Journal of Quantum Informatio n Science, 2011, 1, 105-110
doi:10.4236/jqis.2011.13014 Published Online December 2011 (http://www.SciRP.org/journal/jqis)
105
Perfect Entanglement Transport in
Quantum Spin Chain Systems
Sujit Sarkar
PoornaPrajna Institute of Scientific Research, Bangalore, India
Received June 20, 2011; revised November 24, 2011; accepted December 5, 2011
Abstract
We propose a mechanism for perfect entanglement transport in anti-ferromagnetic (AFM) quantum spin
chain systems with modulated exchange coupling and also for the modulation of on-site magnetic field. We
use the principle of adiabatic quantum pumping process for entanglement transfer in the spin chain systems.
We achieve the perfect entanglement transfer over an arbitrarily long distance and a better entanglement
transport for longer AFM spin chain system than for the ferromagnetic one. We explain analytically and
physically—why the entanglement hops in alternate sites. We find the condition for blocking of entangle-
ment transport even in the perfect pumping situation. Our analytical solution interconnects quantum many
body physics and quantum information science.
Keywords: Quantum Many Body Models, Entanglement Physics, Quantum Antiferromagnetic Spin Chain
Model, Adiabatic Transport
1. Introduction
Quantum communication between distant co-ordinates in
a quantum network is an important requirement for quan-
tum computation and information. One can construct the
quantum network in different ways. Optical systems are
typically employed in quantum communication and cry-
ptography applications to transfer the state between two
distinct co-ordinates directly via photons [1]. Quantum
computing applications work with trapped atoms to
transfer information between distant sites—photons in a
cavity QED [2-6]. However, we would like to study the
entanglement transfer through the quantum spin chain
systems. The potentiality of the spin chain system—anti-
ferromagnetic (AFM) and ferromagnetic (FM)—as a
network of quantum state and entanglement transport has
already been studied by many groups as referred in the
literature [7-22]. The experimental evidence of nanoscale
spin chain and their properties have been discussed in
Ref. [23].
It is well known that entanglement is the manifestation
of quantum correlations between two systems when they
are in inseparable state. We consider the spin singlet
state as an example of an entangled state.
0,00 000
101 10
2


 

Typically, the sen-
der holds one member of the state of the pair of qubits
while putting the other member at the near end of the
AFM spin chain of length N. The spin chain is in the
ground state. When the spin 0 starts to interact with the
first spin of the chain then the Hamiltonian includes this
additional interaction term (001 IJ


), where 0

and 1

are the Pauli spin operators for the 0 and 1 sites
respectively and
J
is the exchange coupling. The ini-
tial state being


0,0
0  

 
g
 where
>
g is the ground state wave function of the AFM
Hamiltonian and



0>  is the ground state wave
function of the total Hamiltonian. This initial state starts
to evolve and from that one can compute the density
matrix and concurrence to measure the entanglement and
purity of states. This is conventional wisdom of entangle-
ment transport in the existing literature.
Motivations: But the goal of this letter is different: to
solve the problem of entanglement transport orthogonally.
Our main motivation is to interconnect the quantum
many body physics and quantum information science. It
is common practice in quantum many body physics to
create a particle at any point in the system and study the
dynamics of that particle to understand the physical
behavior of the system. Therefore, we consider that one
of the spin (

or ) of the singlet interacts with the
spin chain and this spin itself transports through the


106 S. SARKAR
chain medium due to the adiabatic variation of exchange
couplings of the Hamiltonian, and reaches the other end
of the chain.
We will explain the entanglement transport by calcula-
ting the entanglement current not by calculating concur-
rence and fidelity that reflect the orthogonality to solve
this problem [11]. We achieve the perfect entanglement
transport and also find a condition for blocking the
entanglement transport even in the perfect pumping
condition. These rigorous analytical solution and physical
explanation are absent in all previous studies [7-22].
The other goal of this letter is to provide the correct
analytical and physical explanation of entanglement
hopping in alternate sites. It discusses why the entangle-
ment transport in AFM spin chain outperforms the FM
spin chain and also why the entanglement transport is
better for a longer spin chain than for the shorter one.
Before we proceed further we would like to state the
basic aspects of adiabatic pumping process (one of the
elegant process of quantum many-body physics). An
adiabatic parametric quantum pump is a device that
generates a dc current by a cyclic variation of system
parameters, the variation being slow enough that the
system remains close to the ground state throughout the
pumping cycle [24,25]. It is well known that when a
quantum mechanical system evolves, it acquires a time
dependent dynamical phase and time independent
geometrical phase [26]. The geometrical phase depends
on the geometry of the path in the parameter space. In
the adiabatic entanglement pumping process, the locking
potential well carries a spin of the singlet pairs. As the
locking potential well slides through the adiabatic
variation of system parameters, it induces a current (
I
)
in the system. In this study we calculate the current of
this spin transport, which transports a spin from one end
of the chain to the other and as a result of which
entanglement is transported (because the spin 0

