Journal of Quantum Information Science, 2013, 3, 34-41
http://dx.doi.org/10.4236/jqis.2013.31008 Published Online March 2013 (http://www.scirp.org/journal/jqis)
Measurement-Induced Nonlocality and Geometric
Discord in the Spin-Boson Model
Guoyou Wang, Zilong Fan, Haosheng Zeng*
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of
Education, Department of Physics, Hunan Normal University, Changsha, China
Email: *hszeng@hunnu.edu.cn
Received December 9, 2012; revised January 22, 2013; accepted February 1, 2013
ABSTRACT
Dynamics of measurement-induced-nonlocality (MIN) and geometric measure of discord (GD) in the spin-boson model
are studied. Analytical results show that for two large classes of initial states, MINs are equal but GDs are different. At
the end of evolution, MIN and GD initially stored in the spin system transfer completely to reservoirs. The quantum
beats for MIN and GD are also found, which are the results of quantum interference between two local non-Markovian
dynamics via quantum correlation.
Keywords: Measurement-Induced Nonlocality; Geometric Discord; Quantum Beat
1. Introduction
Quantum correlation arises from noncommutativity of oper-
ators representing states, observables, and measurements
[1]. Quantum entanglement, as the earliest known quan-
tum correlation, has acquired extensive research and is
found to be an useful resource in quantum communica-
tion and quantum computation [2]. However, entangle-
ment is not the unique kind of quantum correlation. A
more general concept, quantum discord [3], was found,
which is regarded as the measure of all nonclassical cor-
relations in a bipartite system, being the entanglement of
a particular case of it. It was shown that there exists sepa-
rable states with nonzero discord which can be used to
perform quantum computation [4,5]. Quantum discord is
also useful in the studies of quantum phase transition [6,7]
and estimation of quantum correlations in Grover search
algorithm [8]. Unfortunately, evaluation of quantum dis-
cord in general requires considerable numerical minimi-
zation and analytical results that are known only for cer-
tain classes of states [9-11]. More recently, two measures
out of geometric perspective in measurements, GD [12]
and MIN [13], were proposed, which as the authors shown
have the merit of easier evaluation. In fact, Luo and Fu
[14,15] have evaluated the GD for some typical classes of
states and found the tight lower bound. The dissecting
about the meaning of GD was also done [16,17].
Realistic quantum systems cannot avoid interactions
with their environments, leading to the change of quan-
tum correlation. In the last decade, the influences of en-
vironments to quantum entanglement [18] and quantum
discord [19] have been investigated extensively. An in-
teresting phenomenon, named as “entanglement sudden
death” (ESD) [20,21] for a pair of entangled qubits ex-
posed to local Markovian environments was found. In
contrast, quantum discord in similar conditions decays
only in asymptotic time [22], which signifies that quan-
tum discord is more robust against Markovian noise than
entanglement. There are also many works involved in the
evolutions of quantum entanglement and discord in the
non-markovian environments [23-29]. Especially, trap-
ping [30] and quantum beats [31] for quantum entangle-
ment and discord for a pair of qubits in local structured
environments were found.
In this paper, we investigate the dynamics of GD and
MIN in a system that consists of two independent spins
(qubits) coupled respectively to their local environments.
Our motivation is to find the evolutional properties of GD
and MIN and the difference between their evolutions and
make a comparison between the evolutions of quantum
entanglement or/and discord. The paper is constructed as
follows. In Section 2, we introduce the original definitions
of GD and MIN. In Section 3, we first introduce the inter-
action model, and then study the evolution of GD and
MIN of different partitions for two classes of initial states.
Section 4 is devoted to the study of quantum beats for GD
and MIN. And the conclusion is arranged in Section 5.
2. GD and MIN
Let
be a bipartite state shared by parties A and B.
The GD of
is defined as [12],
*Corresponding author.
C
opyright © 2013 SciRes. JQIS
G. Y. WANG ET AL. 35
 
2
min ,
A
A
D

 (1)
where the min is taken over all von Neumann measure-
ments on subsystem A, and the Hilbert Schmidt
A
norm defined by 2
X
trX X. On the other hand, the
MIN is defined as [13]
 
