Thông tin tài liệu
ENTANGLEMENT IN QUANTUM SPIN CHAINS
YUKI TAKAHASHI
(B.Sc, Rikkyo University, Tokyo, Japan)
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF SCIENCE
DEPARTMENT OF PHYSICS
NATIONAL UNIVERSITY OF SINGAPORE
2007
Acknowledgements
I would like to thank my supervisor Professor Oh Choo Hiap and Associate
Professor Kwek Leong Chuang for their guidance.
Also, I would like to show my appreciation to all the members in Quantum
Information Technology Lab at National University of Singapore.
Last but not least, I would like to express my thanks to my friends and
family who supported me to study in Singapore for 2 years.
i
Contents
1 Introduction
1
1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Superdense Coding . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . .
5
2 Entanglement in Quantum Spin Chain
8
2.1
Measure of Entanglement
. . . . . . . . . . . . . . . . . . . . .
8
2.2
Some General Arguments . . . . . . . . . . . . . . . . . . . . . .
10
2.3
XXX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
XX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3 Entanglement in Next-Nearest-Neighbor Models
3.1
3.2
23
Heisenberg XXX Model with Next-Nearest-Neighbor Interaction
26
3.1.1
Periodic Boundary Condition . . . . . . . . . . . . . . .
27
3.1.2
Open Boundary Condition . . . . . . . . . . . . . . . . .
33
Heisenberg XX chain . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2.1
45
Periodic Boundary Condition . . . . . . . . . . . . . . .
ii
3.2.2
3.3
Open Boundary Condition . . . . . . . . . . . . . . . . .
48
Summary and Conclusion . . . . . . . . . . . . . . . . . . . . .
55
4 Quantum State Transfer
56
4.1
State Transfer Scheme . . . . . . . . . . . . . . . . . . . . . . .
57
4.2
State Transfer under White Noise . . . . . . . . . . . . . . . . .
62
4.3
Recent Proposals for High Fidelity State Transfer . . . . . . . .
63
5 Engineering Quantum Teleportation through State Transfer
66
5.1
Teleportation Scheme . . . . . . . . . . . . . . . . . . . . . . . .
67
5.2
Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.2.1
Entangled Pair in the Initial Chain . . . . . . . . . . . .
72
5.2.2
Separable Initial Chain . . . . . . . . . . . . . . . . . .
72
5.2.3
Local Phase Acquisition . . . . . . . . . . . . . . . . . .
73
5.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.4
Teleportation under White Noise . . . . . . . . . . . . . . . . .
77
5.5
Extension to Next-Nearest-Neighbor Model . . . . . . . . . . . .
79
5.5.1
Scheme
. . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
Summary and Discussion . . . . . . . . . . . . . . . . . . . . . .
83
5.6
Appendices
91
A Exactly Solvable Models
91
A.1 Heisenberg XXXs=1/2 -Algebraic Bethe Ansatz . . . . . . . . .
91
A.2 XY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
iii
Summary
Entanglement plays a central role in quantum information processing and its
usefulness as a resource for quantum information processing in quantum spin
chain models has been vastly studied. In this thesis, as an extension of previous
studies, quantitative feature of entanglement in Heisenberg XXX and XX model
with next-nearest-neighbor interaction is studied in full detail. Moreover, it is
shown that spin chains can act as a good resource for quantum teleportation
and a quantum teleportation scheme along a Heisenberg XX open chain is
proposed.
iv
List of Figures
2.1
Temperature dependence of nearest-neighbor concurrence in the
periodic Heisenberg XXX model for N = 2 to N = 5. At T = 0,
from top-to-bottom, N = 2, 4, 5. . . . . . . . . . . . . . . . . . .
2.2
17
Temperature dependence of nearest-neighbor concurrence in the
periodic Heisenberg XX model for N = 2 to N = 5. At T = 0,
from top-to-bottom, N = 2, 4, 3, 5. . . . . . . . . . . . . . . . . .
3.1
NN and NNN concurrence for the XXX model(J1 = 1) with
PBC(even N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
29
NN and NNN concurrence for the XXX model(J1 = −1) with
PBC(even N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
29
N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XXX model(J1 = 1) with PBC . . . . . . . . . . . . . . .
3.4
28
NN and NNN concurrence for the XXX model(J1 = 1) with
PBC(odd N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
22
31
NN and NNN concurrence for the XXX model(J1 = −1) with
PBC(odd N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
31
3.6
N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XXX model(J1 = −1) with PBC . . . . . . . . . . . . . .
32
3.7
C12 and C13 for the XXX model(J1 = 1) with OBC(even N ) . .
34
3.8
C12 and C13 in for the XXX model(J1 = 1) with OBC(odd N ) .
35
3.9
C23 and C24 for the XXX model(J1 = 1) with OBC(even N ) . .