and 0
are singlet and monogonus in nature) from one side to
the other. This quantization is topologically protected
against the other perturbation as long as the gap along
the loop remains finite [27,28].
2. Model Hamiltonians, Quantum Field
Theoretical Studies and Physical
Interpretations
Here we consider two different Hamiltonian: 1
H
with
modulated exchange coupling in
X
Y plane and 2
H
for onsite magnetic field modulation. Hamiltonians of
the systems are the following.
 


11
11
11
nnnn n
n
H
JtSSS 

 
 

1nn
zz
n
SS 

(1)
This model Hamiltonian has some experimental rele-
vance [27]. The other model Hamiltonian with onsite
magnetic field modulation is


 

11
2
02
1
11
2
nn nnnn
xx yyzz
nn
nn
z
n
1
H
JSS SSSS
BtS 


 


 (2)
Here we consider that the modulation is periodic over
two lattice sites. We see that this model have essential
ingredients to capture the adiabatic entanglement pum-
ping.
Here, we would like to present the basic analytical
methods to study the entanglement transport physics. In
one dimensional quantum many body system, the
Luttinger liquid (LL) theory based method is one of the
most successful analytical methods to study the quantum
systems. The analytical technique to implement LL
picture is the Abelian bosonization method. At first, we
express the spin Hamiltonians in terms of spinless
fermion (annihilation and creation) operators. After that
stage, we use the Abelian bosonization method to solve
these Hamiltonians. One can express spin chain systems
to a spinless fermion systems through the application of
Jordan-Wigner transformation. In Jordan-Wigner trans-
formation the relation between the spin and the electron
creation and annihilation operators are
†
1
1
†
12,
exp π,
exp π,
z
nnn
n
nn j
j
n
nn j
j
S
Sin
Si


















n
(3)
[29], where †
j
jj
n

  is the fermion number at site
. Spin operators in terms of bosonic field are the
following.
j






23
23
1
π
cos 2π 1cos ,
π
cos 2π 1sin
π 1cos2π.
n
x
n
n
y
n
n
z
nx
ScKc K
ScKcK
ScK
K

,












 






(4)
K
is the Luttinger liquid parameter, and are
the constants. The fermionic fields,
2
c3 c

  
(5) e
2π
ir xx
r
rU
x



 S
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107 S. SARKAR
r denotes the chirality of the fermionic fields, right
(1) or left movers (–1). The operators r (Klein opera-
tors) preserve the anti-commutivity of the fermionic
fields
U

field corresponds to the quantum fluctuations
(bosonic) of spin and

is the dual field of

.
Using the standard machinery of continuum field
theory [29], we finally get the bosonized Hamiltonians as
 

1
0
10 22
22
d:cos2
2π
d:cos4
2π
J
Et
H
HxK
xKx





 




x
(6)
 
 
02
20
0
22
d:cos2
2π
dd:cos4
22π
x
Bt
HHx Kx
B
x
xxK





 

 


x 
(7)
0
H
is the gapless Tomonoga-Luttinger liquid part of
the Hamiltonian.
We consider spin singlet as a reference entangled state.
Therefore, we would like to explain the basic aspects of
quantum entanglement pumping in terms of spin pumping
physics of our model Hamiltonians. An adiabatic sliding
motion of one dimensional potential, in gapped Fermi
surface (insulating state), pumps an integer numbers of
particle per cycle. In our case the transport of Jordan-
Wigner fermions (spinless fermions) is nothing but the
transport of spin from one end of the chain to the other
end because the number operator of spinless fermions is
related to the z-component of spin density [30]. We will
see that the non-zero and introduce the
gap at around the Fermi point and the system is in the
insulating state (Peierls insulator).

