Preparation of a 2n-qubit W state via entanglement transfer

Physics Letters A 361 (2007) 59–62 www.elsevier.com/locate/pla

Preparation of a 2n-qubit W state via entanglement transfer Jun-Gang Li, Jian Zou ∗ , Jin-Fang Cai, Bin Shao Department of Physics, Beijing Institute of Technology, Beijing 100081, China Received 11 August 2006; received in revised form 7 September 2006; accepted 14 September 2006 Available online 25 September 2006 Communicated by P.R. Holland

Abstract We propose a scheme to generate a W state of n + n qubits trapped in two spatially separated cavities respectively by means of entanglement transfer. We find that the 2n qubits will be entangled in the W state periodically. The influence of cavity dissipation is also analytically demonstrated. © 2006 Elsevier B.V. All rights reserved. PACS: 03.67.Mn; 42.50.Dv; 03.65.Ud Keywords: W state; Entanglement transfer; Dissipation

1. Introduction Entanglement is the key resource of quantum computation and quantum information processing [1]. There are numerous applications of spatially separated entangled pairs of qubits such as quantum teleportation [2], broadcasting of entanglement [3] and distributed quantum computation [4]. Then it is necessary to build a reliable channel to entangle spatially separated qubits. Due to its handiness in generating and propagating entanglement [5], the usage of a optical field to implement a quantum channel becomes a natural choice. On the other hand, static qubits, such as the Josephson charge qubits, are easily accessible and manipulative by means of external excitations. Therefore, it may be an optimal strategy to use an optical quantum channel to bring quantum correlation to two remote sites of static qubits, where the entanglement is subsequently utilized for quantum information processing. The entanglement transfer between the field and the qubit system has been widely studied [6–15]. It has been shown that the efficiency of the entanglement transfer from continuous-variable to finite-dimensional systems can be enhanced by employing mul* Corresponding author.

E-mail address: [email protected] (J. Zou). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.09.029

tiple qubits [11]. Entanglement reciprocation between qubits and continuous variables [12] and the connection of entanglement transfer to one-way quantum computation have been studied [13]. Moreover, very recently, there have been some studies of the relation between energy and entanglement in the entanglement transfer [14,15]. Entanglement of multiparticle system is fascinating and has been widely studied, for example, the generation of multiparticle W states and their applications have been extensively discussed [16–24]. √ The general form of the W state for n particles is |W n = (1/ n )|n − 1, 1 where |n − 1, 1 denotes the symmetric state involving n − 1 zeros and 1 one. For example, when n = 4, we obtain |W 4 = 1/2(|1000 + |0100 + |0010 + |0001). An interesting property of the W state is that the entanglement of this state is very robust against particle losses, i.e., the state |W n remains entangled even if any (n − 2) parties lose the information about their particles [16]. The W state is not only of intrinsic interest itself but also of practical importance in quantum information processing. It has been applied to quantum teleportation [17,18], quantum key distribution [19] and quantum telecloning [20,21]. Several schemes for generation of the W state have been proposed [22–24]. In these schemes, the W states are often generated in a local cavity. The entanglement of spatially separated qubits is necessary in all the schemes of quantum computer based on distributed quantum informa-

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tion processing [25–29]. Then, it is very interesting to study the generation of multiparticle entangled state of spatial separated qubits. In this Letter, we propose a scheme to generate W state of n + n qubits, which are trapped in two spatially separated cavities by means of entanglement transfer. We find that the 2n qubits will be entangled in the state |W 2n periodically. Practically, the scheme cannot be completely isolated from the environment, the coupling of the quantum system to the environment, resulting in decoherence, should be taken into account. We also study the effect of cavity dissipation on the entanglement transfer. We find that, in the long time limit, the cavity dissipation brings the whole system to the ground state, but for the first a few periods, we can obtain the W state with very high fidelity; and for the definite cavity decay rate κ, the fidelity of the W state, which we can obtain, increases with the increasing of n. 2. The generation of the W state Our system consists of n + n identical qubits and two spatially separated cavities. We denote the two cavities A and B when the context requires us to differ them, but otherwise they are supposed to be identical. The first n qubits are trapped in cavity A, and the second n qubits are trapped in cavity B. A two-mode nonclassical state field is injected respectively into these two separate cavities. We assume that the dipole–dipole interactions between the qubits in each cavity can be neglected. The ground and excited states for the qubits are respectively denoted by |gj and |ej (j = 1, 2, . . . , n). For the sake of simplicity, we suppose that these qubits be coupled to the field with the same coupling constant g and the field mode be resonant with the transition frequency of qubits in both cavities. Then the Hamiltonian for the interaction between cavity A and the first n qubits in the interaction picture and rotation wave approximations is (h¯ = 1) HA = −ig

n  

aσj+ − a † σj−



(1)