2
max
A
A
N

, (2)
here the max is taken over the von Neumann measure-
ments which do not disturb A
locally, that is,
AA AA
kk
k

.
For a two-qubit system, the general state may be writ-
ten as,
3
1
33
1,1
1
4
.
AB AB
ii
i
AB AB
ii ijij
iij
x
yt



 
 

11 1
1
(3)
Here i
are the Pauli operators,
and

AB
ii
xTr

1
AB
i
y
i
Tr
1
ij
tTr
are the components of the local
Bloch vectors,
AB
i j

ij
Tt
are the components
of the correlation tensor
. For this state, we
have [12,13]

22
max
1,
4
DxTk

(4)
and


2
3
11
if 0,
4
1if 0.
4
ttt
t
trTTx TTxx
x
N
trTT x






(5)
where 22
i
i
x
x with
123
,, t
x
xxx
t
, is the
max
k
t
maximum eigenvalue of matrix
x
xTT, and 3
the
minimum eigenvalue of matrix .
t
TT
3. Model and Dynamics of MIN and GD
The model we consider consists of two independent spins
interacting respectively with their local boson reservoirs.
The total Hamiltonian is,

2
0
,,,
1
2
,, ,
1
2
.
z
iikiki
ik
iki iki ik
ik
Hb
gb b









k
b
(6)
Here
z
i
and
ii

are the Pauli and raising (low-
ering) operators for the spin. is the crea-
ith

,,ikik
bb
ith
tion (annihilation) operator of the mode in the
reservoir with corresponding frequency ,ik
ith
. The cou-
pling strength between spin and its reservoir mode
is denoted by ,ik
i k
g
. For simplicity, we assume the two
spins have equal Zeeman splitting 0
, and the reservoirs
are initially in vacuum states and not correlated to the
spins. Under these conditions, the model can be solved
exactly for any initial state of the spins and any form of
spectral reservoirs. In this paper, we will mainly discuss
two types of reservoirs—unstructured flat reservoirs and
structured Lorentzian reservoirs, and also two large types
of initial states of spins—double-excitation and one-exci-
tation states.
3.1. Double-Excitation Dynamics of MIN and
GD
Let us first study the double-excitation case, i.e., the joint
initial state for the whole system is,

12 2
00 1100,
121
0
s
sssrr
 
 (7)
where 22
1

, and the collective state
00
ii
k
rr
k
represents vacuum state of reservoir .
i
r
Due to the conservation of excitation number under
the evolution of Jaynes-Cummings model, the dynamical
state of the whole system has the form,
  
 
11
12
2
10
10
sr
sr
ss
t
11
22 22
12
11
2
01
01
0000.
sr
sr
rr
tt t
t
 

 (8)
Here we introduce the collective state
 
1
1
ii
kk
k
rr
i
t
t
1 to denote the one-excitation
state of the reservoir i [19,32]. The coefficients r
it
and
it
, which are determined by the quantum dy-
namical equation, satisfy

111
0d
t
ii
ttfttt
 
,
(9)
and
 
22
1.
ii
tt


,
(10)
Here the kernel function is defined by
 


01
1de
itt
ii
ftt J




with spectral density


2
,,iik
k
Jg


ik
.
The reduced density matrix for the two spins reads
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL.
36

12
22
12 12
2
12
2
12
2
12 12
00
00
00 0
00
ss
 
 
 
 
0
(11)
From Equations (4) and (5), the analytical expressions
for MIN and GD can be written as

12
2
12
2
ss
N

, (12)
and




12
2
22
112
2
22
121 2
2
22
12 1
2
22
121 2
1128
4
122
max4, 12
12 2.
ss
D
 
 
 
 
 
 

 
(13)
Similarly, the MIN and GD for the reservoirs reads,

12
2
12
2
rr
N

, (14)





12
2
22
11
2
22
12 12
2
22
12 1
2
22
12 12
112 8
4
12 2
max4, 12
12 2.
rr
D2
 
 
 
 
 
 

 
(15)
In order to further demonstrate the dynamical features
of MIN and GD, we specify our study to two exemplary
reservoir spectra widely used in the literature, flat spec-
trum and Lorentzian spectrum. First, we consider the
case that both spins are respectively embedded into two
equal but independent flat spectral reservoirs,

