/6=0.833.
| i
effectofaspontaneousemission. Aftertheamplification, Recently, a different implementation of the 1 2
→
a balanced polarization-insensitive beam splitter BS2 in SUQCM was experimentally performed [13]. It is based
thesignalmodeseparatestwophotonsinthestates 2,0 on the following joint projection
| i
or 0,2 totwodistinguishablespatialmodescorrespond-
| i
ingtothedisturbedphotonS andcloneS′. Anactionof ΠS =(1SS′ Ψ SS′ Ψ ) 1I (6)
−| −i h −| ⊗
the beam splitter BS2 on a pair of photons is as follows
ofanunknownpolarizationstate Ψ S′ =aV +B H of
1 | i | i | i
1,1 S 0,0 S′ (1,1 S 0,0 S′ + 0,0 S 1,1 S′ + theinputphotonandtheantisymmetricpolarizationBell
| i | i → 2 | i | i | i | i state Ψ = 1 (VH HV ) of two photons pro-
|1,0iS|0,1iS′ +|0,1iS|1,0iS′), duced|by−itShIe sp√on2ta|neouis−p|aramietric down-conversion
1 from the same BBO nonlinear crystal as in the previous
2,0 S 0,0 S′ (2,0 S 0,0 S′ + 0,0 S 2,0 S′)+
| i | i → 2 | i | i | i | i experiment. Thecorrespondingexperimentalsetupisde-
1 1,0 S 1,0 S′, pictedinFig.2. Theinitialstate|ΨiS′ =a|ViS′+b|HiS′
√2| i | i is prepared by the same method. The projection ΠS to
1 the symmetric subspace [14] on the state Ψ S′ Ψ SI
0,2 S 0,0 S′ (0,2 S 0,0 S′ + 0,0 S 0,2 S′)+ canbe accomplishedby asequenceoftwobe|ami sp|li−ttiers
| i | i → 2 | i | i | i | i
3
BS1,BS2andasingle-photondetectorD1. Iftwoinput HV S′I T HV S′I RVH S′I,
| i → | i − | i
cpohnosttornusctinivethlyeisnatmerefesrteatoen|0th,1eifiSr|0st,1biaSl′aonrce|d1,b0eiSam|1,s0pilSit′)- |VHiS′I → T|VHiS′I −R|HViS′I, (8)
ter BS1 and no photon is detected by the detector D1, where T + R = 1. It can be simply proved that this
then a twin of photons in the state 0,2 S or 2,0 S is transformation can be expressed in a covariant way
| i | i
produced in the mode S with a success probability 1/2.
The twins are further divided on the second balanced ΨΨ S′I (T R)ΨΨ S′I,
| i → − | i
beam splitter BS2 to separate the photons to the differ- ΨΨ S′I T ΨΨ S′I RΨ Ψ S′I. (9)
entspatialmodesandifweselectedonlysuchcaseswhen | ⊥i → | ⊥i − | ⊥ i
exactlyansinglephotonisineverymodeS,S ,I thesym- Itisanasymmetricalprojectioncontrolledbytheparam-
′
metric states 0,1 S 0,1 S′ or 1,0 S 1,0 S′ with proba- eter R, in a contrastto the symmetrizing projection (7).
bility 1/4 are|obtaiin|ed ait a re|sult.i O| n tihe other hand, If an output state is selected only when there is an sin-
iftwoorthogonalbasisstates 0,1 S 1,0 S′, 1,0 S 0,1 S′ gle photonineachmode S,S′,I,we obtainthe following
are mixed at the beam splitte|r BiS1|, thiey d|o niot|mutiu- transformation for the polarization states of the photon
ally interfere and in addition, if no photon is registered
1
onthedetectorD1,thenthestate 1,1 S withthesuccess H ((2 R)HHV SS′I
probability 1/2 is within the mod|e S.i Thus after split- | i → N(R) − | i −
ting the photons by BS2 to separate spatial modes, a (p1+R)HVH SS′I (1 2R)VHH SS′I),
tsayimnemdewtriitchstthateeto√1t2a(l|s1u,c0cieSs|s0,p1rioSb′a+bi|l0it,y1i1S/|21.,0TihSu′)s,iswoitbh- |Vi → N1(R)|((2−iR)|VV−HiS−S′I −| i
the probability 1/4 the following transformation
(p1+R)VHV SS′I (1 2R)HVV SS′I)
| i − − | i
ΨΨ SS′ √2ΨΨ SS′, (10)
| i → | i
1
|ΨΨ⊥iSS′ → √2(|ΨΨ⊥iSS′ +|Ψ⊥iSS′), (7) wdehteercetiNon(Rof)t=he6p(1ho−toRn(1in−thRe))m.oIdneaIrceaanl ebxeppereirmfoernmt,eda
destructivelybyasingle-photondetectorD2whereasthe
of the states of the photons S,S is in fact conditionally
′ signal photons from total cloning operation are detected
implemented. Assuming that a state of the idler photon
in the state analyzers. It can be simply proved that the
I is selectedonly ifthis procedureis successful,the total
transformation(11) is covariantand it can be written in
p(5r)o.jeTcthiouns (t6h)etorapntismfoarlmSsUtQheCiMnpuistcsotantdeit|iΨoniSa′l|lΨy−aicScIomto- a form
plished however now a spontaneous emission of maxi- 1
Ψ ((2 R)ΨΨΨ SS′I
mally entangled pairs is used rather than a stimulated | i → N(R) − | ⊥i −
emission in the previous experiment. In the experiment (p1+R)ΨΨ Ψ SS′I (1 2R)Ψ ΨΨ SS′I)
basedonthis idea[13],the fidelities ofthe cloneanddis- | ⊥ i − − | ⊥ i
(11)
turbed original are FS,FS′ 0.826 which are even more
≈
closer to the theoretical value 5/6 = 0.833 than it has
which corresponds to the following projection
been in the previous case.