1t


2t

Here, we would like to discuss the physics of geome-
tric phase related to our model Hamiltonians and its
relation to the entanglement current. It is well known that
the physical behavior of the system is identical at these
two Fermi points. We would like to analyses these
double degeneracy points following the seminal paper of
Berry [26]. In our model Hamiltonian there are two
adiabatic parameters and . The Hamilto-
nian starts to evolve under the variation of these two
adiabatic parameters. When the Hamiltonian returns to
its original form after a time T, the total geometric
phase acquired by the system is

1t


2t


d
2π
nnR
C
i
TR  
n
, (8)
A line integral around a closed loop in two dimen-
sional parameter space. Using Stokes theorem, one can
write

= d
2π
nRnR
i
T
  

The flux

through a closed surface C is,
d
BS


. Therefore, one can think of the Berry phase
as flux of a magnetic field. Now we express,


1
11
nKn
BK AK 

and
 
1
11
2π
nK
i
AKnK nK1 ,
where






12
1, ,
K
kt t  ,


1t
 and


2t
 are
the real and imaginary part of the fourier transform of


1t
 . Similarly for the Hamiltonian 2
H
, the adiabatic
parameters will be different. Here and
n
Bn
A
are the
fictitious magnetic field (flux) and vector potential of the
nth Bloch band respectively. The degenerate points
behave as a magnetic monopole in the generalized
momentum space (1
K
) [26], whose magnetic unit can be
shown to be 1 analytically as where
positive and negative signs of the above equations are
respectively for the conduction and valance band
meeting at the degeneracy points [26,27]. represent
an arbitrary closed surface which enclose the degeneracy
point. In the adiabatic process the parameters
1S
dSB
 1
1

S


1t
 or


2t
 are changed along a loop () enclosing the
origin (minima of the system). We define the expression
for spin current (

I
) from the analysis of Berry phase. It
is well known in the literature of adiabatic quantum
pumping physics that two independent parameters are
needed to achieve the adiabatic quantum pumping in a
system [31]. Here one may consider these two para-
meters as the real and imaginary part of the fourier
transform of a modulated coupling induced potential.
When the shape of the potential will change in time, then
it amounts to changing the phase and amplitude in time.
We define the expression for spin current (
I
) from the
analysis of Berry phase. Then according to the original
idea of quantum adiabatic particle transport [24,25,
27,28], the total number of spinless fermions (
I
) which
are transported from one side of this system to the other
is equal to the total flux of the valance band, which
penetrates the 2D closed sphere () spanned by the
2
S

and Brillioun zone [27].
1
2
d
S
ISB
1


 (10)
n
S
 . (9)
1
B

is directly related with the Berry phase (


nT
 )
which is acquired by the system during the adiabatic
variation of the exchange couplings during the time
period of the adiabatic process. This quantization is
topologically protected against the other perturbation as
long as the gap along the loop remains finite [27,28].
Therefore, the adiabatic entanglement pumping is cons-

108 S. SARKAR
tant over the arbitrarily long distance of the system. This
result is in contrast with the existing results in the
literature [8,9,20]. Therefore we have solved the problem
of entanglement transport in terms of entanglement
current but not in terms of conventional wisdom of the
literature [8-22]. The authors of Ref. [23] have found that
the entanglement decays exponentially after a certain
distance.
Now we explain the quantum entanglement transfer
for 1
H
. The second term of the Hamiltonian for NN
exchange interaction has originated from the
x
and
component of exchange interaction. This term implies
that infinitesimal variation of coupling in lattice sites and
is sufficient to produce a gap around the Fermi points. So
when
y
12<<1K, only these time dependent exchange
couplings are relevant and lock the phase operator at
π
0n
K
  . Now the locking potential slides adiabati-
cally. The speed of the sliding potential is low enough
such that the system stays in the same potential valley,
i.e., there is no scope to jump onto the other potential
valley. The system will acquire phase during one
complete cycle of the adiabatic process. This expectation
is easily verified when we notice the physical meaning of
the phase operator (
2π

x
 ). Since the spatial derivative of
the phase operator corresponds to the z-component of
spin density, this phase operator is nothing but the minus
of the spatial polarization of the z-component of spin, i.e.,
1
1
z
N
z
j
j
S
P
N
 jS. During the adiabatic process t

changes monotonically and acquires – phase. In this
process
2π
z
s
P increases by 1 per cycle. We define it
analytically as [27, 32]

1
dd
2π
zz
ss x
PP xx 

 
 1
 (11)
This physics always hold as far as the system is locked
by the sliding potential and 1

 [27]. The change of
the spatial polarization by unity during a complete
evaluation of adiabatic cycle implies the transport of
entanglement across the system. The entanglement
transport of this scenario can be generalized up to the
value of for which 
K
is greater than 1/2 . In this
limit, z-component of the exchange interaction has no
effect on the entanglement pumping of our system. But
when 12K, then the interaction due to becomes
relevant and creates a gap in the excitation spectrum.
This potential profile is static. Therefore, there is no
scope to slide the potential and to get a adiabatic pumping
across the system.