12,JJ

 (16)
where
is a constant that is commonly used as the
Markov approximation with the interval of the spectral
density much broader than the corresponding energy
scale of the system. For this set of spectra, we obtain the
time-dependent coefficients,

12
2
121 2
e,1e . (17)
tt

 

 
We plot the time evolution of and for differ-
ent partitions as in Figure 1, where we take two initial
states for spins: Bell state with
ND
12

 and Bell-
like state with 18, 78

. From the figure
along with the analytical expressions of Equations
(12)-(15), we can find the following features. Firstly,
there is clearly correlation transference between the spins
and reservoirs [Figure 1(a) and 1(b)]. Both and
initially stored in the spins run into reservoirs gradually.
N D

And in the equilibrium,


0
ss
121 2
limtrr
Nt
 N

and



1 20
ss
D
12
limtrr
Dt
 , i.e., the transfer-
ence is complete. Secondly, both and of the
spins deplete gradually and no sudden death occurs in the
process of transference, which is different from the con-
currence [20]. Indeed, the concurrence of the spin system
ND
is

2
max 0,1ee
t
C

 




t
. When
(i.e., the Bell-like initial state in the figure), it
will occur sudden death. Thirdly, due to the interaction of
the spins with their own reservoirs, any spin
12
s
s
becomes correlated with its reservoir

1
ras shown in
Figure 1(c). Also, 1
2
r
s
tangles with 2
r [Figure 1(d)]
due to the initial correlation of the spins. However, at the
end of evolution there is no correlation between any one
of the spins and reservoirs. Therefore, the spin system
can finally be discarded without any effect, and all corre-
lations transfer to the reservoirs. Lastly, we find that the
0 2 4 6
0
0.2
0.4
0.6
0.8
γt
Correlations
0 2 4 6
0
0.2
0.4
0.6
0.8
γt
Correlations
0 2 4 6
0
0.1
0.2
0.3
0.4
γt
Correlations
0 2 4 6
0
0.05
0.1
0.15
0.2
γt
Correlations
(b)
(a)
(c) (d)
Figure 1. (Color online) Evolution of and N
D
among
different partitions with flat spectral density and for the
double-excitation initial spin state

0
. Solid black line
and purple plus sign denote respectively and
N
D
for
Bell state. Dotted red line and dashed green line denote
respectively and
N
D
for Bell-like state with
,18 78

. (a) Spins with ; (b) Reservoirs
with ; (c) Spin with reservoir ; (d) Spin
s1s2
r1
r1
with
r2s1s1
reservoir r2.
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL. 37
time evolution of s and in this flat Markovian
se
of
N
D
dreservoir are roughly in accor ance, only in few time
intervals they behave large difference. This result high-
lights in some sense the relation between the two defini-
tions. Note that there are clearly some non-smooth points
(sudden changes) on the evolution of D (more obvious
in Figure 1(c)), which are due to theax operation in
the expressions of D [see Equations (13) and (15)].
Up to now, we have mainly concentrated on the ca
m
flat reservoirs where the dynamics is Markovian.
There will be different features for structured reservoirs
which will lead to obvious non-Markovian dynamics. As-
sume that the two spins are now embedded in the local
Lorentzian reservoirs with spectra,


21W
22
0
.
π
i
i
J

  (18)
This spectrum is commonly used to describe a two-
level atom in an imperfect cavity. Where 0
is the Bohr
frequency of the spins, and 0iic
ts c
  is the fre-
quency detuning between spin avity mode.
The quantity
i and i
is the photon-leakage rate of the cavity
whose inversenotes the reservoir correlation time. The
ideal cavity limit is obtained for 0
de
, where


2
0
JW
 
, displaying a strong non-M
resonance
12
0 , we have,
arkovian
effect. For thof e case
 
