Physica A xx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
Dynamics of bipartite and tripartite entanglement in a
dissipative system of continuous variables
Q1
Yang Zhao a,∗ , Fulu Zheng a , Jing Liu a,b , Yao Yao a,c
∧
a
∧
∧
∧
∧
∧
∧
∧
Division of Materials Science, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
b
Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
c
State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China
highlights
•
•
•
•
•
An analytical criterion was derived for bipartite entanglement in Agarwal bath.
Entanglement phase diagram constructed showing regions of and without sudden death.
Using basis transformation, master equations for common bath cases are derived.
W and GHz type three modes states are investigated for common and separate baths.
Common bath mediated inter-mode interactions lead to persistent entanglement.
article
info
Article history:
Received 25 July 2014
Received in revised form 15 December 2014
Available online xxxx
Keywords:
Dissipative baths
Quantum entanglement
Continuous variable
1
abstract
With the help of the characteristic function and the covariance matrix, we study the effect
of a dissipative bath on bipartite entanglement in a system of continuous variables with no
direct inter-mode coupling. Under the dissipation of an Agarwal bath, the time evolution of
entanglement between two oscillators initially in a squeezed state has been examined, and
an analytical criterion, as a function of the squeezing parameter s and the bath temperature
n̄, has been derived for bipartite entanglement at long times, resulting in an entanglement
phase diagram comprising of a sudden death region and a no-sudden death region. If (s,
n̄) lies in the no-sudden death region, there will be sustained bipartite entanglement in
the long-time limit. The analysis has been extended to three-mode systems including fully
symmetric tripartite states and bisymmetric states, which are invariant with the exchange
of two modes (out of the three). If two symmetric modes are propagated in separate baths,
the bipartite entanglement between one of the two and the third mode will vanish in a
finite time. The disappearance of the entanglement leads to a decrease in cloning fidelities
to below the classical value of 0.5. For fully symmetric tripartite states, both the separateand common-bath cases have been considered. It is found that entanglement vanishes in a
finite time in the presence of separate baths, while it persists for a long time in the presence
of a common bath.
© 2015 Published by Elsevier B.V.
1. Introduction
Q2
2
3
In the existing literature [1–7] of quantum information and communication, a tremendous amount of interest has been
raised in bipartite and multipartite systems of continuous variables (CV), which are often synonymous to a set of harmonic
∗
Corresponding author.
E-mail address: yzhao@ntu.edu.sg (Y. Zhao).
http://dx.doi.org/10.1016/j.physa.2014.12.028
0378-4371/© 2015 Published by Elsevier B.V.
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Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
Fig. 1. (Color online) Schematic for the experimental realization on ZnO whispering gallery microcavity (left panel) and the corresponding parametric
scattering process for the polariton condensate (right panel).
oscillators. Generating, preserving and quantifying CV entanglement, an issue of fundamental importance, have been theoretically formulated and experimentally tested [8–11,1,5,12]. For example, a recent effort was focused upon entanglement
generation from two remote microscopic systems with no direct interactions, which was said to be realized with the assistance of a bosonic bath [12]. On the other hand, technological breakthroughs in quantum optics has renewed interest
in the basic subject of entanglement [13–16]. For example, the polariton, an entity mixing light and matter, can be a suitable vehicle for experimental realization of CV entanglement. In particular, Liew and Savona [16] extensively studied the
bipartite and multipartite entanglement between polariton modes with the consideration of parametric scattering as an
effective interaction among the modes. Thanks to both parametric scattering and repulsive potential induced by the exciton
reservoir and the pump field, a strong coupling microcavity in the form of a ZnO whispering gallery structure, was demonstrated to achieve a polariton condensate at room temperature [17–19]. This achievement affords us opportunities to study
∧
CV entanglement embedded in a heat bath from a new prospective, which is the main focus of this work.
Generally speaking, a separable density matrix can be represented by a mixture of direct-product states. A bipartite
density matrix is separable if it can be written in the form
ρ=

pi ρAi ⊗ ρBi ,
(1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
i
where pi > 0 and i pi = 1. An inseparability criterion concerning partial transpose of the density matrix was proposed
by Peres [2], and later shown by Horodecki to be a necessary and sufficient condition for inseparability of 2 × 2 or 2 × 3
systems [3]. Although the majority of the work in the field is concerned with Hilbert spaces with finite dimension, interests
have also expanded to CV systems. It has been shown by Simon [7] that the Peres–Horodecki criterion is both necessary
and sufficient for inseparability in systems of two harmonic oscillators as well. The bipartite entanglement initiated by
a common dissipative bath is another interesting topic in this field [20–22,12]. Braun showed that entanglement can be
generated between two qubits in a common bath in thermal equilibrium in the absence of direct inter-qubit coupling, and
the entanglement can persist for an arbitrarily long time [4]. Very recently, indirect inter-qubit coupling attributed to a
common boson bath has also been discussed in an extension of the spin–boson model [23,24] and in a scenario in which a
∧
two-level system is coupled simultaneously to a spin bath and a boson bath [25].
In this work we investigate bipartite and tripartite CV entanglement of oscillator systems coupled to separate as well as
common bosonic reservoirs. For an initially squeezed two-mode state, the entanglement is found to persist for a long time in
the presence of a common bath, and entanglement in the long time limit depends on the initial squeezing parameter and the
bath temperature. For systems of three oscillators, we study the bipartite and tripartite separability of the bisymmetric and
fully symmetric states. For the bisymmetric state, the tripartite entanglement is more stable than the bipartite entanglement
in the presence of the heat baths. The fully symmetric state has also been considered. This state will keep entangled if
three modes are coupled to a common bath, while the entanglement will dissipate to zero if each mode is coupled to an
∧
independent bath.
The remaining part of the paper is organized as follows. In Section 2, we introduce the Brownian motion of two oscillators
∧
in a common bath of the Agarwal type with the goal of probing the entanglement evolution of the oscillators. In Section 3, we
expand our analysis to tripartite systems and investigate the two-mode and three-mode separability of the bisymmetric and
fully symmetric states. Further more, the influence of the common bath and independent baths on the dynamical behavior
of the entanglement are also considered for the fully symmetric states. Section 4 is the conclusion of this paper.

