Large scale quantum entanglement in de Sitter spacetime
Akira Matsumura Yasusada Nambu
gr-qc quant-ph





matsumura.akira@h.mbox.nagoya-u.ac.jp

nambu@gravity.phys.nagoya-u.ac.jp

Department of Physics, Graduate School of Science, Nagoya 
University, Chikusa, Nagoya 464-8602, Japan

 




  We investigate quantum entanglement between two symmetric spatial
  regions in de Sitter space with the Bunch-Davies vacuum.  As a
  discretized model of the scalar field for numerical simulation, we
  consider a harmonic chain model. Using the coarse-grained variables
  for the scalar field, it is shown that the multipartite
    entanglement on the superhorizon scale exists by checking the
    monogamy relation for the negativity which quantifies the
    entanglement between the two regions. Further, we consider the
  continuous limit of this model without coarse-graining and
  find that non-zero values of the logarithmic negativity exist even
  if the distance between two spatial regions is larger than the
  Hubble horizon scale.



04.62.+v, 03.65.Ud
Large scale quantum entanglement in de Sitter spacetime
    Yasusada Nambu
    May 29, 2018
=======================================================






§ INTRODUCTION


Quantum entanglement is an interesting aspect of quantum physics,
which has recently received remarkable attention from various fields. 
As is well known in the quantum theory or quantum
information theory, quantum entanglement represents a non-local correlation and leads to
the violation of the Bell-CHSH
inequality <cit.>. This quantum correlation is
needed as a resource to carry out some protocols, e.g. quantum
teleportation, superdense coding, quantum error correction and so on
<cit.>. For quantum many-body systems or quantum field
theories, the entanglement of the ground state or vacuum state
characterizes the structure of its wave function. In a previous work
by S. Marcovitch et al. <cit.>, the quantum
entanglement of a free Klein-Gordon field in the 1+1-dimensional
Minkowski spacetime was examined in terms of the logarithmic
negativity which is a useful measure to quantify the entanglement for
a mixed state <cit.>. They focused on two spatially
separated regions and numerically investigated the logarithmic
negativity for the Minkowski vacuum as a function of the ratio of the
separation between the regions to the size of the each region. It was
shown that the logarithmic negativity decays exponentially as the
ratio becomes large. This behavior is consistent with the
Reeh-Schlieder theorem <cit.> which implies that the quantum
entanglement persists for all scales for the Minkowski vacuum.

The nonlocal quantum correlation also was examined for
cosmological
situations <cit.>.
In a previous work by Y. Nambu <cit.>, the quantum
entanglement of a massless scalar field in de Sitter space was
investigated using the spatially coarse grained scalar field. It was
revealed that the logarithmic negativity for the coarse-grained field
vanishes, when the physical size of each region exceeds the Hubble
horizon and becomes causally disconnected due to the de Sitter
expansion. This behavior of entanglement is consistent with the
scenario of quantum to classical transition of the primordial
fluctuation generated by inflationary expansion in the early universe
<cit.>. The similar transition to zero negativity state is
known for a model of 1+1-dimensional harmonic chain with a finite
temperature <cit.>. For sufficiently high
  temperatures, the negativity between two spatial regions becomes
zero. Qualitatively, this transition can be understood as follows;
above the critical temperature, the wavelength of the thermal
fluctuations becomes shorter than the lattice spacing and the quantum
correlation between adjacent lattice sites is destroyed by the thermal
fluctuations. Concering the quantum field in de Sitter space, the
effective comoving lattice spacing for the coarse-grained field
becomes larger than the wavelength of quantum fluctuations which is
equal to the Hubble length of de Sitter space, and the negativity
becomes zero.  We expect that appearance of zero negativity state is
related to the coarse-graining treatment of the quantum field. As
mentioned above, for the Minkowski vacuum in quantum field theory, the
Reeh-Schrieder theorem is known and the theorem also holds for
thermal states <cit.>. It seems that there
is a difference between the feature of entanglement in the
coarse-grained field and its continuous limit. Thus for complete
understanding the property of the entanglement in de Sitter space, we need to
analyze the quantum entanglement between two spatial regions using
both a model with coarse-graining and one without coarse-graining.