2
ecosh
t
12
2
12 1
2 sinh2,
1,
tt
 

 
 (19)
with
22
4W
 . The time evolution of and
es , we obtain,
N
D for different partitions in this case is presented in
ure 2. For clarity, we only present the evolution of
N and D for the initial state of the spins to be in Bell
te. Theistinctive property compared with the case of
flat reservoirs is the oscillation in the evolution of N
and D, which is the result of non-Markovian effe
Other alike properties include: complete correlation trans-
ference between the spins and reservoirs, no sudden
death of correlations for the spins in the transferring pro-
cess, the vanishing correlations between spins and reser-
voirs at the end of evolution, and the roughly overlapping
evolutional curves for N and D.
For the case of non-ronance 
Fig
sta d
ct.
0
i

 
2
i
it
 

2
ecosh2sinh2,
1,
i
iii
i
ii
i
tt






(20)
with

22
4
ii
iW
 
01234
0
0.2
0.4
0.6
0.8
. The time evolution of
dfor different parti
N antions is plotted in Figure 3,
ere we also choose 20W
D
wh
and 12
20
 .
Similarly, the oscillation due to the non-Markovian effect
λt
Correlations
0.8
01234
0
0.2
0.4
0.6
λt
Correlations
01234
0
0.05
0.1
0.15
0.2
λt
Correlations
0 12 3 4
0
0.05
0.1
0.15
0.2
λt
Correlations
(a) (b)
(c) (d)
Figure 2. (Color online) Dynamics of and N
D
for the
double-excitation initial Bell state of s ande Lorpins in th-
entz spectrum with ,
W12
20 0
 . Red dotted and
blue lines denote respectively N and
D
. (a) Spins s1 with
s2; (b) Reservoirs r1 with2ith reservoir r1;
(d) Spin s1 with reservoir r2
r; (c) Spin
.
s1 w
01234
0
0.2
0.4
0.6
0.8
λt
Correlations
0.8
0246
8
0
0.2
0.4
0.6
λt
Correlations
02468
0
0.05
0.1
0.15
0.2
λt
Correlations
0 2 4 6
8
0
0.05
0.1
0.15
0.2
λt
Correlations
(b)
(a)
(c) (d)
Figure 3. (Color online) Evolution of and
N
D
among
different partitions with non-resonanntectrut Lorez spm
,
W12
20 0
 and for the doub-excitatn initial
Bell state of spins. Red dotted and blue lines denote respec-
le io
tively N and
D
. (a)
ir
Spins s1 with s2; (b) Reservoirs r1
with r2; (c) Spin s1 with reservoir r1; (d) Spin s1 with
reservo r2.
also exist, bsut the amplitude of oscillation becomes
aller compared with the resonant case. From Equa-
tween the spin and its reservoir. The steady values of
sm
tions (12)-(15), we can also find that the correlation
transference between the spins and reservoirs is also
complete, but the transferring time becomes longer com-
pared with the corresponding resonant case. This is be-
cause the non-resonant effect decreases the coupling be-
N
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL.
38
and D for 11
s
r
and 12
s
r
are also zero [not show com-
pletely in Figure 3(c) and (d)]. Note that the evolutions
of N and D also roughly coincide.
3.2. Single-Excitation Dynamics of MIN and GD
Wew stud the correlation evolution noy for another type
of initial state which has only one excitation in the spin
system,


12121 2
0011000,
s
sssrr
 
 (21)
with 22
1

. The dynamical evolution of the
overall system in this case can be solved as,

 

 

12 121212
12121212
11
01
10 000010,
rr
ssrrssrr
tt


(22
here and are also determined by Equa-
0). for different partitions
22
0100 00
ssrrss
t
tt
 

)
are,

it
tions (9) and (1

it
N and D

12
2
12
2,N

(23)

ss



12
2
22
112 21
2
22 22
1212 1
112812 2
4
max4, 12122,
ss
 

 

(24)
and
2
2
22
D

12
2
12
2,
rr
N

(25)




12
22
2222
2

11
2 21
22
22 22
1212 1
112 812
4
max4,12122
rr
 
 
 




