Int J Theor PhysDOI 10.1007/s10773-013-1743-8

Deterministic Generation of Genuine Six-QubitEntangled State in Cavity QED System

Hu-ping Zhu · Mei-li Tang

Received: 25 April 2013 / Accepted: 12 July 2013© Springer Science+Business Media New York 2013

Abstract Recently, a genuine six-qubit entangled state has been proposed by Tapiador etal. (J. Phys. A 42: 415301, 2009). In this paper, we present a deterministic scheme forgenerating such a state in cavity QED system, where the atoms interact simultaneouslywith the highly detuned single-mode cavity and the strong classical driving field. Thus ourscheme is not sensitive to both the cavity decay and thermal field.

Keywords Quantum information · Cavity QED · Generation · Genuine six-qubit entangledstate

1 Introduction

Entanglement is an important physical resource in the context of quantum information the-ories. As shown for last two decades it plays a crucial role in quantum teleportation [1],quantum information splitting [2–4], quantum superdense coding [5], quantum cloning [6],quantum computation [7–9]. Recently, different types of multipartite entangled states, in-cluding GHZ state [10], W state [11] and cluster state [12], have been extensively exploredin different systems [13–15].

In 2009, Tapiador et al. found a genuine six-qubit entangled state [16], which is given by

|ψ〉123456 =1

4

[(|0000〉 + |1111〉)(|00〉 − |11〉) + (|0011〉 + |1100〉)(|00〉 + |11〉)

+ (|0110〉 + |1001〉)(|01〉 − |10〉) + (|0101〉 + |1010〉)(|01〉 + |10〉)]123456

.

(1)

H.-p. Zhu (B)College of Computer Information Engineering, Jiangxi Normal University, Nanchang 330022, Chinae-mail: hupingz@yahoo.com

M.-l. TangSchool of Business, Jiangxi Normal University, Nanchang 330022, China

Int J Theor Phys

This state can not decompose into pairs of Bell states and therefore it exhibits genuinemultipartite entanglement according to negative partial transpose measurement. In addition,all the partial density matrices are completely mixed for this state [16]. More importantly,such a genuine six-qubit entangled state has been found wide application in quantum infor-mation processing, including both quantum teleportation and quantum information splitting[17].

In this work, an efficient scheme is proposed to prepare the genuine six-qubit entangledstate by using the atoms in cavity QED system. Furthermore, we investigate that two andmore atoms interact simultaneously with the thermal cavity field and the strong classicaldriving field, thus our scheme is insensitive to both the cavity decay and thermal field [18].

2 The Model

Let us firstly consider N identical two-level atoms in a single-mode cavity with a strongclassical driving field. In rotating-wave approximation, the Hamiltonian is given by [19]

H = ω0Sz + ωaa†a + g

2

(a†S−

j + aS+j

) + Ω

2

(S+

j e−iωt + S−j eiωt

), (2)

where

Sz = 1

2

N∑

j=1

(|1j 〉〈1j | − |0j 〉〈0j |), S+

j =N∑

j=1

(∣∣1j 〉〈0j

∣∣),

and S−j = ∑N

j=1(|0j 〉〈1j |), while |0j 〉 and |1j 〉 are the ground and excited states of the j -thatom with energy level spacing ω0, a(a†) is the annihilation (creation) operator of the cavity-field mode. g represents the atom-cavity-field coupling strength, and Ω denotes the Rabifrequency of classical field. ωa is cavity frequency,ω0 is atomic transition frequency, and ω

is the frequency of the classical field. In the large detuning δ � g between ω0 and ωa , andstrong driving field Ω � δ, g, the energy exchange may not take place between the atomsand the cavity. Under the case of ω0 = ω, thus, the evolution operator of the system can begiven by [19]

U(t) = e−iΩtSx−iλtS2x , (3)

where Sx = 12

∑N

j=1(S+j + S−

j ) and λ = g2/4δ.Using the representation of the operator Sz, the atomic states |0102 . . .0N 〉 and |1112 . . .1N 〉

can be expressed as |N/2,−N/2〉 and |N/2,N/2〉. Thus the above states can be expandedin terms of the eigenstates of Sx [20]

|N/2,−N/2〉 =N/2∑

M=−N/2

CM |N/2,M〉x, (4)

|N/2,N/2〉 =N/2∑

M=−N/2

CM(−1)N/2−M |N/2,M〉x . (5)

Int J Theor Phys

Suppose that the N atoms are initially in the state |0102 . . .0N 〉. At any time, the systemwill become

|0102 . . .0N 〉N/2∑

M=−N/2

CMe−i(ΩM+λM2)t |N/2,M〉x . (6)

When N is even, M is an integer. By choicing λt = π/2 and Ωt = 2nπ , we can obtain

√2

2

[e−iπ/4|0102 . . .0N 〉 + eiπ/4(−1)N/2|1112 . . .1N 〉]. (7)

For the case where N is odd, on the other hand, by choosing λt = π/2 and Ωt = (2n +3π/2), we can obtain

