Tetrapartite entanglement of fermionic systems in noninertial frames

Optik 127 (2016) 9788–9797

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Optik journal homepage: www.elsevier.de/ijleo

Original research article

Tetrapartite entanglement of fermionic systems in noninertial frames Yazhou Li a , Cunjin Liu a , Qi Wang a , Haoliang Zhang a,b , Liyun Hu a,b,∗ a b

Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China Key Laboratory of Optoelectronic and Telecommunication of Jiangxi, Nanchang 330022, China

a r t i c l e

i n f o

Article history: Received 26 May 2016 Accepted 25 July 2016 Keywords: Entanglement Noninertial frame GHZ state Tangles

a b s t r a c t Under the single-mode approximate, we theoretically investigate the properties of entanglement for four-mode Fermi GHZ state when some observers are in uniform acceleration with respective to others. We use two different definitions for calculating the degree of entanglement, i.e., the arithmetic and geometric average of tangles. It is shown that a maximally entangled state in an inertial frame shall become one with less entanglement in noninertial frame and the partial and the whole entanglements decrease as the number of accelerated observers. In particular, for 1–3 accelerated observers, there is still entanglement in the limit of infinite acceleration for all tangles. However, the sudden death shall be present for both the whole and the part one only when four observers are accelerated. In addition, we make a comparison for these two definitions of entanglement. It is interesting to notice that, for three accelerated observers, the difference first increases and then decreases as the increasing parameter r. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction As a property of multipartite quantum states, entanglement of quantum states is not only the central concept but also the most desirable physical resources for a variety of quantum information processing tasks [1–3], such as quantum teleportation [4], quantum control [5] and quantum simulation [6]. For this reason, much attention has recently been paid to investigating the properties of entanglement for various states [7]. As a combination of general relativity, quantum field theory and quantum information theory, the quantum information in an accelerated frame has been a focus of research on quantum information science over recent years for both theoretical and experimental reasons. Doubtlessly, it is more important to research the revolution of quantum states in a general frame. Because it can not only help us grasp the quantum information of the general reference frame and provide a more complete framework for the quantum information theory, but also play an important role in the understanding of the entropy and information paradox of black holes [8–10]. As a result, there are an increasing number of articles discussing the entanglement in the accelerated setting, in particular on how the Unruh and Hawking effect affect the degree of entanglement [11–23], including quantum systems with continuous-variable [24]. The bipartite entanglement in noninertial frames was initially investigated by Fuentes-Schuller and Mann [25]. They showed that the maximal bipartite entanglement is degraded when the observers are relatively accelerated. As the increasing

∗ Corresponding author at: Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China E-mail address: [email protected] (L. Hu). http://dx.doi.org/10.1016/j.ijleo.2016.07.069 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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acceleration, the degradation of entanglement becomes larger and larger, and the bipartite state eventually reduces to a separable state at infinite acceleration. The phenomenon is called as Unruh decoherence which is closely related to the Unruh effect [26]. Then the tripartite entanglement of scalar field in noninertial frames was examined by Hwang et al. [27]. They showed that the tripartite entanglement decreases as the increasing acceleration but it is different from bipartite one where one observer has an infinite acceleration. The case is true for the tripartite entanglement of Fermi field. In general, the entanglement of both bosonic modes and fermionic fields decreases from the perspective of observers being uniformly accelerated. However, it is interesting to notice that the entanglement completely disappears in infinite-acceleration limit for bosonic cases, while a nonzero finite amount of entanglement can be remained in the limit for Fermi fields. In addition, the entanglement is dependent on observers and can also be transferred to others in noninertial frames. However, the entanglement in inertial frames can be preserved regardless of its transformation among different degrees of freedom [28–30], whose reason is still not clear. Although many efforts on the properties of entanglement have been made, a good understanding of such resource is only limited in bipartite systems and tripartite partly. For higher dimensional or multipartite quantum systems, the property and quantification of entanglement is still an important topic. In this paper, we will investigate the tetrapartite entanglement of Fermi fields when some of observers are under the case of acceleration for the initial Greenberger–Horne–Zeilinger (GHZ) states. In particular, we will focus our attention on how the acceleration affect the degree of tetrapartite entanglement and examine the properties of entanglement, such as entanglement sudden death. It is shown that the tetrapartite entanglement decreases as the increasing accelerated observers and the sudden death of entanglement happens only when all observers −1/2

