Experimental asymmetric phase-covariant quantum cloning of polarization qubits
Jan Soubusta, Lucie Bartuskova, Antonin Cernoch, Miloslav Dusek, Jaromir Fiurasek
aa r X i v : . [ qu a n t - ph ] O c t Experimental asymmetric phase-covariant quantum cloning of polarization qubits
Jan Soubusta, Lucie Bart˚uˇskov´a, Anton´ın ˇCernoch, Miloslav Duˇsek, and Jarom´ır Fiur´aˇsek Joint Laboratory of Optics of Palack´y University and Institute of Physics of Academy of Sciences of the Czech Republic,17. listopadu 50A, 779 07 Olomouc, Czech Republic Department of Optics, Faculty of Science, Palack´y University,17. listopadu 50, 772 00 Olomouc, Czech Republic (Dated: October 29, 2018)We report on two optical realizations of the 1 → PACS numbers: 03.67.-a, 03.67.Hk, 42.50.-p
I. INTRODUCTION
Optimal copying of quantum states is an importantprimitive in quantum information processing [1, 2]. Sinceexact copying of unknown quantum states is forbiddendue to the linearity of quantum mechanics [3] this taskcan be accomplished only approximately. A figure ofmerit commonly employed to quantify the performanceof quantum cloners is the fidelity of the clones [1, 2].Optimal cloning machines that maximize the cloning fi-delity have been identified theoretically for a wide rangeof classes of input states and numbers of copies [1, 2].The universal quantum cloners copy all states from theunderlying Hilbert space with the same fidelity [4, 5, 6, 7].Sometimes, however it is more beneficial to clone opti-mally only a certain subset of states. A particularly im-portant example is the phase-covariant quantum cloner[8, 9, 10, 11] that optimally copies all qubits from theequator of the Bloch sphere, i.e. all balanced superposi-tions of the computational basis states. The advantage ofsuch dedicated cloning machine is that it reaches highercloning fidelities than the universal machine.Phase-covariant cloning represents an optimal individ-ual eavesdropping attack on BB84 quantum key distri-bution protocol [12, 13]. In this context, the asymmetriccloning machines that produce two copies with differentfidelities [13, 14, 15] are particularly important. Tuningthe asymmetry of the cloning operation enables to con-trol the trade-off between information on a secret crypto-graphic key gained by the eavesdropper and the amountof noise added to the copy which is sent down the channelto the authorized receiver.For potential applications in quantum communica-tion, such as eavesdropping on quantum key distribu-tion, cloning of the quantum states of single photons isof great interest [1, 2]. Universal cloning of polariza-tion states of single photons has been implemented ex-perimentally using either stimulated parametric down-conversion [16, 17] or bunching of photons on a balanced beam splitter [18, 19, 20]. Asymmetric universal cloning[21] and symmetric 1 → → II. OPTIMAL PHASE-COVARIANT CLONING
We are interested in copying of a polarization stateof a single photon | ψ i . This single-qubit state can beconveniently parametrized by two Euler angles θ and φ , | ψ i = cos θ | V i + e iφ sin θ | H i . (1)Here the two orthogonal computational basis states | V i and | H i represent the vertical and horizontal linear polar-ization states, respectively. In this paper we focus on thecloning of the polarization states situated on the equatorof the Bloch sphere ( θ = π ), | ψ i = 1 √ (cid:0) | V i + e iφ | H i (cid:1) . (2)The optimal asymmetric phase-covariant cloning trans-formation reads [25], | V i| V i → | V i| V i , | H i| V i → √ q | V i| H i + √ − q | H i| V i , (3)where q ∈ [0 ,
1] is the asymmetry parameter. Note thatthis unitary transformation requires only two qubits, thesignal whose state we want to clone and an ancilla qubit(a blank copy) prepared in a fixed state | V i . The secondline of Eq. (3) means creation of a superposition of theinput state with a state where the two photons have beenexchanged. Such states are naturally produced by a beamsplitter with splitting ratio depending on the asymmetryparameter q .The quality of the clones is quantified by their fidelity,which is defined as the overlap of each clone state withthe original state (2). The fidelities of the two clones pro-duced by the optimal asymmetric phase-covariant cloningtransformation (3) read, F = 12 (cid:16) p − q (cid:17) , F = 12 (1 + √ q ) . (4) In the case of symmetric cloning ( q = 1 /
2) both fidelitieshave the same value F sym , pc ≈ . F u1 = 1 − (1 − p ) − p + p ) , F u2 = 1 − p − p + p ) , (5)where parameter p ∈ [0 ,
1] controls the asymmetry of thetwo clones. An universal cloner copies all states (not onlythe equatorial ones) with the same fidelities F u1 and F u2 .The fidelity of the symmetric universal cloner ( p = 1 / F sym , univ ≈ .
