Multipartite quantum coherence under electromagnetic vacuum fluctuation with a boundary

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Nuclear Physics B ••• (••••) ••••••

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www.elsevier.com/locate/nuclphysb

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Multipartite quantum coherence under electromagnetic vacuum fluctuation with a boundary

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Zhiming Huang

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School of Economics and Management, Wuyi University, Jiangmen 529020, China

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Received 4 January 2019; received in revised form 11 October 2019; accepted 3 November 2019

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Editor: Stephan Stieberger

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Abstract

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We investigate the dynamics of multipartite quantum coherence (QC) for accelerated detector qubit coupled with fluctuating electromagnetic field by presence of a perfectly reflecting boundary. We firstly derive the master equation that the detector evolution obeys. We find that without boundary, QC declines under the impact of Unruh thermal effect and vacuum fluctuation. However, with a boundary, the degradation, floating and protection of QC are closely related to boundary effect, detector polarization and acceleration. The presence of boundary can effectively protect QC under the influence of vacuum fluctuation and Unruh thermal effect in some certain conditions and give us more freedom of adjusting the QC behaviors. Besides, it is shown that the QC of W state manifests better robustness than that of GHZ state under the influence of the vacuum fluctuation and acceleration. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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1. Introduction

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QC derived from quantum state superposition is a crucial physical resource for quantum physics and various quantum technologies, such as quantum optics [1–6], quantum thermodynamics [7–11], quantum information [12,13], and quantum biology [14–17]. QC is intrinsically associated with other quantum resources, such as entanglement [18,19] and quantum correlation [19–21]. Recently, Baumgratz et al. [22] present a theoretic framework for measuring coherence based on the resource theory. Based on such a paradigm, some measures for quantifying

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E-mail address: [email protected]. https://doi.org/10.1016/j.nuclphysb.2019.114832 0550-3213/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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coherence are proposed which satisfy the defining conditions of coherence measure, such as nonnegativity, monotonicity and convexity [22]. Some measures, such as the relative entropy of coherence and l1 norm of coherence [22] meet the conditions. l1 norm of coherence is a very intuitive coherence measure since coherence properties of a quantum state are determined by the off-diagonal elements of density matrix, which is defined as  Q= |ρi,j |. (1) i=j

However, the unavoidably interaction between quantum system and surrounding environment would induce decoherence. For example, the coherence affected by vacuum fluctuation of quantum field would be damaged [23,24]. Moreover, QC can be degraded by the Unruh thermal noise in accelerated frames [25,26]. A reflecting boundary in the quantized field would modify the vacuum fluctuation [27,28]. In this work, we explore how the vacuum fluctuation, acceleration and boundary affect the tripartite QC of W state and GHZ state when one of them is modeled an uniformly accelerated detector coupled with electromagnetic vacuum fluctuation with a perfectly reflecting boundary. Multipartite QC affected by electromagnetic field and Unruh noise is seldom studied, and the quantum system affected by electromagnetic field is a more practical model compared with the massless scalar field [25]. It is found that QC decreases with the acceleration and evolution time in the free space. With the presence of a reflecting boundary, the QC may increase or decline dependent on the detector position, polarization and acceleration, and QC can be preserved in some special cases with a boundary, which affords a way to manipulate QC. It is found that that W state is more robust than GHZ state against Unruh thermal noise and electromagnetic vacuum fluctuation, which may be helpful to quantum information processing. Our research will deepen our understanding for the theory of open quantum system and relativistic quantum information. The organization of this paper is as follows. In Section 2, we introduce the model of a qubit modeled by accelerated detector interacting with quantized electromagnetic field. In Section 3, we firstly solve the master equation that describes the system evolution and then we discuss the QC behaviors in detail. A brief conclusion is given in the last section. 2. Interaction model between detector and electromagnetic field We consider that a semiclassical Unruh-Dewitt detector weakly couples with quantized electromagnetic field. The detector Hamiltonian can be written as HS = ω2 σ3 with ω being the energy gap. HI = −er · E(x(τ )) is the interaction Hamiltonian between detector and fluctuating electromagnetic field, with er being the electric dipole moment and E(x) denoting the electric field strength. Thus, the whole Hamiltonian can be written as H = HS + HF + HI , where HF is the Hamiltonian of the electromagnetic field. Supposing that the initial state of the total system takes the form ρtot (0) = ρ(0) ⊗ |00|, where ρ(0) denotes the initial reduced density matrix of the detector and |0 is the vacuum state of the external electromagnetic field. Under the BornMarkov approximation, by tracing over the field degrees of freedom, the evolution equation of the detector can be described with the Kossakowski-Lindblad form [29–31] ∂ρ(τ ) (2) = −i[Heff , ρ(τ )] + L[ρ(τ )], ∂τ where 1 1 i Heff = σ3 = {ω + [K(−ω) − K(ω)]}σ3 , (3) 2 2 2

