Optics Communications 283 (2010) 1558–1560

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Optics Communications

journal homepage: www.elsevier .com/ locate/optcom

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

Yong He, Nian-Quan Jiang *

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

a r t i c l e i n f o a b s t r a c t

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

Keywords:Cavity QEDGenuine four-qubit entangled statesOrthonormal basis states

0030-4018/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.optcom.2009.11.076

* Corresponding author.E-mail address: jiangnq@wzu.edu.cn (N.-Q. Jiang).

We propose a scheme to generate a type of genuine four-qubit entangled states, which were firstly intro-duced by Yeo et al. [Y. Yeo, W. K. Chua, Phys. Rev. Lett. 96 (2006) 060502]. These states have many inter-esting entanglement properties and possess possible applications in quantum information processing andin fundamental tests of quantum physics. We show that such a type of 16 orthonormal basis states can bedeterministically distinguished by a cavity QED system.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Quantum entanglement, one of the most striking features ofquantum mechanics, plays an important role in quantum compu-tation and quantum information processing (QIP) [1], such asdense coding [2], quantum teleportation [3] and quantum cryptog-raphy [4]. So, it is important to explore entangled states. Up tonow, bipartite entangled states have been well understood [5],but multipartite entangled states are still under extensive explora-tions and have attracted much attention [6,7]. In recent years,different types of multipartite entangled states, such as Greenber-ger–Horne–Zeilinger (GHZ) state [8], W state [9] and linear clusterstate [10], have been explored in different systems [11–15]. Lately,to faithfully teleport an arbitrary two-qubit state, Yeo and Chua[16] introduced a new type of genuine four-qubit entangled state

jv00i3214 ¼1

2ffiffiffi2p ðj0000i � j0011i � j0101i þ j0110i þ j1001i

þ j1010i þ j1100i þ j1111iÞ3214: ð1Þ

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

ll rights reserved.

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

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 field. Theatomic transition frequency between the excited state j1i andground state j0i is largely detuned from cavity frequency and isequal to that of the classic field. In the interacting picture, theinteracting Hamiltonian is [19,20] (Let �h = 1)

H ¼X2

j¼1

g aSþj eidt þ aþS�j e�idt� �

þX Sþj þ S�j� �h i

; ð2Þ

where a+ and a are the creation and annihilation operations of thecavity mode, Sþj ¼ j1jih0jj; S�j ¼ 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 field, and d is thedetuning between the atomic transition frequency and cavityfrequency. 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 ¼ H0 þ He; H0 ¼ XX2

j¼1

Sxj ;

He ¼k4

X2

j¼1

ðj1jih1jj þ j0jih0jjÞ þ 2Sx1Sx

2

" #; ð3Þ

where k ¼ g2=d; Sxj ¼ Sþj þ S�j .

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.

Y. He, N.-Q. Jiang / Optics Communications 283 (2010) 1558–1560 1559

Now, 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 ¼ k

4

P2j¼1ðj1jih1jj þ j0jih0jjÞ þ 2Sx

1Sx2

h iand

HC10 ¼ XðSx

1 þ Sx2Þ. After a period of interaction time t1, the initial

state j0102i of the atoms 1 and 2 is evolved intoe�iH

C10 t1 cos kt1

2

� �j0102i � i sin kt1

2

� �j1112i

� �, where a common phase

factor e�ikt1=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 e�iH

C20 t2 cos kt2

2

� �j0304i � i sin kt2

2

� ��j1314i�, where a common phase factor e�ikt2=2 has been discarded.Then, the atoms 2 and 3 simultaneously enter the cavity C3, theeffective Hamiltonians are HC3

e ¼ k4

P3j¼2ðj1jih1jj þ j0jih0jjÞ þ 2Sx

2Sx3

h iand HC3

0 ¼ XðSx2 þ Sx

3Þ. After a period of interaction time t3, the evo-lution of 2 and 3 obeys

j1203i ! e�iHC30 t3 cos

kt3

2

j1203i � i sin

kt3

2

j0213i

� �;

j1213i ! e�iHC30 t3 cos

kt3

2

j1213i � i sin

kt3

2

j0203i

� �;

j0203i ! e�iHC30 t3 cos

kt3

2

j0203i � i sin

kt3

2

j1213i

� �;

j0213i ! e�iHC30

t3 coskt3

2

j0213i � i sin

kt3

2

j1203i

� �;

where a common phase factor e�ikt32 has been discarded. After the

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

jvi1234 ¼ e�iHC10 t1 e�iH

C20 t2 e�iH

C30 t3 cos

kt1

2

cos

kt2

2

�

� coskt3

2

j01020304i � i cos

kt1

2

cos

kt2

2

� sinkt3

2

j01121304i � i cos

kt1

2

sin

kt2

2

� coskt3

2

j01021314i � cos

kt1

2

sin

kt2

2

� sinkt3

2

j01120314i � i sin

kt1

2

cos

kt2

2

� coskt3

2

j11120304i � sin

kt1

2

cos

kt2

2

� sinkt3

2

j11021304i � sin

kt1

2

sin

kt2

2

� coskt3

2

j11121314i þ i sin

kt1

2

sin

kt2

2

� sinkt3

2

j11020314i

�: ð4Þ

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

jv00i03214 ¼1

2ffiffiffi2p ½j03020104i � ij13120104i � ij13020114i

� j03120114i � ij03121104i � j13021104i� j13121114i þ ij03021114i�: ð5Þ

It is obviously that jv00i03214 ¼ 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 atomsbe 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 into

jvm1m2m3m4i1234 ¼

12ffiffiffi2p ðjm1m2m3m4i � ijm1 �m2 �m3m4i

� ijm1m2 �m3 �m4i � jm1 �m2m3 �m4i� ij �m1 �m2m3m4i � j �m1m2 �m3m4i� j �m1 �m2 �m3 �m4i þ ij �m1m2m3 �m4iÞ; ð6Þ

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 ¼ jv00i03214. From Eq. (6), we can also find thathvm01m02m03m04

jvm1m2m3m4i ¼ dm1m01

dm2m02dm3m03

dm4m04, so fjvm1m2m3m4

i1234;

m1;m2;m3;m4 ¼ 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 ¼ 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

jvm1m2m3m4i1234 ! ij �m1m2m3 �m4i; ð7Þ

where 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 ¼ 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 � 10�2 s. The coupling constant of the atoms to the cavityfield is g = 2p � 24 kHz [23,24]. With the choice d = 5 g andX = 20 g, the required atom-cavity-field interaction time is on theorder of T 6 2p/k � 2 � 10�4 s, the total time to perform thescheme is t 6 3T � 6 � 10�4 s. Therefore, the time needed to

1560 Y. He, N.-Q. Jiang / Optics Communications 283 (2010) 1558–1560

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.

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