No survival of Nonlocalilty of fermionic quantum states with alpha vacuum in the infinite acceleration limit

Physics Letters B 748 (2015) 204–207

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Physics Letters B www.elsevier.com/locate/physletb

No survival of Nonlocalilty of fermionic quantum states with alpha vacuum in the infinite acceleration limit Younghun Kwon Department of Physics, Hanyang University, Ansan, Kyunggi-Do, 425-791, South Korea

a r t i c l e

i n f o

Article history: Received 24 January 2013 Received in revised form 14 June 2015 Accepted 2 July 2015 Available online 3 July 2015 Editor: S. Dodelson

a b s t r a c t In this article, we investigate the nonlocal behavior of the quantum state of fermionic system having the alpha vacuum. We evaluate the maximum violation of CHSH inequality in the quantum state. Even when the maximally entangled quantum state is initially shared it cannot violate the CHSH inequality, regardless of any alpha vacuum, when the infinite acceleration is applied. It means that the nonlocality of the quantum state in fermionic system with the alpha vacuum cannot survive in the infinite acceleration limit. © 2015 The Author. 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 .

1. Introduction The quantum information in a relativistic regime has been extensively investigated recently [1–12]. One of the main researches is focused on the case that one of the parties sharing the entangled state travels in an accelerated frame. In fact, it was expected that the entanglement of the bipartite system might be degraded due to the acceleration that a party undergoes. So when one of the parties sharing the maximally entangled state travels in the infinite acceleration, the entanglement of the bipartite system was expected to vanish. However, it turned out that the entanglement behavior depends on the physical system, whether given systems are bosonic or fermionic. In other words, the entanglement of bosonic system vanishes in the limit of the infinite acceleration but entanglement of fermionic system shows a convergent behavior, which means the survival of entanglement even in that limit [3,6,8,9]. So it was argued that the existence of entanglement in the infinite acceleration limit seems to be caused due to the particle’s statistical behavior. Another interesting observation on nonlocality was made when one of the parties sharing the entangled state travels in an accelerated frame. It was known that in the quantum state, the property of nonlocality behaves differently from entanglement. In fact the nonlocal behavior of the Werner state is different from its entanglement behavior. Recently, the nonlocal behavior was studied when one of the parties sharing the quantum state travels in an accelerated frame. Surprisingly, it was shown that even in fermionic system, the CHSH inequality, which is the measure of nonlocality,

E-mail address: [email protected].

cannot be violated in the infinite acceleration limit, even when the method beyond single-mode approximation was used [13]. The nonlocal behavior of quantum correlation at the infinite acceleration is extremely important in studying the capability of quantum communication. It is known that no-violation of CHSH inequality implies no improvement in communication, compared to a classical method. Therefore, if CHSH inequality is not violated at the infinite acceleration, quantum communication at the infinite acceleration cannot surpass a classical one. In the other line, it is interesting to understand the behavior of quantum correlation of inflation theory in terms of relativistic quantum information. One should note that inflation in view of de Sitter spacetime [14,15] has the peculiar property, called alpha vacuum [16–18]. The alpha vacuum, understood as modifications to UV of Bunch–Davies vacuum, may provide alternative initial conditions in inflation. Since the alpha vacuum depends on the value of α , a choice of α may cause a different scenario of inflation. Recently, the entanglement behavior in de Sitter spacetime has been studied [19]. Furthermore, the entanglement entropy in de Sitter space has been evaluated [20]. In fact, there exists the information paradox related with the existence of horizons. In case of de Sitter space, de Sitter horizons may appear. Even though classical information cannot be transmitted beyond the horizon, the quantum information can be encoded in Hawking radiation and can be detected by an observer outside the horizon [21]. Recently, a blackhole was investigated in terms of the early and the late Hawking radiations, which caused the firewall controversy [22]. The monogamy property in quantum information can explain why the firewall controversy may occur. An interesting point is that the nonlocality also has a monogamy property.

http://dx.doi.org/10.1016/j.physletb.2015.07.005 0370-2693/© 2015 The Author. 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 .