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and

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3 1  L[ρ] = Sij [2σj ρσi − σi σj ρ − ρσi σj ]. 2

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i,j =1

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Sij = Aδij − iB ij k δ3k − Aδ3i δ3j ,

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(5)

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1 A = [ G(ω) + G(−ω)], 4

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1 B = [ G(ω) − G(−ω)], 4

(6)

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G(ω) =

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(7)

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Gij (λ) =

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dτ eiλτ Gij (τ ),

(8)

P Kij (λ) = πi

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−∞

∞ −∞

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Gij (ω) dω , ω−λ

(9)

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τ )

(x  )|0

where Gij (τ − = 0|Ei (x)Ej is the field correlation function and P is the principal value. Note that c = h¯ = e = 1 throughout this article.

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3. Multipartite QC dynamics

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In this section, we discuss the tripartite QC behaviors. We consider three observers, Alice, Bob, and Charlie, each of them owns an Unruh-DeWitt detector modeled as a two-level noninteracting qubit. Only Alice´s detector moves with uniform acceleration a along the x axis and is coupled to electromagnetic vacuum fluctuation in the presence of a reflecting boundary. Supposing that the reflecting boundary is placed at y = 0 and the distance between the detector and the boundary is y, then the trajectory of Alice’s detector can be described as 1 t (τ ) = sinh aτ, a

1 x(τ ) = cosh aτ, a

y(τ ) = y,

(10)



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With the method of images [32], the two point correlated function has the form 

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z(τ ) = 0.

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0|Ei (x(τ ))Ej (x(τ ))|0 = 0|Ei (x(τ ))Ej (x(τ ))|00 + 0|Ei (x(τ ))Ej (x(τ ))|0b , (11)

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∞

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where di = 0|ri |1 is the electric dipoles of the detector, and Gij (ω) and Kij (ω) denote Fourier and Hilbert transforms of field correlation function respectively, defined as

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di dj∗ Gij (ω),

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with

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where

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The Kossakowski matrix Sij can be written explicitly as

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where 0|Ei (x(τ ))Ej (x(τ  ))|00 is the electric two-point function in the unbounded space:

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0|Ei (x(τ ))Ej (x(τ  ))|00 =

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1 , ×  2  2 (x − x ) + (y − y ) + (z − z )2 − (t − t  − iε)2

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1 (∂0 ∂0 δij − ∂i ∂j ) 4π 2

∂

with ε → +0, being the differentiation with respect to boundary 0|Ei (x(τ ))Ej (x(τ  ))|0b is given by

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(12)

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The two-point functions with the

1 0|Ei (x(τ ))Ej (x(τ ))|0b = − [ (δij − 2ni nj ) ∂0 ∂0 − ∂i ∂j ] 4π 2 1 . × (x − x  )2 + (y + y  )2 + (z − z )2 − (t − t  − iε)2

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Here n = (0, 0, 1) is the unit normal vector. With a Lorentz transformation from the laboratory frame to the frame of the detector, according to the contour integral, the Fourier transforms (8) of the two point functions can be calculated,

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ω3 Gij (ω) = [gij − hij ], 3π(1 − e−2πω/a ) where gij =

a2 ω2

(14)

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2ω sinh−1 ay + 3[a y (4y ω a + 4y ω − 5) − 2] sin }, a  1 2ω sinh−1 ay 2 y 2 + 1(2a 2 y 2 + 1) cos h33 = {6yω a 16y 3 ω3 (a 2 y 2 + 1)3/2 a

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2 2

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2ω sinh ay }, a  −1 2ω sinh−1 ay 2 y 2 + 1(2a 2 y 2 − 1) cos h12 =h21 = {6yω a a 16y 2 ω3 (a 2 y 2 + 1)5/2

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2ω sinh−1 ay + 3[4y 2 (a 2 y 2 ω2 + a 2 + ω2 ) + 1] sin }, a h13 =h31 = 0, h23 =h32 = 0.