2. Two oscillators
In a one-dimensional microcavity with strong coupling, such as the ZnO whispering gallery structure, the polariton condensate undergoes parametric scattering from one transverse electric mode to another [18]. As shown in Fig. 1, two balanced
polariton pair (signal and idler) could be generated by the parametric scattering, with the pair initially being entangled with
each other. Due to the strong binding energy of polaritons and the whispering gallery structure, these two polaritons will
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Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
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be trapped for a long time and move along different directions in the microcavity with the presence of dissipation induced
by the exciton reservoir. Then, there arises an interesting question: can the entangled polaritons in a dissipative bath with
no direct coupling preserve their entanglement after a sufficient long time? To this end, we adopt a model Hamiltonian describing two identical primary oscillators of frequency ω0 and mass m, with each oscillator denoting a polariton which has
no direct coupling with each other. The oscillators are coupled to a common bath of secondary oscillators of frequency ωk
and mass mk (k ̸= 0). As in this work the entanglement dynamics of oscillators coupled to a common bath and to independent baths are both included, we adopt the convention of labeling the oscillators coupled to a common bath as x oscillators,
and those coupled to independent baths as y oscillators. As will be shown, the two sets of oscillators are related by a simple
rotation. Here, as an extension of our previous work [26] a dissipative bath first proposed by Agarwal [27] is utilized for
reasons that will be elaborated later. The system Hamiltonian for two oscillators coupled to a common bath then reads

Ĥ =
h̄ω0 aĎxi axi +
13
14
15
16
17
18
19
20
21
Ď

i,k
23
qsxi

=
1/2
h̄
2mω0
(aĎxi
+ axi ),
27
28
29
ĤRWA =

h̄ω0 aĎxi axi +
31
33
34
35
36
=
1/2
h̄
2mk ωk
(bĎk + bk ),
(3)

Ď
h̄ωk bk bk +

i,k
Ď
gk (bk axi + bk aĎxi ).
(4)
To obtain the master equation for the x oscillators in a common bath, we perform a basis transformation from x oscillators
to y oscillators
qsy1


1
= √
qsy2
2
1
1
1
−1
qsx1


qsx2
,
(5)
where
qsyi =

1/2
h̄
2mω0
(aĎyi + ayi )
(6)
Ď
are the coordinate observables and ayi=1,2 (ayi=1,2 ) are the creation (annihilation) operators for y oscillators. Then the total
Hamiltonian (2) for x oscillators becomes
Ĥ =

h̄ω0 aĎyi ayi +

Ď
h̄ωk bk bk +

k
√
32

k
i
30
qbk
Ď

26
(2)
and gk are the coupling coefficients between the x oscillators and the common bath. axi=1,2 (axi=1,2 ) and bĎ (b) are the creation
(annihilation) operators for the x oscillators and the bath oscillators, respectively.
Now, we come to study the Brownian motion of polaritons in a dissipative bath where the system and the bath are weakly
coupled. In this weak system–bath coupling regime, the rotating-wave approximation (RWA) which neglects the rapidly
∧
oscillating terms of Eq. (2) is valid and has been widely used in quantum optics. Adopting RWA, the model Hamiltonian
reduces to
24
25
gk qsxi qbk ,
where qsxi and qbk are the coordinate observables for the x oscillators and the bath oscillators, respectively, which are related
to the corresponding boson operators by
i
22

h̄ωk bk bk +
k
i
12
3
gk′ qsy1 qbk ,
(7)
k
with gk′ = 2gk . The transformed Hamiltonian (7) describes two y oscillators among which only oscillator y1 coupled to a
bath with coupling strength gk′ while y2 is free.
Agarwal has obtained a Schrödinger-representation
 master equation for the reduced system density operator ρ in the

limit of an infinite number of bath oscillators ( k → dωk f (ωk )) [27]. As y oscillators dissipated by independent Agarwal
type baths, the master equation for the reduced density matrix ρ can be written as



 
 
∂ρ
= −iω
[aĎyi ayi , ρ] −
γyi n̄ aĎyi + ayi , aĎyi + ayi , ρ
∂t
i
i
  
  
 

Ď
−
γyi ayi ayi + ayi , ρ − aĎyi + ayi , ρ aĎyi − 2ρ ,
(8)
i
37
38
39
40
where γyi = π f (ω0 )|gcyi (ω0 )|2 is the damping constant, f (ω) is the density of bath oscillators, gcyi (ω) is the continuum
form of gkyi , n̄ = (eh̄ω/kB T − 1)−1 , with T the temperature and kB the Boltzmann constant. ω is the renormalized frequency
of ω0 [28],
ω = ω0 + P
∞

dω
0
41
f (ω)|gcyi (ω)|2
ω − ω0
,
(9)
and P stands for the Cauchy principal part. As the system and the dissipative bath are weakly coupled, the correlation time
of the bath is shorter than the characteristic time of the system and the influence of the system–bath interactions on the bath
∧
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Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
is negligible. Therefore, approximations including the short memory hypothesis for the bath and the Born approximation
which treats the bath effects in the lowest order are applied in deriving Eq. (8).
For the y oscillators after transformation, there are γy1 = 2γx , γy2 = 0 with γx = π f (ω0 )|gcx (ω0 )|2 and gcx (ω) being the
continuum form of gk . Then Eq. (8) reduces to





∂ρ
= −iω
aĎyi ayi , ρ − γy1 n̄ (aĎy1 + ay1 ), (aĎy1 + ay1 ), ρ
∂t
i


 
  Ď
− γy1 ay1 (ay1 + ay1 ), ρ − (aĎy1 + ay1 ), ρ aĎy1 − 2ρ .
Ď
ay i ,
Replacing
ayi and γy1 with
in a common bath
Ď
ax i ,
2

i,j
4
6
axi and γx , respectively, in Eq. (10), one obtains the master equation for two x oscillators
7
8
9
(11)
Agarwal master equation (8) has been applied to a system of two oscillators [26,29,30]. Despite widespread interest in
quantum dissipation across many disciplines [31–34], a quantum description of dissipation has been a challenging task. The
choice of baths remains at the heart of a successful dissipation theory. We choose the Agarwal bath because of its appearance
of the perfect bath from the physical perspective: it preserves the translational invariance of the system, which is mostly
important for the 1D microcavity, and delivers the eventual thermal equilibrium to the dissipative process while maintaining
density matrix positivity for almost all initial conditions [29].
Gaussian states evolving under the Agarwal bath are adequately described by the normal-ordered characteristic functions. Instead of resorting to numerical integration which may involve continuous evolution of the density matrix as well as
jumps at random instances [35,36], we adopt a method of solution utilizing the quantum characteristic function χ (λ, t ) [37]
−λ∗ a
3
(10)
 
  

 
γx axi aĎxj + axj , ρ − aĎxi + axi , ρ aĎxj − 2ρ .
Ď
λ1 a1
2
5
2
2

 


 
∂ρ
= −iω
[aĎxi axi , ρ] −
γx n̄ aĎxi + axi , aĎxj + axj , ρ
∂t
i
i,j
−
1
Ď
λ 2 a2
−λ∗ a
χ(λ, t ) = Tr(ρ e e 1 1 e e 2 2 ),
(12)
where λ = (λ1 , λ∗1 , λ2 , λ∗2 )T and the trace is taken over the system of two oscillators. Instead of dealing directly with the
density matrix, we derive from Eqs. (11) and (12) an equation of motion for the characteristic function χ (λ, t ) for two x
oscillators [26]
10
11
12
13
14
15
16
17
18
19
20
21
22
23
∂χ
= iω
∂t
2

