Disentanglement in a quantum critical environment
Zhe Sun and Xiaoguang Wang∗
Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
C. P. Sun†
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China
(Dated: October 25, 2018)
We study the dynamical process of disentanglement of two qubits and two qutrits coupled to
an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the
concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively.
Explicit connections between the concurrence (negativity) and the decoherence factors are given for
two initial states, the pure maximally entangled state and the mixed Werner state. We find that the
concurrence and negativity decay exponentially with fourth power of time in the vicinity of critical
point of the environmental system.
PACS numbers: 05.40.-a, 03.65.Yz, 32.80.-t,03.67.Mn
I. INTRODUCTION
Entanglement is one of the most essential features in
quantum mechanics [1] and in recent decades has been
focused by people in many fields of physics. Motivated by
the progress of quantum information, entanglement has
become a basic resource in the quantum technologies such
as quantum teleportation and quantum cryptography [2]-
[4]. On the other hand, generally a realistic system is
surrounded by an environment. The coupling between a
quantum system and its environment leads to decoher-
ence of the system. Thus, it is natural for us to consider
the process of degradation of entanglement due to the
decoherence. More recently, Yu and Eberly [5] showed
that two entangled qubits become completely disentan-
gled in a finite time under the influence of pure vacuum
noise. Surprisingly, they found that the behaviors of lo-
cal decoherence is different from spontaneous disentan-
glement. The decoherence effects take an infinite time
evolution under the influence of vacuum while the entan-
glement vanishes suddenly in a finite time. Some other
researchers also investigated the process of disentangle-
ment in the open quantum systems [6]-[9]. The problem
of decoherence from spin environments was studied by
Cucchietti et al [10], while they considered the spin en-
vironments consisting of N independent other than cor-
related spins .
In most of the previous studies, uncorrelated environ-
ments are usually considered, and modelled by a reser-
voir consists of harmonic oscillators. Although a collec-
tion of harmonic oscillators is a well approximated mod-
elling to represent the environment weakly coupled to
system, however, in the practical situation, particles in
the environment may have interactions with each other.
Consequently, a problem comes out: How does the en-
∗Electronic address: xgwang@zimp.zju.edu.cn
†Electronic address: suncp@itp.ac.cn
tanglement evolves in a correlated environment? In this
paper, we consider this problem and choose a correlated
spin chain, the Ising model in a transverse field, as the
surrounding system. Moreover, this surrounding system
displays quantum phase transition (QPT) at some criti-
cal point and thus it possesses the dynamic hypersensi-
tivity with respect to the perturbation even induced by
a single qubit [11].
As a quantum critical phenomenon, QPT happens at
zero temperature, at which the thermal fluctuations van-
ish. Thus, QPT is driven only by quantum fluctuation.
Usually, at the critical point there exists degeneracy be-
tween the energy levels of the systems when QPT hap-
pens. Therefore, it can be excepted that, when we study
the dynamic evolution of the system coupled to a envi-
ronment with QPT, some special dynamic features will
appear at the critical point. Quan et al [11] have studied
the decoherence induced by the correlated environment.
It was shown that at the critical point of a QPT the de-
coherence is enhanced. Following this work, Cucchietti
et al [12] discovered that the decoherence induced by the
critical environment possesses some universality with the
Boson-Hubbard model as an illustration.
Now, we consider two spins coupled to the Ising spin
chain in a transverse field, and the purpose is to reveal
the effect of the correlated environment on the dynamic
evolution of the two-spin entanglement. We will study
different cases including two qubits and qutrits. More-
over, we will consider cases that the two spins initially
start from a pure maximally entangled state and a mixed
Werner state [13]. The ‘sudden death’ of entanglement
is found to be a quite common phenomenon.
This paper is organized as follows. In Sec. II, we intro-
duce the model of two-spin system coupled to Ising spin
chain with a transverse field. By exactly diagonalizing
the Hamiltonian, we give expression of the time evolu-
tion operator. In Sec. III, the analytical results of the
concurrence [14] of the two qubits are calculated to show
the dynamics of entanglement. Numerical results are also
given to illustrate the details of the dynamical behaviors
http://arxiv.org/abs/0704.1172v1
mailto:xgwang@zimp.zju.edu.cn
mailto:suncp@itp.ac.cn
of entanglement. In Sec. IV, two qutrits are coupled to
the Ising spin chain. The analytical and numerical re-
sults of the negativity [15, 16] are given. At last we give
the conclusion in Sec. V.
II. MODEL HAMILTONIAN AND EVOLUTION
OPERATOR
We choose the engineered environment system to be
an Ising spin chain in a transverse field which displays a
QPT. Two spins are transversely coupled to the chain.
The corresponding Hamiltonian reads
σxl σ
l+1 +
(s1z + s2z)
, (1)
where λ characterizes the strength of the transverse field,
g denotes the coupling strength between the Ising chain
and the two spins, s1 and s2, σ
(α = x, y, z) are the
Pauli operators defined on the l-th site, and the total
number of spins in the Ising chain is L = 2M + 1. The
Ising model is the simplest model which exhibits a QPT,
and can be exactly calculated.
In order to diagonalize the Hamiltonian, firstly we no-
tice that [s1z + s2z, σ
] = 0, thus it is convenient to define
an operator-valued parameter
Λ̂ = λ+
(s1z + s2z), (2)
which is a conserved quantity. When we diagonalize the
Ising spin chain, the parameter Λ̂ can be treated as a c-
number with different values corresponding to the eigen-
values of s1z + s2z in the two-spin subspace.
By combining Jordan-Wigner transformation and
Fourier transformation to the momentum space [17], the
Hamiltonian can be written as [18]
σkx (Ωkσkz) e
σkx +
σ0z (3)
where we have used the following pseudospin operators
σkα (α = x, y, z) [18]
σkx = d
+ d−kdk, (k = 1, 2, ...M)
σky = −id†kd
+ id−kdk,
σkz = d
dk + d
d−k − 1,
σ0z = 2d
0d0 − 1, (4)
and d
, dk{k = 0, 1, 2, ...} denote the fermionic creation
and annihilation operators in the momentum space, re-
spectively. Here,
−Λ̂+2 cos (2πk/L)
+ 4 sin2 (2πk/L), (5)
θk = arcsin
−2 sin
. (6)
From Eq. (3) and the units where h̄ = 1, the time evolu-
tion operator is obtained as:
U(t) = e−i(−
+1)σ0zt
σkxe−itΩkσkze−i
σkx . (7)
Having explicitly known the evolution operator, we now
consider the entanglement dynamics of the two qubits
and two qutrits.
III. DYNAMICAL DISENTANGLEMENT OF
TWO QUBITS
A. The case with initial pure entangling state
We investigate the dynamic evolution of two-qubit en-
tanglement and assume that the two qubits initially start
from a maximally entangled state.
|Φ〉 = 1√
(|00〉+ |11〉) . (8)
Here, |0〉 and |1〉 denote the spin up and down, respec-
tively. The initial state of environment is assumed to
be the vacuum state in the momentum space, namely,
|ψE〉 = |0〉k=0 ⊗k>0 |0〉k|0〉−k, and the vacuum state
|0〉k satisfies dk|0〉k = 0. We may write a more general
initial state of this composite system as
|Ψ(0)〉 = (a |00〉+ b |11〉)⊗ |ψE〉 . (9)
From the evolution operator (7), the state vector at time
t is given by
|Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉 , (10)
where the unitary operator U0 and U1 can be obtained
from the unitary operator U(t) by replacing operator Λ̂
with number λ+g/2 and λ−g/2, respectively.
