Nearly Deterministic Teleportation of a Photonic Qubit with Weak Cross-Kerr Nonlinearities

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CHIN. PHYS. LETT. Vol. 26,No. 10 (2009) 100301

Nearly Deterministic Teleportation of a Photonic Qubit with Weak Cross-KerrNonlinearities *

ZHOU Jian(周建)1,2, YANG Ming(杨名)1**, LU Yan(卢艳)1, CAO Zhuo-Liang(曹卓良)3,1

1Key Laboratory of Opto-electronic Information Acquisition and Manipulation (Ministry of Education), School ofPhysics and Material Science, Anhui University, Hefei 230039

2Anhui Xinhua University, Hefei 2300883Department of Physics and Electronic Engineering, Hefei Teachers College, Hefei 230061

(Received 23 March 2009)Scheme for teleporting an unknown single-qubit photonic state is revisited. With the help of quantum nonde-

molition measurements, the scheme now can be achieved in an almost deterministic way. The weak-cross-Kerr-nonlinearities-based quantum nondemolition measurement acts as an entangler as well as the Bell-state analyzer,and thus plays a similar role as the CNOT gate in the teleportation process. This improvement makes the presentscheme more efficient than the schemes using nonunitary projective measurements and feasible with the currentexperimental technology.

PACS: 03. 67. Hk, 03. 67. Mn, 03. 67. Pp, 42. 50. Dv

Entanglement plays an important role in quantuminformation and computation because of its intrin-sic nonlocality.[1] Quantum teleportation is a paradig-matic example using this nonlocality.[2] It is a processto transmit quantum states to a remote receiver via aquantum channel (i.e. entangled states) with the helpof classical communication. The original protocol isproposed by Bennett et al.[3] and realized in experi-ment later in a photonic system.[4] Generally speak-ing, a quantum teleportation protocol can be achievedbased on the following conditions: (1) the quantumchannels can be prepared, (2) an implementable mea-surement of the involved qubits, and (3) a physicallyfeasible way of distinguishing the outcomes of thecollapsed wave function. However, currently exist-ing teleportation schemes conventionally adopted thejoint Bell-state measurement (BSM), which is not easyto be realized experimentally. Usually, joint BSM isconverted into the product of separate measurementson single qubits using linear optical elements.[4,5] Thiswill inevitably result in the probabilistic nature of theteleportation process even though the quantum chan-nel is a maximally entangled state because of the usedcoincidence count when detecting the collapsed wavefunction.

Now, along with the development of quantumnondemolition (QND) technology,[6,7] the realizationof CNOT gate,[8] entangled states generation[9] andBSM[10] based on weak cross-Kerr nonlinearities wereachieved. The idea of using weak cross-Kerr non-linearities combined with strong coherent fields hasbeen developed by researchers and applied in variousways.[10−14] Recently, schemes for implementing gates,entanglement purification and concentration[15−19]

and continuous-variable quantum teleportation[20] viaQND were also proposed. In this Letter, we pro-pose an almost deterministic quantum teleportationfor photonic state with QND technology. Acting as anentangler as well as the analyzer of the Bell states, thecross-Kerr-nonlinearity-based QND makes one free ofthe CNOT gate and BSM in realizing a teleportationprotocol. In addition, the scheme now can be achievedin a nearly deterministic way, instead of the originalprobabilistic way. These distinct features make ourscheme more efficient and feasible.

Before presenting our protocol, we first considerthe process of QND measurement for the photon num-ber state using cross-Kerr nonlinearity. The cross-Kerr nonlinearities can be described by the Hamil-tonian

𝐻QND = ℎ𝜒��𝑠��𝑝, (1)

where ��𝑠(��𝑝) denotes the number operator for themode signal (probe), and 𝜒 is the coupling strengthof the nonlinearity, which is determined by the prop-erty of the used material. If the signal state is |𝜓⟩ =𝑐0|0⟩𝑠 + 𝑐1|1⟩𝑠 while the probe beam is initially in acoherent state of |𝛼⟩𝑝, then the cross-Kerr interactiondrives the combined signal-probe system to evolve as

𝑈 |𝜓⟩𝑠|𝛼⟩𝑝 = 𝑒𝑖𝐻QND𝑡/ℎ(𝑐0|0⟩𝑠 + 𝑐1|1⟩𝑠)|𝛼⟩𝑝= 𝑐0|0⟩𝑠|𝛼⟩𝑝 + 𝑐1|1⟩𝑠|𝛼𝑒𝑖𝜃⟩𝑝, (2)

where the Fock state |𝑛⟩𝑠 is unaffected by the inter-action while the coherent state |𝛼⟩𝑝 picks up a phaseshift 𝜃 = 𝜒𝑡. Generally, this phase factor is propor-tional to the number of photons 𝑛 in the |𝑛⟩𝑠 state,i.e., for 𝑛 photons in the signal mode, the probe beam