3% [4, 5].
III. FREE SPACE REALIZATION WITH ASPECIAL BEAM SPLITTER PCPC SBS D D PBSGP D D PBS/2/4/2/4 /2/4/2/4 GP s i g n a l a n c ill a ++ FIG. 1: (Color online) Scheme of the cloning setup based onthe special beam splitter and polarization dependent losses.PC - polarization controller, SBS - special beam splitter, GP η ,GP ν - polarization dependent losses, PBS - polarizing cubebeam splitter, λ/ , λ/ The first setup for the optimal asymmetric phase-covariant cloning of polarization states of single photonsis shown in Fig. 1. This setup is based on an interferenceof two photons on a special unbalanced beam splitter(SBS) with splitting ratios different for vertical and hori-zontal polarizations. The interference on SBS is followedby polarization filtration performed on each output portof the beam splitter. The polarization filters are realizedby tilted glass plates (GP), where the tilt angle deter-mines the ratio of transmittances for the horizontal andvertical polarizations. As we shall show below, by tiltingthe plates we are able to control the asymmetry of thecloner. The device operates in the coincidence basis andsuccessful cloning is heralded by the presence of a singlephoton in each output port of the cloning machine. Inpractice, we postselect only the cases when we observecoincidence between photon detections in the upper andlower output arms. All other events are discarded.Let us describe the experimental setup in more details.A non-linear crystal of LiIO is pumped by cw Kr + laserat 413 nm to produce pairs of photons in the type I pro-cess of spontaneous parametric down conversion. Pho-tons comprising each pair exhibit tight time correlationsand are horizontally polarized. The photons are coupledinto single mode fibers that serve as spatial filters. Thepolarization controllers (PC) are adjusted such as to en-sure horizontal linear polarization of the two photons atthe outputs of the fibers. The polarization state of eachphoton is set by means of half- and quarter-wave plates( λ/ , λ/ | V i , c.f. Eq. (3). Both photons enter the special beamsplitter which forms the Hong-Ou-Mandel interferometer[26]. The pairs of tilted glass plates introduce differentamplitude transmittances for horizontal and vertical po-larizations, ( η V , η H for GP η ; and ν V , ν H for GP ν ). It isconvenient to define the intensity transmittance ratios,Σ η = (cid:18) η V η H (cid:19) , Σ ν = (cid:18) ν V ν H (cid:19) . (6)GP η dominantly attenuates vertical polarization, henceΣ η ≤
1, while GP ν imposes higher losses for horizontalpolarization, and Σ ν ≥
1. We use two glass plates ineach arm to reach higher transmittance ratios for the twopolarizations. Moreover, since the two plates are tiltedin opposite directions, the beams are not transversallydisplaced by the filtration.The transformation introduced by the setup shown inFig. 1 can be written in the form | V i sig | V i anc → η V ν V ( r V − t V ) | V i | V i , | H i sig | V i anc → η H ν V r V r H | V i | H i − η V ν H t V t H | H i | V i , (7)where r V , r H ; t V , t H are the (fixed) real amplitude re-flectances and transmittances of the SBS. We use no-tation R j = r j and T j = t j , j = H, V , for intensityreflectances and transmittances and we have R j + T j = 1for a lossless beam splitter.Mapping (7) becomes equivalent to the unitary cloningtransformation (3) up to an overall prefactor representingthe amplitude of the probability of success of the cloningif the following two conditions are satisfied simultane-ously, η H ν V r V r H = √ q η V ν V ( r V − t V ) , − η V ν H t V t H = p − q η V ν V ( r V − t V ) . (8)After some algebra we obtain the transmittance ratiosof the polarization filters expressed as functions of theasymmetry parameter q ,Σ η = R V R H (2 R V − q , Σ ν = (1 − R V )(1 − R H )(2 R V − − q . (9) q T r a n s m itt a n ce r a ti o -1 FIG. 2: (Color online) Transmittance ratios Σ η and Σ − ν for the asymmetric cloner with the special unbalanced beamsplitter. Plotted dependences were calculated according toEq. (9) using experimentally determined parameters of SBS: R V = 75 .