35
3.10 C23 and C24 for the XXX model(J1 = 1) with OBC(odd N ) . . .
36
3.11 C34 and C35 for the XXX model(J1 = 1) with OBC(even N ) . .
36
3.12 C34 and C35 for the XXX model(J1 = 1) with OBC(odd N ) . . .
37
3.13 N dependence of C12 (J2 = −1) and C13 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
37
3.14 N dependence of C23 (J2 = −1) and C24 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
38
3.15 N dependence of C34 (J2 = −1) and C35 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
38
3.16 C12 and C13 for the XXX model(J1 = −1) with OBC(even N ) .
40
3.17 C12 and C13 in for the XXX model(J1 = −1) with OBC(odd N )
40
3.18 C23 and C24 for the XXX model(J1 = −1) with OBC(even N ) .
41
3.19 C23 and C24 for the XXX model(J1 = −1) with OBC(odd N ) . .
41
3.20 C34 and C35 for the XXX model(J1 = −1) with OBC(even N ) .
42
3.21 C34 and C35 for the XXX model(J1 = −1) with OBC(odd N ) . .
42
3.22 N dependence of C12 (J2 = −1) and C13 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC . . . . . . . . . . . . . . . . . . . . .
43
3.23 N dependence of C23 (J2 = −1) and C24 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC . . . . . . . . . . . . . . . . . . . . .
vi
43
3.24 N dependence of C34 (J2 = −1) and C35 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC . . . . . . . . . . . . . . . . . . . . .
44
3.25 NN and NNN concurrence for the XX model(J1 = 1) with PBC(even
N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.26 NN and NNN concurrence for the XX model(J1 = 1) with PBC(odd
N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.27 N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XX model(J1 = 1) with PBC . . . . . . . . . . . . . . . .
47
3.28 NN and NNN concurrence for the XX model(J1 = −1) with
PBC(odd N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.29 N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XX model(J1 = −1) with PBC . . . . . . . . . . . . . . .
49
3.30 C12 and C13 for the XX model(J1 = 1) with OBC(even N ) . . .
50
3.31 C12 and C13 for the XX model(J1 = 1) with OBC(odd N ) . . . .
51
3.32 C23 and C24 for the XX model(J1 = 1) with OBC(even N ) . . .
51
3.33 C23 and C24 for the XX model(J1 = 1) with OBC(odd N ) . . . .
52
3.34 C34 and C35 for the XX model(J1 = 1) with OBC(even N ) . . .
52
3.35 C34 and C35 for the XX model(J1 = 1) with OBC(odd N ) . . . .
53
3.36 N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
53
3.37 N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
54
3.38 N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC . . . . . . . . . . . . . . . . . . . . . .
vii
54
4.1
Time dependence of the state transfer fidelity F (t); N = 3(solid
line) and N = 4(dotted line) . . . . . . . . . . . . . . . . . . . .
4.2
Time dependence of the state transfer fidelity F (t); N = 9(solid
line) and N = 10(dotted line) . . . . . . . . . . . . . . . . . . .
4.3
61
Fidelity of the state transfer with different amount of noise levels;
F = 0, F = 0.01, and F = 0.05 . . . . . . . . . . . . . . . . . . .
5.1
61
63
Teleportation fidelity via a Heisenberg XX spin chain of various
number of sites from N = 2 to N = 20 in scenario A(solid line)
and scenario B(dotted line) with optimal time and optimal phase
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Teleportation fidelity with 4 different noise levels (from the top,
F = 0, 0.001, 0.01, 0.05) up to N = 20 in scenario A. . . . . . . .
5.3
78
Maximum concurrence between the 1st and (N/2 + 1)th site,
C1,N/2+1 up to N = 20 with HJ (thick line) and H0 . . . . . . .
5.5
78
Teleportation fidelity with 4 different noise levels (from the top,
F = 0, 0.001, 0.01, 0.05) up to N = 20 in scenario B. . . . . . . .
5.4
76
82
Maximum concurrence between the 1st and N th site(C1,N ) up
˜ J (thick line) and H0 together with the conto N = 12 with H
currence C1,N with optimally chosen initial state under the usual
Heisenberg XX Hamiltonian H0 (dotted line) . . . . . . . . . . .
viii
84
Chapter 1
Introduction
1.1
Overview
Quantum information science or quantum information technology is an interdisciplinary area of science, located at the intersection of fundamental physics,
mathematics, and information technology. In the past two decades, the world
has witnessed a rapid development of this emergent field with the development
of algorithms that showed that with a quantum computer, one could perform
faster computation in the quantum regime. Two such algorithms are the Shor
factorization algorithm and the Grover search algorithm. These algorithms
have been shown to provide faster and more efficient computation compared
to existing ’classical’ algorithms. Apart from advances in algorithms, quantum
information technology allows us to carry out innovative information processing such as quantum teleportation [6], superdense coding [7], and quantum
cryptography [8].