Y. Kwon / Physics Letters B 748 (2015) 204–207

205

However when the space expands with an acceleration, the inertial observer with static coordinates describes the de Sitter metric as

ds2 = −(1 − r 2 H 2 )dt 2 + (1 − r 2 H 2 )−1 dr 2 + r 2 d2

(6)

The metric covers only region I of the Carter–Penrose diagram in Fig. 1. The cosmological horizon for the observer at r = 0 is given by the hypersurface r = H1 . A free fermionic quantum field with mode expansion becomes

ψ(x) =

 I† I (+) (ckI φk (x) + dk φ I (−) (x) k II†

+ ckII φkII(+) (x) + dk φ II(−) (x))

Fig. 1. (Color online.) The de Sitter space of the Carter–Penrose diagram.

(7)

j†

Therefore, the understanding the firewall controversy in terms of nonlocality may be interesting. In this article, we investigate the nonlocal correlation of fermionic quantum states in alpha vacuum. Especially we are interested in the nonlocal behavior of quantum state in fermionic system with the alpha vacuum when one of the parties experiences the infinite acceleration. The present article is organized as follows. In Section 2, we will give a brief description for fermionic system with an alpha vacuum. In Section 3, we investigate the nonlocal correlation of pure and mixed quantum states in fermionic system. In Section 4, we conclude and discuss our result. 2. Alpha vacuum

1

(−dτ 2 + dς 2 + ς 2 d2 ) ( H τ )2

(1)

Here τ = −e − Ht / H is the conformal time. The coordinates covers a part (region I and II) of the Carter–Penrose diagram, shown in Fig. 1. A free fermionic quantum field with mode expansion becomes

ψ(x) =

 † (ck φk+ (x, τ ) + dk φk− (x, τ ))

(2)

k †

Here ck (dk ) is the annihilation operator for the particle (the creation operator for the antiparticle) satisfying anticommutation re†

†

lations such as {ck , ck } = {dk , dk } = δk,k . The Bunch–Davies vacuum is defined as

ck∞ |0∞  = dk∞ |0∞  = 0

(3)

where ∞ denote the case when τ goes −∞ [23–25]. Mottola and Allen considered the transformation such as ∗

∞†

∗

∞†

c α = N α (ck∞ − e (α ) d−k ) dα = N α (dk∞ + e (α ) c −k ) Here N α = 

1

1+e α +α

∗

(4)

. The newly defined operators (ckα and dkα )

satisfy ckα |0α  = dkα |0α  = 0. The alpha vacuum can be defined by applying S (α ) operator to the Bunch–Davies vacuum as follows

|0α  = S (α )|0∞  Here S (α ) = e

† ∞ ∞ α (ck∞† d∞ −k +ck d−k )

(5) .

j†

j†

satisfying anticommutation relations such as {cki , ck } = {dki , dk } =

i, j δk,k .

If one takes τ to be backwardly long enough, one may not make any distinction between de Sitter space and Minkowski space. Then we can define the Bogoliubov transformations between I ,II I ,II ck∞ (dk∞ ) and ck (dk ) such that

ck∞ = cos γ ckI − sin γ d−k II†

∞†

dk

II†

I = cos γ dk + sin γ c − k

(8)

Here tan γ = e −π |k|/ H . Then the Bunch–Davies vacuum and the alpha vacuum for the particle become

|0k∞ + = cos γ |0kI ; 0II−k  + sin γ |1kI ; 1II−k 

The metric of de Sitter spacetime can be described as

ds2 =

where cki (dk ) is the annihilation operator for the particle in the region i (the creation operator for the antiparticle in the region j)

|0kα + = 

1 1 +  2 tan2 γ

(|0kI ; 0II−k  +  tan γ |1kI ; 1II−k )

(9)

eα 1+ tan γ

Here  = 1−eα tan γ . 3. Nonlocality of quantum states in fermionic system with alpha vacuum 3.1. 2 party pure entangled states In this section we will investigate the nonlocal behavior of fermionic quantum system. At first let us consider the case when two parties Alice and Bob share a pure entangled quantum state in fermionic system having the alpha vacuum. When facing the fermionic quantum states, one may argue the ambiguity in fermionic system. In our paper we can treat the quantum state in fermionic system without ambiguity since the quantum state we handle is free of ambiguity. Let us consider the initial quantum state of fermionic system between Alice and Bob in Minkowski spacetime

| + (α ) = cos θ|0 M |0α + sin θ|1 M |1+ α .