λi
i =1
∂χ
∂χ
− λ∗i ∗
∂λi
∂λi
2

−
2





γ n̄ λi + λ∗i λj + λ∗j χ −
γ (λi + λ∗i )
i,j=1
i,j=1

∂χ
∂χ
+ ∗
∂λj
∂λj

,
(13)
which is to be solved by method of characteristics. Now we assume that the characteristic function has the form [26,29]
χ(λ, t ) = exp


24
25

∗ n k
∗ l
Cmn,kl (t )λm
.
1 (−λ1 ) λ2 (−λ2 )
(14)
26
mnkl
The Gaussian wavepackets are obtained by restricting m + n + k + l ≤ 2 in the summation over m, n, k, l. The second order
term in the exponent of Eq. (14) can be put in a matrix form
1 ∗T
λ L(t )λ,
2
in which

(15)
2C02,00
−C11,00
C01,01
−C01,10
−C01,10
C10,10
−C00,11
2C00,20
1 T
ξ M (t )ξ,
2
where the new basis ξ = (λR1 , λI1 , λR2 , λI2 )T is related to the old one by
Λ
0
0
29

C01,01
−C10,01 
.
2C00,02 
−C00,11
(16)
Another equivalent form of the second order term in the exponent is
ξ=
28
30
−C11,00
 2C
L(t ) =  20,00
−C10,01
C10,10

27

Λ
λ
31
32
(17)
33
34
(18)
with
35
36
Λ=

1/2
−i/2
1 /2
,
i/2

(19)
37
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
1
and M (t ) is a real 4 × 4 symmetric matrix
M (t ) =
2
3
4
5
6
7
V1 (t )
V12 (t )
T
V12
(t )
V2 ( t )

T
Ξ (t ) ≡ Det[I − V1 ]Det[I − V2 ] + (1 − |Det[V12 ]|)2 − Tr[(I − V1 )KV12 K (I − V2 )KV12
K]
− Det[I − V1 ] − Det[I − V2 ]
12
with the matrix K
13
K =

16
17
18
19
21
0
−1
(22)

1
.
0
(23)
Here Det(·) represents the determinant and I is the identity matrix. As shown by Simon [7], the Peres–Horodecki condition
is both necessary and sufficient to determine the inseparability of the two-oscillator system in an Agarwal bath, for which
the wave packet is Gaussian for all times.
Given the equation of motion and quantification of entanglement, we are now on the stage to discuss the evolution of
bipartite entanglement. As the main objective of this work, we consider the initial state of the two polaritons as the entangled
∧
two-mode squeezed state
|ψ⟩s = exp[−s(aĎ1 aĎ2 − a1 a2 )]|0⟩
20
22
(21)
where the auxiliary function Ξ (t ) is defined as
11
14
(20)
Ξ (t ) < 0,
10
15

with V1 (t ), V2 (t ), and V12 (t ) labeling 2 × 2 matrix of function of second order moments. Because the dynamics of the second
order term is irrelevant to the first order term [38,39], thus, if one only focuses on the dynamical information of the second
T
order term, it is allowed to substitute exp[ 21 ξ M (t )ξ] into Eq. (13) and solve the corresponding differential equations.
The Peres–Horodecki condition for inseparability concerns the positivity of partial transpose (PPT) of the density matrix.
When applied to Gaussian states in a two-harmonic-oscillator system, the system is inseparable when [7]
8
9
5
(24)
with s a real number and |0⟩ the vacuum state for both oscillators. Expanding the exponent of the two-mode squeezed state,
one can obtain an equivalent form
|ψ⟩s = (cosh s)−1
23

(− tanh s)n |n, n⟩ .
(25)
n
24
Then, it follows that the characteristic function at time t = 0 has the form


1
χ (λ, 0) = exp − sinh 2s(λ1 λ2 + λ∗1 λ∗2 ) − sinh2 s(|λ1 |2 + |λ2 |2 ) .
25
26
2
Comparing above expression with Eq. (14), one can find that there are only four nonzero second moments, they are
C11,00 (0) = C00,11 (0) = sinh2 s,
27
28
29
30
31
32
33
34
35
36
η = U ξ,
(29)
1
1 0
U = √ 
2 1
0
39
41
42
1
where the transformation matrix U reads

40
(27)
sinh 2s.
(28)
2
The two primary oscillators initially prepared as |ψ⟩s are coupled to a common bath and labeled as the x1 and x2 oscillators. After the basis transformation (5), the system can also be represented by two new oscillators y1 and y2 . Comparing
Eqs. (2) and (7), the system in which two oscillators x1 and x2 are coupled to a common bath is equivalent to the system in
which only oscillator y1 is coupled to the bath while y2 is free. Utilizing the transformation (5), we have successfully reduced
the problem of two oscillators coupled to a common bath to a special scenario of two oscillators coupled to independent
baths, as shown in Fig. 2.
The phase-space parameter vector η = (η1R , η1I , η2R , η2I )T in the y1 –y2 basis can be transformed from the x1 –xx basis
∧
through
C10,10 (0) = C01,01 (0) = −
37
38
(26)
0
1
0
1
1
0
−1
0

0
1 
.
0 
−1
(30)
It is easy to find that U = U T = U −1 . We can also obtain the transformed matrix L′ (t ) and M ′ (t ), which are the counterpart
of L and M in y1 –y2 basis, as below,
∧
L′ (t ) = UL(t )U ,
(31)
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Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
Fig. 2. (Color online) Schematic for the transformation between x1 –x2 pair and y1 –y2 pair.
and
1
M ′ (t ) =

V1 ( t )
′T
V12
(t )

V12 (t )
.
′
V2 ( t )
′
′
(32)
′
The three 2 × 2 matrices, V1′ (t ), V2′ (t ) and V12
(t ), contain elements that are functions of second order moments. For an initial
two-mode squeezed state in the x1 –x2 basis, the corresponding initial L′ (0) reads
1

− sinh2 s

 1
− sinh 2s
 2
L′ (0) = 


0


− sinh 2s
0
2
2
0
0
− sinh s
0
0
− sinh2 s
(33)
sinh 2s
2
Evoking the PPT criterion, it is obvious that the y1 and y2 oscillators are separable at t = 0. Based on the Eqs. (8) and (12),
one can have the differential equation on χ in y1 –y2 basis, which reads
∂χ
= iω
∂t
2

i =1
λi
∂χ
∂χ
− λ∗i ∗
∂λi
∂λi

3
4




0

.