8% and R H = 17 . We have chosen the splitting ratios of the SBS such thatsymmetric cloning could be realized without any furtherpolarization filtration. If we set Σ η = Σ ν = 1 and q = 1 / R V = (1 + √ ) ≈ . R H = 1 − R V [23]. The experimentally determinedparameters of the custom-made SBS manufactured byEkspla read R V = 75 .
8% and R H = 17 .
9% which is closeto the desired values. Figure 2 shows the theoretical de-pendence of the transmittance ratios on q calculated ac-cording to Eq. (9) using the experimentally determinedvalues of R H and R V . In particular, note that the sym-metric operation would be achieved for Σ − ν = 0 .
67 andΣ η = 1 .
02. The probability of success of the cloning isgiven by P SBSsucc = η V ν V ( r V − t V ) . It can be shown that P succ is highest for the symmetric cloner and decreaseswith increasing asymmetry, because the losses introducedby polarization filters are increasing. Comparison of themeasured and theoretically attainable P succ is given atthe end of Sec. IV for both setups.As we already mentioned, the cloning procedure is suc-cessful only if there is one photon in each output arm ofthe device. The performance of the cloning machine isprobed by polarization analysis of the two clones. Thesetting of wave plates at the output is inverse with respectto the signal photon preparation. This means that thephotons with the same polarization as the signal pho-ton are transmitted through the PBS to the detectorD + whereas the photons with orthogonal polarizationare reflected to the detector D − . This allows us to inferthe cloning fidelities from the four measured coincidencerates C ±± between detectors at the two output arms. Forinstance, coincidence rate C ++ represents the number ofsimultaneous clicks of detectors D +1 and D +2 per secondand the other coincidence rates are determined similarly.The fidelities of the clones are calculated as the ratio ofcoincidences corresponding to the projection of the first
50 60 70 80 90 F (%) F ( % ) FIG. 3: (Color online) Fidelities F vs F of clones measuredwith the setup based on the special beam splitter and polar-ization dependent losses. The full line denotes the theoreticallimit for the fidelities of the phase-covariant cloner, the dot-ted line shows the limit of the universal cloner. The dashedline shows the symmetric line ( F = F ). q Σ − ν Σ η F [%] F [%] P SBSsucc [%]0.93 0.10 0.55 64 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
2- - - 84 . ± . . ± . . ± . P SBSsucc for the specified asymmetry parameter q for the setup based on the SBS. The last row represents themeasurement without glass plates. (second) clone onto the input state and the sum of allcoincidences C sum = C ++ + C + − + C − + + C −− , F = C ++ + C + − C sum , F = C ++ + C − + C sum . (10)The probability of success of the device is determined asa fraction of the sum of all measured coincidences to thetotal number of the photon pairs entering the cloner C tot , P succ = C sum /C tot .We selected seven representative values of asymmetryparameter q and set the angles of the GP η and GP ν ac-cordingly. Then we measured clone fidelities for a set ofnine states √ ( | V i + e ikπ/ | H i ), k = − , . . . ,
4, located onthe equator of the Bloch sphere. These states span overcircular and diagonal linear polarization states. Result-ing mean fidelities averaged over the nine states are plot-ted in Fig. 3 and listed in Tab. I. The statistical errorswere calculated from 10 ten-second measurements. Theyreach values ∼