An important resource in many of these quantum information processes is
1
the existence a purely quantum correlation called ’entanglement’, a notion that
was first discussed by Einstein, Podolsky, and Rosen and now commonly known
as the EPR paradox. Entanglement is perhaps one of the most striking and
peculiar property of quantum mechanics. Entanglement is not the only property
that has been exploited in quantum information processing. Another important
quantum mechanical property is the superposition or coherent interference.
For quantum information processing to be possible, it is essential to realize and construct quantum logic gates. Many potential platforms have been
proposed and demonstrated at the single logic gate level: atoms in ion trap,
nuclear magnetic resonance device, superconducting qubits, linear optics implementation, and so forth. In recent years, there has been an intense effort
at the realization of gates through solid state devices. An initial effort in this
direction is the study of quantum spin chains. Quantum spin chain models,
such as the Heisenberg XXX model, Heisenberg XXZ model, and Heisenberg
XY model, have always been useful as natural theoretical models for studying
magnetism. Surprisingly, these one-dimensional models are exactly solvable,
that is, the full spectrum of the Hamiltonian can be obtained, by means of
special techniques such as Bethe ansatz and Wigner-Jordan transformation. A
brief review of exactly solvable models is given in Appendix A and a detailed
study of exactly solvable models can be found in Ref.[3]. There exists many
quasi one-dimensional systems that provide invaluable test-beds for many theoretical predictions, ranging from inorganic compounds like SrCu2 O3 , VO2 P2 O7 ,
and CuGeO3 to organic compounds like TTF-CuS4 C4 (CF3 )4 .
Recently, many studies have started investigating quantum spin chains from
an information perspective. It has also been demonstrated that quantum spin
2
chains are potentially useful resources for quantum information processing.
Moreover, the effective Hamiltonian for many realistic systems such as quantum dots and cavity QED systems can be treated as simple one-dimensional
or two dimensional quantum spin chains [54, 55, 56]. These studies have given
rise to a vast amount of literature devoted to the study of entanglement and
states associated with quantum spin chain systems[40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53]. In particular, it has been shown that one-dimensional
Heisenberg spin chains can act as a quantum wire for the transmission of an
unknown quantum state from one site to another [18].
In this thesis, we study one-dimensional quantum spin chain models in the
context of quantum information science. Specifically, we fully analyze the properties of entanglement in the ground state of the Heisenberg XXX and XX models with next-nearest-neighbor interaction. In addition, we will also propose a
new scheme of quantum teleportation in a quantum spin chain via measurement
process.
This thesis is structured as follows. Firstly, we review some key ideas in
quantum information technology including simple quantum information processes like quantum teleportation and superdense coding in Chapter 1. In
Chapter 2, some results regarding the properties of entanglement in spin chain
models will be reviewed. In Chapter 3, we extend the study of spin chains to
systems possessing next-nearest-neighbor interactions in addition to the usual
nearest-neighbor interactions and study the effect of next-nearest-neighbor interactions on the entanglement properties of the system. The discussion then
moves on to a study of the dynamics of the spin chain models for the rest of
3
the chapter. In Chapter 4, we review quantum state transfer scheme along a
spin chain and we propose a quantum teleportation scheme using a Heisenberg
spin chain as a resource in Chapter 5.
1.2
Notations
In this section, we summarize some frequently used notations used in this thesis.
• Qubit
0
1
|0 = | ↓ = , |1 = | ↑ =
1
0
(1.1)
• Pauli matrices
0
1 0
0 1
0 −i
1
I=
, σx =
, σy =
, σz =
0 1
1 0
i
0
0 −1
(1.2)
1
1
1
sx = σx , sy = σy , sz = σz
2
2
2
(1.3)
• Raising and lowering operator
1 x
(σ + iσ y )
2
1 x
(σ − iσ y )
=
2
σ+ =
(1.4)
σ−
(1.5)
• Bell states
|Ψ±
|Φ±
√
≡ (|00 ± |11 )/ 2
√
≡ (|01 ± |10 )/ 2
4
(1.6)
(1.7)
1.3
Superdense Coding
In essence, superdense coding is a procedure to communicate two bits of classical information using a single shared qubit [1, 7]. Let us suppose that Alice
and Bob share one of the Bell state |Ψ+ and Alice wishes to send 2 bits of
classical information to Bob. Alice is only allowed to operate on the qubit in
her possession. To complete her transmission, Alice performs a local operation
on her qubit, depending on the information she wants to send to Bob. For
instance, if she wishes to send ’00’ to Bob, she does nothing, and if ’01’, she
performs σx on her qubit. She then sends her single qubit to Bob. Since the
four Bell states are mutually orthogonal, Bob is able to distinguish and tell
precisely the information Alice sent by performing Bell state measurements on
both qubits. In summary,
Inf ormation : Alice′ s operation = F inal state
1.4
00 :
I ⊗ I|Ψ+
= |Ψ+
01 :
σx ⊗ I|Ψ+
= |Φ+
10 :
iσy ⊗ I|Ψ+
= |Φ−
11 :
σz ⊗ I|Ψ+
= |Ψ−
Quantum Teleportation
Quantum teleportation is probably one of the most striking achievement of
quantum information technology [1, 6]. Suppose Alice wishes to teleport a
quantum state to Bob and they share a maximally entangled pair, say |Ψ+ , at
5
the beginning. To start the teleportation procedure, she attaches her unknown
qubit |φ = α|0 + β|1 which she wishes to send to Bob to her portion of the
entangled qubit through a Hadamard gate. Here, α and β are some unknown
parameters and the Hadamard gate H is defined as
1
1 1
H=√
2
1 −1
(1.8)
After undergoing a Hadamard gate and rearranging the terms, the total state
of |φ and |Ψ+ becomes
|ψtotal
=
1
|00 (α|0 + β|1 ) + |01 (α|1 + β|0 )
2
+|10 (α|0 − β|1 ) + |11 (α|1 − β|0 )
(1.9)
Now she performs a Bell state measurement on her 2 qubits in her possession.