In this paper, we use a 1+1-dimensional lattice model of a massless
scalar field whose mode equation is equivalent to that of the
1+3-dimensional de Sitter spacetime and numerically evaluate the
entanglement between two spatial regions.  As a quantum state, we
assume the Bunch-Davies vacuum, which is a vacuum state in de Sitter
spacetime and corresponds to the Minkowski vacuum in the far remote
past. We introduce coarse-grained variables in the lattice model and
the negativity between two spatial regions is calculated.  To
understand the connection between the coarse-grained system and the
continuous one, the effect of multipartite entanglement is considered
using the monogamy relation for the negativity. We also compute the
logarithmic negativity between the two regions without the
coarse-graining and its behavior is discussed especially focusing on
the continuous limit and the existence of the Hubble horizon.

This paper is organized as follows: we introduce our 1+1-dimensional
lattice model of a minimal coupled massless scalar field in de Sitter
spacetime and the entanglement measures (the negativity and the
logarithmic negativity) for the Gaussian state in Sec. II.  In
Sec. III, we define the coarse-grained variables in the lattice system
and calculate the negativity numerically. Then we discuss the
monogamous property of the negativity. Also, we present our main
numerical result of the logarithmic negativity without coarse-graining
and provide the fitting formula for the super horizon scale. Sec. IV
is devoted to summary and conclusion.




§ HARMONIC CHAIN MODEL AND ENTANGLEMENT MEASURES FOR  GAUSSIAN
  STATE



The Hamiltonian for a minimal coupled massless scalar field q in de
Sitter spacetime with a spatially flat slice is given by


    ℋ=∫ d^3 x  1/2[p^2+(∂_i
    q)^2+1/ad a/dη (qp+pq) ],   
    a=-1/Hη,


where a is the scale factor,  η<0
is the conformal time and  H represents the Hubble constant. For
simplicity of numerical analysis, we assume that the field depends
only on the conformal time η and one spatial coordinate. By
introducing a lattice spacing Δ x of the spatial direction, the
dimensionless form of the Hamiltonian of this model (harmonic
chain) <cit.> is expressed as


    ℋ=∑_j=1^N[ 1/2  p_j^2+q_j^2-α 
     q_j q_j-1+1/2ada/dτ (p_jq_j+q_jp_j) ],


where we impose a periodic boundary condition on q_j and p_j
to respect the translational invariance of the model. N denotes the
total number of lattice sites. τ and α are the
dimensionless conformal time and the IR cutoff parameter,
respectively. These two parameters are given by


    τ=η/Δ x,    (m Δ x)^2=2(1-α),


where  m is the mass of the
scalar field corresponding to the IR cutoff. We quantize this model as follows:


    q̂_j=1/√(N)∑_k=0^N-1[ â_k 
      f_k+â_N-k^†  f_k^*]e^iθ_kj,   p̂_j=-i/√(N)∑_k=0^N-1[ â_k  g_k-â_N-k^†  g_k^*]e^iθ_kj,


where θ_k=2π k/N. â_k and â_k^†
are annihilation and creation operators, which obey the
commutation relations


    [â_k,â_k'^†]=δ_kk',   [â_k,â_k']=[â_k^†,â_k'^†]=0.


The mode functions f_k and g_k satisfy


    f̈_k+(ω^2_k-ä/a)f_k=0,  
    f_kḟ^*_k-ḟ_kf^*_k=i,  
    g_k=i (ḟ_k-ȧ/a f_k),


where “·" is the derivative with respect to τ and
ω_k^2=2(1-αcosθ_k). Although this lattice model is
introduced in the 1+1-dimensional spacetime, we assume that the equation
of the mode functions has the same form as that in the 1+3-dimensional de Sitter space. 

For the investigation of quantum entanglement of this system with the
Bunch-Davies vacuum which belongs to a family of Gaussian
states, we present a brief review of the negativity and the
logarithmic negativity for a Gaussian state. Let us consider a phase
space composed of canonical variables
{q̂_j,p̂_j}_j=1,...,N. The canonical commutation
relations are


    [ R̂_j, R̂_k]=i Ω_jk,   Ω=⊕_j=1^N J,  
    J=[  0  1; -1  0 ],


where R̂_j represent canonical variables defined by


    R̂=[ q̂_1, p̂_1,...,q̂_N,p̂_N]^T.


A Gaussian state ρ̂ is determined by its first  moment 
⟨R̂_j⟩=Tr[ρ̂ R̂_j] and the
covariance matrix 


    V_jk=1/2 Tr[ (ΔR̂_j ΔR̂_k+
    ΔR̂_k ΔR̂_j)ρ̂],  ΔR̂_j=R̂_j-⟨R̂_j⟩.