have the same parameter of acceleration r ≈ 0.417, where cos r = (e−2ωc/a + 1) , and the constants ω, c and a in the exponential item stand for Fermi particle’s frequency, the speed of light in vacuum and observer’s acceleration, respectively. This paper is arranged as follows. In Section 2 we introduce some concepts about quantum entanglement theory, in particular the concurrence and negativity, which will be used to measure the degree of entanglement. In Section 3 we briefly discuss the essential features of the Fermi fields in the inertial frame. In Section 4 we study the entanglement of four-body in the noninertial frame. And we make a discussion and conclusion in the last section. 2. Measures of tetrapartite entanglement It is all known that, for an arbitrary bipartite state AB , there are two usual measures of entanglement: Concurrence and Negativity NAB [31–33]. Here we will use the negativity as a measurement of entanglement. The negativity NAB is defined as TA NAB = ||AB || − 1, TA || = Tr where TA means the partial transpose of AB and ||AB



(1) TA )† TA is the trace norm of a matrix TA . (AB AB AB

For tripartite quantum system, there are many different criteria for quantifying tripartite entanglement. Among them, three-tangle [31] and -tangle [34] can be considered as the most popular ones. For an arbitrary tripartite state ABC , there is Coffman–Kundu–Wootters (CKW) monogamy inequality [31] 2 2 2 NAB + NAC ≤ NA(BC) ,

(2)

where NA(BC) and NAB (NAC ) are ‘two-tangle’ and ‘one-tangle’ entanglement, which are defined as the negativity of the mixed TA || − 1, respectively. Based on the difference between the two sides of Eq. (2), one can state AB = trC (ABC ) and NA(BC) = ||ABC also introduce the three-tangle entanglement, which has three different forms, i.e., 2 2 2 A = NA(BC) − NAB − NAC , 2 − NAB

(3)

2 − NBC ,

(4)

2 2 2 C = NC(AB) − NAC − NBC .

(5)

B =

2 NB(AC)

Thus the -tangle of ABC is defined as an average value of these three three-tangles: ABC =

1 (A + B + C ) . 3

(6)

The -tangle is a good quantifier for the entanglement of tripartite states, which can describe the entanglement of three qubits state. In a similar way to defining the measures of entanglement above, we can extend directly the definitions to the four qubit states [35,36]. That is to say, we can still discuss the degree of entanglement by using negativity. For four qubit states, we may define TA NA(BCD) = ||ABCD || − 1,

NA(BC) = NAB =

TA ||ABC || − 1,

TA ||AB || − 1,

(7) (8) (9)

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Y. Li et al. / Optik 127 (2016) 9788–9797 TAB NAB(CD) = ||ABCD || − 1,

(10) √ where TA and TAB indicate the partial transpose over the first qubit A and AB, respectively, and ||O|| = tr O† O stands for the trace norm of an operator O. And NA(BCD) , NA(BC) , NAB , and NAB(CD) describe the entanglement between two parts, such as 1–3 tangle, 1–2 tangle, 1–1 tangle, and 2–2 tangle entanglement. For instance, NA(BCD) describes the entanglement between the part A and the other part of B, C and D, and NA(BC) describes the entanglement between the A and B, C after tracing over qubit D. The definition is similar to NAB , NAB(CD) (here AB = trCD (ABCD )). Here, the 1–1 tangle and 1–3 tangle satisfy the following CKW monogamously inequality relation [31]: 2 2 2 2 NA(BCD) ≥ NAB + NAC + NAD .