There are four possible outcomes: namely, 00,01,10,11 and the state of the Bob’s
qubit will be changed to 00:(α|0 + β|1 ), 01:(α|1 + β|0 ), 10:(α|0 − β|1 ),
11:(α|1 − β|0 ) accordingly depending upon the measurement result. After
Alice has performed a Bell state measurement, she tells Bob the result of her
measurement through a classical channel. With Alice’s measurement result,
Bob applies one of the Pauli operators depending on the classical information,
that is, 00:I, 01:σx , 10:σz , 11:σy . These operations allows Bob to recover the
unknown state |φ perfectly. It is of course possible to perform quantum teleportation using other Bell state, however, the Pauli operators that Bob has to
apply must be changed accordingly.
6
Quantum teleportation is not merely a theoretical construct. It has been
demonstrated experimentally using entangled photons [9, 10, 11] as well as
atoms [12, 13]. Moreover, quantum teleportation plays an important role not
only in quantum communication but also in quantum computation - quantum
teleportation can act as a universal computational primitive [14]. In addition
to quantum teleportation with qubits, quantum teleportation with continuous
variables has also been studied and demonstrated via squeezed states [15].
7
Chapter 2
Entanglement in Quantum Spin
Chain
In this chapter, we begin a brief review some quantitative measures of entanglement. Using these measures, we will look at some analytical results obtained so
far in quantum spin chain systems such as the Heisenberg XXX and XY model.
2.1
Measure of Entanglement
Entanglement is a crucial component of quantum information processing. Primitives for quantum information processing such as quantum teleportation and
superdense conding can be carried out with the help of entangled states. It
is then natural to ask about the entanglement of an arbitrary given state and
how useful it is for quantum information processing with the given amount of
entanglement. Violation of the Bell inequality is one criterion to distinguish
entangled states from separable states. Several types of Bell inequalities have
8
been studied. It has been shown that for bipartite system with dimensions below 2 × 3, a necessary and sufficient condition for a given state to be separable
is that the partial transpose of the system is positive. This condition is often
called the Peres-Horodecki criteria[39].
We still do not understand how to obtain a good measure of entanglement
for the most general situation such as a multipartite system with arbitrary
spin dimension. A good review of multipartite entanglement can be found
in Ref.[58]. However, for bipartite systems, various quantitative measures of
entanglement have been proposed. One of the most useful and widely used
measure of entanglement for spin-1/2 bipartite systems is concurrence, first
developed by Wootters [37].
To see how we can get the concurrence of a given state, consider the density
matrix of a bipartite spin-1/2 system ρ˜:
ρ˜ = (σ y ⊗ σ y )ρ∗ (σ y ⊗ σ y )
(2.1)
The concurrence C is given by
C = max 0,
λ1 −
λ2 −
λ3 −
λ4
(2.2)
where λ1 ≥ λ2 ≥ λ3 ≥ λ4 are the eigenvalues of the matrix product ρ˜ρ. The
value of concurrence lies between 0 ≤ C ≤ 1; C = 0 corresponds to an unentangled state and C = 1 corresponds to a maximally entangled state.
Concurrence also possesses a feature of entanglement called monogamy,
which basically means that it is not allowed to have entanglement with a third
9
party if the system is maximally entangled. This monogamy is a purely quantum phenomena in the sense that there exists no such limitation on correlations
in the classical regime.
Negativity is also a useful measure of entanglement for bipartite systems
developed by Vidal and Werner [38]. For a given density matrix ρ, negativity
is defined as
N (ρ) =
where ρTA
ρTA − 1
2
(2.3)
denotes the sum of the absolute values of the spectrum of ρ
with partial transpose with respect to system A being taken. A significant
difference between concurrence and negativity is that although concurrence is
applicable only for spin-1/2 birpartite systems, negativity can be applied to
pairwise entanglement of higher spin systems, e.g. for bipartite spin-1 systems.
In the following analysis, I will mainly focus on pairwise entanglement in a
system and concurrence will be employed as a quantitative measure of entanglement.