The negativity 𝒩 of a bipartite Gaussian state
ρ̂_AB is given by using the symplectic eigenvalues of the
partially transposed covariance matrix Ṽ_AB
<cit.> obtained from V_AB by replacing p̂_jA
with -p̂_jA <cit.>:


    𝒩=1/2(∏_j=1^N1/min(2ν̃_j,1)-1 ),


where ν̃_j are positive eigenvalues of
i Ω Ṽ_AB. The sufficient condition for the
entangled state is that the negativity 𝒩 of the state does
not vanish. The logarithmic negativity E_𝒩 is defined by
the negativity as 


    E_𝒩=ln[ 2𝒩+1 ].


The logarithmic negativity E_𝒩 provides an upper bound
of the distillable entanglement (the number of the Bell pairs
extractable from a bipartite state) <cit.>, and if
E_𝒩 is nonzero then the bipartite state is
entangled. However, there exists an entangled state whose the
entanglement of distillation vanishes and such a state is called a
bound entangled state. Fortunately, no bound entangled state exists
for a bipartite Gaussian state with E_𝒩=0
<cit.>.

To compute the negativity and the logarithmic negativity, we need the two-point functions of canonical variables on each site for the vacuum state. The correlation functions of the vacuum state are 


    1/2⟨ 0|(q̂_jq̂_l+ q̂_jq̂_l)|0⟩ =
    1/N∑_k=0^N-1 |f_k|^2cos[ θ_k(j-l)
      ],   
    
    
       1/2⟨ 0|(p̂_jp̂_l+p̂_lp̂_j)|0⟩=
      1/N∑_k=0^N-1|g_k|^2cos[ θ_k(j-l)
      ], 
    
    
       1/2⟨ 0|(q̂_jp̂_l+p̂_lq̂_j)|0⟩=
      1/N∑_k=0^N-1i/2(f_kg^*_k-f^*_kg_k )cos[ θ_k(j-l) ].


We choose the mode functions f_k and g_k which correspond to the Bunch-Davies vacuum as follows: 


    f_k=1/√(2ω_k)(1+1/iω_kτ)e^-iω_kτ,  
    g_k=√(ω_k/2)  e^-iω_kτ.


 In the continuous limit of the lattice model, the correlation functions of the Bunch-Davies vacuum are given as


    1/2⟨ 0|(q̂(x,η)q̂(y,η)+ q̂(y,η)q̂(x,η))|0⟩   = ∫^∞_-∞dk/2π |f_k|^2cos [k(x-y)] 
       ∼1/2π(-γ-ln[m|x-y|])+1/2π (mη)^2, 
    
    
    1/2⟨ 0|(p̂(x,η)p̂(y,η)+p̂(y,η)p̂(x,η))|0⟩   =∫^∞_-∞dk/2π |g_k|^2cos [k(x-y)] 
       ∼ -1/2π |x-y|^2, 
    
    
    1/2⟨ 0|(q̂(x,η)p̂(y,η)+p̂(y,η)q̂(x,η))|0⟩   =
     ∫^∞_-∞dk/2πi/2(f_kg^*_k-f^*_kg_k )cos [k(x-y)] 
       ∼1/2πη(-γ-ln [m|x-y|]),
 

where the above approximated formulas are obtained for m|x-y| ≪ 1,
and γ is the Euler constant. The two-point correlation of our effective
model in a 1+1-dimensional spacetime decrease with the distance |x-y|
more slowly compared to that in the 1+3-dimensional de Sitter space
case.




§ BEHAVIOR OF ENTANGLEMENT

Because the scalar field is a many-body system, 
we expect that the multipartite entanglement is a key property to
understand behavior of entanglement between two spatial regions in the
de Sitter space. For this purpose, we introduce the coarse-grained
field and the monogamy inequality of entanglement to quantify the
multipartite entanglement of the scalar field.




 §.§ Coarse-grained field and entanglement monogamy

To focus on the behavior of the multipartite
entanglement, we introduce the coarse-grained variables
(Q̂_i,P̂_i) as follows:


    Q̂_i=1/√(n_c)∑_j=0^n_c-1q̂_n_ci+j, P̂_i=1/√(n_c)∑_j=0^n_c-1p̂_n_ci+j,
 

where we denote n_c as the coarse-graining size of the
canonical variables. The coarse-grained variables satisfy the
canonical commutation relations


    [Q̂_i, P̂_j]=i δ_ij, [Q̂_i, Q̂_j]=[P̂_i, P̂_j]=0.
 