1
sinh 2s

2

− sinh2 s
1
0
2

2


− γy1 n̄ λ1 + λ∗1 χ − γy1 λ1 + λ∗1

∂χ
∂χ
+ ∗
∂λ1
∂λ1

.
(34)
5
6
7
8
9
In Fig. 3, we compare the bipartite separabilities in the two pairs of oscillators at two temperatures, n̄ = 0.1 and n̄ = 0.5.
We consider two polarization directions of the mode, x and y, where the x1 –x2 pair is initially in a two-mode squeezed state
while the y1 –y2 pair is separable. Fig. 3(a) and (b) show the respective time evolution of Ξ (t ) in x1 –x2 pair and y1 –y2 pair at
∧
n̄ = 0.1. Fig. 3(c) and (d) show the respective time evolution of Ξ (t ) in x1 –x2 pair and y1 –y2 pair at n̄ = 0.5. From the four
plots one can see that the x1 –x2 pair retains entanglement for a much longer time than the y1 –y2 pair. This is attributed to
that despite the lack of direct coupling between the two x oscillators, the common bath provides an indirect inter-oscillator
interaction, which facilitates the preservation of bipartite entanglement in the x oscillators.
Alternatively, it is possible to change the basis as follows
10
11
12
13
14
15
16
17
z1 = x1 cos θ + x2 sin θ ,
(35)
18
z2 = x1 sin θ − x2 cos θ,
(36)
19
where θ is the angle between the x1 and z1 axes. At t = 0, the entanglement between the two z oscillators can be expressed
as a function of the angle θ and the squeeze parameter s:
Ξ (θ , s) = −4 cos (2θ ) sinh (2s).
2
2
(37)
The y oscillators can be obtained through transformations (35) and (36) by θ = π /4 from the x oscillators. In the long-time
limit, the matrix L′ (t = ∞) for the y1 –y2 pair settles into
−n̄
0

′
L (t = ∞) = 
0

0
0
−n̄
0
0
0
0
2
0
− sinh s
0
1 −2iωt
e
sinh 2s
2
1
2
e
2iωt
21
22
23
24



sinh 2s
.

− sinh2 s
20
(38)
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Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
a
b
c
d
7
Fig. 3. (Color online) The bipartite separabilities evolving with time in x1 –x2 pair and y1 –y2 pair. (a) The evolution of Ξ (t ) in x1 –x2 pair at n̄ = 0.1;
(b) the evolution of Ξ (t ) in y1 –y2 pair at n̄ = 0.1; (c) the evolution of Ξ (t ) in x1 –x2 pair at n̄ = 0.5; (d) the evolution of Ξ (t ) in y1 –y2 pair at n̄ = 0.5.
Other parameters are set as s = 0.5 and γy1 = 2γx = 0.01.
1
Therefore, for the x1 and x2 oscillators, L (t = ∞) takes the form

2
3
4
5
−n̄∞
 ce−2iωt

L (t = ∞) = 
−n̄−
−ce−2iωt
ce2iωt
−n̄∞
−ce2iωt
−n̄−
−n̄−
−ce−2iωt
−n̄∞
ce−2iωt
−ce2iωt

−n̄− 
,
ce2iωt 
−n̄∞
(39)
where n̄∞ = n̄ + sinh2 s /2, n̄− = n̄ − sinh2 s /2 and c = sinh(2s)/4. Consequently, an analytical expression for the
inseparability of L in the long-time limit can then be derived as




Ξ (t = ∞) = 2 + 4n̄ (1 + n̄) − 2 (1 + 2n̄) cosh 2s < 0.
(40)
19
From Eq. (40), we can also plot the phase diagram for two oscillators dissipated by an Agarwal bath. Fig. 4 shows the
two-dimensional phase diagram of the bipartite separability of the x1 and x2 oscillators in the long-time limit as spanned
by squeezing parameter s and the bath temperature n̄. In this phase diagram, the white region represents the so-called sudden death (SD) phase, and the shaded region, the no-sudden death (NSD) phase. If the parameters s and n̄ are initialized
in the NSD region of the phase diagram, the entanglement can persist for arbitrarily long times. On the other hand, if the
∧
parameters s and n̄ are in the SD region, the entanglement will disappear in a finite time. For a fixed temperature, one can
see that the increase of the s will facilitate the preservation of bipartite entanglement, while for a fixed squeezed parameter,
lower temperatures allow a better chance for entanglement to survive at long times. Paz and Roncaglia [40] investigated
the entanglement dynamics between two oscillators in a non-Markovian bath. A similar boundary between the SD and NSD
phases was recovered.
Furthermore, Eq. (37) allows a study of the entanglement as a function of θ . As shown in Fig. 5(a), the entanglement
oscillates between zero and a maximal value with respect to the variation of angle θ . The period of oscillation is π /4. The
time evolution of entanglement is also analyzed. As shown in Fig. 5(b), it is found that the entanglement, while persisting
for a long time, is in general a decreasing function of time.
20
3. Three oscillators
6
7
8
9
10
11
12
13
14
15
16
17
18
21
22
23
24
As shown in Fig. 1, additional polaritons may be generated by a pump beam onto the sample, resulting in a multi-polariton
system. To study entanglement evolution in systems of many polaritons, our preceding analysis needs to be extended to a
∧
multipartite scenario, in which the particles are entangled in complicated ways. For a tripartite system, for example, there
exist two different kinds of entanglement. One is the three-mode entanglement between one mode and the subsystem
8
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
10
8
6
4
2
0
0
0.5
1
S
1.5
2
Fig. 4. The phase diagram of the bipartite separability of the x1 and x2 oscillators at long time limit as spanned by squeezing parameter s and the bath
temperature n̄. The horizontal axis represents the squeezing parameter s from 0 to 2, and the vertical axis represents the bath temperature n̄ from 0 to 10.
In the black region, the bipartite oscillators will keep inseparability.
Fig. 5. (Color online) (a) The bipartite separabilities oscillating with angle θ in z1 –z2 pair. (b) The bipartite separabilities in z1 –z2 pair evolving with time
t for the different values of angle θ .
consisting of the other two modes. The other is bipartite entanglement between two modes obtained by tracing over the
remaining mode. Similar to a two-mode system, the separability of the three-mode system can be determined by the positivity of the partially-transposed density matrices [7,38,41,42]. Based on previous studies [39], tripartite entangled states
can be classified into five different classes from fully inseparable states to fully separable states:
Class 1. Fully inseparable states, i.e., not separable for any grouping of the parties.
Class 2. One-mode biseparable states, which are separable if two of the parties are grouped together, but inseparable
with respect to the other groupings.
Class 3. Two-mode biseparable states, which are separable with respect to two of the three bipartite splits but inseparable
with respect to the third.
Class 4. Three-mode biseparable states, which are separable with respect to all three bipartite splits but cannot be written
as a mixture of tripartite product states.
Class 5. The fully separable states, which can be written as a mixture of tripartite product states.
In this section, we focus on the separability of two types of tripartite states: bisymmetric states, which are invariant
under the exchange of two given modes, as well as the fully symmetric states, which are invariant under the exchange of
any two modes.
3.1. Bisymmetric states

|Ψ ⟩ = exp −s aĎ1 aĎ3 + aĎ2 aĎ3 − a1 a3 − a2 a3
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
First, we focus our attention on the three-mode and two-mode entanglement of bisymmetric tripartite states. We choose

