Perfect Entanglement Transport in Quantum Spin Chain Systems

doi:10.4236/jqis.2011.13014

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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

E-mail: [email protected]

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





 



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



), where 0



and 1



are the Pauli spin operators for the 0 and 1 sites

respectively and

is the exchange coupling. The ini-

tial state being





0,0

 



 

 where

g is the ground state wave function of the AFM

Hamiltonian and









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 (

)

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

with

modulated exchange coupling in

Y plane and 2

for onsite magnetic field modulation. Hamiltonians of

the systems are the following.

 



nnnn n

JtSSS





 

 



1nn

SS





(1)

This model Hamiltonian has some experimental rele-

vance [27]. The other model Hamiltonian with onsite

magnetic field modulation is





 



nn nnnn

xx yyzz

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

†

12,

exp π,

nnn

nn j

Sin

































n

(3)

[29], where †



 is the fermion number at site

. Spin operators in terms of bosonic field are the

following.



cos 2π 1cos ,

cos 2π 1sin

π 1cos2π.

ScKc K

ScKcK

ScK

























 











(4)

is the Luttinger liquid parameter, and are

the constants. The fermionic fields,



 



(5)

2π

ir xx









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



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

 



10 22

d:cos2

2π

d:cos4

2π

HxK

xKx











 







(6)

 

d:cos2

2π

dd:cos4

22π

HHx Kx

xxK











 



 







(7)

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).









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











2π

nnR





n

, (8)

A line integral around a closed loop in two dimen-

sional parameter space. Using Stokes theorem, one can

write



= d

2π

nRnR



 



The flux



through a closed surface C is,





. Therefore, one can think of the Berry phase

as flux of a magnetic field. Now we express,





nKn

BK AK 



and

 

2π

AKnK nK1

where













1, ,

kt t



,







and







are

the real and imaginary part of the fourier transform of







. Similarly for the Hamiltonian 2

, the adiabatic

parameters will be different. Here and

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

) [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

dSB

 1















are changed along a loop () enclosing the

origin (minima of the system). We define the expression

for spin current (



) 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 (

) 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 (

) 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



and Brillioun zone [27].

ISB

1





 (10)



. (9)



is directly related with the Berry phase (







)

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

. The second term of the Hamiltonian for NN

exchange interaction has originated from the

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

12<<1K, only these time dependent exchange

couplings are relevant and lock the phase operator at



 . 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





). 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.,

N

 jS. During the adiabatic process t



changes monotonically and acquires – phase. In this

process

2π

P increases by 1 per cycle. We define it

analytically as [27, 32]



2π

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



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

, the second term of

the Hamiltonian produces the gap and the pumping

process is the same as that of 1

. 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-

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



.

However, even then the entanglement pumping is not

perfect due to the non vanishing

. 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-

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

P and wrong entanglement transport is

Q. The total pumping error,





 , 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.

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