We introduce two spatial symmetric regions A and B by choosing the following coarse-grained variables in the harmonic chain: 


    R̂^A=[ Q̂_1, P̂_1,...,Q̂_n,P̂_n]^T, R̂^B=[ Q̂_n+1+d/n_c, P̂_n+1+d/n_c,...,Q̂_2n+d/n_c,P̂_2n+d/n_c]^T


where n is the number of modes of the coarse-grained variables
contained in each region, and d is the comoving separation between A
and B. The comoving size of each region is denoted as
l=n× n_c (FIG. <ref>). 



Since the vacuum state is
Gaussian, the quantum bipartite entanglement can be completely
characterized by the covariance matrix defined as


    V_AB=[   A   C; C^T   B ],  
          A^T=A,   
          B^T=B


where A, B, C are 2n× 2n matrices
given by


    A_ij=1/2⟨ 0| (R̂^A_iR̂^A_j+R̂^A_jR̂^A_i) |0 ⟩, 
     
    B_ij=1/2⟨ 0| (R̂^B_iR̂^B_j+R̂^B_jR̂^B_i) |0 ⟩, 
     
    C_ij=1/2⟨ 0| (R̂^A_iR̂^B_j+R̂^B_jR̂^A_i)|0 ⟩,


with ⟨ 0|R̂_i^(A,B) |0 ⟩=0.

In Ref. <cit.>, the logarithmic negativity between A and B
were calculated with the number of each mode n=1, which corresponds to assigning a pair of
canonical variables to each region.



The following results are based on the numerical calculation with the
number of lattice sites N=2× 10^4 and the IR cutoff parameter
α=1-10^-12. FIG. <ref> shows behavior of the
  negativity with different size of coarse-graining. The left panel
of FIG. <ref> presents the time evolution of the negativity with
fixed the comoving distance d=0 and the comoving size l=60. As the
number of modes of the coarse-grained variables n in the system A
increases, the time at which the negativity vanishes becomes
later. The right panel of FIG. <ref> gives the distance
dependence of the negativity with fixed l=60 and τ=-80. For a
larger n, the negativity increases and vanishes at a larger distance
d.




To
compare the previous work <cit.> with our results, we
introduce


    l_p=-l/τ,  
    d_p=-d/τ,


where l_p and d_p represent the proper (physical) size of each
region and the distance between the two regions in the unit of the
Hubble length H^-1, respectively. In Ref. <cit.>, it is
found that the quantum entanglement between A and B with the
number of each mode n=1 disappears when the proper size of each region
is comparable to the Hubble horizon, that is, at l_p=1. This
means that the quantum fluctuation for the super horizon scale behaves
classically in terms of bipartite entanglement. However, according to
the left panel of FIG. <ref>, we find that the quantum
entanglement with the number of each mode n≥2 in the system A and B does not vanish even if
the proper size of each region l_p is larger than 1 (for
example, in the case n=2, the negativity is nonzero at
τ=-10, l=60, that is, l_p=6) . Hence we expect there
exists the multipartite entanglement even for the quantum fluctuation
in the super horizon scale. Also, in the right panel of
FIG. <ref>, the maximum distance that the negativity exists
increases monotonically as n increases (in Ref. <cit.>, the
negativity vanishes trivially for d≥1). It seems that the
multipartite entanglement between casually disconnected regions also
survives for a larger n (in the following section B, we clarify the distance dependence
 for the continuous limit of our model).

To get a clear intuition for the behavior of FIG. <ref>, we
introduce the monogamy relation of the negativity for this model. We
consider a tripartite state ρ_ABC and the negativity
𝒩_AB|C between AB and C. It is conjectured that
𝒩_A|C, 𝒩_B|C and 𝒩_AB|C
obeys the following inequality


    𝒩^2_AB|C≥𝒩^2_A|C+𝒩_B|C^2,


where 𝒩_A|C and 𝒩_B|C are the
negativity between AC or BC, respectively. This is called the monogamy
relation of the negativity, which is an crucial property of the
quantum entanglement. The monogamy relation is proved for a
multi-qubit system <cit.>, and the similar relation holds
for the entanglement measure defined in the Gaussian system
<cit.>. However, there is no proof of the monogamy
relation of the negativity for the Gaussian system. If the monogamy
relation holds then the multipartite entanglement can be expressed by
the quantity


    𝒩_A|B|C:=𝒩^2_AB|C-𝒩^2_A|C - 𝒩_B|C^2.