(26)
It is interesting to find that and
.
D

12
ss
N
12
rr
N
have the same expressions for one- and two-excit
sp(23) 5) with
ation
in states [compare Equations and (2
Equations (12) and (14)]. So do 11

s
r
N
and
12
sr
N
[not given in the text]. While for D, there is not similar
result.
When the two spins coupled reserv to flatoirs, the
parameters i
and i
are still gen by Equation (17).
We sho
iv
w e corrlation transference of this case in
Fi
g
th e
gure 4, which has roughly the similar properties as the
correspondin two-excitation case.
If the spins couple to resonant and non-resonant Lo-
rentz reservoirs, the corresponding parameters i
and
i
are given respectively by Equations (19) an20).
ca
d (
The corresponding correlation transference depicted in
Figures 5 and 6 are still similar to that of two-excitation
se.
0 2 4 6
0
0.2
0.4
0.6
0.8
γt
Correlations
0.8
0.6
0 2 4 6
0
0.2
0.4
γt
Correlations
0 2 4 6
0
0.1
0.2
0.3
0.4
γt
Correlations
0 2 4 6
0
0.05
0.1
0.15
0.2
γt
Correlations
(a) (b)
(c) (d)
Figure 4. (Color online) Dynamics of and
N
D
for the
single-excitation initial spin state
0
andpectral
density. Solid black line and purple plus sign denote re
flat s
s-
pectively N and
D
for initial Bell sta, while tted red
line and dashed green line for inell-like state with
te
itial Bdo
18
, 78
. (a) Spins s1 with s2; (b) Reser-
voirs r1 with r2; ( Spin s1 with reservoir r1; (d) Spin
s1 with reservoir r2.
c)
01234
0
0.2
0.4
0.6
0.8
λt
Correlations
0.8
01234
0
0.2
0.4
0.6
λt
Correlations
01234
0
0.05
0.1
0.15
0.2
λt
Correlations
01234
0
0.05
0.1
0.15
0.2
λt
Correlations
(a) (b)
(c) (d)
Figure 5. (Color online) Dynamics of and N
D
for the
single-excitation initial Bell state of spdesonant
Lorentz spectra ins an for r

12
20 ,W
0. Red and d dotte
blue lines denote respectively N and
D
. (a) Spins s1 with
s2; (b) Reservoirs r1 with r2; (c) Spin s1 with reservoir r1;
(d) Spin s with res
1ervoir r2.
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL. 39
01234
0
0.2
0.4
0.6
0.8
λt
Correlations
0 2 4 6 8
0
0.2
0.4
0.6
0.8
λt
Correlations
02468
0
0.05
0.1
0.15
0.2
λt
Correlations
0 2 4 6 8
0
0.05
0.1
0.15
0.2
λt
Correlations
(a) (b)
(c) (d)
Figure 6. (Color online) Evolution of and N
D
among
different partitions for non-resonanntectrum t Lorez sp


12
20 ,20W
 and for the sinle-exc initiag
s d
itation
es
l
Bell state of spins. Red dotted and blue l ineenote rpec tively
N and
D
. (a) s1 with s2; (b) r1 with r2; (c) s1 with r1;
4. Quant Beat for MIN and GD
Quantum beat is a very inte
(d) s1 with r2.
resting phenomenon in quan-
tum optics. We discussed the entanglement and discord
n two-level sys-quantum beats in detail early [31] in ope
tems. Here, we find that for the time evolution of N
and D a similar phenomenon also appears. Let us still
assume that the two spins are plugged into their own
Lorentz reservoirs with spectral density given by Equa-
tion (18). For simplicity, we only discuss the non-reso-
nant and two-excitation case. For resonant or/and one-
excitation case, similar phenomenon of quantum beat for
N and D also exists. The quantum correlations N
and D for the spins and the reservoirs are described
respectively by Equations (12)-(15) with parameters i
and i
given by Equation (20). The corresponding time
evolutions are depicted in Figure 7, where
12

 and 12
50 ,45
. The tiny dif-
ference between 1
and 2
is the demand of observ-
ing quantum beat. Only in this way can we induce two
harmonic oscillations with tiny different frequencies,
whose interfering superpoion forms quantum beat.
These quantum beats originate from non-Markovian ef-
fect, as no any direct or mediated interaction exists be-
tween the two spins or the two reservoirs. It is the result
of both non-Markovian effect and quantum interference.
The detailed mathematical analysis may be consulted in
Reference [31].
5. Conclusions
We have studied
sit
the dynamics of both and in
the spin-boson model, where two independent spins re-
N D
01234
0.2
0.25
0.3
0.35
0.4
0.45
0.5
λt
Correlations
0.06
01234
0
0.01
0.02
0.03
0.04
0.05
λt
Correlations
(a) (b)
Figure 7. (Color online) Dynamics for and N
D
for the
double-excitation initial Bell state of spi non-
resonant Lorentz spectrum ns and for
10 ,50 ,45W
12