(11)

Thus, based on the difference between two sides of Eq. (11) we can introduce the residual tangles (four-tangle entanglement) as 2 2 2 2 A = NA(BCD) − NAB − NAC − NAD ,

(12)

2 2 2 2 B = NB(ACD) − NBA − NBC − NBD ,

(13)

2 2 2 2 − NCA − NCB − NCD , C = NC(ABD)

(14)

2 2 2 2 − NDA − NDB − NDC . D = ND(ABC)

(15)

It is clear that there are four different forms for j . Thus we may introduce 4 tangle to measure the whole degree of entanglement for four-mode Fermi fields, define as follow: 4 =

1 (A + B + C + D ) , 4

(16)

which will be used to measure the degree of entanglement for the following four-mode systems. In addition, as a comparison, here we introduce another whole entanglement [37], defined as √ ˘4 = 4 A B C D . (17) In the follows, we will examine the whole entanglement of four qubits system by using these two definitions of entanglement. 3. Entanglement of GHZ state In order to realize our discussion about the entanglement of four-mode GHZ in noninertial frame, here we first consider the entanglement of GHZ state by using the above entanglement criteria. Doing so, we can form a clear comparison between the entanglement in inertial frame and that in noninertial frame. Thus we first introduce the initial maximally entangled state (GHZ state), which is shared by Alice, Bob, Charlie and Daniel, defined as 1 |GHZABCD = √ [|0000 + |1111], 2

(18)

where |0000 denotes to |0A ⊗ |0B ⊗ |0C ⊗ |0D , so as to the |1111 and the following cases. The density operator for Eq. (18) can be easily obtained by using ABCD = |GHZABCD GHZ| = 12 [|0000 0000| + |0000 1111| + |1111 0000| + |1111 1111|]. Thus we can calculate those needed partial transpose density operator. For instance, we have TA = A(BCD)

and

1 [|0000 0000| + |1000 0111| + |0111 1000| + |1111 1111|], 2



TA ||A(BCD) || = tr

TA TA (A(BCD) )† A(BCD) = 2,

(19)

(20)

thus the 1–3 tangle of NA(BCD) is given by TA NA(BCD) = ||A(BCD) || − 1 = 1.

(21)

In a similar way, we can also get other 1–3 tangles, i.e., NB(ACD) = NC(ABD) = ND(ABC) = 1.

(22)

From Eqs. (22), we can clearly see that there are the same entanglement between one part and the other parts (say A and BCD). The physical reason lies in the equivalent property among the four observers. In addition, we can also calculate the 2–2 tangle. For the density operator ABCD , the corresponding partial transpose TAB can be calculated as, for example, density operator ABCD TAB AB(CD) =

1 [|0000 0000| + |1100 0011| + |0011 1100| + |1111 1111|], 2

(23)

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where TAB denotes the partial transpose of AB(CD) . Then the 2–2 tangles are given by NAB(CD) = NAC(BD) = NAD(BC) = 1.

(24)

However, our calculation shows that both 1–2 tangles and 1–1 tangles are zero, i.e., NA(BC) = NB(CD) = NC(AD) = ND(AB) = 0,

(25)

Nij = 0, (i, j = A, B, C, D, i = / j),

(26)

and

which indicate that there is no entanglement between arbitrary observer from the point of negativity, and only for all of them there are entanglement which is maximum value of 1. On the other hand, using Eqs. (12)–(17), the residual tangles and the whole degree of entanglement for four-mode are calculated as A = B = C = D = 1, √ 4 = ˘4 = 4 A B C D = 1.