This is the difference between the two side of Eq. (<ref>)
and can be interpreted as the residual entanglement. If this quantity
vanishes, the entanglement between AB and C can be decomposed to the
entanglement between A and C, and the entanglement between B and
C. Thus the entanglement between AB and C is written as sum of pure
bipartite entanglement between sub system. In such a case, there is no
multipartite entanglement.  

To understand the multipartite entanglement in the Bunch-Davies
vacuum, we consider the negativity with the number of each mode n=3
in the systems A and B (FIG. <ref>).  The density operator
ρ̂_AB is a 3 × 3 mode Gaussian state and the system A is
composed of 3 subsystems A_1A_2A_3 (similarly, B is
B_1B_2B_3). The monogamy relation is written as


    𝒩^2_A^⊗ 3|B≥𝒩_A^⊗ 2|B^2+𝒩^2_A^⊗ 1|B,


where A^⊗ 3=A_1A_2A_3 and
(A^⊗ 2,A^⊗
  1)=(A_1A_2,A_3), (A_1A_3,A_2), (A_2A_3,A_1).
For example, each negativity
𝒩_A_2A_3|B, 𝒩_A_1A_3|B and
𝒩_A_3|B which appears on the right side for
(<ref>) corresponds to the entanglement between the two regions A
and B shown in FIG. <ref>.



Let us check the monogamy relation of the negativity for the
Bunch-Davies vacuum in the case n=3. The left and right panel of FIG. <ref>
present the time dependence with d=0 and the distance dependence at
τ=-80 of the quantity
𝒩_A^⊗ 2|A^⊗ 1|B, respectively.



From these results, we confirm that the monogamous relation of the
negativity 𝒩_A^⊗ 2|A^⊗ 1|B≥ 0 holds for our model. Hence we can characterize the multipartite
entanglement in de Sitter space by the negativity.

In the left panel of FIG. <ref>, the time dependence of
𝒩^2_A_1A_2A_3|B, 𝒩^2_A_2A_3|B, 𝒩^2_A_1A_3|B
and 𝒩^2_A_3|B for d=0 is shown (the other
cases 𝒩_A_1A_2|B, 𝒩_A_1|B and
𝒩_A_2|B are trivially zero).  We observe that the
negativities 𝒩^2_A^⊗ 1|B and
𝒩^2_A^⊗ 2|B decay faster than the negativity
𝒩^2_A^⊗ 3|B for the 3 × 3 mode
Gaussian system. This behavior guarantees the monogamy inequality
(<ref>).



The right panel of FIG. <ref> shows that the distance dependence
of
𝒩^2_A^⊗ 3|B, 𝒩^2_A^⊗
    2|B
and 𝒩^2_A^⊗ 1|B for τ=-80 (the other
cases 𝒩_A_1A_2|B, 𝒩_A_1|B and
𝒩_A_2|B are trivially zero again). As in the case of
the left panel of FIG. <ref>, the negativities
𝒩^2_A^⊗ 1|B and
𝒩^2_A^⊗ 2|B decrease more than
𝒩^2_A^⊗ 3|B with the distance d to keep
the monogamy relation (<ref>). The behaviors observed in 
FIG. <ref> also suggest that the multipartite entanglement remains in the 
super horizon scale when the number of modes n becomes large.  





 §.§ Continuous limit

To investigate the entanglement for the super horizon scale, we
consider the continuous limit of our lattice model, where multipartite
entanglement plays an important role.  For realization of the
continuous limit of our lattice model, we use the canonical variables
with n_c=1 (no coarse-graining), and choose each parameter as
N=2× 10^4 and α=1-10^-12, again. It is also assumed
that each region contains l harmonic oscillators and their comoving
separation is d (FIG. <ref>). In the appendix, we present the
convergence check and the small violation of the uncertain relation
due to numerical error to confirm that our numerical calculation
really corresponds to the continuous limit and is stable. We 
 compare the previous
works <cit.> with our numerical results. In
Ref <cit.>, the authors considered a massless scalar
field in the 1+1-dimensional Minkowski space and numerically showed
that the logarithmic negativity of a massless scalar field between two
spatially regions decays exponentially as the ratio d/l
increases. The property that the logarithmic negativity depends only
on the ratio d/l is derived from the scale invariant for the
massless theory. In the following, we investigate the entanglement for
the super horizon scale and how it depends on the Hubble scale H.



