Schemes to generate and distinguish a type of genuine four-qubit entangled states in a cavity QED system

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  • pa

    nerW.rtientushed

    the most striking features oftant rorocesstion [3plore ee beestill uention

    This state, which does not belong to the above well-known threetypes of states, has many interesting entanglement properties. Forexamples, a new Bell inequality is optimally violated by jv00i[17], there exists maximum entanglement between qubits (3, 2)and (1, 4), (3, 1) and (2, 4) of the state. More importantly, it hasmany applications in QIP, such as teleportation and dense coding[16]. The scheme to prepare this state in an ion-trap system wasintroduced by Wang and Yang [18]. In this paper, we propose a

    scheme to generate such a type of states and show how to discrim-inate them in a cavity QED system.

    frequency. In the case of 2X d and d g, there is no energyexchange between the atomic system and the cavity mode. Theeffective Hamiltonian is given by [21]

    H H0 He; H0 XX2j1

    Sxj ;

    He k4X2j1

    j1jih1jj j0jih0jj 2Sx1Sx2" #

    ; 3

    where k g2=d; Sx S S.* Corresponding author.

    Optics Communications 283 (2010) 15581560

    Contents lists availab

    m

    elsE-mail address: jiangnq@wzu.edu.cn (N.-Q. Jiang).different types of multipartite entangled states, such as Greenber-gerHorneZeilinger (GHZ) state [8], W state [9] and linear clusterstate [10], have been explored in different systems [1115]. Lately,to faithfully teleport an arbitrary two-qubit state, Yeo and Chua[16] introduced a new type of genuine four-qubit entangled state

    jv00i3214 1

    22

    p j0000i j0011i j0101i j0110i j1001i

    j1010i j1100i j1111i3214: 1

    interacting Hamiltonian is [19,20] (Let h = 1)

    H X2j1

    g aSj eidt aSj eidt

    X Sj Sj

    h i; 2

    where a+ and a are the creation and annihilation operations of thecavity mode, Sj j1jih0jj; Sj j0jih1jj; j1ji and j0ji are excited andground states of the jth atom, g is the atom-cavity couplingstrength, X is the Rabi frequency of the classic eld, and d is thedetuning between the atomic transition frequency and cavity1. Introduction

    Quantum entanglement, one ofquantum mechanics, plays an importation and quantum information pdense coding [2], quantum teleportaraphy [4]. So, it is important to exnow, bipartite entangled states havbut multipartite entangled states aretions and have attracted much att0030-4018/$ - see front matter 2009 Elsevier B.V. Adoi:10.1016/j.optcom.2009.11.076le in quantum compu-ing (QIP) [1], such as] and quantum cryptog-ntangled states. Up ton well understood [5],nder extensive explora-[6,7]. In recent years,

    2. Generating of genuine four-qubit entangled states

    We consider that two identical two-level atoms simultaneouslyinteract with a single-mode cavity driven by a classic eld. Theatomic transition frequency between the excited state j1i andground state j0i is largely detuned from cavity frequency and isequal to that of the classic eld. In the interacting picture, theSchemes to generate and distinguish a tystates in a cavity QED system

    Yong He, Nian-Quan Jiang *

    College of Physics and Electric Information, Wenzhou University, Wenzhou 325035, Chin

    a r t i c l e i n f o

    Article history:Received 28 July 2009Received in revised form 8 November 2009Accepted 27 November 2009

    Keywords:Cavity QEDGenuine four-qubit entangled statesOrthonormal basis states

    a b s t r a c t

    We propose a scheme to geduced by Yeo et al. [Y. Yeo,esting entanglement propein fundamental tests of quadeterministically distingui

    Optics Com

    journal homepage: www.ll rights reserved.e of genuine four-qubit entangled

    ate a type of genuine four-qubit entangled states, which were rstly intro-K. Chua, Phys. Rev. Lett. 96 (2006) 060502]. These states have many inter-s and possess possible applications in quantum information processing andm physics. We show that such a type of 16 orthonormal basis states can beby a cavity QED system.