1% in the symmetric case. For higherdegrees of asymmetry, the polarization filtration resulted in higher losses, leading to decrease of the success prob-ability and increase of statistical errors of fidelities up to ∼ F is greater thanthe error of F . This is due to more pronounced oscilla-tions of F when scanning over the equatorial states. Thiseffect is caused by residual uncompensated phase shiftsinduced by the special beam splitter. The reflected andtransmitted photons acquire different phase shifts. Fig-ure 3 shows the comparison of our measurements withthe theoretical limits of the universal asymmetric cloner(5) and of the phase-covariant asymmetric cloner (4). Allmeasured points are very close to the universal asymmet-ric cloning limit but do not reach the theoretical phase-covariant cloning limit on fidelity. The main effect thatreduces the fidelity of the two clones and prevents us tosurpass the universal cloning limit is the non-ideal over-lap of the spatial modes of the two photons on the SBS. IV. HYBRID FREE-SPACE AND FIBER SETUP PC PC PC FC/2 /4/2 /4 BS D D PBS GP GP D D PBS /2/4/2/4 s i g n a l a n c ill a ++ FIG. 4: (Color online) Scheme of the hybrid cloning setup. FC- fiber coupler, BS - nonpolarizing beam splitter, PBS - po-larizing beam splitter, PC - polarization controller, λ/ , λ/
4- wave plates, GP η , GP ν - polarization dependent losses, D -detector. In order to increase the cloning fidelity we have builtan alternative setup which combines advantages of bothfiber and free space propagation, see Fig. 4. Fiber cou-pler (FC) ensures perfect overlap of spatial modes of sig-nal and ancilla photons. The free space part allows touse simple encoding of information into the polarizationstates of the photons. We can use wave plates ( λ/ , λ/ η and GP ν ensure implementation of phase-covariantcloning transformation, compensate for non-ideal split-ting ratio of the BS and allow to tune the asymmetry T r a n s m itt a n ce r a ti o q FIG. 5: (Color online) Transmittance ratios Σ η and Σ ν forhybrid asymmetric cloning setup. Plotted dependences werecalculated according to Eq. (13) using experimentally deter-mined parameters of BS: R V = 50 .
9% and R H = 46 . of the cloner. The device again operates in the coinci-dence basis and the cloning is successfully accomplishedif a single photon is detected in each output port of thecloner. The transformation realized by the whole devicecan be written as, | V i sig | V i anc → rtη V t V r V ν V | V i | V i , | H i sig | V i anc → rtη V η H ( t V r H ν V | H i | V i + t H r V ν H | V i | H i ) , (11)where coefficients r and t represent reflectance and trans-mittance of the FC; r j and t j , j = V, H , denote re-flectance and transmittance of the BS, which are slightlypolarization dependent ( R V = 50 .
9% and R H = 46 . η V η H t V r H ν V = p − q η V t V r V ν V ,η V η H t H r V ν H = √ q η V t V r V ν V . (12)After some algebra we arrive at the dependence of thetransmittance ratios of the polarization filters on thesetup parameters and the asymmetry parameter q ,Σ η = R H R V − q ) , Σ ν = R V (1 − R H ) R H (1 − R V ) 1 − qq . (13)Dependences of Σ η and Σ ν on q calculated for the pa-rameters of our setup are plotted in Fig 5.The measurement routine starts with an adjustment ofthe HOM interference dip in the fiber coupler FC. In thispreliminary stage two outputs of the FC are connecteddirectly to the detectors and the overlap of the two pho-tons is maximized finding a minimum of the coincidencecounts. Optimal overlap of the polarization states onthe FC is achieved by adjusting polarization controllersPC and PC . Then one output of the FC is directed - - /2 0 /2 [rad] F i d e lit y F F F F |-45 o | R |45 o | L |-45 o } q = 0.60} q = 0.70 FIG. 6: (Color online) Fidelities F and F of cloning of equa-torial qubits with the hybrid cloning setup for two asymmetryparameters, q = 0 .
60 and q = 0 .
70. The top axis shows thesignal qubit state corresponding to the phase φ (see Eq. (2)). to the free space part of the setup. The last polarizationcontroller PC is used to compensate polarization trans-formation induced in the fibers. The BS splits the photonpair into two paths with probability . GP η and GP ν aretilted to provide demanded polarization state filtration.Input polarization states and measurement bases are setby half- and quarter-wave plates.As in the previous section we performed cloning of ninepolarization states of a signal qubit distributed over theequator of the Bloch sphere. Figure 6 shows two typicalexamples of the experimentally measured fidelities for theequatorial qubits ( q = 0 .
60 not oscillating, q = 0 .