In Ref. <cit.>, the logarithmic negativity of the
coarse-grained field in de Sitter space vanishes when the two regions
are causally disconnected and FIG. <ref> also shows the
  negativity with the coarse-grained field becomes zero for
  sufficiently large scales or late times. On the other hand,
the negativity obtained without coarse-graining
  (FIG. <ref>) does not vanish even when the distance between
two regions is larger than the horizon scale (d_p=1 corresponds to
the Hubble horizon scale). This observation confirms that the
multipartite entanglement remains on the super horizon scale as
expected above. For a vacuum state in the quantum field theory, the
Reeh-Schrieder theorem characterizes the (multipartite) entanglement
of quantum field <cit.>. Our numerical results suggest that
the Reeh-Schrieder theorem also holds for the Bunch-Davies vacuum in de
Sitter space.



As the Bunch-Davies vacuum approaches to the Minkowski vacuum in the
remote past, the behavior of the logarithmic negativity for l_p <1
and d_p <1 is expected to be same as that for the Minkowski
case. To focus on the entanglement peculiar to de Sitter space, we
consider the behavior of the logarithmic negativity for l_p≥1
and d_p≥1.  FIG. <ref> shows the logarithmic negativity
E_𝒩 as a function of d_p in this case.



The logarithmic negativity for l_p≥1 and d_p≥1 behaves
as almost linear functions in log plot. We use the fitting
function of the exponential factor with the power-law correction to
compare with the logarithmic negativity in the Minkowski vacuum
<cit.>. The solid lines in FIG. <ref> represent
the fitting result


    E_𝒩^(fit)≈ d_p e^-k d_p,


where k is a real parameter whose values depend on the ratio
l_p. FIG. <ref> shows k as a function of l_p and the
solid line in the figure represents a function k=a_1+a_2 l_p^-1
where a_1 and a_2 are  O(1) constants given by
a_1∼1.08 and a_2∼ 4.35 . 



Combining these, we obtain the following fitting formula of the
logarithmic negativity for d_p≥ 1 and l_p≥ 1:
E_𝒩^(fit)≈ d_p e^-a_1d_p-a_2
  (d_p/l_p).
By restoring the dimension of the variables, this is rewritten as


    E_𝒩^(fit)≈D_p/H^-1exp[-a_1D_p/H^-1-a_2D_p/L_p],


where H^-1 is the Hubble length and the proper size of the region
L_p and distance D_p are given by
L_p=a(η) l Δ x, D_p=a(η) d Δ x. In the
  formula (<ref>) we observe that the logarithmic negativity decays
  exponentially with respect to H. This means that the quantum
  entanglement is degraded by the thermal noise with the Hawking
  temperature H. Thus we consider the value of the numerical factor
  a_1 is related to the thermal noise which is independent of
  details of the theory. On the other hand, the interpretation of the
  factor a_2 is not so clear because a value of this coefficient
  of D_p/L_p depends on the theory. To make clear its physical
  interpretation, we will need further analysis of quantum
  entanglement using other theories of the scalar field in de Sitter
  space. For the super horizon regions L_p ≫ H^-1, the
logarithmic negativity (<ref>) becomes independent of the size
L_p of the two symmetric spatial regions and its value is determined
only by the ratio D_p/H^-1. We expect that this property can be
understood as follows: the physical wavelength of quantum fluctuation
in the considering regions (FIG. <ref>) is initially smaller than
the Hubble horizon. As the universe expands, the wavelength exceeds
the Hubble horizon. After the horizon exit, the scale of fluctuations
in each region is determined only by the Hubble horizon scale. Hence,
the amount of entanglement depends only on D_p/H^-1.


The above property of the logarithmic negativity (<ref>) 
is expected to be true for the 1+3-dimensional de Sitter space. This
is because the feature of the entanglement is determined by the mode
function and we use the same mode function as the 1+3-dimensional
model. As the quantum fluctuation of super horizon mode is scale
independent, the quantum entanglement is determined only by the Hubble scale and the separation between the considering regions. 