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(15)

Note that gij and hij correspond to the unbounded case and bounded case respectively. According to Eqs. (6) and (7), we can obtain the coefficients Sij (5)

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2ω sinh−1 ay − 3[2y 2 (2y 2 a 4 − 2y 2 ω2 a 2 + a 2 − 2ω2 ) + 1] sin }, a  1 2ω sinh−1 ay 2 y 2 + 1(2a 4 y 4 + a 2 y 2 + 2) cos h22 = {6yω a a 16y 3 ω3 (a 2 y 2 + 1)5/2

+ 3[4y 2 ω2 (a 2 y 2 + 1) − 1] sin

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+ 1 for i = j , gij = 0 for i = j , and

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3   coth πω a A= (gij − hij )dˆi dˆj∗ , 4 i,j =1

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(16)

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with  = ω3 |d|2 /3π denoting the spontaneous emission rate of the detector qubit, and dˆi = di /|d| being a unit vector. Now, we consider to solve the master equation. Note that we only consider Alice´s detector moves with uniform acceleration and interacts with external field with a boundary. Bob´s and Charlie´s detectors at rest can be regarded as a closed system, so there are not field correlation functions associated with Bob´s and Charlie´s detectors. For this reason, only adding one 4 ⊗ 4 identity operator I into to the master equation (2), the master equation that governs the evolution process of the three detectors becomes

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∂ρ(τ ) = −i[Heff , ρ(τ )] + L [ρ(τ )], ∂τ

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(17)

 where = 12 (σ3 ⊗ I ) = 12 {ω + 2i [K(−ω) − K(ω)]}(σ3 ⊗ I ), L [ρ] = 12 3i,j =1 Sij [2(σj ⊗ I )ρ(σi ⊗ I ) − (σi ⊗ I )(σj ⊗ I )ρ − ρ(σi ⊗ I )(σj ⊗ I )]. Any three-qubit states can be represented  Heff

with Pauli matrices

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ρ=

3 3 3 1 

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bi,j,k (τ )σi ⊗ σj ⊗ σk .

(18)

i=0 j =0 k=0

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By plugging Eq. (18) into Eq. (17), the non-trivial coupled differential equations are given  b0,j,k (τ ) = 0,  b1,j,k (τ ) = −2Ab1,j,k (τ ) − b2,j,k (τ ),  b2,j,k (τ ) = b1,j,k (τ ) − 2Ab2,j,k (τ ),  b3,j,k (τ ) = −4(Ab3,j,k (τ ) + Bb0,j,k (τ )).

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(19)

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(20) (21)

√1 (|000 + |111) and |W = √1 (|100 + |010 + |001). 3 2

We consider two initial states |GHZ = By solving above differential equations with these initial states, the final evolution states of detectors respectively are ⎛ ⎞ T1 0 0 0 0 0 0 T2 ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ 1 ⎜ 0 0 0 T3 0 0 0 0 ⎟ ⎟, ρGHZ = ⎜ (22) ⎟ 2⎜ ⎜ 0 0 0 0 T4 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 0 0 0 ⎠ T5 0 0 0 0 0 0 T6 ⎛ ⎞ T3 0 0 0 0 0 0 0 ⎜ 0 T1 T1 0 T2 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 T1 T1 0 T2 0 0 0 ⎟ ⎜ ⎟ 1⎜ 0 0 0 0 0 0 0 0⎟ ⎜ ⎟, (23) ρW = ⎜ 3 ⎜ 0 T5 T5 0 T6 0 0 0 ⎟ ⎟ ⎜ 0 0 0 0 0 T 4 T4 0 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 T4 T4 0 ⎠ 0 0 0 0 0 0 0 0

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Fig. 1. QC of (a) W state and (b) GHZ state as function of τ and a/ω for the case without boundary. e−2Aτ [A cosh(2Aτ )−B sinh(2Aτ )] , 2 (A+B)e−2Aτ sinh(2Aτ ) iτ −2Aτ , T5 = e , and T6 A

where T1 =

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T2 = e−2Aτ −iτ , T3 =

= we can obtain QC for GHZ state and W state

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QGHZ = e−2Aτ , (24) 2 B sinh(2Aτ ) (A + B) sinh(2Aτ ) |+ + 2]. (25) QW = e−2Aτ [| cosh(2Aτ ) − 3 A A When Alice´s detector is very far from the boundary (y → ∞, without boundary), we get A=

(a 2

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+ ω2 ) coth( πω a ) , 4ω2

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(a 2

+ ω2 )