1

|0⟩
(41)
17
18
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
1
9
as the initial state which is invariant under the exchange of modes 1 and 2. This state [43–46] can also be expressed as
m+n 
∞ 

(m + n)!
tanh α
|m, n, m + n⟩
(42)
|Ψ ⟩ =
√
cosh α m,n=0
m!n!
2
√
with α = − 2s. This entangled state is comprised of three modes: modes 1 and 2 are two remote polaritons, while mode
1
2
3
4
5
6
7
8
3 acts as a polariton from the pumping field entangled with the other two modes. It will be shown that this state is fully
inseparable, and exhibits both three-mode and two-mode entanglement.
We now consider the situation in which modes 1 and 2 are coupled to two identical but separate baths of the Agarwal
type during their propagation, and mode 3 is free. The master equation can be written as


3
2
 
 



∂ρ
Ď
= −iω
ai ai , ρ −
γ n̄ aĎi + ai , aĎi + ai , ρ
∂t
i
i =1
−
9
2
 

  
 

γ ai aĎi + ai , ρ − aĎi + ai , ρ aĎi − 2ρ .
(43)
i=1
10
Similar to a two-mode system, the quantum characteristic function for a three-mode system can be defined as
Ď
11
12
13
Ď
∗
∗
χ (λ, t ) = exp

Cm1 n1 ,m2 n2 ,m3 n3 (t )
mi ni
15
16
17
18
19
21
(44)
3


mi
i
λ (−λi )
∗ ni
,
(45)
i=1
the Gaussian wave packets are obtained by restricting m1 + n1 + m2 + n2 + m3 + n3 ≤ 2 in the summation over all
the subscripts. Apart from the first- and second-order moments introduced for each individual oscillator, twelve more cross
moments, namely, C10,10,00 (t ), C01,01,00 (t ), C10,00,10 (t ), C01,00,01 (t ), C00,10,10 (t ), C00,01,01 (t ), C10,01,00 (t ), C01,10,00 (t ), C10,00,01 (t ),
C01,00,10 (t ), C00,10,01 (t ) and C00,01,10 (t ), are added to account for cross correlations of these three oscillators. The second-order
terms in the exponent of Eq. (45) can be put in a matrix form
 −C
20
∗
where λ = (λ1 , λ∗1 , λ2 , λ∗2 , λ3 , λ∗3 )T . The equation of motion of this characteristic function is similar with Eq. (13). Assuming
that the characteristic function has the form

14
Ď
χ (λ, t ) = Tr(ρ eλ1 a1 e−λ1 a1 eλ2 a2 e−λ2 a2 eλ3 a3 e−λ3 a3 ),
11,00,00
2C02,00,00
−C01,10,00
C01,01,00
−C01,00,10
 2C20,00,00
−C10,01,00
L3 ( t ) = 
 C10,10,00

−C10,00,01
C10,00,10
−C11,00,00
C10,10,00
−C00,11,00
2C00,20,00
−C00,10,01
C00,10,10
−C10,01,00
C10,00,10
−C00,01,10
C00,10,10
−C00,00,11
2C00,00,20
C01,01,00
−C01,10,00
C01,00,01
−C01,00,10
2C00,02,00
−C00,11,00
C00,01,01
−C00,01,10
C01,00,01

−C10,00,01 
C00,01,01 
,
−C00,10,01 

(46)
2C00,00,02
−C00,00,11
where the subscript 3 stands for the tripartite state. At t = 0, there are nine nonzero second order moments
√
22
C11,00,00 (0) = C00,11,00 (0) = sinh2 ( 2s)/2,
(47)
√
23
24
25
26
27
28
29
30
31
32
33
C00,00,11 (0) = sinh2 ( 2s),
√
√
√
C10,00,10 (0) = C01,00,01 (0) = − sinh(2 2s)/(2 2),
(49)
C00,10,10 (0) = C00,01,01 (0) = − sinh(2 2s)/(2 2),
(50)
√
C10,01,00 (0) = C01,10,00 (0) = sinh2 ( 2s)/2.
(51)
Following the equation of motion of χ (λ, t ), one can derive a set of differential equations for Cm1 n1 ,m2 n2 ,m3 n3 (t ). The
differential equations for the first and second order moments are given in the Appendix.
After the same basis transformation used in Section 2, one can obtain a real symmetric matrix M3 (t ) from matrix L3 (t ).
Then the Wigner characteristic function of three modes state can be written as


1
χ W (ξ, t ) = exp − ξ T σ (t )ξ ,
2
(52)
T
where ξ = λR1 , λI1 , λR2 , λI2 , λR3 , λI3 and σ (t ) = I − M3 (t ) is the covariance matrix (CM) with I being the 6 × 6 identity
matrix. Inserting the values of the second order moments into CM, we can attain the CM for our initial state |Ψ ⟩ as

α
σ (0) = εT
ϵT

34
(48)
√
ε
α
ϵT

ϵ
ϵ ,
β
(53)
10
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
a
b
Fig. 6. (Color online) Time evolution of η1 (t ) (a) and η3 (t ) (b) with different damping constant γ = 0.02, 0.04, 0.06 and 0.08. n̄ = 0.01, s = 0.5.
√
√
√
where α = cosh2 ( 2s)I, β = cosh(2 2s)I and ε = sinh2 ( 2s)I, and ϵ = diag{ν + , ν − } with
1
√
ν± = ±
sinh(2 2s)
,
(54)
2
which remains the same form under the exchange of modes 1 and 2, and exhibits explicitly the bisymmetric property of |Ψ ⟩.
As we aim to investigate the three-mode separability, we introduce a set of three new matrices [7,38,41,42,46].
√
Γj (t ) = Λj σ (t )Λj + iΩ ,
j = 1, 2, 3,
(55)
where the auxiliary matrices Λ1 = diag(−1, 1, 1, 1, 1, 1), Λ2 = diag(1, 1, −1, 1, 1, 1) and Λ3 = diag(1, 1, 1, 1, −1, 1).
These matrices Γj (t ) are for the mirror reflection in the phase space which correspond to partial transpose of the density
matrices, and
2
3
4
5
6
7
8
3
Ω=