Similarly, for the Hamiltonian 2
H
, the second term of
the Hamiltonian produces the gap and the pumping
process is the same as that of 1
H
. Therefore, we con-
clude that the modulations in plane exchange coupling
and also for the modulations in the on site magnetic field
yield the same adiabatic entanglement pumping. If we
consider the unmodulated exchange coupling in our spin
chain system, then there is no gap in the excitation
spectrum. Therefore, there is no entanglement transport
in our system. As we have been already described, in
adiabatic quantum pumping process gap is absolutely
necessary.
In this pumping process the most favorable states of
the system are the antiferromagnetic configuration
010101... > and 101010,,,> (0 stands for up spin
and 1 stands for down spin). One may start from any
antiferromagnetic states and transfer the spin of every
site to the right by two sites to achieve the pumping.
Therefore, our test spin which we introduce at the one
end of the spin, hops to the right by two sites in every
step. Thus when we study the entanglement transport
between the spin 0

and 0, it is natural that the entan-
glement also is transported through every alternate sites.
This is the first correct and complete analytical and
physical explanation in the literature. The authors of Ref.
[7,22] have observe, the non analytical behavior of
entanglement transport as a function of time. But in our
study the entanglement current is constant and it is
almost perfect entanglement pumping. In their case the
spin chain has the spin rotational symmetry. When one
member of an entangled pair of qubits is transmitted
through such a channel , then the two qubits states evolve
to a Werner state [33]. But our spin chain systems has no
spin rotational invariant symmetry and the transport
mechanism is also different.
Here, we would like to explain the difference of
entanglement transport between the FM and AFM spin
chain. It is mentioned in the literature but the complete
physical explanation is not upto the mark [7-18,22]. Here
we consider the AFM/FM spin chain with static exchange
coupling to use the result of Bethe ansatz calculations.
As we know that entanglement is a quantum mechanical
property, Schrodinger singled out many decades ago as
the characteristic of quantum mechanics [34] and that
has been studied extensively in connection with Bell’s
inequality [35]. In FM ground state, there is no difference
between the classical and quantum mechanical ground
state and the low lying excitations are spin-1 magnons.
The AFM ground state has a complex structure specified
by the Bethe-ansatz solution. There are no similarities
between classical and quantum mechanical ground state
and first excited state of the AFM chain and as a result of
the quantum mechanical property of the system, the
entanglement manifests prominently in the AFM spin
chain. This is the only clear reason why AFM outper-
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109 S. SARKAR
forms the FM spin chain as far as entanglement related
properties are concerned. This correct physical analysis
was absent in all previous studies.
Here we discuss possible sources of imperfections in
the entanglement pumping process of our system. The
non-adiabatic contributions leave the system in an
unknown superposition of states after the full cycle. Also,
the appearance of Landau-Zener transition in the pum-
ping system should be negligible so that the system is in
the ground state. This condition limits the pumping rate
of entanglement by the mathematical relation h
J

.
However, even then the entanglement pumping is not
perfect due to the non vanishing
J
. Our effort also
should take the elimination of entanglement pumping in
the wrong directions. The residual exchange coupling
may lead to a different spin state. An entangled spin
transported through a correct exchange coupling modula-
tion with probability and through the residual ex-
change coupling with the probability . There-
P
1QP
fore, the pumping error in each site is Q
P. During this
analysis we assume that . Our system consists of
sites. So the probability of correct entanglement
transport is
PQ
N
2
~N P and wrong entanglement transport is
2
~N Q. The total pumping error,
2N
Q
P