2.2
Some General Arguments
To quantify the amount of entanglement present in a system, for instance to
evaluate concurrence, one in general needs to construct the density matrix of
the system from the eigenvalues and eigenvectors of the Hamiltonian followed
by tracing out the system matrix except the subsystem in consideration. From
a computational point of view, this procedure is indeed a hard task to carry
out since the full knowledge of eigenvalues and eigenvectors is necessary and
10
the dimension of the Hilbert space which we have to deal with blows up exponentially as the number of sites in the spin chain grows. However, if the
system possesses some symmetries, several of the general conclusions can be
obtained regarding the state and concurrence and the effort of computation
can be reduced significantly.
Now we consider a system in which Hamiltonian H posesses the following
properties,
• Translation invariance
z
• [H, σtotal
] = 0.
z
z
where σtotal
is defined as σtotal
≡
i
σiz . The second condition implies the
conservation of total z-component spin. Note that at this moment, we do not
specify any particular Hamiltonian. If the two conditions above are satisfied, it
is guaranteed that the reduced density matrix of nearest neighbor site has the
following form [52].
u
0
ρ=
0
0
+
0
0
w1
z
z ∗ w2
0
0
0
0
0
(2.4)
u−
In fact, the Heisenberg XXX, XXZ, and XY model with periodic boundary
condition belong to this class of Hamiltonian. The elements of the matrix are
1
¯ + Gzz )
(1 ± 2M
4
1
z =
(Gxx + Gyy + iGxy − iGyx )
4
u± =
11
(2.5)
(2.6)
¯ denotes the magnetization per site and Gαβ =< σ1α σ2β > is the corwhere M
relation function between nearest neighbor. Internal energy and magnetization
are defined as usual
U =−
1 ∂Z
1 ∂Z
, M =−
Z ∂β
Zβ ∂B
(2.7)
The elements in the matrix can be checked in the following way
¯ + Gzz = < (1 + σz )(1 + σz ) >
1 + 2M
= Tr[ρ(1 + σz )(1 + σz )]
u+ 0 0
0
0 w 1 z
0
= Tr
0
z ∗ w2
0
0
0 0 u−
= 4u+
4 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
(2.8)
In the same way, u− can be obtained using the relation
¯ + Gzz =< (1 − σz )(1 − σz ) >
1 − 2M
12
(2.9)
Now for z
Gxx + Gyy + iGxy − iGyx = < (σ x − iσ y )(σ x + iσ y ) >
= Tr[ρ(σ x − iσ y )(σ x + iσ y )]
+
0 0
0
u
0 w1 z
0
= Tr
0
z ∗ w2
0
0
0 0 u−
0 0 0 0
0 4z 0 0
= Tr
0 4w 0 0
2
0 0 0 0
= 4z
0 0 0 0
0 0 0 0
0 4 0 0
0 0 0 0
(2.10)
From the expression of the density matrix (2.4), concurrence is readily obtained as
C = 2 max[0, |z| −
√
u+ u− ]
(2.11)
Notice that, so far, the discussions merely rely on the symmetry of the system
and the results given here do not depend on the specific Hamiltonian. In the
following sections, I will study the properties of concurrence based on specific
Hamiltonians; Heisenberg XXX and XY model.
13
2.3
XXX Model
The Hamiltonian of the Heisenberg XXX model reads
y
x
z
(σix σi+1
+ σiy σi+1
+ σiz σi+1
)
H=J
(2.12)
i
where J is the coupling constant between neighboring sites. Wang and Zanardi
showed that concurrence is directly related to statistical quantities such as internal energy, magnetization, and two-point correlation functions [49]. Considering the fact that magnetization is zero at any temperature in the Heisenberg
model and by using Eq.(2.5), (2.6) and (2.11), concurrence can be written as
C=
1
max[0, |Gxx + Gyy | − Gzz − 1]
2
(2.13)
In addition, because of the global SU(2) symmetry and translation invariance,
we also have Gxx = Gyy = Gzz and Gzz = U/(3JN ), which reduces the expression of concurrence to
C=
1
max[0, 2|U¯ /J| − U¯ /J − 3]
6
(2.14)
i.e. concurrence is solely determined by internal energy or partition function.
This is a very lucky case since one only needs the knowledge of eigenvalues,
whereas usually it is necessary to have a full knowledge of both eigenvalues and
eigenvectors.
From Eq.(2.14), it directly follows that concurrence between nearest-neighbor
14
sites can be written as
U
1
max[0, −
− 1] (AF M )
2
JN
1
U
C =
max[0,
− 1] (F M )
2
3JN
C =
(2.15)
for antiferromagnetic(AFM) and ferromagnetic(FM) cases respectively. Here,
an antiferromagnetic system corresponds to a positive coupling constant(J > 0)
and a ferromagnetic system corresponds to a negative coupling constant(J < 0).