(27) (28)

From Eqs. (24)–(26) we can see that, for the four qubits GHZ state, the negativity between each qubit and the rest of the system is 1, and all the pairwise negativities are zero which indicates that the qubits in each pair are classically correlated but not entangled. That is to say, among arbitrary reduced two bodies or reduced three bodies, there is no the entanglement, while the entanglement only exist among these four qubits. In addition, from Eq. (28) it is clear that the GHZ state is a maximally entangled one, as expected. 4. Entanglement of GHZ state in noninertial frames In this section, we consider the entanglement properties of four-mode GHZ state in noninertial frames. Let’s assume that Alice, Bob, Charlie, and Daniel share a four-mode GHZ entangled state initially. In this section and subsequence we will use the following notation: A, B, C, D indicate the observers Alice, Bob, Charlie, and Daniel, respectively. When one of them moves with respect to others in a uniform acceleration a, then its Fermi vacuum and one-particle states under the single-mode approximation [15] can be given by |0M → cos r|0I |0II + sin r|1I |1II ,

(29)

|1M → |1I |0II ,

(30) −1/2

where the subscript M stands for Minkowski and cos r = (e−2ωc/a + 1) . The constants ω, c and a, in the exponent stand, respectively, for Fermi particle’s frequency, the speed of light in vacuum and observer’s acceleration, and the acceleration parameter r is in the range 0 ≤ r < (/4) for 0 ≤ a <∞. Here, I and II indicate the uniformly accelerated observer Alice(AI ) confined to the region I, and the fictitious complementary observer AII in region II, respectively. 4.1. One observer with acceleration In this subsection, we first consider that Alice moves with respect to other three observers with a uniform acceleration a. Using Eqs. (29) and (30), the Alice’s acceleration transforms the initial GHZ state in Eq. (18) into the following form: 1 |GHZAI,II BCD = √ [cos r|0AI |0AII |000BCD + sin r|1AI |1AII |000BCD + |1AI |0AII |111BCD ]. 2

(31)

Note that Alice is actually disconnected from the region II, this is to say, |0II , |1II are a physically inaccessible state for other observers, the only information that is physically accessible to the observers is encoded in modes B, C, D described by other three observers and the mode I described by Alice. Thus we need take the trace over the state of region II. After doing so, we can obtain the reduced quantum state shared by A, B, C and D as the following mixed state: AI BCD =

1 [|1111 1111| + cos2 r|0000 0000| + sin2 r|1000 1000| + cos r (|0000 1111| + |1111 0000|)], 2

(32)

where for simplicity, we have used the |mnpq m n p q | to denote |mnpqAI BCD m n p q |. Using Eq. (32), the tangle of two part of the systems can be calculated as NAI (BCD) =

1 1 cos 2r + 2 2



cos2 r + sin4 r,

(33)

and NB(AI CD) = NC(AI BD) = ND(AI BC) = cos r, Nij = 0, (i, j = AI , B, C, D, i = / j) as well as N(AI B)(CD) = cos r, which lead to the following residual tangles: A =



1 cos 2r + 4



cos2 r + sin4 r

2

,

(34)

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Fig. 1. The partial and whole entanglement of the four-qubit state as a function of the acceleration parameter r. Here, for simplicity, we denote AI as A.

B = C = D = cos2 r,

(35)

thus the whole four-qubit state entanglement is given by

  1

4 =

1 4

˘4 =

cos3/2 r √ 2

4

cos 2r +

and



cos2 r + sin4 r

 cos 2r +



2



+ 3cos2 r ,

(36)

cos2 r + sin4 r.

(37)

In Fig. 1, we plot the entanglement of the four-qubit state as a function of the acceleration parameter r. From Fig. 1, we can clearly see that (i) the entanglement described by both tangles and the whole system decrease √ with the increasing r; (ii) the 1–3 tangle NB(AI CD) equals to that of the 2–2 tangle NAI B(CD) as expected, which approaches 2/2 0.707 in the limit of a → ∞ corresponding to r → (/4); (iii) however, the 1–3 tangle NAI (BCD) degrades quicker than NB(AI CD) (or NAI B(CD) ) as the √ increasing r and finally approaches 3/4 0.433 in the limit of a → ∞ . These results show that the entanglement of both 1–3 and 2–2 tangles will never disappear no matter what the acceleration is. That is to say, the tetrapartite entanglement among the observers is still present when one of them falls into a black hole. In addition, noticing the difference between NAI (BCD) and NB(AI CD) , we can know clearly that the entanglement can be changed among the observers when one observer has an acceleration. However, at the point of r = 0 corresponding to the case without acceleration, different tangles share the same value of entanglement in inertial frame. In addition, by comparing the two whole entanglement defined by Eqs. (36) and (37) (see Fig. 1), we can see that ˘ 4 tangle decreases quicklier than 4 tangle as the increasing acceleration. 4.2. Two observers with acceleration Now, we consider the case of A and B have the same acceleration of a. In this case, the initial GHZ state can be transformed into the following form 1 1 | GHZ = √ [cos2 r|0000|00 + sin2 r|1111|00 + sin 2r (|1100|00 + |0011|00) + |1010|11]. 2 2

