Quantum Inf Process (2013) 12:1787–1795DOI 10.1007/s11128-012-0481-9

A nearly deterministic scheme for generating χ -typeentangled states with weak cross-Kerr nonlinearities

Li Dong · Xiao-Ming Xiu · Ya-Jun Gao · X. X. Yi

Received: 29 June 2012 / Accepted: 28 August 2012 / Published online: 5 October 2012© Springer Science+Business Media, LLC 2012

Abstract We propose a generation scheme of χ -type entangled states based onweak cross-Kerr nonlinearities. After Homodyne measurement, the individual pho-tons can be entangled together with the help of coherent states. Assisted with classicalfeed-forward, the generation efficiency of the entangled state nearly approaches unity.Depending on the currently available optical elements and techniques such as beamsplitters, reflection mirrors, polarization beam splitters and classical feed-forward, thepresent scheme is expectable to realize in experiment.

Keywords Quantum entanglement · Cross-Kerr nonlinearities ·χ -type entangled state

1 Introduction

Quantum entanglement has been a hot issue due to its charming and appealing proper-ties, which find various applications in quantum information processing, a field totallybeyond classical physics.

It is well known that an Einstein–Podolsky–Rosen state is the maximal entangledstate in two-qubit entanglement. Furthermore, Greenberger–Horne–Zeilinger (GHZ)

L. Dong (B) · X.-M. Xiu · Y.-J. GaoCollege of Mathematics and Physics, Bohai University, Jinzhou 121013, People’s Republic of Chinae-mail: donglixm@163.com

L. Dong · X.-M. Xiu · X. X. YiSchool of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024,People’s Republic of China

X.-M. Xiue-mail: xiuxiaomingdl@126.com

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states and W states are two classes of the three-qubit entangled states under sto-chastic local operations and classical communication (SLOCC) [1]. However, theresearchers encounter troubles when the characterization and classification of multi-qubit entangled states, as an important subject, comes to the agenda. Despite it, numer-ous researchers devoted to study the properties of multi-qubit entanglement and makesome progress [2–8]. For four-qubit entangled states, Verstraete et al. [2] divided themto nine families under SLOCC. Bai et al. also discussed the multipartite entangle-ment of the four-qubit pure states [3] and cluster states [4]. Additionally, Lee et al.[5] constructed a kind of four-qubit genuine entangled states, χ -type entangled states,different from the four-qubit GHZ states and W states [6], which can be denoted as

|χ〉= 1

2√

2(|0000〉−|0011〉−|0101〉 + |1001〉+|0110〉 + |1010〉+|1100〉 + |1111〉).

(1)

The entangled state is optimally violated a new Bell inequality [7], by which manycommunication tasks can be implemented.

Owing to potential and important applications in quantum information processing,χ -type entangled states draw considerable researchers’ interests. Exploiting it, Yeoand Chua [8] proposed a teleportation protocol of an arbitrary and unknown two-qubitstate and discussed dense coding [9] scheme based on it. Lin et al. [10] conceived atwo-party quantum secure direct communication protocol with dense coding. Addi-tionally, some quantum communication protocols with dense coding [11–13] were putforward. Assisted with entanglement swapping [14], two-party deterministic quantumcommunication and quantum key distribution protocols were presented [15–18]. UsingBell-state measurement and GHZ basis measurement, Zhan et al. [19] showed that thetwo-qubit states can be split. Employing quantum teleportation [20], Wang et al. [21]put forth a protocol for hierarchical quantum state sharing. For reducing the exper-imental complexity, an incomplete teleportation was used to perform deterministicquantum communication [22]. Moreover, a four-site distribution and the correspond-ing communication protocol were designed [23].

There exist so many applications, so the preparation of χ -type entangled statesis studied in various physics systems. In a controllable ion-trap system, Wang andYang [24] proposed a scheme for preparing this entangled state. Assisted with cavityquantum electrodynamics, Wang et al. [25] presented the generation scheme of theentangled state through large-detuning interaction of atoms with a single-mode cavityfield. In addition, the construction of the entangled state of four atoms in separatedoptical cavities is designed by Liu and Kuang [26]. Moreover, Lin [27] proposed ageneration scheme of the entangled state among four atoms in cavity QED. Recently,Shi et al. [28] presented the scheme for generating the four-qubit entangled state in anion trap system.

In virtue of the features of simple interaction and good coherence time, the photonicquantum information processing holds on the vast majority of research interests. Inthe context, the implementation of optical four-qubit χ -type entangled states becomesa preliminary and important task. Wang and Zhang [29] showed that the χ -type entan-gled states can be prepared with simple linear optical elements and conventional photon

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A generation scheme of χ -type entangled states 1789

detectors. However, limited by natural properties of photons, the generation proba-bility of the state based on linear optics is largely far from unity. With the advantageof weak nonlinearity, hybrid systems are good candidates for generating the entan-gled state among photons with high probability, resulting from the bridge functionof coherent states with the assistance of Homodyne measurement. Employing cross-Kerr nonlinearities and Homodyne detection, a probabilistic preparation scheme ofthe χ -type entangled states and GHZ states is constructed by Zhao et al. [30].