(10)

Two parties share a pure entangled state in fermionic system and Bob moves in an acceleration by expansion of de Sitter spacetime, where the Bob’s vacuum becomes α vacuum. Then the one particle state can be obtained by applying the suitable creation operator to α vacuum. The non-inertial motion causes the accessible spacetime to be limited, which means that one should trace out the unaccessible region. The quantum state after tracing out the unaccessible region becomes

206

Y. Kwon / Physics Letters B 748 (2015) 204–207

cos2 θ

+

ρ A BI =

1 +  2 tan2

+ 

γ

|0000| +

sin 2θ

2 1 +  2 tan2 γ

 2 cos2 θ tan2 γ |0101| 1 +  2 tan2 γ

(|0011| + |1100|)

+ sin2 θ|1111|

(11)

In fact we focus the nonlocal behavior of the quantum state when Bob is in the infinite acceleration limit. When Bob moves in an accelerated frame, the quantum correlation of the quantum state between Alice and Bob shows its maximum, since the quantum state shared between Alice and Bob is obtained from the global mode. So if the maximum of nonlocal correlation between Alice and Bob shows upper bound, even though we use the global mode, it is meaningful. The nonlocality of the quantum state can be found by considering Bell inequalities. When one needs to understand the nonlocality of bipartite quantum state, the CHSH inequality may be found to be suitable for the purpose. If a quantum state turns out to be local, it satisfies the bound

| B CHSH | ≤ 2

(12)

 ⊗ (b + b ) · σ + Here B CHSH is the operator given by B CHSH = a · σ

 ⊗ (b − b ) · σ . And a, a , b and b are the unit vectors and a · σ σ are the vectors of Pauli matrices. In other words if the CHSH inequality for the quantum state violates the above inequality the quantum state shows the nonlocal behavior. The maximal violation of CHSH inequality for a given quantum state ρ is found to be [26]

√  B max (ρ ) = 2 ν1 + ν2

(13)

Here ν1 and ν2 are the two largest eigenvalues of T t T , where T is the matrix whose element is given by t i j = Tr[ρσi ⊗ σ j ]. In our case T t T matrix is given by

⎛

A TtT = ⎝ 0 0 Here A =

0 A 0

Fig. 2. (Color online.) Violation of CHSH inequality. The maximally entangled state of fermionic system with the alpha vacuum cannot violate the CHSH inequality in the infinite acceleration limit. In bottom, the lines from top to bottom denote the cases of α = −10, α = −1 and α = −0.1, respectively.

⎞

0 0⎠ B

sin2 2θ 1+ 2 tan2 γ 2

sin2 θ

∗

(14)

ρ A BI =

+ 

and B = {(2 + 2 cos 2γ +  cos 2(γ − θ) − 2

(15)

 B max (ρ A B I ) ≤ 2

(16)

So the CHSH inequality cannot be violated in the infinite acceleration limit even when Alice and Bob share the maximally entangled state in fermionic system with the alpha vacuum. It implies that in the infinite acceleration limit, there exists no nonlocality for the quantum state, regardless of α . Let us now consider another case that Alice and Bob prepare the pure entangled states as follows:

| ∗ (α ) = cos θ|0 M |1+ α + sin θ|1 M |0α .