Tracing out the environment, in the basis spanned by
{|00〉 , |11〉 , |01〉 , |10〉}, the reduced density matrix of the
two-spin system is obtained as
ρ1,2 =
|a|2 ab∗F (t)
a∗bF ∗(t) |b|2
⊕ Z2×2, (11)
where F (t) = 〈ψE |U †1U0 |ψE〉 is the decoherence factor,
and Z2×2 denotes the 2×2 zero matrix. Now, the concur-
rence [14] of the reduced density matrix can be readily
given by
C = 2|ab∗F (t)| = C0|F (t)|, (12)
where C0 is the concurrence of the initial state. We see
that the concurrence is proportional to the norm of the
decoherence factor, and when the initial state is in a max-
imally entangled state (8), C = |F (t)|, namely, the con-
currence is equal to the norm of the decoherence factor.
Let us consider the decoherence factor
F (t) = 〈ψE |U †1U0 |ψE〉 =
Fk, (13)
where Un(n = 0, 1) is generated from Hamiltonian Hn
with Λ̂ = Λn(a number). From the unitary operator (7)
and the initial vacuum state, we obtain
|F (t)| =
sin(Ω
t) cos(Ω
t) sin θ
− cos(Ω(0)
t) sin(Ω
t) sin θ
− sin2(Ω(0)
t) sin2(Ω
t) sin2(θ
− θ(1)
where Ω
and θ
are obtained by replacing Λ̂ with Λn
in Eqs. (5) and (6), respectively. Here, Λ0 = λ+g/2 and
Λ1 = λ−g/2. This is one of our main results. We see
that the zero mode (k = 0) has no contribution to the
decoherence factor. Clearly, every factor Fk is less than
unit. So it can be well expected that in the large L limit,
|F (t)| will go to zero under some reasonable conditions.
By carrying out similar analysis of Ref. [11], we intro-
duce a cutoff number Kc and define the partial product
for the decoherence factor
|F (t)|
Fk ≥ |F (t)| , (15)
from which the corresponding partial sum
S (t) = ln |F (t)|
|lnFk| . (16)
For the case of small k and large L, we have Ω
|2− Λn|, consequently
− θ(1)
≈ 16k
2π2 (Λ0 − Λ1)2
L2 (2− Λ0)2 (2− Λ1)2
. (17)
As a result, if L is large enough and Λ0−Λ1 is very small
perturbation the approximation of S can be obtained as
S (t) ≈ −2E (Kc) (2− Λ0)−2 (2− Λ1)−2
×{(Λ0 − Λ1)2 sin2 (|2− Λ0| t) sin2 (|2− Λ1| t)
+[sin (|2− Λ0| t) cos (|2− Λ1| t) |2− Λ1|
− sin (|2− Λ1| t) cos (|2− Λ0| t) |2− Λ0|]2},
where
E (Kc) = 4π
2Kc (Kc + 1) (2Kc + 1) /
. (19)
In the derivation of the above equation, we have used
ln(1− x) ≈ −x for small x and
k2 = n(n+ 1)(2n+
1)/6.
0 50 100
0 2 4 6 8 10
λ=0.1
(a) (b)
FIG. 1: (a) Concurrence versus time t with different λ in
the case of weak coupling strength g = 0.1. The size of the
environment is L = 300. (b) shows the cases of larger λ.
0 20 40 60 80
L=200
L=600
L=1000
FIG. 2: Concurrence versus time with different environment
size L = 200, 600 and 1000. The transverse field λ = 4, and
the coupling strength g = 0.1.
For our two-qubit case, Λ0 = λ+g/2, Λ1 = λ−g/2.
When λ→ 2, and with a proper small g we have
|F (t)|
≈ e−γt
with γ = 2E (Kc) g
2. Notice that |F (t)|
is larger than
|F (t)| = C. Therefore, from the above heuristic analysis
we may expect that when the parameter λ is adjusted to
the vicinity of the critical point λc = 2, the concurrence
(or the decoherence factor) will exponentially decay with
the fourth power of time. Moreover, for short times, from
Eq. (14), the concurrence becomes
C ≈ e−Γt
with Γ = 1/2
sin2(θ
− θ(1)
Now we resort to numerical analysis of the dynamical
sensitivity and the concurrence decay. In the Fig. 1 (a)
and (b), we plot the concurrence versus time for different
λ. We find that in the vicinity of the critical point about
λ ∈ [2 − 0.3, 2 + 0.3], concurrence decays monotonously
with time. And extending the time range, however there
are not the revivals of concurrence. Figure 1 (a) shows
the cases of λ ≤ 2. We can see that concurrence for
0 2 4 6 8 10
g=0.1
g=100
FIG. 3: Concurrence versus time at the critical point λ = 2
with different coupling strength g.
the case λ = 2 decays more rapidly than other cases. It
should be noted that, the dynamics of the two-qubit en-
tanglement in Eq. (12) is absolutely determined by the
decoherence factor in Eq. (14), thus from a theoretical
point of view, the complete disentanglement cannot be
realized in a finite time. When parameter λ becomes
larger than λc,(g = 3, 4 and 5), the numerical results of
the concurrence are shown in Fig. 1 (b). The concur-
rence oscillates with time, and collapses and revivals are
observed. This is in contrast with the case of small λ,
where no revivals are found.
The surrounding system displays a QPT near the crit-
ical point, and there exists a competition between differ-
ent order tendencies [17]. From another point of view,
near the critical point quantum chaotic behaviors may
emerge [19]. For a system with quantum chaos, though
it is prepared in identical initial state, two slightly differ-
ent interactions can lead to two quite different quantum
evolutions. In our system the decoherence factor can act
as a fidelity and quantify the difference between the two
states which are produced through two different evolu-
tions. Decay of the fidelity can indicate the presence of
the quantum chaos [20], and here the monotonous de-
cay of the decoherence factor (concurrence) at the crit-
ical point may be considered as a signature of quantum
chaos.
In Fig. 2, for weak coupling g = 0.1 and λ = 4, the
oscillation of concurrence is suppressed by enlarging the
size of environment. The larger environment prevents
the revival of entanglement. In the short-time region, we
can see the larger size of environment will accelerate the
monotonous decay of concurrence. From Eq. (14), each
factor Fk is smaller than 1, thus it is reasonable that large
size of environment will be more effective to suppress the
factor F (t), and consequently suppress the concurrence.
In Fig. 3, we consider the effects of coupling g on the
dynamics of entanglement. At the critical point λ = 2,
we adjust g from a small one g = 0.1 to a strong one
g = 100. It can be found that when we properly en-
large the coupling, e.g. g = 1, the concurrence decays
more sharply than the case g = 0.1. However, when we
continue enlarging the coupling to about g > 10, e.g.
g = 25, concurrence will oscillate quickly and does not
decay monotonously to zero any more. For the case of
very large coupling g = 100, concurrence behaves as a
weak oscillation near the initial value of C = 1. It can be
expect that to the strong coupling limit of g, the concur-
rence will stay at C = 1 without changing with time. The
above behaviors remind us of the quantum Zeno effects
in process of quantum measurement [21]. The phenom-
ena shown in Fig. 3 is similar to the decay probability
which can be suppressed by the increasing coupling be-
tween system and measuring apparatus in quantum Zeno
effects.
B. The case of mixed state
Now, we study the dynamics of disentanglement of
mixed entangled state and assume the two qubits being
initially in a Werner state [13], which is given by
ρs = P |Φ〉 〈Φ|+
I4×4, (22)
where |Φ〉 is the maximally entangled state given by Eq.
(8), the parameter P ∈ [0, 1], and I4×4 denotes a 4 × 4
identity matrix. This state is a mixed state except the
extreme case of P = 1. Only when P > 1/3, the Werner
state ρs is entangled.
We assume the initial state of the whole system ρtot is
in a direct product form as
ρtot = ρs ⊗ |ψE〉 〈ψE | , (23)
where |ψE〉 is the initial state of the environment. Af-
ter the time evolution, we can obtain the reduce density
matrix of the two-qubit system in the basis spanned by
{|00〉 , |11〉 , |01〉 , |10〉} as follows
ρ1,2 =
PF (t)
PF ∗(t) 1+P
I2×2, (24)
where the decoherence factor F (t) is the same as Eq. (14).