    2009 Elsevier B.V. All rights reserved.

    le at ScienceDirect

    unications

    evier .com/ locate/optcomj j j

  • jvm1m2m3m4 i1234 ! ij m1m2m3 m4i; 7where mj 2 {0,1}, j = 1,2,3,4., and mj is the counterpart of the binarynumber mj. Eq. (7) shows that each of the 16 basis statesfjvm1m2m3m4 i1234;m1;m2;m3;m4 0;1g can be transformed into acorresponding product state of particles 1, 2, 3, and 4. So a deter-ministic FQBB measurement can be achieved by individual detec-tion of the related qubits.

    4. Experimental feasibility

    Now, we give a brief discussion for the experimental matters.Based on the current cavity QED techniques [21], the cavity canhave a photo storage time 1 ms, and the radiative time of the Ryd-berg atoms with principal quantum numbers 50 and 51 is aboutTr = 3 102 s. The coupling constant of the atoms to the cavityeld is g = 2p 24 kHz [23,24]. With the choice d = 5 g and

    unicNow, we let four two-level atoms, which are initially in theground states, cross the corresponding cavities in Fig. 1. Whenthe atoms 1 and 2 simultaneously enter the cavity C1, the effective

    Hamiltonians are HC1e k4P2

    j1j1jih1jj j0jih0jj 2Sx1Sx2h i

    and

    HC10 XSx1 Sx2. After a period of interaction time t1, the initialstate j0102i of the atoms 1 and 2 is evolved intoeiH

    C10 t1 cos kt12

    j0102i i sin kt12 j1112i , where a common phasefactor eikt1=2 has been discarded. Similarly, when 3 and 4 simulta-neously enter the cavity C2, after a period of interaction time t2, theinitial state j0304i is evolved into eiH

    C20 t2 cos kt22

    j0304i i sin kt22 j1314i, where a common phase factor eikt2=2 has been discarded.Then, the atoms 2 and 3 simultaneously enter the cavity C3, theeffective Hamiltonians are HC3e k4

    P3j2j1jih1jj j0jih0jj 2Sx2Sx3

    h iand HC30 XSx2 Sx3. After a period of interaction time t3, the evo-lution of 2 and 3 obeys

    j1203i ! eiHC30 t3 cos

    kt32

    j1203i i sin kt32

    j0213i

    ;

    j1213i ! eiHC30 t3 cos

    kt32

    j1213i i sin kt32

    j0203i

    ;

    j0203i ! eiHC30 t3 cos

    kt32

    j0203i i sin kt32

    j1213i

    ;

    j0213i ! eiHC30

    t3 coskt32

    j0213i i sin kt32

    j1203i

    ;

    where a common phase factor eikt32 has been discarded. After the

    four atoms have passed all the three cavities, the nal state is

    jvi1234 eiHC10 t1eiH

    C20 t2eiH

    C30 t3 cos

    kt12

    cos

    kt22

    cos kt32

    j01020304i i cos kt12

    cos

    kt22

    sin kt32

    j01121304i i cos kt12

    sin

    kt22

    cos kt32

    j01021314i cos kt12

    sin

    kt22

    sin kt32

    j01120314i i sin kt12

    cos

    kt22

    cos kt32

    j11120304i sin kt12

    cos

    kt22

    sin kt32

    j11021304i sin kt12

    sin

    kt22

    cos kt32

    j11121314i i sin kt12

    sin

    kt22

    sin kt32

    j11020314i

    : 4

    Assuming X = 100k and choosing t1 = t2 = t3 = p/(2k), we obtain

    jv00i03214 1

    22

    p j03020104i ij13120104i ij13020114i

    j03120114i ij03121104i j13021104i j13121114i ij03021114i: 5

    It is obviously that jv00i03214 l:u:jv00i3214, where l.u. indicatesthat the equality holds up to a local unitary transformation onone or more of the qubits [22]. Generally, let four two-level atoms