√2

2ei7π/8

[e−iπ/4|0102 . . .0N 〉 + eiπ/4(−1)(1+N)/2|1112 . . .1N 〉]. (8)

As an example, the two atoms (for the case where N = 2) are initially in one of fourproduct states |00〉, |01〉, |10〉 and |11〉. If λt = π/2 and Ωt = 2π are chosen and the twoatoms are sent into the single-mode cavity, the four product states will become the followingstates, respectively,

|00〉 →√

2

2

(|00〉 − i|11〉), (9)

|01〉 →√

2

2

(|01〉 − i|10〉), (10)

|10〉 →√

2

2

(|10〉 − i|01〉), (11)

|11〉 →√

2

2

(|11〉 − i|00〉). (12)

3 Generation of Genuine Six-Atom Entangled State

In order to generate the genuine six-atom entangled state, we prepare the six atoms that areinitially in the state |001100〉123456 and let the atoms 1, 4 and 6 interact with the single-mode cavity I with a classical driving field. By controlling the interaction time and the Rabifrequency, one has λt1 = π/2 and Ωt1 = 3π/2. Thus the state |001100〉123456 will becomethe following state

|ψ1〉123456 = 1√2

(|001100〉 + i|101001〉)123456

. (13)

Secondly, we make a single-atom operation |1〉 → i|1〉 on the atom 4, and then the atoms1, 2, 3, 4, 5, 6 are simultaneously sent to another singe-mode cavity II in which are drivenby the same classical field. By choosing λt2 = π/2 and Ωt2 = 2π , the state |ψ1〉123456 willevolve into the following state,

|ψ2〉123456 = 1

2

(i|001100〉 + i|101001〉 + |110011〉 + |010110〉)

123456. (14)

Int J Theor Phys

In the third step, we let the atoms 3 and 4 interact with the third single-mode cavityIII under the same classical driving field. In the situation of λt3 = π/2 and Ωt3 = 2π , thetime-dependent evolving state |ψ2〉123456 is

|ψ3〉123456 = 1

2√

2

[|00〉(|0000〉 + i|1100〉) + |11〉(|0011〉 − i|1111〉)

+ |01〉(|0110〉 − i|1010〉) + |10〉(|0101〉 + i|1001〉)]123456

.

At last, we let that the atoms 3, 4, 5 and 6 interact with the fourth single-mode cavity IVin which are driven by the classical field. If we choose λt4 = π/2 and Ωt4 = 2π , the state|ψ3〉123456 is given by

|ψ4〉123456 = 1

4

[|00〉(|0000〉 − |0011〉 + i|1100〉 + i|1111〉)

+ |11〉(|0011〉 + i|1100〉 − i|1111〉 + |0000〉)

+ |01〉(|0110〉 + i|1001〉 − i|1010〉 + |0101〉)

+ |10〉(|0101〉 + i|1001〉 + i|1010〉 − |0110〉)]123456

. (15)

where we have discarded the overall phase factor. By performing another single-qubit oper-ation |1〉 → −i|1〉 on the atom 3, we find

|ψ5〉123456 = 1

4

[(|0000〉 + |1111〉)(|00〉 − |11〉) + (|0011〉 + |1100〉)(|00〉 + |11〉)

+ (|0110〉 + |1001〉)(|01〉 − |10〉) + (|0101〉 + |1010〉)(|01〉 + |10〉)]123456

,

which is just the genuine six-atom entangled state [16]. Thus, a scheme to prepare the gen-uine six-atom entangled state has been realized.

4 Discussions and Conclusions

Next we give a brief discussion on the experimental matters. For the Rydberg atoms withprincipal quantum numbers 50 and 51, the radiative time is about Tr = 3 × 10−2 s, and thecoupling constant is g = 2π × 24 kHz [21]. For a three-atom system, the damping time isT ′

r = 1 × 10−2 s [22]. With the choice of δ = 10g, the required atom-thermal-cavity-fieldinteraction time is on the order of 10−4 s, the total time to perform the scheme is 10−3 s.Therefore, the time required to complete the whole procedure is much shorter than T ′

r .The scheme needs to control two or three atoms to simultaneously interact with a cavity.

But in real case, we cannot achieve simultaneousness in perfect precise. Considering theimperfection that there exist temporal fluctuations when the atoms interact with a cavitysuggests that it only slightly affects the fidelity of the reconstruct state [23]. Meanwhile,it is noted that the atomic state evolution is independent of the cavity field state, thus theproposed scheme might be realizable with current cavity QED techniques [24, 25].

In conclusion, we presented a novel scheme for generating the genuine six-atom entan-gled state in cavity QED system. Such an entanglement state has an creasing interesting inpossible applications of quantum information processing and fundamental tests of quantumphysics. Because two and more atoms interact with a thermal cavity under the assistance ofthe strong classical driving field, especially, our scheme is not sensitive to both the cavitydecay and the thermal field.

Int J Theor Phys

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