From Eq.(24), the concurrence is derived as
C = max
|F |+ 1
. (25)
When P = 1, it reduces to Eq. (12) for the pure maxi-
mally entangled state. While in the region 1/3 < P < 1,
the concurrence vanishes when the decoherence factor
|F | ≤ (P−1 − 1)/2. (26)
Thus there exists a finite disentanglement time td, after
which the entanglement is zero. According to the results
of heuristic analysis in Eq. (20), |F (t)|
≈ e−γt4 , in the
condition of weak coupling and λ → 2, we can approxi-
mately give the disentanglement time
. (27)
0 2 4 6 8 10
P=0.5
P=0.7
FIG. 4: Concurrence versus time at the critical point λ = 2
and coupling strength g = 0.1 for parameters P = 0.5, 0.7
and 1.
Then, the disentanglement time increases as the proba-
bility P increases from 1/3 to 1.
In Fig. 4, we also numerically calculate the concurrence
versus time for different probabilities. For the mixed
states corresponding to P = 0.5 and 0.7, disentangle-
ment process takes only a finite time, while for the pure
state case (P = 1), disentanglement is only completed
asymptotically, and it will take an infinite time. Numeri-
cal results are consistent with the above analytical results
that the disentanglement time increases with the increase
of P .
IV. DYNAMICAL ENTANGLEMENT
EVOLUTION OF TWO QUTRITS
Now, we consider the case of two qutrits and use the
negativity [15] to quantify entanglement. For the sys-
tems with spin larger than 1/2, a non-entangled state
has necessarily a positive partial transpose (PPT) ac-
cording to the Peres-Horodecki criterion [15]. In the case
of two spin halves, and the case of (1/2,1) mixed spins, a
PPT is also sufficient. Vidal and Werner [16] developed
the Peres-Horodecki criterion and presented a measure of
entanglement called negativity that can be computed ef-
ficiently, and the negativity does not increase under local
manipulations of the system. The negativity of a state ρ
is defined as
N (ρ) =
|µi|, (28)
where µi is the negative eigenvalue of ρ
T2 , and T2 denotes
the partial transpose with respect to the second subsys-
tem. If N > 0, then the two-spin state is entangled.
The negativity has been used to characterize the entan-
glement in large spin system very well [22]-[24]. And
by means of negativity, Derkacz et al. have studied the
process of disentanglement in a pair of three-level atoms
interacting with the vacuum [8].
A. The case with initial pure state
In a similar vein as the study of two-qubit case, we
write a general initial state of the many-body system as
|Ψ(0)〉 = (a |00〉+ b |11〉+ c|22〉)⊗ |ψE〉 . (29)
where |0〉, |1〉 , |2〉 denote the spin-one state with mag-
netic quantum number 1, 0, -1 respectively. From the
evolution operator (7), the state vector at time t is given
|Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉
+c|22〉 ⊗ U2 |ψE〉 , (30)
where the unitary operator U0, U1,and U2 are obtained
from the unitary operator U(t) by replacing operator Λ̂
with number λ+g, λ and λ− g, respectively.
In the basis spanned by {|00〉, |11〉, |22〉, |01〉, |10〉,
|02〉, |20〉, |12〉, |21〉}, the reduced density matrix of the
two-qutrit system is
ρ1,2 =
|a|2 ab∗F1(t) ac∗F2(t)
a∗bF ∗1 (t) |b|2 bc∗F3(t)
a∗cF ∗2 (t) b
∗cF ∗3 (t) |c|2
⊕Z2×2 ⊕ Z2×2 ⊕ Z2×2, (31)
where
F1(t) = 〈ψE |U †1U0 |ψE〉 ,
F2(t) = 〈ψE |U †2U0 |ψE〉 ,
F3(t) = 〈ψE |U †2U1 |ψE〉 (32)
are the decoherence factors.
The partial transpose with respect to the second sys-
tem gives
ρT21,2 = diag(|a|2, |b|2, |c|2)⊕B1 ⊕B2 ⊕B3, (33)
where the three 2× 2 matrices
0 ab∗F1(t)
a∗bF ∗1 (t) 0
0 ac∗F2(t)
a∗cF ∗2 (t) 0
0 bc∗F3(t)
b∗cF ∗3 (t) 0
. (34)
Then, from the above matrix ρT21,2, one can obtain the
negativity as
N = |ab∗F1(t)|+ |ac∗F2(t)|+ |bc∗F3(t)|. (35)
For the maximally entangled state, a = b = c = 1/
and the negativity simplifies to
N = 1
(|F1(t)|+ |F2(t)|+ |F3(t)|) . (36)
0 2 4 6 8 10
λ=0.1
0  20 40 60 80 100
(a) (b)
FIG. 5: (a) Negativity versus time with different cases of λ =
0.1, 1 and 2. The coupling g = 0.1 and the size of environment
L = 300. (b) shows the cases of λ = 3, 4 and 5. The highest
one (solid line with up triangles) corresponds to the case λ =
5, and the lowest one (dashed line with points) corresponds
to λ = 3.
0 2 4 6 8 10
g=0.1
g=100
FIG. 6: Negativity versus time with different coupling
strengths g = 0.1, 1, 15 and 100 at the critical point λc = 2.
From the above equation, we can find the negativity is
a linear combination of three decoherence factors. Also
with the vacuum state of environment, the decoherence
factors |Fν(t)| = 〈ψE |U †jUi |ψE〉 are given by Eq.(14)
by the replacements Ω
→ Ω(i)
→ Ω(j)
→ θ(j)
. Here, Fν(t) denotes the three factors
F1(t), F2(t) and F3(t). U
jUi correspond to U
1U0, U
and U
2U1 in the three factors Eq. (32). The parameters
and θ
(n = 0, 1, 2) can be obtained by substitut-
ing Λ0 = λ+g, Λ1 = λ and Λ2 = λ−g into Eq. (5) and
During the similar analysis in the case of two qubits,
we can also introduce the cutoff number Kc and de-
fine the partial product for the three decoherence fac-
tors. Through the small k approximation, we can obtain
the three partial sums corresponding to the three fac-
tors. Therefore, under the condition of weak coupling g
and λ→ 2, in a finite time the three factors F1(t), F2(t)
and F3(t) will decay exponentially with time in a similar
form as Eq. (20).
We numerically calculate the dynamics of negativity.
In Fig. 5 (a), it shows the similar phenomena in Fig. 1
(a). When the coupling g is weak and λ→ 2, the dynam-
ical behaviors of the three decoherence factors in nega-
tivity (36) are nearly identical. Each of the factors decay
with time just as in Eq. (20), thus it can be understood
that negativity also decays monotonously with time in
the vicinity of λ = 2. In Fig. 5 (b), we consider the cases
of larger couplings. Comparing it with Fig. 1 (b), the
behaviors of negativity have some differences with con-
currence. More revivals are found in the behavior of the
negativity, and they result from the linear superposition
of the three decoherece factors.
In Fig. 6, we numerically study the effects of different
couplings g on the dynamics of negativity. Similar to the
dynamic behaviors of the concurrence. With a properly
large coupling such as g = 1, the decay of negativity
will be much sharper. But very strong coupling (g =
15) will make negativity oscillate rapidly. To the strong
coupling limit case of g = 100, negativity decays from
the initial value N = 1 to a steady value 1/3, which is
different from the concurrence of the two qubits. Let us
carry out the approximate analysis just like in the case
of two qubits. We can obtain three partial sum S1, S2
and S3, corresponding to the three decoherence factors
in Eq. (32), which are similar to Eq. (18). When g → ∞
and λ → 2, we have S2 → 0 and S1 = S3 ≈ −2E (Kc) t2
where E (Kc) is in Eq. (19), thus negativity will decay
sharply to a steady value of 1/3. We can see that different
dynamic properties of the factors cause the behaviors of
negativity shown in Fig. 6 is different from concurrence
in Fig. 3.