Let us now apply the above general results to an explicit number of sites; N = 2
and N = 3.
• N =2
From the eigenvalues of the Hamiltonian (2.12) for N = 2, it is straighforward to compute the partition function and internal energy. They are
given by
Z = 3e−2βJ + e6βJ
U = 6J(e−2βJ − e6βJ )
(2.16)
Therefore, from Eq.(2.15) concurrence reads
C = max 0,
C = 0 (F M )
e8βJ − 3
e8βJ + 3
(AF M )
(2.17)
Note that there is a threshold temperature above which entanglement
vanishes. In this case, the threshold temperature is given by T = 8J/ ln 3.
15
• N =3
In the same manner, the partition function and internal energy are computed as
Z = cosh (3βJ)
U = −3J tanh (3βJ)
(2.18)
Again with the use of Eq.(2.15), concurrence reads
C = max 0,
tanh (3βJ) − 1
2
C = 0 (F M )
(AF M )
(2.19)
However, even in the AFM case, it is clear that the function in the expression of the concurrence is never positive because of the tanh function,
which gives C = 0 in the AFM as well. Therefore, there is no pairwise
entanglement in 3-qubit Heisenberg XXX ring at any temperature.
In fact, it is possible to deduce a more general conclusion; the ground
state of the AFM Heisenberg XXX ring always has non-zero concurrence
except for N = 3 below critical temperature and concurrence is always
zero at any temperature for the FM Heisenberg XXX ring. Notice however
that this conclusion is only applicable to the case of periodic boundary
condition and the situation is totally different when the system is under
open boundary condition.
To elaborate the behaviour of concurrence , we plot the temperature dependence of the nearest-neighbor concurrence in Fig.2.1 for N = 2, 3, 4 and 5. It
16
can be seen that the threshold temperature slowly decreases as the number of
sites increases. Note that the value of J is taken to be unity and concurrence
is absent in any temperature for N = 3 as expected from the analysis given
above.
1
Concurrence
0.8
0.6
0.4
0.2
0
0
2
4
6
Temperature
8
10
Figure 2.1: Temperature dependence of nearest-neighbor concurrence in the
periodic Heisenberg XXX model for N = 2 to N = 5. At T = 0, from top-tobottom, N = 2, 4, 5.
2.4
XX Model
The Hamiltonian of the Heisenberg XX model reads
y
x
+ σiy σi+1
)
(σix σi+1
H=J
(2.20)
i
In this case, the concurrence can not be written simply with respect to internal
energy and the knowledge of correlation function is also necessary to calculate
17
its concurrence. Specifically, concurrence is given as
C=
1
max[0, |U/J| − Gzz − 1]
2
(2.21)
However for small number of sites, it is relatively easy to obtain an analytical
expression for concurrence by explicitly working on eigenvalues and eigenvectors. In the following, I will study the case of N = 2 and N = 3.
• N =2
A detailed study of thermal entanglement in XX model with external
magnetic field has been given by Wang in Ref.[40]. The Hamiltonian can
be written as
H=
B z
(σ1 + σ2z ) + J(σ1+ σ2− + σ2+ σ1− )
2
(2.22)
For this simple Hamiltonian, the eigenvalues and its corresonding eigenvectors are readily obtained as
|00 : −B, |11 : B, |Ψ+ : J, |Ψ− : −J,
(2.23)
which can be checked easily by simply operating the Hamiltonian (2.22)
on the above states. Therefore, it is easy to obtain the density matrix
18
using ρ(T ) = exp(−βH)/Z. The density matrix can be written as
1 Bβ
e |00 00| + e−Bβ |11 11| + eJβ |Φ+ Φ+ | + e−Jβ |Φ− Φ− |
Z
B/T
e
0
0
0
1
1
0
cosh
−
sinh
0
1
T
T
(2.24)
=
Z 0
1
1
− sinh T cosh T
0
−B/T
0
0
0
e
ρ =
where Z = 2(cosh J/T + cosh B/T ). From this expression and the definition of concurrence, it is straightforward to show that concurrence is
given as
C = max 0,
sinh J/T − 1
cosh J/T + cosh B/T
(2.25)
There exists a critical temperature at T = 1.13J, above which entanglement vanishes. It is worth noting that entanglement is present for
both AFM(J > 0) and FM(J < 0) interaction. This is different from the
case of XXX model where entanglement is absent if the interaction is FM.
Now, it is a very important observation that the ground state undergoes
a so-called quantum phase transition [4]. A quantum phase transition
is a purely quantum phenomenon in which a qualitative change of state
occurs at T = 0 by changing an external parameter(in this case, external
magnetic field B), while in the classical regime, phase transitions are usually triggered by thermal fluctuation. In this case, a qualitative change
of state occurs at B = J. At T = 0, C = 0(separable) when B > J, and
19
C = 1(maximally entangled) when B < J.
