SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
March 2013 Vol.56 No.3: 600–605
doi: 10.1007/s11433-012-4870-y
Quantum discord of thermal states of a spin chain
with Calogero-Moser type interaction
MA XiaoSan1*, QIAO Ying1, ZHAO GuangXing1 & WANG AnMin2
1
School of Electric Engineering and Information, Anhui University of Technology, Ma’anshan 243002, China;
2
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
Received January 1, 2012; accepted April 6, 2012; published online September 3, 2012
In this paper, we have investigated the effect of Calogero-Moser type interaction on the quantum discord of thermal states of a
spin chain. Our results imply that the quantum discord depends on the relative distance between the spins, the external magnetic field, and the temperature. By a comparison between the quantum discord and the entanglement of formation, the quantum discord is more robust than the entanglement of formation in the sense that the latter takes a zero value in a large range of
the parameters, while the former takes a nonzero value.
quantum discord, entanglement of formation, Calogero-Moser interactions
PACS number(s): 03.65.Ud, 03.67.-a, 03.67.Hk, 75.10.Pq
Citation:
Ma X S, Qiao Y, Zhao G X, et al. Quantum discord of thermal states of a spin chain with Calogero-Moser type interaction. Sci China-Phys Mech
Astron, 2013, 56: 600605, doi: 10.1007/s11433-012-4870-y
Entanglement attracts much attention from physicists in
theory and in experiment [1–5] as it plays an important role
in quantum information processing such as quantum communication [2] and quantum cryptography [6]. However,
there are some exceptions. For instance, the quantum computation [7] based on one pure qubit does not employ entanglement but it needs other non-classical resources. This
suggests that entanglement is not a unique resource but just
one kind of quantum correlation. In fact, there is another
kind of quantum correlation termed quantum discord (QD)
[8–11]. QD has gained much attention recently and some
work concerning QD has been done [12–16]. In ref. [12],
the author gave some analysis and calculation of the QD for
a large family of quantum states. The sufficient and necessary condition for completely positive maps by using QD is
discussed in ref. [13]. The robustness of QD to sudden death
is investigated by the authors in ref. [14]. The authors in ref.
[15] analyzed the sudden transition between classical and
*Corresponding author (email: mxiaosan@mail.ustc.edu.cn)
© Science China Press and Springer-Verlag Berlin Heidelberg 2012
quantum decoherence. The geometry and QD of two-qubit
states is studied by the authors in ref. [16]. These studies
suggest that QD is important and is of interest to be investigated extensively.
One important aspect of QD is to evaluate the QD of a
thermal density matrix. The basic motive to study the QD of
quantum systems in equilibrium is to find properties of
thermal states to construct quantum gates and to realize solid quantum information processing as the quantum systems
cannot be at absolute zero temperature but are always in an
equilibrium. For instance, the system for nuclear magnetic
resonance (NMR) is a thermal density matrix [17]. A lot of
work concerning QD of a thermal density matrix has been
done [18–25]. The QD of XX Heisenberg spin chains has
been analyzed by the authors in refs. [18–20]. The effect of
quantum phase transition of quantum spin chains is analyzed by the authors in ref. [21]. We analyzed the QD of a
spin-star model in ref. [22]. The DM interaction on the QD
is studied by the authors in refs. [23–25]. In fact, in realistic
systems, there is distance between spins as well as other
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Ma X S, et al.
Sci China-Phys Mech Astron
physical parameters. For example, short range quantum
correlation between spins or charge degrees of freedom in
quantum dots, nanotubes or molecules has been observed
[26,27]. For the long-range quantum correlation, there is no
experiment to prove its existence without inclusion of some
medium between the spins. In order to obtain the long-range
quantum correlation between spin qubits, some kind of mediating interaction between the spins is needed. Up to now,
many kinds of spin Hamiltonian have been investigated
theoretically. However, very few investigations have been
made to consider the effect of distance [28,29] between
spins on quantum correlation. Obviously, in realistic spin
lattices, the positions of spins could oscillate due to lattice
photons at a finite temperature. In this case, the dependence
of the exchange interaction addressed with distance between
spins on the quantum correlation should be analyzed and
such a type of interaction is known as a Calogero-Moser
type interaction [28,29]. The consideration of the effect of
the Calogero-Moser interaction on the QD should be made
in order to contribute the realistic assessment of the potential of such spin systems in solid quantum information processing. Therefore, in this paper, we study the effect of the
Calogero-Moser interaction on the QD of a spin chain.
The content is arranged as follows. In sect. 1, we introduce the model and the definition of the QD. In sect. 2, the
main results are presented in detail. Finally, we conclude
our results in sect. 3.
1 A quantum spin chain with Calogero-Moser
interaction
The Hamiltonian for the spin chain with Calogero-Moser
interaction [28,29] reads
H   Be iz  J ( R ) ( ix ix1   iy iy1 ).
i
(1)
i
From the expression of the Hamiltonian, the CalogeroMoser interaction is similar to the usual Heisenberg XX
interaction. However, the physical content is different. Here,
we concentrate ourselves on the effect of the distance of
spins on the QD. Specially, there are two types of CalogeroMoser interaction and they read J(R)=1/R2 and J(R)=1/
sin2(R). It should be noted that the relative distance between
the spins is very small and 1/R2 is equivalent with 1/sin2(R) in
this limiting case. Therefore, we consider the case of
J(R)=1/R2 in this paper. The parameters of the Hamiltonian
in eq. (1) are explained as follows. Be denotes the external
magnetic field, R represents the distance between spins, and
 ix ( y , z ) are the familiar Pauli matrices. In order to analyze
the QD and entanglement of formation of the thermal density matrix of such a spin chain with Calogero-Moser interaction, we should give the thermal density matrix of the
spin chain in equilibrium and consider the two-spin density
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matrix of the nearest neighbor spin chain in the following
expression:

1
Z
4
 e 

i 1
i
i i ,
(2)
where Z is the partition function and it reads Z=tr(eH).  is
the inverse of the Boltzman constant multiplied by the temperature, that is =1/kT. The parameters i are the eigenvalues of the Hamiltonian in eq. (1) and the states  i are
the corresponding eigenstates. With a straightforward calculation, we obtain the eigenvalues and eigenstates in the
following expression:
1  2 Be ,  1  11 , 2  2 Be ,  2  00 ,
3  2 / R 2 ,  3 
4 
1
2


1
01  10 , 4  2 / R 2 ,
2
(3)
 01  10  .
The thermal density matrix of  can be obtained with the
above notations.

 e 2  Be

1 0
Z 0

 0
0
0
0
2
2
cosh(2  / R )  sinh(2  / R )
0
2
2
 sinh(2  / R ) cosh(2  / R )
0
0
0
e 2  Be


 , (4)



where Z=2cosh(2Be)+2cosh(2/R2). From the expression,
we can find that the ground state of the Hamiltonian in eq.
(1) is separable when the magnetic field is strong enough.
Also, the ground state will become an entangled state when
the relative distance of R is small enough. In this sense, we
can say that a strong magnetic field will shrink the quantum
correlation measured by the QD and entanglement of formation, while a small relative distance can contribute to a
large amount of quantum correlation.
In order to analyze the effect of relative distance on the
QD, we should introduce the definition of QD. QD was first
proposed by Zurek in ref. [8] in order to give the quantum
version of mutual information between the two parties of
the density matrix AB with the following two expressions
from the different points of view.
I ( A : B )  S ( A )  S ( B )  S ( AB ),
J ( A : B )  S ( A )  S ( A | B ),
(5)
where I(A:B) is the direct extension from the classical
mutual information to the quantum version and J(A:B) is
another route to generalizing the classical mutual information to the quantum case by using a measurement-based
conditional density operator. S()=Tr( log() is the von
Neumann entropy of , A(B) is the reduced density matrix
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of AB by tracing out B(A), and S(A|B) is quantum conditional entropy. The conditional entropy S(A|B) depends on
the choice of measurement from a set of projectors {j}
as positive operator valued measure (POVM) elements to
act on the party B locally. The quantum state after a measurement changes to j=[(Ij)AB(Ij)]/Tr(IjAB I
j), where I is the identity operator for the party A. The
quantum conditional entropy S(A|B) ≡ min{j}S{j}(A|B)
is defined as:
S{ j } ( A|B )   q j S ( j ).
(6)
j
The QD is defined by the following expression:
Q  I ( A : B )  J ( A : B ).
(7)
The QD Q provides us with information on the quantum
nature of the correlations between two parties and it is zero
only for states with classical correlations and nonzero for
states with quantum correlations.
Beyond QD, we introduce the entanglement measure
based on concurrence [4] for a comparison between QD and
entanglement, which is another kind of measure for quantum correlation. Here, the entanglement measure based on
concurrence is considered and it is entanglement of formation (EoF).
EoF   f (C ) log 2 f (C )  [1  f (C )]log 2 [1  f (C )],
(8)
where f (C )  (1  1  C 2 ) / 2. The concurrence is defined
by C=max{0, 1234}, where 1, 2, 3, and 4 are the
square roots of the eigenvalues in decreasing order, of the
matrix R   . Here  is the time-reversed matrix,
   1y   y2  * 1y   y2 and * denotes complex conjugation. With the definition of QD and EoF, we can analyze the
effect of the Calogero-Moser interaction on QD and entanglement.
2 Main results
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Q1  S ( 1 ) 

e 2  Be
e 2  Be
log 2 2  B
Z
e e  cosh(2  / R 2 )
cosh(2 / R 2 )
cosh(2 / R 2 )
log 2 2  B
Z
e e  cosh(2  / R 2 )
 S ( ) 

e 2  Be
e 2  Be
log 2 2  B
Z
e e  cosh(2 / R 2 )
cosh(2 / R 2 )
cosh(2  / R 2 )
log 2 2  B
,
Z
e e  cosh(2   R 2 )
Q2  S ( 1 )  S (  ) 
1 
1  1 
1 
log 2

log 2
,
2
2
2
2
where 1 is the reduced density matrix of  in eq. (4) by
tracing off the party 2 and  satisfies the relation