. (a) wits121
r2. Red dotted and blue lines denote respectively N and
with s. (b) rh
D
.
al resevoirs. We cons dered
o large types of initial states of the spins, i, one- d
o-excitation states. We found that N of differt
spectively couple to their locri
tw .e. an
entw
partitions is identical for the two types of initial states,
while D depends on the initial states. In the situations
of flat and Lorentz spectral reservoirs, we have both
analytically and numerically simulated the time evolu-
tions of N and D. We found that there exist complete
correlation transference between the system and reser-
voirs, all N and D initially stored in the spins com-
pletely transfer to reservoirs at the end of evolution.
There is no sudden death of correlations for the spin sys-
tem in the transferring process, which forms bright con-
trast to the quantum entanglement. For flat spectral res-
ervoirs, there is no oscillation for N and D in the trans-
ferring process. While for memory Lorentz reservoirs,
oscillation appears which is the symbol of non-Mar-
kovian effect. In particular, when the detunings of the
spins with their reservoirs have tiny difference, the quan-
tum beats for N and D are observed which signify the
quantum interference between the two dynamics of the
spins through quantum correlations. For the Lorentz res-
ervoirs and in the case of resonance between the spins
and their reservoirs, the oscillation amplitudes of N
and D are larger, and the time of the correlation trans-
ference is shorter. While in the detunings (particularly for
large detunings), the oscillation amplitudes become
smaller and the transferring time becomes longer. Finally,
we found that though the different definitions, the evolu-
tions of N and D are very close for most cases under
consideration. This is an astonishing result which high-
lights in some sense the relation between the two defini-
tions. Also note that it may happen sudden change for the
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL.
40
evolutionf D.
Note that a similar work for the evolutions of the quan-
tum and classical correlations was done [29]. The trans-
ference of N and D studied here has some similar
properties to at
o
of quantum and classical correlations.
H
of
n
d finding the
dy
or the Doctoral Program
20124306110003) the Cons-
g Measurement-Induced Disturbance to
racterizeations as Classical or Quantum,” Physica
Review A01.
doi:10.1103/Ph
th
in
Correl
, Vol. 7
owever it also appears many differences, for example,
N is the same for one- and two-excitation cases, and
D may happen sudd change in the evolutional proc-
ess. Especially, N and D have very coincident evolu-
tions in most cases under considerations.
Quantum correlation is a kind of unique characteristics
quantum system which could be a new resource in
quantum information techology. Exploring the relation
between various quantum correlations an
en
namical rules of them in practical environments can
not only contribute to a better understanding of the con-
cepts, but also offer possible references for applications.
6. Acknowledgements
This work is supported by the National Natural Science
Foundation of China (Grant Nos. 11275064, 11075050),
Specialized Research Fund f
Higher Education (Grant No.
of
truct Program of the National Key Discipline, the Pro-