Now let us study the effect of anisotropy in the Hamiltonian on concurrence [40]. Hamiltonian is now
H=
J
[(1 + γ)σ1x σ2x + (1 − γ)σ1y σ2y ]
2
(2.26)
where γ denotes the degree of anisotropy. Note that γ = 0 corresponds
to the XX model and γ = 1 corresponds to the Ising model. Now in
the same manner shown above, that is, by constructing the density matrix from eigenvalues and eigenvectors, it is straightforward to obtain the
concurrence. Concurrence is given as
C = max 0,
sinh J/T − cosh Jγ/T
cosh J/T + cosh Jγ/T
(2.27)
• N =3
3-qubit Heisenberg XY model has been studied in Ref.[42]. The eigenvalues(Ei , i =
0, · · · , 7) and corresponding eigenvectors(|φi , i = 0, · · · , 7) are
E0 = E7 = 0
E1 = E2 = E3 = E4 = E5 = −J
E3 = E6 = 2J
20
(2.28)
|φ0
|φ1
|φ2
|φ3
|φ4
|φ5
|φ6
|φ7
1
= √ (|000
3
1
= √ (q|001 + q 2 |010 + |100
3
1 2
= √ (q |001 + q|010 + |100
3
1
= √ (|010 + |010 + |100 )
3
1
= √ (q|110 + q 2 |101 + |011
3
1 2
= √ (q |110 + q|101 + |011
3
1
= √ (|110 + |101 + |011 )
3
1
= √ |111
3
)
)
)
)
(2.29)
where q = exp (2πi/3). Therefore, the density matrix is by the definition,
ρ(T ) =
1
[|φ0 φ0 | + |φ7 φ7 |
Z
+eJ/T (|φ1 φ1 | + |φ4 φ4 | + |φ2 φ2 | + |φ5 φ5 |)
+e−2J/T (|φ3 φ3 | + |φ6 φ6 |)]
(2.30)
By tracing out one of the three qubits(independent of which qubit to
trace out because of the periodic boundary condition) and computing the
eigenvalues following the definition of concurrence, it is straightforward
to obtain the expression for concurrence as
C = max 0,
2|e−2J/T − eJ/T | − 3 − 2eJ/T − e−2J/T
3(1 + 2eJ/T + e−2J/T )
(2.31)
Notice that concurrence only depends on the ratio of J and T . From
21
Eq.(2.31), it directly follows that for the AFM case, there exists no entanglement and for the FM case, the concurrence reduces to
1 − 4e3J/T − 3e2J/T
C = max 0,
3(1 + 2e3J/T + e2J/T )
(2.32)
Again the temperature dependence of the nearest-neighbor concurrence is
plotted in Fig.2.2 up to N = 5. We observe a similar decrease in the threshold
temperature as the number sites increases in this case as well. Note that we
take J = −1 for N = 3 to give non-zero concurrence.
1
Concurrence
0.8
0.6
0.4
0.2
0
0
1
2
3
Temperature
4
5
Figure 2.2: Temperature dependence of nearest-neighbor concurrence in the
periodic Heisenberg XX model for N = 2 to N = 5. At T = 0, from top-tobottom, N = 2, 4, 3, 5.
22
Chapter 3
Entanglement in
Next-Nearest-Neighbor Models
The study of cooperative behavior in magnets has also led to a prevalent interest
in competing (frustrated) systems. Frustrated systems possess a high degree of
degeneracy at low energy states. Indeed, in the context of magnetic systems,
geometrical frustration often leads to new states such as spin glass and spin
liquid. In fact, a simple and natural way to incorporate frustration into a spin
chain is to consider next-nearest-neighbor interaction of the form
N
H=
J1 Hi,i+1 + J2 Hi,i+2
(3.1)
i=1
where H could be a Heisenberg XXX model or an XY model.
Recently, entanglement properties in these next-nearest-neighbor models
have been studied in the ground state as well as in the thermal state. In
Ref.[59], the authors have studied the properties of entanglement in antiferro-
23
magnetic Heisenberg XXX ring with next-nearest-neighbor interaction in the
ground state as well as the thermal state, through the direct relation of correlation functions with concurrence. Their study has shown that the presence
of next-nearest-neighbor interaction could induce entanglement between nextnearest-neighbor(NNN) sites, while suppressing entanglement between nearestneighbor(NN) sites. This work has been extended subsequently to the spin-1
periodic Heisenberg XXX model with next-nearest-neighbor interaction [60] and
amount of entanglement in the ground state and the thermal state was studied by employing the negativity measure. The study has shown that, as in the
case of spin-1/2 system, the presence of next-nearest-neighbor interaction could
enhance the entanglement between next-nearest-neighbor sites and suppresses
the entanglement between nearest-neighbor sites. In addition, analytical investigation into three-qubit next-nearest-neighbor model has been performed [61].