B. The case of mixed state
We then consider the mixed state, namely, the two-
qutrit Werner state
ρs = P |Φ〉〈Φ|+
I9×9, (37)
where |Φ〉 is the maximally entangled state of two qutrits
and |Φ〉 = (|00〉+ |11〉+ |22〉) /
3. Assume that the
whole system is initially in ρ tot = ρs ⊗ |ψE〉 〈ψE |. After
time evolution operator in Eq. (7), we can obtain the re-
duce density matrix of the two qutrits at arbitrary time t.
Then, we make the partial transpose with respect to the
second system on the reduce density matrix, and obtain
ρT21,2 =
diag(1 + 2P, 1 + 2P, 1 + 2P )
⊕B1 ⊕B2 ⊕B3, (38)
where the three 2× 2 matrices
PFk(t)
(t) 1−P
k = {1, 2, 3} (39)
From partially transposed reduced density matrix, the
negativity is given by
N = 1
|Fk(t)|+
. (40)
Since |Fk(t)| ≤ 1, the existence of nonzero negativity
needs the parameter P satisfying the condition 1/4 <
P ≤ 1. From the above equation, we can also reads
that the disentanglement occurs only when all the three
factors satisfy |Fk(t)| ≤ (P−1 − 1)/3.
Furthermore, we study the case of a d-dimension
Werner state being the initial state. Thus we give the
initial state of the system as
i,j=0
|ii〉 〈jj|+ 1− P
Id2×d2 , (41)
where the basis vector |ii〉 is the eigenvector of sz =
s1z + s2z with the eigenvalue 2i+1− d. Then the initial
state of the whole system is also performed by a direct
product form as ρtot = ρs ⊗ |ψE〉 〈ψE | . After the simi-
lar process mentioned in the former parts, we have the
matrix ρT21,2 denoting the reduce density matrix after the
partial transpose over the second subsystem at time t,
which is shown as:
ρT21,2 =
i,j=0
|ij〉 〈ji|Fi,j(t) +
Id2×d2
diag [1 + (d− 1)P, ..., 1 + (d− 1)P ]
PFi,j(t)
PF ∗i,j(t)
, (42)
where the decoherence factors Fi,j(t) = 〈ψE |U †jUi |ψE〉 ,
and the corresponding time evolution operator Ui can be
obtained from Eq. (7) by replacing operator Λ̂ with value
λ+g/2(2i + 1 − d), respectively. It is apparent that we
should only focus on the 2 × 2 matrices and obtain the
negativity as
N = 1
|Fi,j(t)|+
, (43)
from which we can see that negativity will be complete
vanishes when all the norms satisfy |Fi,j(t)| ≤ (P−1 −
1)/d simultaneously.
V. CONCLUSION
In summary, we have studied the dynamics of entan-
glement in a pure dephasing system. By making use of
the concept of concurrence, we studied two qubits cou-
pled to an Ising spin chain in a transverse field. When
the two qubits initially started from a pure entangled
state, we obtained the analytical results of concurrence
which is just a simple product of the initial concurrence
C(0) and the decoherence factor F (t). Thus the dynamic
properties of concurrence is absolutely determined by the
decoherence factor. Specially, in the case of weak cou-
pling, the concurrence decays exponentially with time
when λ → λc. Moreover, we found the decay of deco-
herence factor is of the form exp(−Γt4), which is not a
Gaussian form like in Ref. [11] and [12]. Certainly this
is due to the initial state of the environment we have
chosen.
Furthermore, when the two qubits are initially in the
Werner state, we have found that the complete disentan-
glement takes place in a finite time just as the ‘sudden
death’ of entanglement discovered in Ref. [5]. In [5], due
to the process of spontaneous emission, the sudden death
of entanglement can occur in an arbitrary entangled state
(pure or mixed). However, in our system with dephas-
ing effects, when the two entangled qubits are in a pure
state, there does not exist such a phenomena.
We also considered two qutrits coupled to the Ising spin
chain. When the qutrits initially start from a pure state,
we have obtained the expression of negativity which is
a linear combination of three decoherence factors. With
weak coupling, negativity also decays monotonously in
the condition λ → 2. When the qutrits are initially in
a Werner state, the complete disentanglement could oc-
cur in a finite time, and then the properties of negativity
are the three decoherence factors. Indeed, the correlated
environment, especially when QPT happens, greatly af-
fects the decoherence and the disentanglement process.
The entanglement decay in other environment which dis-
plays a QPT [25], or quantum chaos [26] deserves further
investigations.
Acknowledgments
This work is supported by NSFC with grant
Nos.10405019 and 90503003; NFRPC with grant No.
2006CB921206; Specialized Research Fund for the Doc-
toral Program of Higher Education (SRFDP) with grant
No.20050335087.
[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,
777 (1935).
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information Cambridge University Press,
Cambridge, England, 2000).
[3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A.
Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895
(1993).
[4] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991)
[5] T. Yu and J.H.Eberly, Phys. Rev. Lett. 93, 140404
(2004).
[6] M. S. Zubairy, G. S. Agarwal, and M. O. Scully, Phys.
Rev. A 70, 012316 (2004).
[7] P. J. Dodd, Phys. Rev. A 69, 052106.
[8] L. Derkacz and L. Jakóbczyk, Phys. Rev. A 74,
032313(2006).
[9] K. Roszak and P. Machnikowski, Phys. Rev. A 73,
022313 (2006).
[10] F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Phys. Rev.
A 72, 052113 (2005).
[11] H. T. Quan, Z.Song, P. Zanardi, and C. P. Sun, Phys.
Rev. Lett. 96, 140604 (2006).
[12] Fernando Martin Cucchietti, Sonia Fernandez Vidal and
Juan Pablo Paz, Phys. Rev. A 75, 032337 (2007).
[13] R. F. Werner, Phys. Rev. A 40, 4277 (1989).
[14] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[15] A. Peres, Phys. Rev. Lett. 77, 1413 (1996); M. Horodecki,
P. Horodecki and R. Horodecki, Phys. Lett. A 223 1
(1996).
[16] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314
(2002).
[17] S. Sachdev, Quantum Phase Transition (Cambridge Uni-
versity Press, Cambridge England, 1999)
[18] Y. D. Wang, Fei Xue and C. P. Sun, quant-ph/0603014.
[19] Clive Emary and Tobias Brandes, Phys. Rev. Lett. 90,
044101 (2003).
[20] Joseph Emerson, Yaakov S.Weinstein, Seth Lloyd and
D.G. Cory, Phys. Rev. Lett. 89, 284102 (2002).
[21] Kazuki Koshino and AkiraShimizu, Physics Reports 412
(2005) 191C275.
[22] J. Schliemann, Phys. Rev. A 68, 012309 (2003).
[23] X. Wang, H. B. Li, Z. Sun and Y. Q. Li, J. Phys. A:
Math. Gen. 38 8703 (2005).
[24] Z. Sun, X. Wang and Y. Q. Li, New J. Phys. 7, 83 (2005).
[25] Neill Lambert, Clive Emary and Tobias Brandes, Phys.
Rev. Lett. 92, 073602 (2004).
[26] H. Fujisaki, T. Miyadera and A. Tanaka, Phys. Rev. E
67, 066201 (2003).
http://arxiv.org/abs/quant-ph/0603014

  We study the dynamical process of disentanglement of two qubits and two
qutrits coupled to an Ising spin chain in a transverse field, which exhibits a
quantum phase transition. We use the concurrence and negativity to quantify
entanglement of two qubits and two qutrits, respectively. Explicit connections
between the concurrence (negativity) and the decoherence factors are given for
two initial states, the pure maximally entangled state and the mixed Werner
state. We find that the concurrence and negativity decay exponentially with
fourth power of time in the vicinity of critical point of the environmental
system.