§ SUMMARY AND CONCLUSION


We investigated the quantum entanglement between two symmetric spatial
regions with the Bunch-Davies vacuum for the 1+1-dimensional effective
harmonic chain model. We introduced the coarse-grained variables and
examined the multipartite entanglement in de Sitter spacetime by the
monogamy relation of the negativity. In the previous work
<cit.>, it has been shown that the bipartite entanglement
disappears on the super horizon scale. In contrast, in this paper, it
was found that the multipartite entanglement of the super horizon
scale remains. This indicates that the multipartite entanglement plays
an important role to characterize the quantum nature of the the super
horizon scale fluctuations.  We also considered the continuum limit of
our model and calculated the logarithmic negativity for the original
canonical variables (without coarse-graining). We confirmed that the
logarithmic negativity remains non-zero even if the distance between
the two regions becomes larger than the Hubble length. That is, there
exists the quantum entanglement between two causally disconnected
regions and the existence of the entanglement means that the
Reeh-Schrieder theorem holds in de Sitter space.

Finally, we comment on the relation between our lattice model and
1+3-dimensional theory. We considered the 1+1-dimensional effective
lattice model of the free massless scalar field in the de Sitter
space. As the behavior of the logarithmic negativity depends on the
spatial dimension, the numerical simulation in our model is not
equivalent to the universe with three spatial dimensions. At the
  end of Sec. II, we observed that the two-point correlation in our
model is larger than it in a 1+3-dimensional de Sitter
space. Hence for the scalar field without coarse-graining
  (continuous limit), if the entanglement disappears for some size and
  separation of each region in 1+1-dimensional model, we expect that
  the entanglement in the corresponding 1+3-dimensional model also
  vanishes. However, according to our numerical calculation, the
  entanglement in 1+1-dimensional model in the continuous limit exists
  in any scale.  Thus our lattice model provides the necessary
condition to judge the existence of quantum entanglement in the
1+3-dimensional de Sitter space. The main features of the logarithmic
negativity found in our analysis is characterized by properties of the
mode function of the scalar field in de Sitter space. The equation in
our model is the same as that in the 1+3-dimensional de Sitter
spacetime, and if we evaluate quantum entanglement between spatial
regions in the 1+3-dimensional de Sitter spacetime, we expect that we
will obtain the similar feature or property of entanglement obtained
this paper.






This work was supported in part by the JSPS KAKENHI Grant Number 16H01094.









§ CONVERGENCE CHECK AND VIOLATION OF UNCERTAINTY RELATION


To confirm that our numerical calculation corresponds to the continuum
limit of the lattice model, we check the convergence of the
logarithmic negativity. By introducing a scaling parameter λ,
we write other parameters contained in the model as


    N=200×λ,  α=1-λ^-2× 10^-8,
        l=λ.


We regard E_𝒩 as a function of λ for fixed l_p and
d_p. λ→∞ corresponds to the continuum limit
of the model. FIG. <ref> shows the result of convergence check.



The logarithmic negativity is independent of the parameter
λ. In the limit λ≫ 1, length scales (for example,
the physical size of the considering region or the distance) is much
larger than the UV cutoff Δ x. Hence, this limit
corresponds to the continuum limit and our numerical calculation well
approaches the continuum limit.

Furthermore, we evaluate  violation of the Heisenberg uncertainty
relation for our numerical calculation by checking a quantity defined
by


    U_N=-∑_j=1^Nlog_2[min(2ν_j,1 ) ],


where ν_j are eigenvalues of i Ω V_AB. If U_N=0
then we get relations ν_j≥ 1/2, which are equivalent
to the uncertainty relation. FIG. <ref> shows U_N as a
function of d_p for fixed l_p.



The uncertainty relation is expressed in terms of the two point
functions ⟨q̂q̂⟩,⟨p̂p̂⟩
and ⟨q̂p̂+p̂q̂⟩.  For the massless
theory, the q̂q̂-correlation has the IR divergence. On the
other hand, there is no the IR divergence in
p̂p̂,q̂p̂+p̂q̂-correlations. In our
numerical calculation, owing to behavior of the mode function in de
Sitter space, there appears the large difference of the magnitude
between the q̂q̂- and
p̂p̂,q̂p̂+p̂q̂-correlations. The violation
of the uncertainty relation due to numerical error tends to become
larger as d_p increases.  According to FIG. <ref>, this
violation of the Heisenberg uncertainty relation is kept small enough
to guarantee accuracy of our numerical calculation.




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