(38)

Here for simplicity, we have denoted |abcd ≡ |abI,II ⊗ |cdI,II , and |ef = |efCD . In a similar way to driving Eqs. (32), the reduced density operator is given by (by tracing over II) AI BI CD =

1 [cos4 r|0000 0000| + cos2 r|0000 1111| + cos2 r|1111 0000|] + |1111 1111| + sin2 rcos2 r|1000 1000| 2 + sin2 rcos2 r|0100 0100| + sin4 r|1100 1100|]

then we can get NBI C(AI D) = tr

(39)



(ATBBI C CD )† ATBBI C CD − 1 = cos2 r, I I

(40)

I I

NAI BI (CD) = ||ATABI BICD || − 1 = I I

 1

1 2 + sin2 r cos2 r + 2 2



cos4 r + sin8 r − 1 ,

(41)

and ND(AI BI C) = NC(AI BI D) = cos2 r, NAI (BI CD) = NBI (AI CD)

1 = 2



cos2 r +





1 + sin4 r cos2 r,

(42) (43)

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Fig. 2. The partial and whole entanglement of the four-qubit state as a function of the acceleration parameter r when A and B are in a uniform acceleration a. Here, for simplicity, we denote AI and BI as A and B, respectively.

as well as Nij = 0;i, j = A, B, C, D, then we have A = B =



1 cos2 r + 4



1 + sin4 r

2

cos4 r,

(44)

C = D = cos4 r,

(45)

thus the four-qubit state entanglement can be calculated as

  1

4 =

1 2

˘4 =

1 (cos2 r + 2

4

cos2 r +

and



1 + sin4 r

2



cos4 r + cos4 r ,

(46)



1 + sin4 r)cos4 r.

(47)

From Eqs. (40)–(43) it is clear that NBI C(AI D) = ND(AI BI C) = NC(AI BI D) , but NAI (BI CD) = NBI (AI CD) = / NAI BI (CD) , which are resulted from the symmetry and asymmetry among four Fermi modes with two acceleration. In addition, it is interesting to notice that from Fig. 2, the 1–3 tangle NBI (AI CD) = NAI (BI CD) equals to the 2–2 tangle NAI BI (CD) in the limit case of a → ∞, which √ approaches (1 + 5)/8 ≈ 0.404. While for the tangle NBI C(AI D) , the corresponding limit value is about 0.5. Similarly, the whole entanglement described by 4 and ˘ 4 degrade to non-zero values of 0.207 and 0.286, respectively, as the increasing acceleration parameter r. The entanglement defined by 4 decreases more rapid than that by ˘ 4 , which is different from the corresponding result shown in Fig. 1. On the other hand, compared to Fig. 1, from Fig. 2 we see that the entanglement with two acceleration decreases quicker than that with one acceleration as the increasing r. This indicates that the entanglement will be reduced due to the increasing acceleration observers. 4.3. Three parts with acceleration Next, we consider the case of A, B and C have the same acceleration of a. In this case, after having traced the II part of the three zones, we can get ABCD =

1  [ cos3 r|0000 0000| + |0000 1111| + |1111 0000| cos3 r + (|0010 0010| + |0100 0100|) sin2 rcos4 r 2 

+ |1111 1111| + sin4 rcos2 r (|0110 0110| + |1010 1010| + |1100 1100|) + sin2 r cos4 r|1000 1000|



+ sin4 r|1110 1110| ].