Disentanglement in a quantum critical environment
Zhe Sun and Xiaoguang Wang∗
Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
C. P. Sun†
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China
(Dated: October 25, 2018)
We study the dynamical process of disentanglement of two qubits and two qutrits coupled to
an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the
concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively.
Explicit connections between the concurrence (negativity) and the decoherence factors are given for
two initial states, the pure maximally entangled state and the mixed Werner state. We find that the
concurrence and negativity decay exponentially with fourth power of time in the vicinity of critical
point of the environmental system.
PACS numbers: 05.40.-a, 03.65.Yz, 32.80.-t,03.67.Mn
I. INTRODUCTION
Entanglement is one of the most essential features in
quantum mechanics [1] and in recent decades has been
focused by people in many fields of physics. Motivated by
the progress of quantum information, entanglement has
become a basic resource in the quantum technologies such
as quantum teleportation and quantum cryptography [2]-
[4]. On the other hand, generally a realistic system is
surrounded by an environment. The coupling between a
quantum system and its environment leads to decoher-
ence of the system. Thus, it is natural for us to consider
the process of degradation of entanglement due to the
decoherence. More recently, Yu and Eberly [5] showed
that two entangled qubits become completely disentan-
gled in a finite time under the influence of pure vacuum
noise. Surprisingly, they found that the behaviors of lo-
cal decoherence is different from spontaneous disentan-
glement. The decoherence effects take an infinite time
evolution under the influence of vacuum while the entan-
glement vanishes suddenly in a finite time. Some other
researchers also investigated the process of disentangle-
ment in the open quantum systems [6]-[9]. The problem
of decoherence from spin environments was studied by
Cucchietti et al [10], while they considered the spin en-
vironments consisting of N independent other than cor-
related spins .
In most of the previous studies, uncorrelated environ-
ments are usually considered, and modelled by a reser-
voir consists of harmonic oscillators. Although a collec-
tion of harmonic oscillators is a well approximated mod-
elling to represent the environment weakly coupled to
system, however, in the practical situation, particles in
the environment may have interactions with each other.
Consequently, a problem comes out: How does the en-
∗Electronic address: xgwang@zimp.zju.edu.cn
†Electronic address: suncp@itp.ac.cn
tanglement evolves in a correlated environment? In this
paper, we consider this problem and choose a correlated
spin chain, the Ising model in a transverse field, as the
surrounding system. Moreover, this surrounding system
displays quantum phase transition (QPT) at some criti-
cal point and thus it possesses the dynamic hypersensi-
tivity with respect to the perturbation even induced by
a single qubit [11].
As a quantum critical phenomenon, QPT happens at
zero temperature, at which the thermal fluctuations van-
ish. Thus, QPT is driven only by quantum fluctuation.
Usually, at the critical point there exists degeneracy be-
tween the energy levels of the systems when QPT hap-
pens. Therefore, it can be excepted that, when we study
the dynamic evolution of the system coupled to a envi-
ronment with QPT, some special dynamic features will
appear at the critical point. Quan et al [11] have studied
the decoherence induced by the correlated environment.
It was shown that at the critical point of a QPT the de-
coherence is enhanced. Following this work, Cucchietti
et al [12] discovered that the decoherence induced by the
critical environment possesses some universality with the
Boson-Hubbard model as an illustration.
Now, we consider two spins coupled to the Ising spin
chain in a transverse field, and the purpose is to reveal
the effect of the correlated environment on the dynamic
evolution of the two-spin entanglement. We will study
different cases including two qubits and qutrits. More-
over, we will consider cases that the two spins initially
start from a pure maximally entangled state and a mixed
Werner state [13]. The ‘sudden death’ of entanglement
is found to be a quite common phenomenon.
This paper is organized as follows. In Sec. II, we intro-
duce the model of two-spin system coupled to Ising spin
chain with a transverse field. By exactly diagonalizing
the Hamiltonian, we give expression of the time evolu-
tion operator. In Sec. III, the analytical results of the
concurrence [14] of the two qubits are calculated to show
the dynamics of entanglement. Numerical results are also
given to illustrate the details of the dynamical behaviors
http://arxiv.org/abs/0704.1172v1
mailto:xgwang@zimp.zju.edu.cn
mailto:suncp@itp.ac.cn
of entanglement. In Sec. IV, two qutrits are coupled to
the Ising spin chain. The analytical and numerical re-
sults of the negativity [15, 16] are given. At last we give
the conclusion in Sec. V.
II. MODEL HAMILTONIAN AND EVOLUTION
OPERATOR
We choose the engineered environment system to be
an Ising spin chain in a transverse field which displays a
QPT. Two spins are transversely coupled to the chain.
The corresponding Hamiltonian reads
σxl σ
l+1 +
(s1z + s2z)
, (1)
where λ characterizes the strength of the transverse field,
g denotes the coupling strength between the Ising chain
and the two spins, s1 and s2, σ
(α = x, y, z) are the
Pauli operators defined on the l-th site, and the total
number of spins in the Ising chain is L = 2M + 1. The
Ising model is the simplest model which exhibits a QPT,
and can be exactly calculated.
In order to diagonalize the Hamiltonian, firstly we no-
tice that [s1z + s2z, σ
] = 0, thus it is convenient to define
an operator-valued parameter
Λ̂ = λ+
(s1z + s2z), (2)
which is a conserved quantity. When we diagonalize the
Ising spin chain, the parameter Λ̂ can be treated as a c-
number with different values corresponding to the eigen-
values of s1z + s2z in the two-spin subspace.
By combining Jordan-Wigner transformation and
Fourier transformation to the momentum space [17], the
Hamiltonian can be written as [18]
σkx (Ωkσkz) e
σkx +
σ0z (3)
where we have used the following pseudospin operators
σkα (α = x, y, z) [18]
σkx = d
+ d−kdk, (k = 1, 2, ...M)
σky = −id†kd
+ id−kdk,
σkz = d
dk + d
d−k − 1,
σ0z = 2d
0d0 − 1, (4)
and d
, dk{k = 0, 1, 2, ...} denote the fermionic creation
and annihilation operators in the momentum space, re-
spectively. Here,
−Λ̂+2 cos (2πk/L)
+ 4 sin2 (2πk/L), (5)
θk = arcsin
−2 sin
. (6)
From Eq. (3) and the units where h̄ = 1, the time evolu-
tion operator is obtained as:
U(t) = e−i(−
+1)σ0zt
σkxe−itΩkσkze−i
σkx . (7)
Having explicitly known the evolution operator, we now
consider the entanglement dynamics of the two qubits
and two qutrits.
III. DYNAMICAL DISENTANGLEMENT OF
TWO QUBITS
A. The case with initial pure entangling state
We investigate the dynamic evolution of two-qubit en-
tanglement and assume that the two qubits initially start
from a maximally entangled state.
|Φ〉 = 1√
(|00〉+ |11〉) . (8)
Here, |0〉 and |1〉 denote the spin up and down, respec-
tively. The initial state of environment is assumed to
be the vacuum state in the momentum space, namely,
|ψE〉 = |0〉k=0 ⊗k>0 |0〉k|0〉−k, and the vacuum state
|0〉k satisfies dk|0〉k = 0. We may write a more general
initial state of this composite system as
|Ψ(0)〉 = (a |00〉+ b |11〉)⊗ |ψE〉 . (9)
From the evolution operator (7), the state vector at time
t is given by
|Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉 , (10)
where the unitary operator U0 and U1 can be obtained
from the unitary operator U(t) by replacing operator Λ̂
with number λ+g/2 and λ−g/2, respectively.