*Supported by the National Natural Science Foundation of China under Grant Nos 10704001 and 60678022, the SpecializedResearch Fund for the Doctoral Program of Higher Education under Grant No 20060357008, Anhui Provincial Natural ScienceFoundation under Grant No 070412060, the Key Program of the Education Department of Anhui Province under Grant NosKJ2008B265, 2008jq1183 and KJ2008A28ZC, and the Talent Foundation of Anhui University.

**To whom correspondence should be addressed. Email: mingyang@ahu.edu.cnc○ 2009 Chinese Physical Society and IOP Publishing Ltd

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CHIN. PHYS. LETT. Vol. 26,No. 10 (2009) 100301

evolves to |𝛼𝑒𝑖𝜙⟩𝑝 with 𝜙 = 𝑛𝜃. For large nonlinear-ity 𝜃 = 𝜋, this cross-Kerr nonlinearity can be useddirectly to implement a phase-CNOT gate. The largenonlinearity is technically difficult,[21,22] thus we hereonly utilize the weak nonlinearity of the Kerr medium.

Firstly, we introduce the generation of the quan-tum channel, i.e., the EPR state, from a product stateof two photonic qubits (here the polarization degreeof freedom of photon is involved):

|𝜓⟩𝑎 = 𝑎0|𝐻⟩𝑎 + 𝑎1|𝑉 ⟩𝑎, (3a)

|𝜓⟩𝑏 = 𝑏0|𝐻⟩𝑏 + 𝑏1|𝑉 ⟩𝑏. (3b)

PBSa

b b

a

Measurement

Classical feedforwardPBS

σx

+θ|α> −θ

Φ↼x↽

Fig. 1. Schematic diagram of QND that distinguishes su-perpositions and mixtures of the states |𝐻𝐻⟩ and |𝑉 𝑉 ⟩from |𝐻𝑉 ⟩ and |𝑉 𝐻⟩ using two cross-Kerr nonlinearitiesand a coherent laser probe beam |𝛼⟩. The measurementbox means homodyne detection or number resolving de-tection, and the section of the dashed line means the ad-justment depending on the measurement result.

These qubit states are split by polarizing beamsplitters (PBS) into spatial modes, then |𝐻⟩𝑎 and |𝐻⟩𝑏will interact with the cross-Kerr nonlinearity medium,as shown in Fig. 1. In this stage, the state of the com-bined system can be expressed as

|𝜓⟩total = [𝑎0𝑏0|𝐻𝐻⟩𝑎,𝑏 + 𝑎1𝑏1|𝑉 𝑉 ⟩𝑎,𝑏]|𝛼⟩𝑝+ [𝑎0𝑏1|𝐻𝑉 ⟩𝑎,𝑏|𝛼𝑒𝑖𝜃⟩𝑝+ 𝑎1𝑏0|𝑉 𝐻⟩𝑎,𝑏|𝛼𝑒−𝑖𝜃⟩𝑝]. (4)

We notice immediately that the even (|𝐻𝐻⟩ and|𝑉 𝑉 ⟩) and odd (|𝐻𝑉 ⟩ and |𝑉 𝐻⟩) parity terms canbe distinguished, without being destroyed, by a highefficiency 𝑋 quadrature homodyne measurement, i.e.,the even parity terms can be split almost determinis-tically from odd parity terms,

|𝜓⟩𝑇1 ∼ 𝑎0𝑏0|𝐻𝐻⟩𝑎,𝑏 + 𝑎1𝑏1|𝑉 𝑉 ⟩𝑎,𝑏, (5a)

|𝜓⟩𝑇2 ∼ 𝑎0𝑏1𝑒𝑖𝜃|𝐻𝑉 ⟩𝑎,𝑏 + 𝑎1𝑏0𝑒

−𝑖𝜃|𝑉 𝐻⟩𝑎,𝑏, (5b)

where ∼ means that there is a small probability oferror (less than 10−5 for 𝛼𝜃2 ≃ 9) to distinguish thestates in Eqs. (5a) and (5b) from each other.[8] In gen-eral, natural Kerr media have extremely small non-linearities with a typical dimensionless magnitude of𝜃 < 10−18,[21,22] thus a large Kerr nonlinearity atthe single-photon level is almost impossible. How-ever, one can make nonlinearities of magnitude up

to 10−2 by using optical fibers,[23] electromagneticallyinduced transparencies technique,[12,24,25] and cavityQED systems.[26,27] Therefore, highly accurate dis-crimination is possible with weak cross-Kerr nonlin-earities 𝜃 ≪ 1 provided that 𝛼 can be made sufficientlylarge.