 , decreases
with the number of sites in nanoscale spin chain.
Therefore, for the spin chain system entanglement trans-
port is better for larger length compared to the smaller
length with the same exchange couplings. This finding is
in contrast with the previous findings.
3. Conclusions
We have presented the theoretical explanation of perfect
adiabatic entanglement pumping for our model Hamilto-
nians. We have solved this problem orthogonally. We
have defined and calculated the entanglement current for
the first time in the literature. We have also found a sub-
class for blocking of entanglement transport, even in the
perfect pumping condition. We have explained many
physical findings of entanglement transport, such as
hopping of entanglement in alternate sites and the fact
that entanglement transport in AFM spin chain outper-
forms the FM spin chain. These facts were subject of
curiosity before our study. We have explained analyti-
cally and physically the reasons why the entanglement
transport is better for the larger length scale compared to
the smaller one. Our rigorous analytical solutions inter-
connect quantum many body physics and quantum infor-
mation science.
The author would like to thank the Center for Con-
densed Matter Theory of IISc for extended facility. The
author would like to give special thanks to Prof. Arnab
Roychoudhuri for constant encourgement to produce bet-
ter quality of work. Finally the author would like to
thank Prof. N. Behera, Prof. R. Srikanth, Dr. T. Tulsi and
Prof. Indrani Bose.
4. References
[1] A. J. Skineer, et al., “Hydrogenic Spin Quantum Com-
puting in Silicon: A Digital Approach,” Physical Review
Letters, Vol. 90, No. 8, 2003, pp. 087901-087904.
doi:10.1103/PhysRevLett.90.087901
[2] A. Zeilinger, “Experiment and the Foundation of Quan-
tum Physics,” Reviews of Modern Physics, Vol. 71, No. 2,
1999, pp. S288-S297.
[3] A. Rauschenbeutel, et al., “Step by Step Engineered Mul-
tiparticle,” Science, Vol. 288, No. 5473, 2000, pp. 2024-
2029. doi:10.1126/science.288.5473.2024
[4] C. A. Sackett, et al., “Experimental Entanglement of Four
Particle,” Nature, Vol. 404, No. 6775, 2000, pp. 256-258.
doi:10.1038/35005011
[5] M. Bayer, et al., “Coupling and Entanglement of Quan-
tum States in Quantum Dot Molecules,” Science, Vol.
291, No. 5503, 2001, pp. 451-453.
doi:10.1126/science.291.5503.451
[6] F. Plastina, R. Fazio and G. M. Palma, “Macroscopic
Entanglement in Josephson Nanocircuits,” Physical Re-
view B, Vol. 64, No. 11, 2001, pp. 113306-113310.
doi:10.1103/PhysRevB.64.113306
[7] S. Bose, “Quantum Computation through Unmodulated
Spin Chain,” Physical Review Letters, Vol. 91, No. 20,
2003, pp. 207901-207904.
doi:10.1103/PhysRevLett.91.207901
[8] V. Subrahmanyam, “Entanglement Dynamics and Quan-
tum State Transport in Spin Chain,” Physical Review A,
Vol. 69, No. 3, 2004, pp. 034304-034307.
[9] A. Kay, “Unifying Quantum State Transfer and State
Amplification,” Physical Review Letters, Vol. 98, No. 1,
2007, pp. 010501-010504.
doi:10.1103/PhysRevLett.98.010501
[10] T. J. Osborne and N. Linden, “Propagation of Quantum
Information through a Spin System,” Physical Review A,
Vol. 69, No. 5, 2004, pp. 052315-052320.
doi:10.1103/PhysRevA.69.052315
[11] L. Amico, et al., “Scaling of Entanglement Close to a
Quantum Phase Transition,” Nature, Vol. 416, No. 6881,
2002, pp. 608-610. doi:10.1038/416608a
[12] A. Bayat and V. Karimipour, “Thermal Effect on Quan-
tum Communication through Spin Chain,” Physical Re-
view A, Vol. 71, No. 4, 2005, pp. 042330-042336.
doi:10.1103/PhysRevA.71.042330