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τ a 2 +ω2 coth πω a − 2ω2

f1 = f 3 =

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τ a 2 +ω2 coth πω a − 2ω2

− 1) sin 2yω + 6yω cos 2yω , 16y 3 ω3

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4 2 QW = e + . (27) 3 3 From above equations, we can easily analyze that QC is independent of the detector polarization, and QC exponentially decreases with the evolution time and acceleration in the case without boundary. The different behaviors of QC for W state and GHZ state are that the former decreases to 23 and the latter reaches a stable value 0 as evolution time and acceleration grow. From Fig. 1, we can clearly see that QC for GHZ vanishes while QC for W state emerges coherence freezing as evolution time and acceleration increase. When a → 0, A = B = 4 (1 − 3i=1 |dˆi |2 fi ) with QGHZ = e

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(26)

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and QC can be obtained

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f2 =

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3(2yω cos 2yω − sin 2yω) . 8y 3 ω3

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(28) The evolution model reduces to the evolution model of static detector [33]. Fig. 2 shows that vacuum fluctuation degrades QC of W state. When acceleration is relatively small, QC oscillatory develops a stable value with the increasing detector distance from the boundary. QC behaves differently with different polarization of detector. When distance from the boundary is very small and detector is polarizable parallel with respect to the boundary (see Fig. 2 (a)), QC is effectively protected from the influences of acceleration and vacuum fluctuation.

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Fig. 2. QC of W state as function of ωy, the detector is polarized along (a) the positive x axis (dˆ = (1, 0, 0)) and (b) the positive y axis (dˆ = (0, 1, 0)).

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Fig. 3. QC of GHZ state as function of ωy, the detector is polarized along (a) the positive x axis (dˆ = (1, 0, 0)) and (b) the positive y axis (dˆ = (0, 1, 0)).

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When the detector is polarized vertical to the boundary (see Fig. 2 (b)), the QC behaviors could be different and exist an opposite changing when detector is near the boundary. Actually, we easily deduce that when detector is not completely vertical polarization, the boundary can protect the QC to some degree. QC of GHZ state has the similar behaviors (see Fig. 3). Specially when detector is placed very close to the boundary (y → 0), we obtain (a 2 + ω2 ) coth( πω (a 2 + ω2 ) ˆ 2 a ) ˆ 2 A= |d2 | , B = |d2 | . (29) 2 2ω 2ω2 QC will remain the maximum value 2 and is totally protected from the impact of acceleration and vacuum fluctuation, since the decay rate becomes zero when the detector is polarizable parallel to the boundary (|dˆ2 | = 0), which stems from the field is completely reflected and the system actually becomes a closed system. When the detector is polarizable vertical to the boundary (|dˆ2 | = 1), comparing Eq. (26) with Eq. (29), it is easily analyzed that QC of W state and GHZ state decays exponentially to 23 and 0 respectively with evolution time and acceleration, and the decay rate doubles and QC is degraded more quickly in the case with a boundary. Fig. 4 shows that in a interval of small accelerations, QC of W state decreases to a stable value with slight oscillations with growing acceleration when the distance from the boundary is relatively large. QC of GHZ state emerges analogous changes except that it declines to zero with increasing acceleration (see Fig. 5).

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Fig. 4. QC of W state as function of a/ω with τ = 1 for different distances from the boundary, the detector is polarized along (a) the positive x axis (dˆ = (1, 0, 0)) and (b) the positive y axis (dˆ = (0, 1, 0)).

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Fig. 5. QC of GHZ state as function of a/ω with τ = 1 for different distances from the boundary, the detector is polarized along (a) the positive x axis (dˆ = (1, 0, 0)) and (b) the positive y axis (dˆ = (0, 1, 0)).

4. Conclusion

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In this paper, we have studied the dynamics of multipartite QC for an uniformly accelerated two-level detector weakly coupled with a bath of fluctuating electromagnetic field with a perfectly reflecting boundary. The master equation that governs the system evolution is derived. It is found that vacuum fluctuation and acceleration decay QC in the case without a boundary. However QC may be effectively protected from the influence of the vacuum fluctuation and acceleration when the detector is very close to the boundary and is not polarizable vertical to the boundary. QC presents oscillatory behaviors within certain range of distance from the boundary and acceleration. The boundary and acceleration provide us more freedom to manipulate the QC behaviors, and QC could offer us a measure to observe the boundary effect and Unruh effect. In addition, QC for GHZ is tending to zero, while QC for W state presents coherence freezing. In this sense, we can say W state is more robust than GHZ state against the influence of vacuum fluctuation and acceleration, which may be useful for quantum information processing. Declaration of competing interest

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There is no conflict of interest.

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References

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