(48)

where for simplicity, we dropped off the subscript I from now on. Note that there is no acceleration only for D, thus the identical 1–3 entanglements are shared by A(BCD), B(ACD) and C(ABD) due to the symmetry among A, B and C. However, the 1–3 entanglement of D(ABC) is different from these above. Using Eq. (34), it is easy to get 1–3 style entanglements as follows NA(BCD) = NB(ACD) = NC(BCD) =

1 1 cos3 r − cos r + 1 cos3 r + 2 2

and ND(ABC) =

1 2







cos4 r + cos r + 3sin2 r cos2 r − 1 +

1 2





cos2 rsin4 r + 1cos3 r,

cos6 r + sin12 r,

(49)

(50)

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Fig. 3. The partial and whole entanglement of the four-qubit state as a function of the acceleration parameter r when A, B and C are in a uniform acceleration a. Here, for simplicity, we denote AI , BI and CI as A, B and C, respectively.

and the 2–2 style entanglement of NAB(CD) is NAB(CD) = NAC(BD) = NAD(BC) =

1 1 2cos2 r + cos r − cos4 r − 1 cos2 r + cos2 r 2 2



cos2 r + sin8 r,

(51)

2 2 as well as Nij = 0;i, j = A, B, C, D. Thus A = B = C = NA(BCD) , D = ND(ABC) , so that the whole entanglement for four-mode is

4 =

1 1 1 cos3 r − cos r + 1 cos3 r+ {3[ 4 2 2 +{

1 2







cos2 rsin4 r + 1cos3 r]

 1

cos4 r + cos r + 3sin2 r cos2 r − 1 +

2

2

2

cos6 r + sin12 r} },

(52)

or

˘4 = [

1 1 cos3 r − cos r + 1 cos3 r+ 2 2

×

1  2



cos2 rsin4 r + 1cos3 r]





cos4 r + cos r + 3sin2 r cos2 r − 1 +

1 2

3/2



cos6 r + sin12 r

1/2

.

(53)

In Fig. 3 we plot the picture of some different angles as a function of the acceleration parameter r. From Fig. 3, it is shown that although these differences among 1–3 tangles and 2–2 tangles, say NA(BCD) , ND(ABC) and NAB(CD) , they still share a common entanglement value 0.3 in the limit of a → ∞ or r → 4 . At the same time, the whole entanglement of four √ 2 bodies approaches to (1 + 2) /64 ≈ 0.09, a non zero value. This fact indicates that the four-partite entanglement does not completely vanish even when all three accelerated parts fall into a black hole. Thus it may be possible to realize the quantum communication between parties even in the presence of the event horizon. This prediction should be checked in the near future by incorporation quantum information theories into black hole physics. In addition, we can also get some similar results to those obtained in one or two acceleration cases not shown here. 4.4. Four accelerators Finally, we consider the case of all observers have the same acceleration of a. After tracing the part II, the reduced density operator is given by ABCD =

1 [cos8 r|0000 0000| + cos4 r|0000 1111| + cos4 r|1111 0000| + |1111 1111| + sin8 r|1111 1111|]. 2

(54)

We can also get the following entanglements NA(BCD) = NAB(CD) = NA(BC) = NA(B) = which lead to A = 4 = ˘4 =

1 1 1 cos8 r + sin8 r + cos4 r − , 2 2 2

(55)

1 1 1 cos8 r + sin8 r − , 2 2 2

1 2

cos8 r +

1 1 sin8 r + cos4 r − 2 2

(56)

2

1

−3

2

cos8 r +

1 1 sin8 r − 2 2

2

.

(57)

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Fig. 4. The partial and whole entanglement of the four-qubit state as a function of the acceleration parameter r when A, B and C, D are in a uniform acceleration a. Here, for simplicity, we denote AI , BI and CI , DI as A, B and C, D, respectively.