Thus we can generate the EPR state Eq. (5a) (if𝑎0 = 𝑎1 = 𝑏0 = 𝑏1 = 1/

√2) via the setup in Fig. 1.

However, for Eq. (5b), the simple local rotations us-ing phase shifters depending on the 𝑋 measurementresult via a feed-forward process (the dashed line inFig. 1) can be performed to transform this state to𝑎0𝑏1|𝐻𝑉 ⟩𝑎,𝑏 + 𝑎1𝑏0|𝑉 𝐻⟩𝑎,𝑏. Furthermore, a simplebit flip on the last two polarization qubits can trans-form it to the form of Eq. (5a), i.e., it is possible tocreate two-qubit arbitrary entangled photonic statesalmost deterministically with the appropriate choiceof 𝑎0, 𝑎1, 𝑏0 and 𝑏1.

1

2

3

BobAlice

EPRsource

QND

Fig. 2. Schematic diagram of quantum teleportation. TheQND box is as the same as shown in Fig. 1.

With the quantum channel, we now turn atten-tion to the detailed quantum teleportation protocolfor photonic state. The unknown single-qubit statefor teleportation is

|𝜓⟩1 = 𝑎|𝐻⟩1 + 𝑏|𝑉 ⟩1. (6)

As shown in Fig. 2, photons 2 and 3, shared by Al-ice and Bob, were in the state of Eq. (5a) for 𝑎0 =𝑎1 = 𝑏0 = 𝑏1 = 1√

2. After the action of the cor-

responding PBS and cross-Kerr nonlinearities on thephotons 1 and 2 in QND box, the coherent state |𝛼⟩𝑝for the odd parity terms of qubits 1 and 2 will pick upa phase shift 𝜃 = 𝜒𝑡, while no phase shift is picked forthe even parity terms [see Eq. (4)]. Similar to the pro-cess of generation of the quantum channel, after theaction of Kerr medium, the result of the 𝑋 quadraturemeasurement of the probe beam and the feed-forwardcircuit, the total state of photons 1, 2 and 3 will reduceto

|𝜓⟩𝑇1 ∼ 𝑎|𝐻𝐻𝐻⟩+ 𝑏|𝑉 𝑉 𝑉 ⟩, (7a)|𝜓⟩𝑇2 ∼ 𝑎|𝐻𝑉 𝑉 ⟩+ 𝑏|𝑉 𝐻𝐻⟩. (7b)

For Eq. (7b), a simple bit flip on the second polar-ization qubit can transform it to the form of Eq. (7a).Then Alice measures the photons 1 and 2 in the bases

|+⟩ =1√2

(|𝐻⟩+ |𝑉 ⟩), (8a)

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|−⟩ =1√2

(|𝐻⟩ − |𝑉 ⟩). (8b)

For the measurement results |+ +⟩ and |−−⟩, thephoton 3 in the Bob’s side will be left in the state|𝜓⟩3 = 𝑎|𝐻⟩3 + 𝑏|𝑉 ⟩3. For the other results | + −⟩and | − +⟩, the photon 3 will be left in the state|𝜓⟩3 = 𝑎|𝐻⟩3 − 𝑏|𝑉 ⟩3, which is related to the origi-nal state up to a single-qubit rotation. The successfulprobability of the present scheme is determined by theprobability of the discrimination of states in Eqs. (7),which is almost a unit.

Now, let us take a brief discussion about ourscheme. In our entanglement generation and telepor-tation process, we just consider the case where we havethe ideal single photon resources. What will happenif we do not have access to the ideal single photonresources? In the real situation, if we do not havethe ideal resources, we can make use of the parame-ter down conversion (PDC) process. One of the twinphotons generated from the PDC process can be usedas the single photon in our current scheme, and theredundant photons can be thrown away by the cor-responding measurement. The strong probe coherentfield with a large amplitude in our scheme will makethe fidelity of the generated states decrease becauseof the photon loss during the nonlinear interaction.In the real situation, the photons will inevitably in-teract with the environment and the initially purephotonic states will evolve to the mixed states dur-ing the transmission process in optical fibre. Fortu-nately, the decoherence can be made arbitrarily smallsimply by an arbitrary strong coherent state associ-ated with a displacement 𝐷(−𝛼) performed on thecoherent state and the QND photon-number-resolvingdetection.[28,29] At the same time, the entanglementof the degraded entangled states after transmissioncan be enhanced by entanglement purification[17−19]

protocol, i.e. we can extract a small number of en-tangled pairs with a relatively high degree of entan-glement from a large number of less entangled pairsusing only local operations and classical communica-tion. Alternatively, we can also distribute the entan-glement resources in a fault-tolerant way.[30] Thus thecurrent weak-cross-Kerr-nonlinearities-based telepor-tation scheme is robust to decoherence.