j =1

(analogously for HB ). Here, a † and a are respectively the creation and annihilation operators for the cavity field A, and σj+ = |ej j g| and σj− = |gj j e| (j = 1, 2, . . . , n) are the transition operators for the qubits. It is noted that similar Hamiltonian has been proposed for two charge qubits coupled to the cavity field [30,31]. The total unitary time-evolution operator, thus can be constructed by a direct product of two unitary time-evolution operators representing evolutions of two independent qubitfield interactions UT (t) = UA (t) ⊗ UB (t). We assume that the n + n qubits are initially prepared in the ground state |gn+n = |gn ⊗ |gn ≡ |g1 · · · gn n ⊗ |g1 · · · gn n and the field be initially prepared in the Bell state |Ψ + A,B = √1 (|10 + |01). The ini2 tial state of the whole state is     Ψ (0) = √1 |10 + |01 ⊗ |gn+n 2  1  = √ |1|gn ⊗ |0|gn + |0|gn ⊗ |1|gn . (2) 2

The composite system of the qubits and the field evolves unitarily and the wave function |Ψ (t) = UT (t)|Ψ (0). The evolution operator for a two-level atom resonantly coupled to a single-mode radiation field inside a cavity has been extensively studied for Jays–Cummings model [32,33]. It is noted that for our model there is no more than one excitation in each cavity and each qubit has an equal probability to absorb a photon, so the qubit subsystem in cavity A must be in the Hilbert subspace spanned by the basis {|W n , |gn }. Using Hamiltonian HA and Taylor expansions of sine and cosine functions, we find the analytical form of the evolution operator of the first n qubits and the cavity A in the basis {|W n , |gn } is √ ⎛ ⎞ √ † cos( naa † gt) − sin( √naa† gt) a aa ⎠. UA (t) = ⎝ (3) √ √ sin( √na † agt) † † a cos( na agt) † a a

Apparently UB (t), the evolution operator for the second n qubits and the cavity B, has the same form as UA (t). Then the total wave function of the whole system at time t is given by     Ψ (t) = UT (t)Ψ (0) 1  = √ UA (t)|1|gn ⊗ UB (t)|0|gn 2  + UA (t)|0|gn ⊗ UB (t)|1|gn √ = − sin( ngt)|W 2n |00A,B   √ + cos( ngt)|g2n Ψ + . (4) A,B

From Eq. (4) we find that the field and the qubits periodically√exchange entanglement as time evolves. √At time √ + mπ/( ng) (m is an integer), sin( ngt) = t = π/(2 ng) √ (−1)m , cos( ngt) = 0, and the wave function reduces to |Ψ (t) = (−1)m+1 |W 2n ⊗ |00, namely √ the n + n qubits √ will ng), sin( ngt) = be in the W state. At time t = mπ/( √ 0, cos( ngt) = (−1)m , and the wave function reduces to |Ψ (t) = (−1)m |g2n |Ψ + A,B , namely the n + n qubits will be in the ground state and the field will return to the Bell state |Ψ + A,B . Here, n = 1 means that there is only one qubit in each cavity, which has been discussed in Ref. [10]. n = 2 means that there are two qubits in each cavity, then Eq. (4) will be √   Ψ (t) = − sin( 2gt)|W 4 |00A,B √   + cos( 2gt)|g4 Ψ + A,B , (5) where  1 (6) |eggg + |gegg + |ggeg + |ggge 2 is the four-qubit W state. When t = √π , the wave function

|W 4 =

2 2g

will be |Ψ ( √π ) = −|W 4 ⊗ |00, and the four-qubit W state 2 2g is achieved. 3. The effects of cavity dissipation To study the effects of cavity dissipation on the entanglement transfer, we denote the cavity decay rate by κA and κB for cavity A and cavity B, respectively. The master equation that governs

J.-G. Li et al. / Physics Letters A 361 (2007) 59–62

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the dynamics of the whole system can be given as  dρ(t) = −i HA + HB , ρ(t) dt   + κA 2aρ(t)a † − a † aρ(t) − ρ(t)a † a   + κB 2bρ(t)b† − b† bρ(t) − ρ(t)b† b .