gram for Changjiang Scholars and Innovative Research
Team in University under Grant No. IRT0964, and Hu-
nan Provincial Natural Science Foundation under Grant
No. 11JJ7001.
REFERENCES
[1] S. Luo, “UsCha-
l
7, No. 2, 2008, Article ID: 0223
ysRevA.77.022301
[2] M. A. Nielsen and I. L. Chuang, “Quantum Computation
and Quantum Information,” Cambridge University Press,
Cambridge, 2000.
[3] H. Ollivier and W. H. Zurek, “Quantum Discord: A Meas-
evLett.88.017901
ure of the Quantumness of Correlations,” Physical Review
Letters, Vol. 88, No. 1, 2001, Article ID: 017901.
doi:10.1103/PhysR
[4] A. Datta, A. Shaji and C. M. Caves, “Quantum Discord and
the Power of One Qubit,” Physical Review Letters, Vol.
100, No. 5, 2008, Article ID: 050502.
doi:10.1103/PhysRevLett.100.050502
[5] B. P. Lanyon, M. Barbieri, M. P. Almeida and A. G.
White, “Experimental Quantum Computing without En-
tanglement,” Physical Review Letters, Vol. 101, No. 20,
2008, Article ID: 200501.
doi:10.1103/PhysRevLett.101.200501
[6] R. Dillenschneider, “Quantum Discord and Quantum Phase
Transition in Spin Chains,” Physical Review B, Vol. 78,
No. 22, 2008, Article ID: 224413.
doi:10.1103/PhysRevB.78.224413
[7] M. S. Sarandy, “Classical Correlation and Quantum Dis-
cord in Critical Systems,” Physical Review A, Vol. 80, No.
2, 2009, Article ID: 022108.
doi:10.1103/PhysRevA.80.022108
[8] J. Cui and H. Fan, “Correlations in the Grover Search,”
Journal of Physics A: Mathematical and Theoretical, Vol.
43, No. 4, 2010, Article ID: 045305.
doi:10.1088/1751-8113/43/4/045305
[9] S. Luo, “Quantum Discord for Two-Qubit Systems,” Phy-
sical Review A, Vol. 77, No. 4, 2008, Article ID: 042303.
doi:10.1103/PhysRevA.77.042303
[10] G. Adesso and A. Datta, “Quantum versus Classical Cor-
1
relations in Gaussian States,” Physical Review Letters,
Vol. 105, No. 3, 2010, Article ID: 030501.
doi:10.1103/PhysRevLett.105.03050
[11] P. Giorda and M. G. A. Paris, “Gaussian Quantum Dis-
cord,” Physical Review Letters, Vol. 105, No. 2, 2010,
Article ID: 020503.
doi:10.1103/PhysRevLett.105.020503
[12] B. Dakic, V. Vedral and C. Brukner, “Necessary and Suf-
ficient Condition for Nonzero Quantum Discord,” Phy-
sical Review Letters, Vol. 105,
190502.
No. 19, 2010, Article ID:
190502doi:10.1103/PhysRevLett.105.
[13] S. Luo and S. Fu, “Measurement-Induced Nonlocality,”
Physical Review Letters, Vol. 106, No. 12, 2011, Article
ID: 120401. doi:10.1103/PhysRevLett.106.120401
[14] S. Luo and S. Fu, “Geometric Measure of Quantum Dis-
cord,” Physical Review A, Vol. 82, No. 3, 2010, Article ID:
034302. doi:10.1103/PhysRevA.82.034302
[15] S. Luo and S. Fu, “Evaluating the Geometric Measure of
Quantum Discord,” Theoretical and Mathematical Phy-
sics, Vol. 171, No. 3, 2012, pp. 870-878.
doi:10.1007/s11232-012-0082-x
[16] B. Bellomo, R. Lo Franco and G. Compagno, “Dynamics
of Geometric and Entropic Quantifiers of Correlations in
Open Quantum Systems,” Physical Revi
No. 1, 2012, Article ID: 012312.
ew A, Vol. 86,
doi:10.1103/PhysRevA.86.012312
[17] B. Bellomo, G. L. Giorgi, F. Galve, R. Lo Franco, G. Com-
pagno and R. Zambrini, “Unified View of Correlations
Using the Square Norm Distance,”
Vol. 85, No. 3, 2012, Article ID: 032
Physical Review A,
104.
doi:10.1103/PhysRevA.85.032104
[18] K. Ann and G. Jaeger, “Finite-Time Destruction of En-
tanglement and Non-Locality by Environmental Influ-