Tracing out the environment, in the basis spanned by
{|00〉 , |11〉 , |01〉 , |10〉}, the reduced density matrix of the
two-spin system is obtained as
ρ1,2 =
|a|2 ab∗F (t)
a∗bF ∗(t) |b|2
⊕ Z2×2, (11)
where F (t) = 〈ψE |U †1U0 |ψE〉 is the decoherence factor,
and Z2×2 denotes the 2×2 zero matrix. Now, the concur-
rence [14] of the reduced density matrix can be readily
given by
C = 2|ab∗F (t)| = C0|F (t)|, (12)
where C0 is the concurrence of the initial state. We see
that the concurrence is proportional to the norm of the
decoherence factor, and when the initial state is in a max-
imally entangled state (8), C = |F (t)|, namely, the con-
currence is equal to the norm of the decoherence factor.
Let us consider the decoherence factor
F (t) = 〈ψE |U †1U0 |ψE〉 =
Fk, (13)
where Un(n = 0, 1) is generated from Hamiltonian Hn
with Λ̂ = Λn(a number). From the unitary operator (7)
and the initial vacuum state, we obtain
|F (t)| =
sin(Ω
t) cos(Ω
t) sin θ
− cos(Ω(0)
t) sin(Ω
t) sin θ
− sin2(Ω(0)
t) sin2(Ω
t) sin2(θ
− θ(1)
where Ω
and θ
are obtained by replacing Λ̂ with Λn
in Eqs. (5) and (6), respectively. Here, Λ0 = λ+g/2 and
Λ1 = λ−g/2. This is one of our main results. We see
that the zero mode (k = 0) has no contribution to the
decoherence factor. Clearly, every factor Fk is less than
unit. So it can be well expected that in the large L limit,
|F (t)| will go to zero under some reasonable conditions.
By carrying out similar analysis of Ref. [11], we intro-
duce a cutoff number Kc and define the partial product
for the decoherence factor
|F (t)|
Fk ≥ |F (t)| , (15)
from which the corresponding partial sum
S (t) = ln |F (t)|
|lnFk| . (16)
For the case of small k and large L, we have Ω
|2− Λn|, consequently
− θ(1)
≈ 16k
2π2 (Λ0 − Λ1)2
L2 (2− Λ0)2 (2− Λ1)2
. (17)
As a result, if L is large enough and Λ0−Λ1 is very small
perturbation the approximation of S can be obtained as
S (t) ≈ −2E (Kc) (2− Λ0)−2 (2− Λ1)−2
×{(Λ0 − Λ1)2 sin2 (|2− Λ0| t) sin2 (|2− Λ1| t)
+[sin (|2− Λ0| t) cos (|2− Λ1| t) |2− Λ1|
− sin (|2− Λ1| t) cos (|2− Λ0| t) |2− Λ0|]2},
where
E (Kc) = 4π
2Kc (Kc + 1) (2Kc + 1) /
. (19)
In the derivation of the above equation, we have used
ln(1− x) ≈ −x for small x and
k2 = n(n+ 1)(2n+
1)/6.
0 50 100
0 2 4 6 8 10
λ=0.1
(a) (b)
FIG. 1: (a) Concurrence versus time t with different λ in
the case of weak coupling strength g = 0.1. The size of the
environment is L = 300. (b) shows the cases of larger λ.
0 20 40 60 80
L=200
L=600
L=1000
FIG. 2: Concurrence versus time with different environment
size L = 200, 600 and 1000. The transverse field λ = 4, and
the coupling strength g = 0.1.
For our two-qubit case, Λ0 = λ+g/2, Λ1 = λ−g/2.
When λ→ 2, and with a proper small g we have
|F (t)|
≈ e−γt
with γ = 2E (Kc) g
2. Notice that |F (t)|
is larger than
|F (t)| = C. Therefore, from the above heuristic analysis
we may expect that when the parameter λ is adjusted to
the vicinity of the critical point λc = 2, the concurrence
(or the decoherence factor) will exponentially decay with
the fourth power of time. Moreover, for short times, from
Eq. (14), the concurrence becomes
C ≈ e−Γt
with Γ = 1/2
sin2(θ
− θ(1)
Now we resort to numerical analysis of the dynamical
sensitivity and the concurrence decay. In the Fig. 1 (a)
and (b), we plot the concurrence versus time for different
λ. We find that in the vicinity of the critical point about
λ ∈ [2 − 0.3, 2 + 0.3], concurrence decays monotonously
with time. And extending the time range, however there
are not the revivals of concurrence. Figure 1 (a) shows
the cases of λ ≤ 2. We can see that concurrence for
0 2 4 6 8 10
g=0.1
g=100
FIG. 3: Concurrence versus time at the critical point λ = 2
with different coupling strength g.
the case λ = 2 decays more rapidly than other cases. It
should be noted that, the dynamics of the two-qubit en-
tanglement in Eq. (12) is absolutely determined by the
decoherence factor in Eq. (14), thus from a theoretical
point of view, the complete disentanglement cannot be
realized in a finite time. When parameter λ becomes
larger than λc,(g = 3, 4 and 5), the numerical results of
the concurrence are shown in Fig. 1 (b). The concur-
rence oscillates with time, and collapses and revivals are
observed. This is in contrast with the case of small λ,
where no revivals are found.
The surrounding system displays a QPT near the crit-
ical point, and there exists a competition between differ-
ent order tendencies [17]. From another point of view,
near the critical point quantum chaotic behaviors may
emerge [19]. For a system with quantum chaos, though
it is prepared in identical initial state, two slightly differ-
ent interactions can lead to two quite different quantum
evolutions. In our system the decoherence factor can act
as a fidelity and quantify the difference between the two
states which are produced through two different evolu-
tions. Decay of the fidelity can indicate the presence of
the quantum chaos [20], and here the monotonous de-
cay of the decoherence factor (concurrence) at the crit-
ical point may be considered as a signature of quantum
chaos.
In Fig. 2, for weak coupling g = 0.1 and λ = 4, the
oscillation of concurrence is suppressed by enlarging the
size of environment. The larger environment prevents
the revival of entanglement. In the short-time region, we
can see the larger size of environment will accelerate the
monotonous decay of concurrence. From Eq. (14), each
factor Fk is smaller than 1, thus it is reasonable that large
size of environment will be more effective to suppress the
factor F (t), and consequently suppress the concurrence.
In Fig. 3, we consider the effects of coupling g on the
dynamics of entanglement. At the critical point λ = 2,
we adjust g from a small one g = 0.1 to a strong one
g = 100. It can be found that when we properly en-
large the coupling, e.g. g = 1, the concurrence decays
more sharply than the case g = 0.1. However, when we
continue enlarging the coupling to about g > 10, e.g.
g = 25, concurrence will oscillate quickly and does not
decay monotonously to zero any more. For the case of
very large coupling g = 100, concurrence behaves as a
weak oscillation near the initial value of C = 1. It can be
expect that to the strong coupling limit of g, the concur-
rence will stay at C = 1 without changing with time. The
above behaviors remind us of the quantum Zeno effects
in process of quantum measurement [21]. The phenom-
ena shown in Fig. 3 is similar to the decay probability
which can be suppressed by the increasing coupling be-
tween system and measuring apparatus in quantum Zeno
effects.
B. The case of mixed state
Now, we study the dynamics of disentanglement of
mixed entangled state and assume the two qubits being
initially in a Werner state [13], which is given by
ρs = P |Φ〉 〈Φ|+
I4×4, (22)
where |Φ〉 is the maximally entangled state given by Eq.
(8), the parameter P ∈ [0, 1], and I4×4 denotes a 4 × 4
identity matrix. This state is a mixed state except the
extreme case of P = 1. Only when P > 1/3, the Werner
state ρs is entangled.