K
(56)
9
1
where the matrix K is defined by Eq. (23) and
denotes a direct sum. If Γj (t ) ≥ 0, the jth mode is separable from the
subsystem spanned by the other two modes at time t. The positivity of Γj (t ) is equivalent to that its minimum eigenvalue
ηj (t ) ≥ 0. Thus, ηj (t ) will be used for as entanglement detection.
As shown in Fig. 6, at t = 0, both η1 (0) and η3 (0) are negative, which indicates that modes 1 and 3 are inseparable from
the tripartite system. Because |Ψ ⟩ is bisymmetric, modes 1 and 2 are identical rendering the tripartite state fully inseparable
and belonging to Class 1. As a result of the bath dissipation, the system gradually become more separable, and an increase
in the damping constant γ speeds up the disappearance of the entanglement.
As for the two-mode separability, first we can calculate the two-mode entanglement by partially tracing over the third
mode. The covariance matrices of three possible partial traces Trl (ρ) (l = 1, 2, 3) can be obtained from σ (t ) by deleting
the corresponding (2l − 1)th and 2lth rows (and columns) of σ (t ) [43,44]. The Gaussian character of the partial traces also
permits us to investigate the two-mode separability by checking the positivity of the matrices Γjk (t ) (j ̸= k) obtained by
deleting the corresponding (2l−1)th and 2lth rows and columns of Γj (t ) or Γk (t ) [43]. Similar to the three-mode case, we also
calculate the minimum eigenvalue ηjk (t ) of Γjk (t ). Furthermore, another quantity we can calculate to describe two-mode
separability is the clone fidelity [47,48]. Ref. [48] shows that a nonclassical optimal fidelity in CV teleportation is available to

10
11
12
13
14
15
16
17
18
19
20
21
22
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
11
a
b
Fig. 7. (Color online) (a) Evolution of the bipartite separability of modes 1 and 3; (2) time evolution of clone fidelity F31 (t ) as a function of ωt. Both of the
quantities are plotted with different damping constant γ = 0.02, ∧
0.04, 0.06 and 0.08. Other parameters are setting as n̄ = 0.01, the squeezing parameter
s = 0.5.
1
2
3
4
5
6
7
estimate multipartite entanglement. The bisymmetric tripartite state |Ψ ⟩ in our work can be used as the entangled state in
the continuous variable telecloning. In this process, the first step is a joint measurement of the input state ρin , the target of
the telecloning, and mode 3. Then, two unitary operations, performed onto the remaining two modes of |Ψ ⟩, can generate
∧
two clones of ρin . The clone fidelity represents the similarity between ρin and the clone state, of which the corresponding
mode is i = 1, 2. The clone fidelity can be calculated as [45]
F3i (t ) =
1
,
√
√
1 + ( N3 (t ) + 1 − Ni (t ))2
with Ni (t ) the average photon number for ith mode
Ď
8
N1 (t ) = ⟨a1 a1 ⟩ = C11,00,00 (t ) + C10,00,00 (t )C01,00,00 (t ),
9
N2 (t ) = ⟨a2 a2 ⟩ = C00,11,00 (t ) + C00,10,00 (t )C00,01,00 (t ),
10
N3 (t ) = ⟨a3 a3 ⟩ = C00,00,11 (t ) + C00,00,10 (t )C00,00,01 (t ).
11
12
13
14
15
16
17
18
19
20
21
22
23
24
(57)
Ď
Ď
(58)
Here the subscript 3 in F3i (t ) denotes that the joint measurement is performed on the ρin and mode 3 of |Ψ ⟩. From the above
∧
equation, one can find that the clone fidelity is entirely quantified by the photon number Ni (t ). For simplicity, we set the
first order moments of the initial state to be zero which have no influence on the separability of the state.
We investigate the two-mode separability between modes 1 and 3 by calculating the minimum eigenvalue η13 (t ) (because of the symmetry, the separability between modes 2 and 3 is the same as that between modes 1 and 3), and the clone
∧
fidelity F31 (t ) with various damping constants γ . As shown in Fig. 7(a), η13 (t ) is negative at t = 0, which means that the
reduced two-mode density matrix Tr2 ρ is entangled for the bisymmetric three-mode state |Ψ ⟩. As expected, increasing γ
leads to a rapid decrease in the bipartite entanglement and the clone fidelity. F31 (t ) shown in Fig. 7(b) also decreases below
the classical value of 0.5 due to the fact that Tr2 ρ will become separable during the evolution.
Furthermore, we investigate the influence of the initial photon number on the entanglement in the optimal symmetric
case, i.e., N1 (0) = N2 (0) = N3 (0)/2. For instance, we calculate η13 (t ), and the clone fidelity F31 (t ) with three initial values
N3 (0) = 0.5, 1.0 and 3.0. Fig. 8(a) shows that modes 1 and 3 become separable at about ωt = 100 in all three cases, while
from Fig. 8(b), one can see that the clone fidelity F31 (t ) decreases more rapidly with the increase of the average photon
number.
12
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
Fig. 8. (Color online) (a) Evolution of the bipartite separability of modes 1 and 3; (b) the clone fidelity F31 (t ) with different initial mean photon numbers
∧
in an Agarwal bath. Both plots are drawn at n̄ = 0.01 and γ = 0.02.
∧
To wrap up this subsection, we study the relationship between the two-mode entanglement and three-mode entangle< τ1 < τ3 (shown in the insert), the entanglement between modes 1
∧
∧
and 3 in the two-mode state Tr2 ρ vanishes faster than the three-mode entanglement between mode 1 and the remaining
∧
subsystem, while the three-mode entanglement between mode 3 and the remaining subsystem has the longest lifetime in
this case, a result that concurs with those in Ref. [38].
Q3 ment, as shown in Fig. 9. Based on the fact that τ13
3.2. Fully symmetric states
1
|Ψ ⟩W = √ (|001⟩ + |010⟩ + |100⟩),
3
1
|Ψ ⟩GHZ = √ (|000⟩ + |111⟩),
2
ε
α
εT
3
4
5
ε
ε ,
α
7
8
(59)
9
(60)
10
where the basis states |0⟩ and |1⟩ represent spin ‘‘up’’ and ‘‘down’’, respectively. These two pure states are fully symmetric, i.e., invariant under the exchange of any two modes. The W state contains the maximal bipartite entanglement in any
reduced two-mode state, tracing over the third mode. In the mean time, the GHZ state possesses maximal tripartite entanglement, but there is no entanglement in any reduced two-mode state. For CV systems, there also exist W-type and GHZ-type
tripartite states. The influence of thermal baths on the entanglement in these states will be the central issue of this section.
Following the work of Adesso et al. [38,51], we use a standard covariance matrix (CM) to describe the initially fully
symmetric CV state. The standard CM can be expressed in a fully symmetric form as
α
σS (0) = ε T
εT
2
6
The fully symmetric tripartite states are invariant under the exchange of any two modes. Two examples are the W state
and the GHZ state [49,50], which are defined as