ences,” Foundations of Physics, Vol. 39, N
790-828.
o. 7, 2009, pp.
295-8doi:10.1007/s10701-009-9
7-8, 2011, pp.
ysRevLett.93.140404
[19] B. Bellomo, G. Compagno, R. Lo Franco, A. Ridolfo and
S. Savasta, “Dynamics and Extraction of Quantum Dis-
cord in Multipartite Open Systems,” International Jour-
nal of Quantum Information, Vol. 9, No.
1665-1676.
[20] T. Yu and J. H. Eberly, “Finite-Time Disentanglement via
Spontaneous Emission,” Physical Review Letters, Vol. 93,
No. 14, 2004, Article ID: 140404.
doi:10.1103/Ph
[21] B. Bellomo, G. Compagno, A. D’Arrigo, G. Falci, R. Lo
Franco and E. Paladino, “Entanglement Degradation in the
Copyright © 2013 SciRes. JQIS
G. Y. WANG ET AL.
Copyright © 2013 SciRes. JQIS
41
and Quantum Noise,”
, Article ID: 062309.
Solid State: Interplay of Adiabatic
Physical R evi ew A, Vol. 81, No. 6, 2010
doi:10.1103/PhysRevA.81.062309
[22] T. Werlang, S. Souza, F. F. Fanchini and C. J. Villas Boas,
“Robustness of Quantum Discord to Sudden Death,” Phy-
sical Review A, Vol. 80, No. 2, 2009, Article ID: 024103.
doi:10.1103/PhysRevA.80.024103
[23] B. Bellomo, R. Lo Franco and G. Compagno, “Non-Mar-
kovian Effects on the Dynamics of Entanglement,” Phy-
sical Review Letters, Vol. 99, No. 16, 2007, Article ID:
160502.
doi:10.1103/PhysRevLett.99.160502
ol. 27, No.
ID: 014019.
[24] R. Lo Franco, B. Bellomo, S. Maniscalco and G. Com-
pagno, “Dynamics of Quantum Correlations in Two-Qu-
bit Systems within Non-Markovian Environments,” In-
ternational Journal of Modern Physics B, V
1-3, 2013, Article ID: 1345053.
[25] R. Lo Franco, A. D’Arrigo, G. Falci, G. Compagno and E.
Paladino, “Entanglement Dynamics in Superconducting
Qubits Affected by Local Bistable Impurities,” Physica
Scripta, Vol. T147, 2012, Article
doi:10.1088/0031-8949/2012/T147/014019
[26] B. Bellomo, G. Compagno, R. Lo Franco, A. Ridolfo and
S. Savasta, “Entanglement Dynamics of Two Independent
Cavity-Embedded Quantum Dots,” Physica Sc
T143, 2011, Article ID: 014004.
ripta, Vol.
doi:10.1088/0031-8949/2011/T143/014004
[27] R. Lo Franco, B. Bellomo, E. Andersson and G. Com-
pagno, “Revival of Quantum Correlations without Sys-
tem-Environment Back-Action,” Physical Review A, Vol.
85, No. 3, 2012, Article ID: 032318.
doi:10.1103/PhysRevA.85.032318
[28] Q. J. Tong, J. H. An, H. G. Luo and
nism of Entanglement Preservation
C. H. Oh, “Mecha-
,” Physical Review A,
Vol. 81, No. 5, 2010, Article ID: 052330.
doi:10.1103/PhysRevA.81.052330
[29] R. C. Ge, M. Gong, C. F. Li, J. S. Xu and G
“Quantum Correlation and Classica
. C. Guo,
l Correlation Dynam-
ics in the Spin-Boson Model,” Physical Review A, Vol.
81, No. 6, 2010, Article ID: 064103.
doi:10.1103/PhysRevA.81.064103
[30] B. Bellomo, R. Lo Franco, S. Manis
pagno, “Entanglement Trapping in
calco and G. Com-
Structured Environ-
ments,” Physical Review A, Vol. 78, No. 6, 2008, Article
ID: 060302(R). doi:10.1103/PhysRevA.78.060302
[31] H. S. Zeng, Y. P. Zheng, N. Tang and G. Y. Wang, “Cor-
relation Quantum Beats Induced by Non-Markovian Ef-
fect,” Quantu m Information Process, Vol. 12, No. 4, 2013,
pp. 1637-1650. doi:10.1007/s11128-012-0437-0
[32] C. E. Lopez, G. Romero and J. C. Retamal, “Dynamics of
Entanglement Transfer through Multipartite Dissipative
Systems,” Physical Review A, Vol. 81, No. 6, 2010, Arti-
cle ID: 062114. doi:10.1103/PhysRevA.81.062114