We assume the initial state of the whole system ρtot is
in a direct product form as
ρtot = ρs ⊗ |ψE〉 〈ψE | , (23)
where |ψE〉 is the initial state of the environment. Af-
ter the time evolution, we can obtain the reduce density
matrix of the two-qubit system in the basis spanned by
{|00〉 , |11〉 , |01〉 , |10〉} as follows
ρ1,2 =
PF (t)
PF ∗(t) 1+P
I2×2, (24)
where the decoherence factor F (t) is the same as Eq. (14).
From Eq.(24), the concurrence is derived as
C = max
|F |+ 1
. (25)
When P = 1, it reduces to Eq. (12) for the pure maxi-
mally entangled state. While in the region 1/3 < P < 1,
the concurrence vanishes when the decoherence factor
|F | ≤ (P−1 − 1)/2. (26)
Thus there exists a finite disentanglement time td, after
which the entanglement is zero. According to the results
of heuristic analysis in Eq. (20), |F (t)|
≈ e−γt4 , in the
condition of weak coupling and λ → 2, we can approxi-
mately give the disentanglement time
. (27)
0 2 4 6 8 10
P=0.5
P=0.7
FIG. 4: Concurrence versus time at the critical point λ = 2
and coupling strength g = 0.1 for parameters P = 0.5, 0.7
and 1.
Then, the disentanglement time increases as the proba-
bility P increases from 1/3 to 1.
In Fig. 4, we also numerically calculate the concurrence
versus time for different probabilities. For the mixed
states corresponding to P = 0.5 and 0.7, disentangle-
ment process takes only a finite time, while for the pure
state case (P = 1), disentanglement is only completed
asymptotically, and it will take an infinite time. Numeri-
cal results are consistent with the above analytical results
that the disentanglement time increases with the increase
of P .
IV. DYNAMICAL ENTANGLEMENT
EVOLUTION OF TWO QUTRITS
Now, we consider the case of two qutrits and use the
negativity [15] to quantify entanglement. For the sys-
tems with spin larger than 1/2, a non-entangled state
has necessarily a positive partial transpose (PPT) ac-
cording to the Peres-Horodecki criterion [15]. In the case
of two spin halves, and the case of (1/2,1) mixed spins, a
PPT is also sufficient. Vidal and Werner [16] developed
the Peres-Horodecki criterion and presented a measure of
entanglement called negativity that can be computed ef-
ficiently, and the negativity does not increase under local
manipulations of the system. The negativity of a state ρ
is defined as
N (ρ) =
|µi|, (28)
where µi is the negative eigenvalue of ρ
T2 , and T2 denotes
the partial transpose with respect to the second subsys-
tem. If N > 0, then the two-spin state is entangled.
The negativity has been used to characterize the entan-
glement in large spin system very well [22]-[24]. And
by means of negativity, Derkacz et al. have studied the
process of disentanglement in a pair of three-level atoms
interacting with the vacuum [8].
A. The case with initial pure state
In a similar vein as the study of two-qubit case, we
write a general initial state of the many-body system as
|Ψ(0)〉 = (a |00〉+ b |11〉+ c|22〉)⊗ |ψE〉 . (29)
where |0〉, |1〉 , |2〉 denote the spin-one state with mag-
netic quantum number 1, 0, -1 respectively. From the
evolution operator (7), the state vector at time t is given
|Ψ(t)〉 = a |00〉 ⊗ U0 |ψE〉+ b |11〉 ⊗ U1 |ψE〉
+c|22〉 ⊗ U2 |ψE〉 , (30)
where the unitary operator U0, U1,and U2 are obtained
from the unitary operator U(t) by replacing operator Λ̂
with number λ+g, λ and λ− g, respectively.
In the basis spanned by {|00〉, |11〉, |22〉, |01〉, |10〉,
|02〉, |20〉, |12〉, |21〉}, the reduced density matrix of the
two-qutrit system is
ρ1,2 =
|a|2 ab∗F1(t) ac∗F2(t)
a∗bF ∗1 (t) |b|2 bc∗F3(t)
a∗cF ∗2 (t) b
∗cF ∗3 (t) |c|2
⊕Z2×2 ⊕ Z2×2 ⊕ Z2×2, (31)
where
F1(t) = 〈ψE |U †1U0 |ψE〉 ,
F2(t) = 〈ψE |U †2U0 |ψE〉 ,
F3(t) = 〈ψE |U †2U1 |ψE〉 (32)
are the decoherence factors.
The partial transpose with respect to the second sys-
tem gives
ρT21,2 = diag(|a|2, |b|2, |c|2)⊕B1 ⊕B2 ⊕B3, (33)
where the three 2× 2 matrices
0 ab∗F1(t)
a∗bF ∗1 (t) 0
0 ac∗F2(t)
a∗cF ∗2 (t) 0
0 bc∗F3(t)
b∗cF ∗3 (t) 0
. (34)
Then, from the above matrix ρT21,2, one can obtain the
negativity as
N = |ab∗F1(t)|+ |ac∗F2(t)|+ |bc∗F3(t)|. (35)
For the maximally entangled state, a = b = c = 1/
and the negativity simplifies to
N = 1
(|F1(t)|+ |F2(t)|+ |F3(t)|) . (36)
0 2 4 6 8 10
λ=0.1
0  20 40 60 80 100
(a) (b)
FIG. 5: (a) Negativity versus time with different cases of λ =
0.1, 1 and 2. The coupling g = 0.1 and the size of environment
L = 300. (b) shows the cases of λ = 3, 4 and 5. The highest
one (solid line with up triangles) corresponds to the case λ =
5, and the lowest one (dashed line with points) corresponds
to λ = 3.
0 2 4 6 8 10
g=0.1
g=100
FIG. 6: Negativity versus time with different coupling
strengths g = 0.1, 1, 15 and 100 at the critical point λc = 2.
From the above equation, we can find the negativity is
a linear combination of three decoherence factors. Also
with the vacuum state of environment, the decoherence
factors |Fν(t)| = 〈ψE |U †jUi |ψE〉 are given by Eq.(14)
by the replacements Ω
→ Ω(i)
→ Ω(j)
→ θ(j)
. Here, Fν(t) denotes the three factors
F1(t), F2(t) and F3(t). U
jUi correspond to U
1U0, U
and U
2U1 in the three factors Eq. (32). The parameters
and θ
(n = 0, 1, 2) can be obtained by substitut-
ing Λ0 = λ+g, Λ1 = λ and Λ2 = λ−g into Eq. (5) and
During the similar analysis in the case of two qubits,
we can also introduce the cutoff number Kc and de-
fine the partial product for the three decoherence fac-
tors. Through the small k approximation, we can obtain
the three partial sums corresponding to the three fac-
tors. Therefore, under the condition of weak coupling g
and λ→ 2, in a finite time the three factors F1(t), F2(t)
and F3(t) will decay exponentially with time in a similar
form as Eq. (20).
We numerically calculate the dynamics of negativity.
In Fig. 5 (a), it shows the similar phenomena in Fig. 1
(a). When the coupling g is weak and λ→ 2, the dynam-
ical behaviors of the three decoherence factors in nega-
tivity (36) are nearly identical. Each of the factors decay
with time just as in Eq. (20), thus it can be understood
that negativity also decays monotonously with time in
the vicinity of λ = 2. In Fig. 5 (b), we consider the cases
of larger couplings. Comparing it with Fig. 1 (b), the
behaviors of negativity have some differences with con-
currence. More revivals are found in the behavior of the
negativity, and they result from the linear superposition
of the three decoherece factors.