1
11
12
13
14
15
16
17

(61)
18
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
13
Fig. 9. Comparison of the bipartite separability and tripartite separability. The insert plot shows the details of the green circle region in the main figure.
τ13 , τ1 and τ3 are the points at which η13 , η1 and η3 equal to zero, respectively. The parameters are set as n̄ = 0.1, s = 0.5 and γ = 0.02. (For interpretation
of the references to color in this figure legend, the reader is referred to the web version of this article.)
∧
1
2
where α = a · I with I the identity matrix and a the local mixedness, which is defined as
ε = diag{e+ , e− } with
e± =
3
4
5
6
7
9
11
12
13
14
15
19
21
22
23
24
25
26
4a
.
(62)
j = 1, 2, 3,
(63)
and the two-mode separability is determined by the positivity of ΓSjk (t ) which is obtained by deleting the corresponding
(2l − 1)th and 2lth rows and columns of ΓSj (t ) with j, k, l = 1, 2, 3 and j ̸= k ̸= l. As a result of the symmetry, we only
concern the positivity of ΓS1 (t ) for three-mode separability and ΓS12 (t ) for two-mode separability in the rest of the paper.
Following the procedure in Section 2, we employ the characteristic function (44) to investigate the entanglement dynamics of the fully symmetric tripartite states. Two cases will be studied: (I) every mode of the system interacts with an
∧
independent bath and (II) all the modes are coupled to a common bath. The equation of motion for the characteristic function
∧
for Case (I) can be written as
∧




3


2

 ∂χ
∂χ
∂χ
∂χ
∂χ
=
iω λi
− λ∗i ∗ − γ n̄ λi + λ∗i χ − γ λi + λ∗i
+ ∗ ,
∂t
∂λi
∂λi
∂λi
∂λi
i =1
(64)
while the equation of motion for the characteristic function for Case (II) reads


 
3 
3
3






 ∂χ
∂χ
∂χ
∂χ
∗ ∂χ
∗
∗
∗
= iω
λi
− λi ∗ −
γ n̄ λi + λi λj + λj χ −
γ λi + λi
+ ∗ .
∂t
∂λi
∂λi
∂λj
∂λj
i=1
i,j=1
i ,j = 1
18
20
(a2 − 1)(9a2 − 1)
ΓSj (t ) = Λj σS (t )Λj + iΩ ,
16
17