4sinh 2 (2  Be )  4sinh 2 (2  / R 2 ) / Z . The analytical
expression of QD suggests that the QD depends on the relative distance, the temperature, and the external magnetic
field. The entanglement of formation of the thermal density
matrix in eq. (4) can also be obtained as the concurrence of
the thermal density matrix and takes the expression of
 1

C  max 0, (sinh(2  / R 2 )  1)  .
Z


The analytical expression of QD and entanglement of
formation can provide us with the dependence relation of
quantum correlation on the parameters of the external magnetic field, the relative distance, and the temperature. Here,
we discuss the expression of QD and entanglement of formation. In theory, the threshold temperature of QD is infinity at which the nondiagonal elements as the quantum coherence of the thermal density matrix  in eq. (4) are approximately zero. Only when the nondiagonal elements take
zero values can the QD of the thermal density matrix in eq.
(4) take a zero value. For the entanglement of formation, the
threshold temperature can be calculated according to the
equation sinh(2R2)1=0. The threshold temperature of the
entanglement of formation takes the following expression:
Tc 
With the definition of QD and entanglement of formation,
we can study the thermal QD and entanglement of a spin
chain with Calogero-Moser interaction. Usually, for a general thermal density matrix, the analytical expression of QD
and entanglement of formation is hard to obtain. Given the
density matrix in eq. (4), we can get the analytical expressions of both QD and entanglement of formation as given in
ref. [8]. The QD of the reduced density matrix in eq. (4) can
be calculated straightforwardly and it takes the following
expression:
Q  min{Q1 , Q2 },
where Q1, Q2 read respectively as:
(9)
(10)
2
R k ln(1  3)
2
.
(11)
In the above calculation, we have used the assumption of
ħ=1 and B=1 as the Bohr magneton, that is, we use atomic
units. Here, it should be noted that the following simulation
results are provided with atomic units. The smaller the relative distance between the spins, the higher the threshold
temperature of the entanglement of formation is. In order to
get a detailed description of the effect of the CalogeroMoser interaction on the QD and entanglement of formation, we perform simulations and the results are given in the
following content.
At first, we consider the effect of the relative distance
between spins on the QD and the entanglement of formation
of the thermal density matrix of a spin chain with Calogero-
Ma X S, et al.
Sci China-Phys Mech Astron
Moser interaction. The results are presented in Figure 1.
From Figure 1, we can find that the behaviors of QD and
entanglement of formation are similar to a large extent.
When the temperature kT takes a zero value, both the QD
and the entanglement of formation have a sudden transition
from a value of 1–0. This transition corresponds to the sudden change of the ground state of the Hamiltonian in eq. (2)
from the state  3 to  1 when the relative distance
takes a value larger than
3
2 . It is known that the state
is an entangled state and  1
is a separable one.
When the temperature is not zero, the QD and entanglement
of formation varies with the relative distance and the temperature when the external magnetic field is fixed. The
higher the temperature is, the smaller the QD is, and the
smaller the entanglement of formation is. In this sense, we
can find that high temperature can shrink the QD and the
entanglement of formation. The relative distance affects the
QD and the entanglement of formation too. The shorter the
relative distance is, the larger the value of the QD and the
entanglement of formation is. Beyond the similarity between the QD and the entanglement of formation in terms of
relative distance and the temperature, there are some differences between them. The entanglement of formation decreases faster than the QD does. This point can also be
found from the threshold temperatures as analyzed above
since the threshold temperature of QD is infinity while that
of the entanglement of formation is finite. In fact, from Figure 1, we find that the entanglement of formation decreases
as a function of the relative distance R suddenly to zero for
a fixed temperature. Meanwhile the QD just decreases
monotonically to a very small value that is not zero in theory. In this sense, we can say that the QD is more robust than
the entanglement of formation to long relative distance and
high temperatures.
Secondly, we examine the effect of the external magnetic
field on the QD and the entanglement of formation and the
results are given in Figure 2. From Figure 2, for the case of
kT=0, we can find the transition of QD and entanglement of
formation from 1 to 0 when the external magnetic field
Figure 1
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603
takes a value of about 4. This transition corresponds to the
change of the ground state of the Hamiltonian from the state
 3 to  1 when the external magnetic field takes a
value of about 4. In this sense, we can say that the external
magnetic field can manipulate the QD and the entanglement
of formation of the system under a low temperature. For the
case kT0, the QD and entanglement of formation decreases
with increasing temperature. However, the QD decreases as
a function of increasing temperature more slowly than the
entanglement of formation does. From Figure 2, we can find