In Fig. 6, we numerically study the effects of different
couplings g on the dynamics of negativity. Similar to the
dynamic behaviors of the concurrence. With a properly
large coupling such as g = 1, the decay of negativity
will be much sharper. But very strong coupling (g =
15) will make negativity oscillate rapidly. To the strong
coupling limit case of g = 100, negativity decays from
the initial value N = 1 to a steady value 1/3, which is
different from the concurrence of the two qubits. Let us
carry out the approximate analysis just like in the case
of two qubits. We can obtain three partial sum S1, S2
and S3, corresponding to the three decoherence factors
in Eq. (32), which are similar to Eq. (18). When g → ∞
and λ → 2, we have S2 → 0 and S1 = S3 ≈ −2E (Kc) t2
where E (Kc) is in Eq. (19), thus negativity will decay
sharply to a steady value of 1/3. We can see that different
dynamic properties of the factors cause the behaviors of
negativity shown in Fig. 6 is different from concurrence
in Fig. 3.
B. The case of mixed state
We then consider the mixed state, namely, the two-
qutrit Werner state
ρs = P |Φ〉〈Φ|+
I9×9, (37)
where |Φ〉 is the maximally entangled state of two qutrits
and |Φ〉 = (|00〉+ |11〉+ |22〉) /
3. Assume that the
whole system is initially in ρ tot = ρs ⊗ |ψE〉 〈ψE |. After
time evolution operator in Eq. (7), we can obtain the re-
duce density matrix of the two qutrits at arbitrary time t.
Then, we make the partial transpose with respect to the
second system on the reduce density matrix, and obtain
ρT21,2 =
diag(1 + 2P, 1 + 2P, 1 + 2P )
⊕B1 ⊕B2 ⊕B3, (38)
where the three 2× 2 matrices
PFk(t)
(t) 1−P
k = {1, 2, 3} (39)
From partially transposed reduced density matrix, the
negativity is given by
N = 1
|Fk(t)|+
. (40)
Since |Fk(t)| ≤ 1, the existence of nonzero negativity
needs the parameter P satisfying the condition 1/4 <
P ≤ 1. From the above equation, we can also reads
that the disentanglement occurs only when all the three
factors satisfy |Fk(t)| ≤ (P−1 − 1)/3.
Furthermore, we study the case of a d-dimension
Werner state being the initial state. Thus we give the
initial state of the system as
i,j=0
|ii〉 〈jj|+ 1− P
Id2×d2 , (41)
where the basis vector |ii〉 is the eigenvector of sz =
s1z + s2z with the eigenvalue 2i+1− d. Then the initial
state of the whole system is also performed by a direct
product form as ρtot = ρs ⊗ |ψE〉 〈ψE | . After the simi-
lar process mentioned in the former parts, we have the
matrix ρT21,2 denoting the reduce density matrix after the
partial transpose over the second subsystem at time t,
which is shown as:
ρT21,2 =
i,j=0
|ij〉 〈ji|Fi,j(t) +
Id2×d2
diag [1 + (d− 1)P, ..., 1 + (d− 1)P ]
PFi,j(t)
PF ∗i,j(t)
, (42)
where the decoherence factors Fi,j(t) = 〈ψE |U †jUi |ψE〉 ,
and the corresponding time evolution operator Ui can be
obtained from Eq. (7) by replacing operator Λ̂ with value
λ+g/2(2i + 1 − d), respectively. It is apparent that we
should only focus on the 2 × 2 matrices and obtain the
negativity as
N = 1
|Fi,j(t)|+
, (43)
from which we can see that negativity will be complete
vanishes when all the norms satisfy |Fi,j(t)| ≤ (P−1 −
1)/d simultaneously.
V. CONCLUSION
In summary, we have studied the dynamics of entan-
glement in a pure dephasing system. By making use of
the concept of concurrence, we studied two qubits cou-
pled to an Ising spin chain in a transverse field. When
the two qubits initially started from a pure entangled
state, we obtained the analytical results of concurrence
which is just a simple product of the initial concurrence
C(0) and the decoherence factor F (t). Thus the dynamic
properties of concurrence is absolutely determined by the
decoherence factor. Specially, in the case of weak cou-
pling, the concurrence decays exponentially with time
when λ → λc. Moreover, we found the decay of deco-
herence factor is of the form exp(−Γt4), which is not a
Gaussian form like in Ref. [11] and [12]. Certainly this
is due to the initial state of the environment we have
chosen.
Furthermore, when the two qubits are initially in the
Werner state, we have found that the complete disentan-
glement takes place in a finite time just as the ‘sudden
death’ of entanglement discovered in Ref. [5]. In [5], due
to the process of spontaneous emission, the sudden death
of entanglement can occur in an arbitrary entangled state
(pure or mixed). However, in our system with dephas-
ing effects, when the two entangled qubits are in a pure
state, there does not exist such a phenomena.
We also considered two qutrits coupled to the Ising spin
chain. When the qutrits initially start from a pure state,
we have obtained the expression of negativity which is
a linear combination of three decoherence factors. With
weak coupling, negativity also decays monotonously in
the condition λ → 2. When the qutrits are initially in
a Werner state, the complete disentanglement could oc-
cur in a finite time, and then the properties of negativity
are the three decoherence factors. Indeed, the correlated
environment, especially when QPT happens, greatly af-
fects the decoherence and the disentanglement process.
The entanglement decay in other environment which dis-
plays a QPT [25], or quantum chaos [26] deserves further
investigations.
Acknowledgments
This work is supported by NSFC with grant
Nos.10405019 and 90503003; NFRPC with grant No.
2006CB921206; Specialized Research Fund for the Doc-
toral Program of Higher Education (SRFDP) with grant
No.20050335087.
[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,
777 (1935).
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information Cambridge University Press,
Cambridge, England, 2000).
[3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A.
Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895
(1993).
[4] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991)
[5] T. Yu and J.H.Eberly, Phys. Rev. Lett. 93, 140404
(2004).
[6] M. S. Zubairy, G. S. Agarwal, and M. O. Scully, Phys.
Rev. A 70, 012316 (2004).
[7] P. J. Dodd, Phys. Rev. A 69, 052106.
[8] L. Derkacz and L. Jakóbczyk, Phys. Rev. A 74,
032313(2006).
[9] K. Roszak and P. Machnikowski, Phys. Rev. A 73,
022313 (2006).
[10] F. M. Cucchietti, J. P. Paz, and W. H. Zurek, Phys. Rev.
A 72, 052113 (2005).
[11] H. T. Quan, Z.Song, P. Zanardi, and C. P. Sun, Phys.
Rev. Lett. 96, 140604 (2006).
[12] Fernando Martin Cucchietti, Sonia Fernandez Vidal and
Juan Pablo Paz, Phys. Rev. A 75, 032337 (2007).
[13] R. F. Werner, Phys. Rev. A 40, 4277 (1989).
[14] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[15] A. Peres, Phys. Rev. Lett. 77, 1413 (1996); M. Horodecki,
P. Horodecki and R. Horodecki, Phys. Lett. A 223 1
(1996).
[16] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314
(2002).
[17] S. Sachdev, Quantum Phase Transition (Cambridge Uni-
versity Press, Cambridge England, 1999)
[18] Y. D. Wang, Fei Xue and C. P. Sun, quant-ph/0603014.
[19] Clive Emary and Tobias Brandes, Phys. Rev. Lett. 90,
044101 (2003).
[20] Joseph Emerson, Yaakov S.Weinstein, Seth Lloyd and
D.G. Cory, Phys. Rev. Lett. 89, 284102 (2002).
[21] Kazuki Koshino and AkiraShimizu, Physics Reports 412
(2005) 191C275.
[22] J. Schliemann, Phys. Rev. A 68, 012309 (2003).
[23] X. Wang, H. B. Li, Z. Sun and Y. Q. Li, J. Phys. A:
Math. Gen. 38 8703 (2005).
[24] Z. Sun, X. Wang and Y. Q. Li, New J. Phys. 7, 83 (2005).
[25] Neill Lambert, Clive Emary and Tobias Brandes, Phys.
Rev. Lett. 92, 073602 (2004).
[26] H. Fujisaki, T. Miyadera and A. Tanaka, Phys. Rev. E
67, 066201 (2003).
http://arxiv.org/abs/quant-ph/0603014