(7)

We also assume that the whole system is in the same initial state as Eq. (2). For this initial state the cavities do not behave as a true continuous-variable system, so that we can easily obtain the analytical results in the presence of the dissipation. Otherwise it is very difficult to obtain the analytical results. In our model, there is no more than one excitation in both cavities, and the qubits in each cavity are equivalent, so the final states of the qubits in each cavity must be symmetric. Then for the leaky cavity, at any time t , the density operator ρ(t) can have nonzero matrix elements only in the Hilbert subspace spanned by the basis {|W n |gn |00, |gn |W n |00, |gn |gn |10, |gn |gn |01, |gn |gn |00}, and Eq. (7) can be solved in this subspace. For the sake of simplicity, we let κA = κB = κ. After some calculation, we find that the density operator ρ(t) takes the form ρ(t) = C1 |W 2n2n W | ⊗ |0000| + C2 |g2n2n g| ⊗ |0000| 

 + C3 |W 2n2n g| ⊗ |00 Ψ +     + |g2n2n W | ⊗ Ψ + 00|    + C4 |g2n2n g| ⊗ Ψ + Ψ + , (8) where 2ne−κt  1 − cos(Θgt) , Θ2 e−κt  C2 = 1 + 2 ηΘ sin(Θgt) + η2 cos(Θgt) − 4n , Θ √ e−κt  Θ sin(Θgt) − η + η cos(Θgt) , C3 = n 2 −Θ C1 =

and   e−κt  ηΘ sin(Θgt) + η2 − 2n cos(Θgt) − 2n −Θ 2  with Θ = 4n − η2 , η = κ/g. When κ = 0, Eq. (8) reduces to the density matrix of Eq. (4). When κ = 0, we will use the state preparation fidelity to study the influence of cavity dissipation on the generation of the W state. The so-called state preparation fidelity F of a 2nqubit state ρ is defined as

Fig. 1. The fidelity as a function of κ/g and the scaled time gt for n = 2.

then from Eq. (9) and Eq. (11) the state preparation fidelity is obtained as   F ρq (t) = 2n W |ρq (t)|W 2n = C1 =

(9)

A sufficient condition for 2n-partite entanglement is given by [34,35] 1 F . (10) 2 By tracing over the cavity field variables of Eq. (8), we obtain the reduced density operator of the qubit subsystem as ρq (t) = C1 |W 2n2n W | + (C2 + C4 )|g2n2n g|,

(11)

(12)

From Eq. (11) we can see that the probability of finding the qubit subsystem being in the 2n-qubit W state is also C1 . We choose n = 2 and plot F as a two-variable function of κ/g and the scaled time gt in Fig. 1. We observe that when κ/g = 0 (without cavity dissipation), F oscillates as √ 2g) time evolves and reaches its maximum 1 at t = π/(2 √ + √ √ mπ/( 2g), which means that at t = π/(2 2g) + mπ/( 2g), the four qubits will be in the W state. When κ/g = 0, F also oscillates as time evolves, but the amplitude of the oscillation becomes smaller√and smaller, and it is noted from Fig. 1 that when t π/(2√ 2g) the fidelity is very high. For example, when t π/(2 2g) for κ/g = 0.1, C1 0.9, C2 0.1, C3 0 and C4 0, then from Eq. (11) we get

C4 =

F (ρ) ≡2n W |ρ|W 2n .

2ne−κt  1 − cos(Θgt) . 2 Θ

 ρq

π √ 2 2g

 = 0.9|W 44 W | + 0.1|gggggggg|.

(13)

Thus at this time the four qubits will be in the W state with the probability 0.9. Now we study the effect of qubit number n on the fidelity. Choosing κ/g = 0.1, we plot F as functions of the scaled time gt for different values of n in Fig. 2. It can be seen from Fig. 2 that the fidelity of the W state we can obtain, is increasing with n. This can be understood as follows: From Eq. (12) we can find thatthe fidelity reaches it maximum at time tm π/(Θg) = π/( 4n − η2 g). It is easy to see that tm is a decreasing function of n, and the decaying factor e−κtm in Eq. (12), characteristic of the dissipation, is increasing with n, and when n → ∞, e−κtm → 1. This means that the fidelity, i.e., C1 , is also an increasing function of n. We can also say that the probability of obtaining the W state in the case of large n is higher than that in the case of small n.

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Fig. 2. The fidelity as a function of the scaled time gt for n = 1, n = 4, and n = 20 with κ/g = 0.1.

4. Discussion and conclusions In this Letter we have proposed a scheme to generate W state of 2n qubits by means of entanglement transfer. We have found that the field and the qubits exchange entanglement periodically. We have also found that in the case of dissipation, the 2n-qubit W state can be generated with very high probability, which increases with the increase of n. We should point out that our scheme can be generalized to (n + m)-qubit (m = n) case in which there are n qubits located in cavity A and m qubits in cavity B. Generally we could not get the (n + m)-qubit |W  state exactly like the n + n case, but from numerical calculations we have found that, by choosing the appropriate interaction time, we could also get the (n + m)-qubit |W  state with quite high fidelity. It has been shown [8] that it might be easier to implement the scheme we have discussed for the charge qubits and cavity field system. The charge qubits can be fixed locally and the interaction between the charge qubits and the cavity can be controlled by tuning the charge qubit on/off resonance with the cavity mode [7,36], and in this way the interaction time can be easily controlled. Acknowledgements The work was supported by National Natural Science Foundation of China under Grant No. 10374007. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, 2000.

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