Although the above schemes have been proposed to realize χ -type entangled states,there are few preparation schemes of the entangled state with near determinacy. In thispaper, a nearly deterministic generation scheme of χ -type polarization-entangled statewith cross-Kerr nonlinearities is proposed. First, we describe the generation schemeof the entangled state in Sect. 2. And then, we evaluate the efficiency of critical com-ponents for preparing the entangled state in Sect. 3. Section 4 is devoted to discussionand summary.

2 The generation scheme of the χ -type polarization-entangled state

In what follows, we present a scheme for generating χ -type polarization-entangledstates with the following description and Fig. 1. For facilitating the description, weadopt that |0〉 denotes horizontal polarization mode |H〉, and |1〉 denotes verticalpolarization mode |V 〉 in some places in the paper.

At first, the manipulator prepares photons (1,2,3,4) in the states 1√2(|0〉 + |1〉)1,

1√2(|0〉 − |1〉)2, 1√

2(|0〉 − |1〉)3, |0〉4. Subsequently, the manipulator sends photons

Fig. 1 (color online). An illustration plot for generating theχ -type polarization-entangled state. We assumeall the optical paths which photons pass through are equal length. The mirror (M) only changes the directionwhich photons go forward. The half wave plate (HWP) can delay phase π , and HWP 45◦ functions as NOT(σx ) gate. The BS denotes the beam splitter with (50:50). The polarizing beam splitter (PBS) reflects |V 〉mode and transmits |H〉 mode. The one-way mirror transmits one photon from one side (up side), andreflect another photon from another side (left side) without remodulating

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(3,4) and (2,4) through Controlled NOT (CNOT) gates with the former one of photonpairs as the control qubit and the latter one as the target qubit. Finally, the manipulatorsends photons (1,4) through a two-qubit Controlled NOT-PHASE-FLIP gate (CNPgate) which can be expressed as

CNP =

⎛⎜⎜⎝

1 0 0 00 1 0 00 0 0 −10 0 1 0

⎞⎟⎟⎠ . (2)

Its function is equal to a Controlled Phase-Flip (CPF) gate and a CNOT gate performedin turn, which can be represented as

CPF =

⎛⎜⎜⎝

1 0 0 00 1 0 00 0 1 00 0 0 −1

⎞⎟⎟⎠, CNOT =

⎛⎜⎜⎝

1 0 0 00 1 0 00 0 0 10 0 1 0

⎞⎟⎟⎠. (3)

After that, photons (1,2,3,4) are in the χ -type state which can be described asEq. (1). The whole process can be shown as

1√2(|0〉 + |1〉)1 ⊗ 1√

2(|0〉 − |1〉)2 ⊗ 1√

2(|0〉 − |1〉)3 ⊗ |0〉4

CNOT3,4−−−−−→ 1

2√

2(|0〉 + |1〉)1 ⊗ (|0〉 − |1〉)2 ⊗ (|00〉 − |11〉)34

CNOT2,4−−−−−→ 1

2√

2(|0〉 + |1〉)1 ⊗ (|000〉−| 011〉 − |101〉+| 110〉)234

CNP1,4−−−−→ 1

2√

2

(|0000〉−| 0011〉 − |0101〉+| 0110〉 + |1001〉+| 1010〉

+ |1100〉+| 1111〉)

1234. (4)

The process can also be seen in Fig. 1, where we plot the illustration for generatingthe χ -type entangled state from four individual photons.

3 The critical components: the CNOT gate and the CNP gate

The former two gates in Fig. 1 are CNOT gates. When passing through Kerr media,the probe coherent state and the signal photons are entangled as

|ψ〉 =(|αeiθ 〉 |a, c〉 + |αe−iθ 〉 |b, d〉

)+ |α〉 (|a, d〉 + |b, c〉) , (5)

where the symbols (‘a’, ‘b’,‘c’,‘d’) denote the four paths from top to bottom in thethree shadow rectangle boxes in Fig. 1.

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Under an X quadrature measurement, the coherent state disentangles from the pho-tons, and the individual photons are projected onto the following state,

|ψx 〉 = f (x, α cos θ)(

eiφ(x) |a, c〉 + e−iφ(x) |b, d〉)

+ f (x, α) (|a, d〉 + |b, c〉) ,(6)

where,

f (x, α cos θ) = (2π)−1/4 exp[−(x − 2α cos θ)2/4],f (x, α) = (2π)−1/4 exp[−(x − 2α)2/4],φ(x) = α sin θ(x − 2α cos θ), (7)

the Gaussian distribution curves f (x, α cos θ) and f (x, α) embody the difference ofthe photon paths (b, d) or (a, c) and paths (a, d) or (b, c). If signal photons passthrough paths (b, d) or (a, c), the classical feed-forward is necessary to perform theNOT gate and modulate additional phase shift.