70 themost oscillating one). Statistical error bars were deter-mined from 10 twenty second measurement periods. Dueto the fact, that the oscillations have the sinusoidal char-acter and for both fidelities the sinusoids have the samephase, we suppose that we did not set the ancilla pho-ton polarization exactly on the pole of the Bloch sphere.Higher oscillations lead to greater errors of the mean fi-delities.Note that any cloner can be converted by a twirling op-eration [2] to a truly phase-covariant cloner whose cloningfidelity does not depend on the input state and is equal tothe mean fidelity of the original cloner. The twirling con-sists in application of the random phase shift operation U ( ϑ ) = | V ih V | + e iϑ | H ih H | to the input state and theinverse operation U ( − ϑ ) to each of the clones. The phaseshift ϑ is selected randomly from the interval [0 , π ]. Inthe present implementation, the twirling could be per-formed by using additional wave-plates.The relevant parameters of the cloner are thus themean cloning fidelities which fully quantify its perfor-mance. The mean fidelities are shown in Fig. 7 andare also listed in Table II. As can be seen the resultingmean fidelities are above the universal cloning limit forall asymmetries. Note that due to technical limitations
50 60 70 80 90 F (%) F ( % ) FIG. 7: (Color online) Fidelities F vs F of clones measuredwith the hybrid setup. The full line denotes the theoreticallimit for the fidelities of the phase-covariant cloner, the dottedline shows the limit of the universal cloner. The dashed lineshows the symmetric line ( F = F ). q Σ η Σ ν F [%] F [%] P Hybsucc [%]0.75 1.00 0.39 74.2 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± P Hybsucc for the specified asymmetry parameter q ofthe hybrid setup. on achievable Σ η and Σ ν we can reach only moderateasymmetries q ∈ [0 . , . P Hybsucc =(2 rtη V t V r V ν V ) . For ideal symmetric cloner we obtain P Hybsucc = which should be compared with P SBSsucc = achieved by the setup discussed in Sec. III. The hy-brid setup exhibits lower probability of success mainlybecause there are two post-selection steps. First, the sig-nal and ancilla photon must leave the FC together by theselected output fiber (the upper one in Fig. 4). Second,there must be one photon in each output arm of the bulkBS. For completeness, we plot the measured probabilitiesof success of both cloning setups in Fig. 8. V. CONCLUSIONS
In this paper we described two experimental setupsproposed to realize optimal asymmetric phase-covariantcloning of single-photon polarization qubits. We char-acterized the real experimental operation of both setupsand compared their performances and limitations. Thecloning is based on interference of the signal photon with q P s u cc SBSHybrid
FIG. 8: (Color online) Probability of success: the setup basedon the special beam splitter (circles), hybrid setup (triangles).The errors are smaller than the symbols shown. The linesrepresent theoretical dependences calculated from measuredtransmittances of the tilted glass plates. the ancilla photon on a beam splitter or fiber coupler fol-lowed by polarization filtration on the outputs. The im-plemented cloning machines operate in the coincidencebasis and a successful operation of the device is heraldedby detection of a single photon in each output arm. Animportant feature of both experimental setups is thatthe polarization filtering allows to tune the asymmetryof the cloning operation. Moreover, the same polariza-tion filtering is used to compensate imperfections of beamsplitters whose splitting ratios slightly differed from thedesired ones.The first setup relies on a special unbalanced beamsplitter with different transmittances for vertical and hor-izontal polarizations. The main advantage of this setupis that we can tune the asymmetry of cloning in a broadrange. However, the imperfect overlap of the spatialmodes of the photons on the bulk beam splitter lim-its the achievable fidelity of the clones and prevents usfrom surpassing the limit of optimal universal asymmet-ric cloning with this approach. The second setup is basedon the fiber coupler ensuring practically perfect overlapof spatial modes. With this second approach we wereable to achieve very high mean cloning fidelities exceed-ing the maximum fidelities obtainable by universal clon-ers. To the best of our knowledge, this is the first exper-iment where universal cloning limit has been surpassedfor asymmetric cloning of equatorial polarization statesof single photons. The price to pay for the fidelity im-provement is a smaller probability of success of this latterscheme and also somewhat narrower accessible asymme-try range.
Acknowledgments
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