Det(α) for every mode, and
For fully symmetric states, local mixedness a is constrained by a ≥ 1 (a = 1 for pure fully separable states). In this work,
we are mainly concerned with the separability when a > 1. As stated in the previous section, we shall study the two-mode
and three-mode separability of the fully symmetric CV states. The three-mode separability is determined by the positivity
of the matrices [7,38,41,42,46]
8
10
a2 − 1 ±
√
(65)
We first study the two-mode separability in Case (I), with each mode interacting with an independent bath. This separability can be reflected by the minimum eigenvalue ηS12 (t ) of the matrix ΓS12 (t ). Fig. 10(a) shows the variation of ηS12 (t )
with local mixedness a and the time t. At t = 0, ηS12 (0) decreases quickly to a minimum value and increases in a for small a.
∧
Then it increases gradually with increasing a and approaches zero for large values of the local mixedness a, as any reduced
two-mode subsystem of this state loses entanglement and becomes separable when the local mixedness a goes to infinity.
This phenomenon is also consistent with the fact that the fully symmetric state will become a CV GHZ state which exhibits
no two-mode entanglement [38]. With the increase of time, ηS12 (t ) always turns positive for all the values of a because
system–bath coupling has a negative influence on the entanglement.
∧
Q4
14
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
Fig. 10. (Color online) (a) Evolution of ηS12 (t ) in Case (I) with different values of local mixedness a; (b) Evolution of ηS1 (t ) in Case (I) with the change of a;
Other parameters are set as γ = 0.05 and n̄ = 0.5 in these figures.
Fig. 11. (Color online) (a) Evolution of ηS12 (t ) in Case (II) with the change of a; (b) Evolution of ηS1 (t ) in Case (II) with the change of a. Other parameters
are set as γ = 0.05 and n̄ = 0.5 in these figures.
For the three-mode entanglement at t = 0, ηS1 (0) is negative even when a is large. This means that the tripartite system
is fully inseparable and belongs to Class 1 [38,42]. The dynamics of ηS1 (t ) with the variation of a is shown in Fig. 10(b). One
can see that for all values of a, the system always trends to separable during the evolution.
As a comparison, we also study the dynamical behavior of ηS12 (t ) and ηS1 (t ) for Case (II) with a common bath, as shown
in Fig. 11(a) and (b). From these two figures one can see that, as in Case (I), ηS12 (t ) and ηS1 (t ) increase with the passage of
time. However, for most values of a, ηS12 (t ) and ηS1 (t ) can be negative for a long time, which indicates that the entanglement
in Case (II) can persist for a much longer time than that in Case (I). The common bath has a relatively positive effect on the
entanglement by providing an indirect channel of interaction between the modes.
After the investigation of the effect of local mixedness a on the two-mode and three-mode separability, we are now in
the position to quantify the entanglement and study the dynamical behavior of the entanglement in Cases (I) and (II). The
quantification of the tripartite entanglement of this state is provided by the residual Gaussian contangle [38,42,51]
i|(jk)
Gτ (t ) = Gτ (t ) = min[Gτ
res
i|j|k
(t ) − Gτ (t ) − Gτ (t )],
i|j
i|k
(66)
where the symbols i, j, k denote all the permutations of the three-mode indexes and
i|(jk)
Giτ|(jk) (t ) = Gτ (σS
i|j(k)
Gτ
(t ) = Gτ (σ
i|j(k)
2
Q5
3
4
5
6
7
8
9
10
11
12
13
i|(jk)
(t ))}]2 ,
(67)
14
i|j(k)
(t ))}] ,
(68)
15
(t )) = [max{0, − ln ν̃− (σS
(t )) = [max{0, − ln ν̃− (σ
1
2
i|(jk)
with σ i|j(k) (t ) being the reduced CM of modes i and j(k) at time t. It can be obtained from σS (t ) by deleting corresponding
(2k − 1)th and 2kth (or (2j − 1)th and 2jth) rows and columns. ν̃− (σ (t )) is the minimum symplectic eigenvalue of the
partially transposed CM of σ (t ). When the residual Gaussian contangle is larger than zero, the system is entangled [52,53,
38,42].
16
17
18
19
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
15
Fig. 12. (Color online) Evolution of residual contangle Gres
τ (t ) of fully symmetric state in (a) Case (I) and (b) Case (II) with different local mixedness a. Other
parameters are set as γ = 0.05 and n̄ = 0.5 in these figures.
7
In order to study the influence of bath on the entanglement, we calculate the residual Gaussian contangle with each
mode coupled to an identical, separate Agarwal bath, in Fig. 12(a) (Case (I)). As a comparison, we also show the dynamical
behavior of the residual Gaussian contangle with all modes coupled to a common bath, in Fig. 12(b) (Case (II)). From these
two plots, one can see that the dynamical behavior of the entanglement of the fully symmetric state in Cases (I) and (II) are
∧
∧
very different. Residual Gaussian contangle in Fig. 12(a) will decrease to zero, while Fig. 12(b) shows that, for the common
bath case, although the tripartite entanglement decays with time, it can persist for a long time. This phenomenon is also
consistent with the entanglement degradation behavior when the system is coupled to different kinds of baths [54,55].
8
4. Conclusions and discussion
1
2
3
4
5
6
31
There has been an increasing interest in quantum dissipation across many disciplines, and the choice of baths remains
at the heart of a successful dissipation theory. The Agarwal bath has the appearance of the perfect bath from the physical
perspective, as it preserves the translational invariance of the system, and delivers the eventual thermal equilibrium to
the dissipative process. In this work, starting from a quantum-optics consideration of two polariton modes generated by
parametric scattering, we investigate an initially two-mode squeezed state undergoing dissipation of the exciton reservoir,
which leads to a general discussion of the bath effect on bipartite entanglement in a system of continuous variables with no
direct inter-mode coupling. Separability in oscillator systems is studied with the help of the characteristic function and the
covariance matrix. Under the dissipation of an Agarwal bath, the time evolution of entanglement between two oscillators has
been analyzed in great detail, and it is found that the entanglement at long times is determined by the squeezing parameter
s and the bath temperature n̄. An analytical criterion as a function of s and n̄ has been derived for bipartite entanglement at
long times, resulting in an entanglement phase diagram comprising of a sudden death region and a no-sudden death region:
if (s, n̄) lies in the no-sudden death region, there will be a sustained bipartite entanglement in the limit of t → ∞. It is our
hope that a comprehensive experimental study on the entangled polaritons in ZnO would provide a close comparison with
the theoretical predictions.
Our analysis has been extended to tripartite states. We first study the bisymmetric state, which is invariant with the exchange of modes 1 and 2. If modes 1 and 2 are propagated in separate baths, the bipartite entanglement between mode 3 and
mode 1 or 2 will vanish in a finite time. The disappearance of the entanglement leads to a decrease in the cloning fidelity to
below the classical value of 0.5. For fully symmetric tripartite states, both the separate- and common-bath cases have been
considered. It is found that entanglement vanishes in a finite time in the presence of separate baths of the Agarwal type,
while it persists for a long time in the presence of a common bath. Our results may be useful in a variety of applications. In
particular, the possibility of sustained entanglement in a noisy environment for a long time period is highly relevant to quantum computation at elevated temperatures. As robustness of the quantum channel is essential in view of decoherence, we
hope that our consideration of entanglement dynamics in this work provides useful information for experimental designs.
32
Acknowledgments
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
33
34
35
36
37
Support from the Singapore National Research Foundation through the Competitive Research Programme (CRP) under
∧
Project No. NRF-CRP5-2009-04 and the Singapore Ministry of Education through the Academic Research Fund (Tier 2) under
∧
Project No. T207B1214 is gratefully acknowledged. One of us (YY) is also supported by the NSF of China (Grant Nos. 91333202,
∧
11134002 and 11104035)and the National Basic Research Program of China (Grant No. 2012CB921401). We thank Rui Yu
∧
and Rui Li for useful discussion.
Q6
16
Y. Zhao et al. / Physica A xx (xxxx) xxx–xxx
Appendix. Equation of motion for the bisymmetric state
1
The first-order moments obey six coupled differential equations.
2
Ċ10,00,00 = iωC10,00,00 − γ (C10,00,00 − C01,00,00 ),
(A.1)
3
Ċ01,00,00 = −iωC01,00,00 + γ (C10,00,00 − C01,00,00 ),
(A.2)
4
Ċ00,10,00 = iωC00,10,00 − γ (C00,10,00 − C00,01,00 ),
(A.3)
5
Ċ00,01,00 = −iωC00,01,00 + γ (C00,10,00 − C00,01,00 ),
(A.4)
6
Ċ00,00,10 = iωC00,00,10 + (C00,10,00 ),
(A.5)
7
Ċ00,00,01 = −iωC00,00,01 + (C00,01,00 ).
(A.6)
8
Eqs. (A.1)–(A.6) obey the Ehrenfest theorem which expresses a formal connection between the time dependence of expectation values of canonically conjugate variables.
The more important part—the second order moments including the cross moments is as follows,
10
Q7
∧
9
11
Ċ20,00,00 = 2iωC20,00,00 − γ n̄ − γ (2C20,00,00 − C11,00,00 ),
(A.7)
12
Ċ02,00,00 = −2iωC02,00,00 − γ n̄ − γ (2C02,00,00 − C11,00,00 ),
(A.8)
13
Ċ00,20,00 = 2iωC00,20,00 − γ n̄ − γ (2C00,20,00 − C00,11,00 ),
(A.9)
14
Ċ00,02,00 = −2iωC00,02,00 − γ n̄ − γ (2C00,02,00 − C00,11,00 ),
(A.10)
15
Ċ00,00,20 = 2iωC00,00,20 ,
(A.11)
16
Ċ00,00,02 = −2iωC00,00,02 ,
(A.12)
17
Ċ11,00,00 = 2γ (n̄ − C11,00,00 ) + 2γ (C20,00,00 + C02,00,00 ),
(A.13)
18
Ċ00,11,00 = 2γ (n̄ − C00,11,00 ) + 2γ (C00,20,00 + C00,02,00 ),
(A.14)
19
Ċ10,10,00 = 2iωC10,10,00 + γ (C10,01,00 + C01,10,00 − 2C10,10,00 ),
(A.15)
20
Ċ01,01,00 = −2iωC01,01,00 + γ (C10,01,00 + C01,10,00 − 2C01,01,00 ),
(A.16)
21
and
22
Ċ10,00,10 = 2iωC10,00,10 − γ (C10,00,10 − C01,00,10 ),
(A.17)
23
Ċ01,00,01 = −2iωC01,00,01 − γ (C01,00,01 − C10,00,01 ),
(A.18)
24
Ċ00,10,10 = 2iωC00,10,10 − γ (C00,10,10 − C00,01,10 ),
(A.19)
25
Ċ00,01,01 = −2iωC00,01,01 − γ (C00,01,01 − C00,10,01 ),
(A.20)
26
Ċ10,01,00 = γ (C10,10,00 + C01,01,00 − 2C10,01,00 ),
(A.21)
27
Ċ01,10,00 = γ (C10,10,00 + C01,01,00 − 2C01,10,00 ),
(A.22)
28
Ċ10,00,01 = −γ (C10,00,01 − C01,00,01 ),
(A.23)
29
Ċ01,00,10 = −γ (C01,00,10 − C10,00,10 ),
(A.24)
30
Ċ00,10,01 = −γ (C00,10,01 − C00,01,01 ),
(A.25)
31
Ċ00,01,10 = −γ (C00,01,10 − C00,10,10 ).
(A.26)
32
Q8 Eqs. (A.7)–(A.16) describe individual motion, and Eqs. (A.17)–(A.26) (A.27)–(A.29), cross correlation of the three oscillators.
33
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17
Q9
Q10