(17)

As it is done before, the quantum state that Alice and Bob share when Bob moves in an acceleration by expansion of de Sitter spacetime, can be obtained by tracing the other region,

For

(18)

∗

ρ A B I , T t T matrix is given by ⎛

A T T =⎝ 0 0 t

 2 sin2 θ tan2 γ |1111| 1 +  2 tan2 γ

(|0110| + |1001|)

+ cos2 θ|0101|

The CHSH inequality for the maximally entangled state of fermionic system with the alpha vacuum can be violated at any negative value of α and finite acceleration, shown in Fig. 2. However in the infinite acceleration limit (γ → π4 ), we find  ≥ 1. Therefore in the limit we have +

sin 2θ

2 1 +  2 tan2 γ

2 2 cos 2θ +  cos 2(γ + θ))2 sec4 γ }/{16(1 +  2 tan2 γ )2 }. By Eq. (14), the maximum value  B max ρ becomes

√ +  B max (ρ A B I ) = 2 2 A

1 +  2 tan2 γ

|1010| +

0 A 0

⎞

0 0⎠ C

(19)

2

sin 2θ 2 Here A = 1+ 2 tan2 γ and C = {(−2 − 2 cos 2γ +  cos 2(γ − θ) − 2 2 cos 2θ +  2 cos 2(γ + θ))2 sec4 γ }/{16(1 +  2 tan2 γ )2 }. Even though T t T matrix is different from previous one, the √ ∗ maximum value of  B max ((ρ )) is the same as 2 2 A. AB I

3.2. 2 party mixed entangled state Up to now, we have considered the case when Alice and Bob prepare the pure entangled states in fermionic system. In this subsection, we consider a more complicate scenario when two parties share a mixed state. It is aimed to find how the nonlocal behavior depends on the mixedness property. In particular, the case when a white noise is added to a maximally entangled states, so-called Werner state, is to be considered. The mixedness of Werner states

Y. Kwon / Physics Letters B 748 (2015) 204–207

is parameterized by a single parameter F . So we suppose that two parties Alice and Bob prepare the Werner state expressed as follows,

ρW = F | + (θ = π /4) + (θ = π /4)| +

1− F 4

I,

(20)

where 0 ≤ F ≤ 1 and the maximally entangled state is taken from Eq. (10) when θ = π /4. Suppose that Bob moves in an accelerated frame, where it is caused by expansion of de Sitter spacetime. The state that Alice and Bob share in Bob’s region I is obtained by tracing the unaccessible region, as follows:

1

1 + F + 2 2 tan2 γ

4

1 +  2 tan2 γ

ρ A B W = (−2F cos 2θ + I

+

1 − F + 2 2 (1 + F cos 2θ) tan2 γ 4 + 4 2 tan2 γ F sin 2θ

+ 

2 1 +  2 tan2 γ

)|1111|

|0101|

(|0011| + |1100|)

1− F |1010| 4 + 4 2 tan2 γ 1 + F + 2F cos 2θ + |0000| 4 + 4 2 tan2 γ

+

(21)

When the Werner state is shared between Alice and Bob T t T matrix is given by

⎛

D T T =⎝ 0 0

⎞

0 D 0

t

2

0 0⎠ E

(22)

2

F sin 2θ 2 2 Here D = 1+ 2 tan2 γ and E = { F (2 + 2 cos 2γ +  cos 2(γ − θ) − 2 2 cos 2θ +  2 cos 2(γ + θ))2 sec4 γ }/{16(1 +  2 tan2 γ )2 }. Therefore the maximum value of  B max (ρ A B W ) becomes

√

√

I

2F 2 A, which is F (≤ 1) times 2 2 A. The violation in CHSH inequality for the Werner state √ of fermionic system with the alpha vacuum can be found if F 2 A ≥ 1, which implies that the violation depends on α and the mixedness F . 4. Discussion and conclusion We have investigated the nonlocal behavior of the quantum state in fermionic system having the alpha vacuum. The alpha vacuum might provide alternative initial conditions in inflation. The nonlocal behavior of the quantum state is important in studying the capability of quantum communication. For the purpose, we considered the case that one of two parties sharing the entangled quantum state moves in an accelerated frame, which is caused by expansion of de Sitter spacetime. We evaluated the maximum violation of CHSH inequality for the quantum state. We showed that even when the maximally entangled quantum state is initially shared between Alice and Bob, the nonlocal correlation of the quantum state cannot survive in the infinite acceleration limit. It implies that quantum communication between two parties, at the infinite acceleration, cannot be superior to a classical one.

207

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