    Y. He, N.-Q. Jiang / Optics Commbe initially in one of the 16 basis product states{jm1m2m3m4i,m1,m2,m3,m4 = 0,1}, after they have passed thoughthe setup in Fig. 1, the state will be evolved intojvm1m2m3m4 i1234 1

    22

    p jm1m2m3m4i ijm1 m2 m3m4i

    ijm1m2 m3 m4i jm1 m2m3 m4i ij m1 m2m3m4i j m1m2 m3m4i j m1 m2 m3 m4i ij m1m2m3 m4i; 6

    where mj 2 f0;1g, mj is the counterpart of the binary number mj.When m1 =m2 =m3 =m4 = 0, it follows from Eqs. (5) and (6) thatjv0000i1234 jv00i03214. From Eq. (6), we can also nd thathvm01m02m03m04 jvm1m2m3m4 i dm1m01dm2m02dm3m03dm4m04 , so fjvm1m2m3m4 i1234;m1;m2;m3;m4 0;1g constitutes a basis of 16 orthonormal states,for convenience, we call it four-qubit Bell-type basis (FQBB). Foreach of these states, the amount of entanglement between threepairs of particles (12 and 34, 13 and 24, 14 and 23) is the same asthat of the state jv00i3214 in Ref. [16]. So, each of them is differentfrom the product of a pair of Bell states and then is a genuinefour-qubit entangled state. They are likely candidates for the four-partite analogue to Bell states [16].

    3. Distinguishing between the 16 basis states

    Next, we demonstrate how to implement FQBB measurement,i.e., deterministically distinguish between the 16 basis statesfjvm1m2m3m4 i1234;m1;m2;m3;m4 0;1gwith the setup in Fig. 1. Con-sidering four atoms, which are initially in the state jvm1m2m3m4 i1234,pass though the setup in the way in Fig. 1 and choosing X = 100kand t1 = t2 = t3 = p/(2k), we obtain

    1

    2

    3

    42C

    1C

    3C

    Fig. 1. Schematic setup to generate the four-atom entangled state. The atoms 1 and2 simultaneously pass through cavity C1, 3 and 4 simultaneously pass throughcavity C2. Then, the atoms 2 and 3 pass through cavity C3 but 1 and 4 do not passthrough it.

    ations 283 (2010) 15581560 1559X = 20 g, the required atom-cavity-eld interaction time is on theorder of T 6 2p/k 2 104 s, the total time to perform thescheme is t 6 3T 6 104 s. Therefore, the time needed to

  • complete the whole procedure is much shorter than the radioac-tive time Tr. Meanwhile, it is noted that the evolution of the atomicstate is independent of the cavity mode state, thus the cavity decaywill not affect the generation of the four-atom states. So ourscheme is feasible in current techniques.

    5. Conclusion

    In summary, genuine four-qubit entangled states [16] havemany interesting properties, they may have applications in QIPand in fundamental tests of quantum physics [18]. Therefore, it ismeaningful to explore them in concrete systems. In the paper,we proposed a simple scheme to generate such a type of statesin a cavity QED system. We also demonstrated how to implementFQBB measurement, which shows that 16 FQBB basis states can bedeterministically distinguished only by individual detection of re-lated qubits.

    References

    [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information,Cambridge Univ. Press, Cambridge, 2000.

    [2] T. Schaetz, Phys. Rev. Lett. 93 (2004) 040505.[3] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54

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    Haroche, Science 288 (2000) 2024.[22] N. Linden, S. Popescu, Fortschr. Phys. 46 (1998) 567.[23] G.W. Lin, M.Y. Ye, L.B. Chen, Q.H. Du, X.M. Lin, Phys. Rev. A 76 (2007)

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    1560 Y. He, N.-Q. Jiang / Optics Communications 283 (2010) 15581560

    Schemes to generate and distinguish a type of genuine four-qubit entangled states in a cavity QED systemIntroductionGenerating of genuine four-qubit entangled statesDistinguishing between the 16 basis statesExperimental feasibilityConclusionReferences

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