The last gate is a CNP gate. In addition to a NOT gate in path a, a Phase-Flip gateis necessary to be input before the NOT gate.

If we consider that the perfect condition can be provided in the other optical ele-ments, the error mainly results from the overlapping of two distribution f (x, α cos θ)and f (x, α). The error probability for mistaking two distributions can be calculatedwith [31]

Perr = 1

2er f c

(∣∣αθ2∣∣

2√

2

), (8)

The implementation of the scheme will fail if the signal photons used to generatethe entangled state are lost. The scenario can be found by putting four QND detectorsbefore the output ports of circuit. Ignoring the above issue, in the construction of theχ -type entangled state with polarization degree of freedom, lower efficiency mainlyresults from the imperfect detectors and photon loss of the probe coherent states.

Photon loss of the coherent states is one of the main reasons of signal photondecoherence. Such damping will lead to a complicated mixed state of signal photons.But in the framework of electromagnetically induced transparencies (EIT) based Kerrnonlinearity, the loss could be low to carry a negligible decoherence effect.

In order to overcome the above adverse influence and achieve the higher successprobability, Munro et al. [32–35] applied displacement measurement to replace bal-anced measurement, which can be performed by employing the probe coherent state totransmit a beam splitter with high reflection or high transmission rather than equivalentone, accompanying with a large amplitude coherent state.

In a word, one can overcome the disadvantages through increasing the intensity ofthe probe coherent state, or adopting a larger strength of cross-Kerr nonlinearities, orexperiencing a longer interaction time.

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4 Discussion and summary

The availability of the proper Kerr media is the core issue for the implementation of ourscheme. The strength of such nonlinearity with the normal materials is too weak, barelyachieving a cross phase in the order of θ ≈ 10−18 [36,37]. The researchers attempt toamplify the strength of cross-phase modulation by the assistance of various appropri-ate systems and methods, such as, EIT [38–43], whispering-gallery microresonators[44], cavity QED systems [45–47], and quantum dot systems [48,49]. However, Gea-Banacloche [50] pointed out that large phase shifts based on the giant Kerr effect areimpossible to realize with the currently experimental technology. He and Scherer [51]studied the difficulty of EIT based kerr nonlinearity and showed that a non-zero phaseshift with high fidelity can be achieved at single-photon level by head-on collision oftwo pulses in the limit of large system bandwidth. There is a more promising candi-date for Kerr nonlinearity utilizing both EIT and long-range interaction. Friedler et al.[52] and He et al. [53] studied that the implementation of Kerr nonlinearity and theassociated disadvantage to overcome. Supporting the applications of Kerr media, Heet al. [54] showed that high fidelities, nonzero conditional phases, and high photonnumbers can be achieved at the same time if the pulses can fully pass through eachother and the unwanted transverse-mode effects are suppressed. Moreover, Feizpouret al. [55] showed that a much larger magnitude of cross-Kerr modulation could beobtained in the single-photon regime with the help of weak measurement.

So various applications such as quantum logical gates, entanglement purification,and the GHZ analyzer [31–33,56–79] based on cross-Kerr nonlinearities are feasibleand expectable in experiment.

Using two-photon logical gates and some optical elements, there exist some poten-tial methods to generateχ -type entangled states. Here, we propose a generation schemeof χ -type entangled states, where cross-Kerr nonlinearities are utilized to entangleindividual photons. After the hybrid system of signal photons and the probe coherentstate passed through Kerr media, the coherent state is measured with Homodyne mea-surement, whose outcomes ambiguously inform the information about paths photonspassing through, and the entanglement of individual signal photons can be obtained.With classical feed-forward techniques, the high efficiency can be obtained. In the pres-ent generation scheme, three model blocks are exploited, so the efficiency is (1−Perr )

3

under the ideal condition.In a word, we propose the scheme for preparing the χ -type polarization-entangled

state based on Kerr media and coherent states. In the proposal, single-photon polar-ization states, classical feed-forward techniques, and some currently feasible opticalelements (e.g. beam splitters, polarization beam splitters, reflection mirrors, and halfwave plates) are necessary to create the entangled state. Due to the availability of theelements and techniques involved, we expected the preparation scheme of the χ -typeentangled state can be realized in the experiment.

Acknowledgments The authors would like to thank the anonymous referee for enlightening instructions.Moreover, we acknowledge Dr. Xiaosong Ma and Dr. Yan Xia for helpful discussions on one-way mirrorand cross-Kerr nonlinearities. This study was supported by the Research Programs of the Educational Officeof Liaoning Province of China (Grant No. 2008006).

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