S. SARKAR
110
[13] L. C. Venuti, et al., “Long-Distance Entanglement in
Spin System,” Physical Review Letters, Vol. 96, No. 24,
2006, pp. 247206-247209.
doi:10.1103/PhysRevLett.96.247206
[25] Q. Niu and D. J. Thouless, “Quantized Adiabatic Charge
Transport in the Presence of Substrate Disorder and
Many-Body Interaction,” Journal of Physics A: Mathe-
matical and General, Vol. 17, No. 12, 1984, pp. 2453-
2462. doi:10.1088/0305-4470/17/12/016
[14] O. Romero-Isart, K. Eckert and A. Sanpera, “Quantum
State Transfer in Spin-1 Chain,” Physical Review A, Vol.
75, No. 5, 2007, pp. 050303-050306.
doi:10.1103/PhysRevA.75.050303
[26] M. V. Berry, “Quantal Phase Factor Accompaying Adia-
batic Charges,” Proceedings of the Royal Society A, Vol.
392, No. 1802, 1984, pp. 45-57.
doi:10.1098/rspa.1984.0023
[15] K. Eckert, O. Romero-Isart and A. Sanpera, “Efficient
Quantum State Transfer in Spin Chain via Adiabatic Pas-
sage,” New Journal of Physics, Vol. 9, No. 5, 2007, pp.
155-173. doi:10.1088/1367-2630/9/5/155
[27] R. Shindou, “Quantum Spin Pump in S = 1/2 Antiferro-
magnetic Chain Holonomy of Phase Operators in Sine-
Gordon Theory,” Journal of the Physics Society Japan,
Vol. 74, No. 4, 2005, pp. 1214-1223.
doi:10.1143/JPSJ.74.1214
[16] V. Srinivasa, J. Levy and C. S. Hellberg, “Flying Spin
Qubit,” Exchange Organizational Behavior Teaching Jour-
nal, 2006, pp. 1-13. [28] J. E. Avron, A. Raveh and B. Zur, “Adiabatic Quantum
Transport in Multiple Connected System,” Reviews of
Modern Physics, Vol. 60, No. 4, 1988, pp. 873-915.
doi:10.1103/RevModPhys.60.873
[17] M. J. Hartmann, et al., “Excitations and Entanglement
Transfer versus Spectral Gap,” New Journal of Physics,
Vol. 8, No. 6, 2006, pp. 94-108.
doi:10.1088/1367-2630/8/6/094 [29] T. Giamarchi, “Quantum Physics in One Dimension,”
Clarendon Press, London, 2004.
[18] L. Amico, et al., “Dynamics of Entanglement in One-
Dimensional Spin Systems,” Physical Review A, Vol. 69,
No. 2, 2004, pp. 022304-022327.
doi:10.1103/PhysRevA.69.022304
[30]
 
1
1sin
2ππ
n
z
nxn n
Sx x  · 


 

field corresponds
to the quantum fluctuations (boson) of spin.
[19] C. Di Franco, et al., “Perfect State Transfer of Spin Chain
without State Initialization,” Physical Re vie w Lett ers, Vol.
101, No. 23, 2008, pp. 230502-230503.
doi:10.1103/PhysRevLett.101.230502
[31] P. Sharma and C. Chamon, “Quantum Pump for Spin and
Charge Transport in a Luttinger Liquid,” Physical Review
Letters, Vol. 87, No. 9, 2001, pp. 96401-96405.
doi:10.1103/PhysRevLett.87.096401
[20] F. Plastina, et al., “Local Control of Entanglement in Spin
Chain,” Physical Review Letters, Vol. 99, No. 17, 2007,
pp. 177210-177213. doi:10.1103/PhysRevLett.99.177210
[32] S. Sarkar and C. D. Hu, “Quantum Spin Pumping at a
Fractionally Quantized Magnetization State for a System
with Competing Exchange Interactions,” Physical Review
B, Vol. 77, No. 6, 2008, pp. 064413-064418.
doi:10.1103/PhysRevB.77.064413
[21] M. Horodecki, et al., “General Teleporation Channel,
Singlet Fraction and Quasidistillation,” Physical Review
A, Vol. 60, No. 3, 1999, pp. 1888-1898. [33] C. H. Bennett, et al., “Mixed State Entanglement and
Quantum Error Correction,” Physical Review A, Vol. 54,
No. 5, 1996, pp. 3824-3851.
doi:10.1103/PhysRevA.54.3824
[22] A. Bayat and S. Bose, “Information Transferring Ability
of the Different Phases of a Finite XXZ Spin Chain,”
Physical Review A, Vol. 81, No. 1, 2008, pp. 1-11.
[34] E. Schrodinger, “Discussion of Probability Relation be-
tween Separated System,” Mathematical Proceedings of
the Cambridge Philosophical Society, Vol. 31, No. 4, 1935,
pp. 555-563.
[23] C. F. Hirjibehedin, et al., “Spin Coupling in Engineered
Atomic Structures,” Science, Vol. 312, No. 5776, 2006,
pp. 1021-1025. doi:10.1126/science.1125398
[24] D. J. Thouless, “Quantization of Particle Transport,” Phy-
sical Review B, Vol. 27, No. 10, 1983, pp. 6083-6087.
doi:10.1103/PhysRevB.27.6083
[35] J. S. Bell, “On the Einstein-Podolsky-Rosen Paradox,”
Physics, Vol. 1, No. 3, 1964, pp. 195-202.