Here, noticing the fact that all four observers are in a same acceleration, thus other symmetrical cases are not shown. In addition, it is interesting to notice that 1–1 and 1–3 tangles are equal to 1–2 and 2–2 tangles, respectively, which are similar to the cases of four observers without acceleration. However, the 1–1 tangle and the 1–3 tangle become nonzero and nonunit, respectively, due to the presence of acceleration, which are obviously different from the zero- and unit-value cases in an inertial frame. This may imply that the entanglement is transformed among these parts due to the presence of all four accelerated observers. In addition, these two definitions of entanglement of 4 and ˘ 4 share a same value. In Fig. 4, we plot these three tangles calculated above as a function of parameter r. From Fig. 4 we can clearly see that the whole entanglement and the one among several parts are going to be zero when the parameter r exceeds a certain value, which can be called as sudden death of entanglement. For instance, when r  0.417, 0.65 the whole and the 1–3 entanglement reduce to zero, respectively. It is clear that the whole entanglement decreases more rapidly than the parts entanglement, a similar result to the other cases. Here it should be emphasized that only when all four observers are in acceleration, the sudden death of entanglement shall be present not only for the whole but for the parts. There will never the phenomena of sudden death when parts of four observers are in acceleration. In order to see clearly this point that how the whole entanglement depends on the number of the accelerated observers, we make a comparison about the entanglement by using 4 and ˘ 4 . In Fig. 5, we re-plot the whole entanglement described by 4 and ˘ 4 for different accelerated observers, respectively. It is clear that the entanglement decreases as the increasing parameter r and the increasing number accelerated observers. However, the entanglement described by ˘ 4 is not always less than that by 4 . To see this point, in Fig. 6 we further plot the difference between 4 and ˘ 4 , i.e., 4 − ˘ 4 as a function

Fig. 5. The whole entanglement as function of r, which are described by these two different definitions by 4 and ˘ 4 , for the different number of the accelerated observers.

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Fig. 6. The difference 4 − ˘ 4 as a function of parameter r.

of parameter r for different number of accelerated observers. It is shown that the difference, for one accelerated observer, increases as the increasing r; for two accelerated observers, the difference is negative which means that 4 < ˘ 4 . However, it is interesting to notice that, for three accelerated case, the difference first increases to a maximum and then decreases. The value of r corresponding to the maximum is about 0.512. For all accelerated case, 4 and ˘ 4 share a common value. Thus for the partially accelerated case (not for all accelerated case), there is a certain difference among the whole entanglement described by these two definitions. 5. Conclusion In this paper, we considered the properties of entanglement for four Fermi particles in non-inertial frame. Here for simplicity, we only examined the ideal case without the effect of noise. It is shown that (i) a state which is maximally entangled in an inertial frame becomes less entangled if the observer are relatively accelerated; (ii) for one to three acceleration cases, both the whole entanglement and the one among some parts decrease as the increasing parameter r, and the corresponding values of entanglement decrease as the increasing number of accelerated observers. In the limit of infinite acceleration case, i.e., r → /4, there is still entanglement for all tangles. (iii) Only when are there four accelerated observers, the phenomena of entanglement sudden death will be present not only for the whole entanglement but also for the part one. For instance, when the parameter r exceeds about 0.417, the whole entanglement will be zero. In addition, we made a comparison between two-definition of entanglement. It is shown that the absolute value of differences increase as the parameter r for 1, 2 accelerated observers. However, it is interesting to notice that, for three accelerated observers, the difference increases and then decreases as the increasing parameter r. And there will be the maximum when r ≈ 0.512. Acknowledgements Project supported by the National Natural Science Foundation of China (Grant Nos. 11264018 and 11464018), the Natural Science Foundation of Jiangxi Province of China (Grant No. 20151BAB212006), and the Research Foundation of the Education Department of Jiangxi Province of China (Nos. GJJ14274 and YC2014-S151) as well as Degree and postgraduate education teaching reform project of Jiangxi province (No. JXYJG-2013-027). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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