An extension of both setups to achieve the optimal ΠA(R)=((1 2R)1S′ 1I +2RΨ S′I Ψ ) 1S,
AUQCM can be presented. It is known that the sym- − ⊗ | −i h −| ⊗ (12)
metrical quantum cloning is LOCC reversible [16]. If a on the state (5) produced by the SUQCM. An interpre-
projective measurement Π = Ψ Ψ on one clone tationofthis projectivemeasurementis straightforward:
and ancilla is performed, th−e oth|er−cliohne−r|eturns back to theasymmetricalcloningisobtainedasapartialoptimal
the initial state Ψ . Thus we can guess that if an ap- reverse of the symmetrical one. For R = 0 the SUQCM
| i
propriate projection in a form α1+βΠ is applied on is obtained and for R = 1/2 the SQUCM is reversed
the one clone and the ancilla, an interm−ediate case cor- and an initial state of the original is precisely restored.
respondingtotheoptimalasymmetricalcloningmachine Anoptimaltotalreverseofthe state after the symmetri-
could be obtained. calcloningwaspreviouslytheoreticallyalreadyanalyzed
Nowweshowthatthis projectioncanbeconditionally [16]. After the total reversion, any input state is deter-
implemented if one mixes a pair of photons in the idler ministicallyrevealedbythecompleteBell-statemeasure-
I and signalS′ modes on unbalancedbeam splitter BS3 ment on the clone and ancilla. Thus this obtainedresult
havingavariablereflectivity0 R 1/2andselectonly also representsa solution of the problem of a partial but
≤ ≤
the cases when both photons leaving the beam splitter still optimal reversion of the symmetrical cloning. Fur-
are separated. Then the beam splitter can be simply ther, this reversion is also obtained only using the local
described by transformation operationsontheclonesandclassicalcommunicationbe-
tweenthem. Itenablestheredistributionofthequantum
VV S′I (T R)VV S′I, information encoded in symmetric clones at a distance
| i → − | i
HH S′I (T R)HH S′I, without an additional quantum channel.
| i → − | i
4
selected sub-ensemble have the following fidelities
F (2 R)2+(1+R)2
F = − ,
S
R 6(1 R(1 R))
0.1 0.2 0.3 0.4 0.5 − −
(2 R)2+(1 2R)2
0.9 FS′ = − − , (15)
6(1 R(1 R))
− −
0.8
with an initial state Ψ which vary with increasing
| i
R 0,1/2 from a perfect SUQCM (R=0) to the triv-
0.7
∈h i
ialnon-cloningcase(R=1/2),asdepictedinFig.3. The
0.6 output states of the original and clone can be measured
bythestateanalyzersanalogicallyasitwasdiscussedfor
0.5 the experiments with the SUQCMs. Inserting the fideli-
ties (15) to the cloning inequality (2) which restricts all
the possible AUQCM,the equalityis obtainedinEq.(2)
FIG. 3: Fidelity of original and clone after AUCQM in de-
as can be straightforwardlyproved.
pendenceon beamsplitter BS3reflectivity. Theuppercurve
In this paper an extension of the recent conditional
corresponds to FS and thelower curveto FS′
cloning experiment for a polarizationstate of photon to-
ward the optimal asymmetrical 1 2 quantum cloning
→
Since the photon in the idler mode is detected in such machine is proposed. Our method is based on a condi-
a way that no information about its polarization is ac- tional partial optimal reverse of the SUQCM controlled
quired, we traceoverthe idler mode andobtain the final by an experimental parameter R. We have applied this
output states of the modes S and S method in the recent symmetrical cloning experiments
′
[11, 12, 13] to obtain the optimal asymmetrical cloning.
ρSS′ = 1 (2 R)2 ΨΨ SS′ ΨΨ+ In summary, the AUQCM can be described as the pro-
N(R) − | i h | jection Π(R)=ΠA(R)ΠS, given explicitly by
(cid:2)
((1+R)ΨΨ SS′ +(1 2R)Ψ Ψ SS′)
((1+R)|hΨΨ⊥⊥i|SS′ +(1−−2R)|hΨ⊥⊥Ψi|SS′)]×. Π(R)=((2−R)1S′ ⊗1S −2(1−2R)|Ψ−iS′ShΨ−|)⊗(11I6,)
(13)
onthestate Ψ S′ Ψ SI composedfromaninitialun-
known state|anid t⊗he| a−ntiisymmetric Bell state produced
This state carries both the disturbed original and clone
fromthespontaneousparametricdown-conversion. Since
1 thefidelitiesobtainedintheseexperimentsareveryclose
ρ = (((2 R)2+(1+R)2)Ψ Ψ +
S to the optimal value 5/6 and the proposed modification
N(R) − | ih |
is rather simple, this asymmetrical cloning machine is
(1 2R)2 Ψ Ψ ),
− | ⊥ih ⊥| feasible and it could be straightforwardlyrealized.
1
ρS′ = (((2 R)2+(1 2R)2)Ψ Ψ +
N(R) − − | ih | Acknowledgments The work was supported by the
(1+R)2 Ψ Ψ ) (14) Post-Docproject202/03/D239ofGrantagencyofCzech
| ⊥ih ⊥| RepublicandprojectsLN00A015andCEZ:J14/98ofthe
and is conditioned by a simultaneous detection of a sin- Ministry of Education of the Czech Republic. I would
gle photon in each of the output modes S,S ,I. The like to thank Jarom´ir Fiura´ˇsek and Petr Marek for the
′
marginalstatesofthedisturbedoriginalandcloneinthe stimulating and fruitful discussions.
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