that the QD and the entanglement of formation is not sensitive to the external magnetic field Be when the temperature
takes a value larger than about 5. This point implies that the
quantum correlation measured by the QD and the entanglement of formation may not be manipulated by the external
magnetic field efficiently when the system is under a high
temperature. In order to give a more detailed description
about the sensitivity of the QD and the entanglement of
formation to the external magnetic field, we plot Figure 3.
From Figure 3, we can find that the QD and entanglement
of formation is sensitive to the external magnetic field when
the temperature is low. Meanwhile for the high temperature
of the case kT=10, they are not sensitive anymore but almost remain stable. Additionally, the QD is more robust
than the quantum entanglement of formation to the high
temperature because the entanglement of formation is zero
for kT=10 while the QD is not zero but 0.1 instead.
Finally, we further analyze the sensitivity of the parameters Be and R for the QD and the entanglement of formation
and the results are given in Figure 4. From Figure 4, we can
find that the relative distance and the external magnetic
field play similar roles in the contribution of the quantum
correlation measured by QD and entanglement of formation.
The longer the relative distance between the spins, the
smaller the QD is. The stronger the external magnetic field
is, the smaller the QD is. Specifically, when the relative
distance is short, the change of QD with increasing external
magnetic field is slow. When the relative distance is long,
the QD decreases with increasing external magnetic field
rapidly. In this sense, we can conclud that the external
(Color online) QD (a) and EoF (b) versus the relative distance R and the temperature kT are plotted, respectively, where Be=0.5.
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March (2013) Vol. 56 No. 3
Figure 2
(Color online) QD (a) and EoF (b) versus the external magnetic field Be and the temperature are plotted, respectively, where R=0.5.
Figure 3
(Color online) QD (a) and EoF (b) versus the external magnetic field Be under different temperatures are plotted, where R=0.5.
Figure 4
(Color online) QD (a) and EoF (b) versus relative distance R and the external magnetic field Be are plotted, where kT=1.
magnetic field can be employed to manipulate the QD for
the states of the spins with long relative distance efficiently.
The behavior of the entanglement of formation is similar to
that of the QD. However, there are some differences. When
the external magnetic field takes a value of zero, the QD
takes a nonzero value for the relative distance R ranging
from 1.5 to 2. The entanglement of formation is zero for the
relative distance R ranging from 1.5 to 2. Additionally, the
entanglement of formation takes a value of zero for a large
range of the parameters R and Be, while the QD does not
take a value of zero but takes a small value larger than zero.
In this sense, we can say that the QD is more robust than the
entanglement of formation to long relative distances and
strong external magnetic fields.
In principle, we can also analyze the other types of
Calogero-Moser interaction including J(R)=1/sin2(R) and
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J(R)=1/sinh2(R) in detail. The analysis is straightforward
and is omitted here for convenience.
3 Discussion and conclusions
5
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To conclude, we have investigated the thermal QD and entanglement of formation of quantum states of a spin chain
with Calogero-Moser interaction. Our results imply that the
QD and entanglement of formation depends on the relative
distance between the spins, the external magnetic field, and
the temperature of the system in equilibrium. For the case of
the temperature kT=0, we find that the QD and the entanglement of formation experience a sudden transition when
the relative distance between spins and the external magnetic field increases in a range. This corresponds to the
change of the ground state of the system. For the cases of
the temperature kT0, the behaviors of the QD and entanglement of formation are smooth without sudden transitions.
The shorter the relative distance is, the larger the QD is. The
stronger the external magnetic field is, the smaller the QD
is. With regard to the effect of temperature, we find that the
higher the temperature is, the smaller the QD is. Specifically, the QD is not sensitive to the change of temperature
when the temperature is higher than a value of about 5.
Even though the behavior of the entanglement of formation
is similar to that of the QD to some extent, there are some
differences. The entanglement of formation contains finite
threshold temperatures, while the threshold temperature of
the QD is infinity. Additionally, the entanglement of formation takes a zero value in a large range of the parameters
and the QD does not take a zero value but takes a small
value larger than zero. In this sense, we can say that the QD
is more robust than the entanglement of formation is. In a
word, our study can contribute to some understanding of the
effect of the Calogero-Moser interaction on the QD and the
entanglement of formation.
This work was supported by the National Natural Science Foundation of
China (Grant Nos. 11105001, 10975125 and 11004001).
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