In conclusion, we have proposed a nearly deter-ministic quantum teleportation protocol for photonicstates using weak cross-Kerr nonlinearities. With thehelp of QND techniques, the complicated CNOT op-erations are no longer required here. In particular, wehave realized the generation of entangled states andthe quantum teleportation only by linear optics andQND photon-number-resolving detectors. These fea-tures make our scheme more efficient and feasible.

We would like to thank Dr. Zheng-Yuan Xue forcareful reading of the initial version of the manuscript.

References

[1] Nielsen M A and Chuang I L 2000 Quantum Computationand Quantum Information(Cambridge: Cambridge Univer-sity)

[2] Furusawa A, Sørensen J L, Braunstein S L, Fuchs C A,Kimble H J and Polzik E S 1998 Science 282 706

[3] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A andWootters W K 1993 Phys. Rev. Lett. 70 1895

[4] Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter Hand Zeilinger A 1997 Nature 390 575Boschi D, Branca S, Martini F De, Hardy L and Popescu S1998 Phys. Rev. Lett. 80 1121

[5] Pan J W, Bouwmeester D, Weinfurter H and Zeilinger A1998 Phys. Rev. Lett. 80 3891

[6] Scully M O and Zubairy M S 1997 Quantum optics(NewYork: Cambridge University)

[7] Chai J H and Guo G C 1997 Chin. Phys. Lett. 14 524[8] Nemoto K and Munro W J 2004 Phys. Rev. Lett. 93

250502[9] Wang W F, Sun X Y and Luo X B 2008 Chin. Phys. Lett.

25 839[10] Barrett S D, Kok P, Nemoto K, Beausoleil R G, Munro W

J and Spiller T P 2005 Phys. Rev. A 71 060302(R)[11] Fiurasek J, Mista Jr L and Filip R 2003 Phys. Rev. A 67

022304[12] Munro W J, Nemoto K, Beausoleil R G and Spiller T P

2005 Phys. Rev. A 71 033819Munro W J, Nemoto K and Spiller T 2005 New J. Phys. 7137

[13] Nemoto K and Munro W J 2005 Phys. Lett. A 344 104[14] Jeong H, Kim M S, Ralph T C and Ham B S 2004 Phys.

Rev. A 70 061801(R)Jeong H 2005 Phys. Rev. A 72 034305

[15] Lin Q and Li J 2009 Phys. Rev. A 79 022301[16] Spiller T P, Nemoto K, Braunstein S L, Munro W J, Van

Loock P and Milburn G J 2006 New J. Phys. 8 30Metz J, Trupke M and Beige A 2006 Phys. Rev. Lett. 97040503

[17] Sheng Y B, Deng F G and Zhou H Y 2008 Phys. Rev. A77 042308

[18] Sheng Y B, Deng F G and Zhou H Y 2008 Phys. Rev. A77 062325

[19] Sheng Y B, Deng F G and Zhou H Y 2009 arXiv:0907.0050[20] Mista Jr L and Filip R 2005 Phys. Rev. A 71 032342[21] Boyd R W 1999 J. Mod. Opt. 46 367[22] Kok P, Lee H and Dowling J P 2002 Phys. Rev. A 66

063814[23] Li X, Voss P L, Sharping J E and Kumar P 2005 Phys.

Rev. Lett. 94 053601[24] Harris S E and Hau L V 1999 Phys. Rev. Lett. 82 4611[25] Braje D A, Balic V, Yin G Y and Harris S E 2003 Phys.

Rev. A 68 041801(R)[26] Grangier P, Levenson J A and Poizat J P 1998 Nature 369

537[27] Turchette Q A, Hood C J, Lange W, Mabuchi H and Kimble

H J 1995 Phys. Rev. Lett. 75 4710[28] Jeong H 2006 Phys. Rev. A 73 052320[29] Barrett S D and Milburn G J 2006 Phys. Rev. A 74

060302(R)[30] Childress L, Taylor J M, Sørensen A S and Lukin M D 2006

Phys. Rev. Lett. 96 070504

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