However, previous studies regarding the effect of next-nearest-neighbor interaction focused on Heisenberg XXX antiferromagnetic nearest-neighbor interaction with periodic boundary conditions(PBC). For finite chains, there is
a distinct qualitative difference between open and closed chains with periodic
boundary conditions. In addition, as we have seen in the previous chapter, it is
possible that there exists a qualitative difference between the type of nearestneighbor interaction, whether antiferromagnetic(AFM) or ferromagnetic(FM).
Moreover in the previous studies, the XX model with next-nearest-neighbor
interaction was never considered. It is therefore interesting to investigate both
open chains as well as next-nearest-neighbor XX models.
24
In this chapter, we perform a comprehensive numerical analysis on entanglement property in the Heisenberg XXX and XX model with next-nearestneighbor interaction employing concurrence measure. Specifically, numerical
simulation of ground state concurrence between nearest-neighbor sites and nextnearest-neighbor sites is carried out up to N = 11 including the case where open
boundary condition(OBC) is assumed and the nearest-neighbor interaction is
ferromagnetic.
In general, these models are very difficult to solve and the spectrum of
the Hamiltonian is unknown. However, for the Heisenberg XXX model with
periodic boundary condition and particular values of J1 and J2 , that is, when
2J1 = J2 , the model reduces to so-called Majumdar-Ghosh model
N
H=
i
1
(σi σi+1 + σi σi+2 )
2
(3.2)
and the ground state is known to be a superposition of state |φ1 and |φ2 .
|φ1
= [1, 2][3, 4] · · · [N − 1, N ]
|φ2
= [N, 1][2, 3] · · · [N − 2, N − 1]
(3.3)
where [i, j] denotes singlet
1
[i, j] = √ (|0 i |1 j − |1 i |0 j )
2
at ith and jth site.
25
(3.4)
Therefore, ground state concurrence of nearest-neighbor site for arbitrary
number of sites N at J = 1/2 can be computed as [59]
C=
(−1)N/2
2 + N/2−2
2
1
1
+ N/2
2 2
−1
(3.5)
In the following sections, we present detailed numerical results on the Heisenberg XXX model with next-nearest-neighbor interaction. Subsequently, analysis on the Heisenberg XX model with next-nearest-neighbor interaction will be
presented.
3.1
Heisenberg XXX Model with Next-NearestNeighbor Interaction
The one-dimensional Heisenberg magnet has often served as a prototype model
for studying ferromagnetic and antiferromagnetic properties in spin chain like
system. The generic Hamiltonian for the Heisenberg magnet with next-nearestneighbor interaction is given as
N
y
x
z
(σix σi+1
+ σiy σi+1
+ σiz σi+1
)
H = J1
i=1
N
y
x
z
(σix σi+2
+ σiy σi+2
+ σiz σi+2
)
+ J2
i=1
26
(3.6)
In the next section, we will discuss the behavior of the Heisenberg magnet with
next-nearest-neighbor interaction for the cases of periodic boundary condition
and open boundary condition.
3.1.1
Periodic Boundary Condition
Antiferromagnetic Nearest-Neighbor Interaction(J1 = 1)
By tracing out all sites except two nearest-neighbor qubits, the nearest-neighbor
concurrence is plotted in the upper graphs in Fig.3.1(even N ) and Fig.3.2(odd
N ). The nearest-neighbor concurrence achieves a maximum value at J2 = 0,
indicating that the next-nearest-neighbor coupling J2 suppresses(frustrates) the
amount of nearest-neighbor entanglement regardless of the sign of J2 . However,
for sufficiently large J2 (of the order of unity), the concurrence for nearestneighbor sites rapidly goes to zero.
We plot the next-nearest-neighbor concurrence, i.e. next-nearest-neighbor
entanglement, in the lower graphs in Fig.3.1(even N ) and Fig.3.2(odd N ). We
note that as long as the coupling J2 > 0 is sufficiently large, there is nonzero next-nearest-neighbor concurrence. Note that for N = 6, next-nearestneighbor entanglement is absent in all value of J2 . Such results have already
been observed in Ref.[59].
We also plot the N dependence of concurrence at specific value of J2 in
Fig.3.3. We have chosen the value of J2 in such way that there is no drastic
change in its vicinity. We denote C12 and C13 for nearest-neighbor concurrence
and next-nearest-neighbor concurrence respectively. It can be seen that the
value of concurrence oscillates with increasing N both in C12 and C13 however,
27
we observe that the amplitude of oscilation is smaller in C12 .
Plot of C
12
and C
13
Versus J for J = 1, N=4, 6, 8, 10
2
1
0.7
N=4
N=6
N=8
N=10
0.6
0.5
C
12
0.4
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J
0.5
1
1.5
2
0.5
1
1.5
2
2
1
N=4
N=6
N=8
N=10
0.8
C
13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J
2
Figure 3.1: NN and NNN concurrence for the XXX model(J1 = 1) with
PBC(even N )
28
Plot of C
12
and C
13
Versus J for J = 1, N=5, 7, 9, 11.
2
1
0.4
N=5
N=7
N=9
N=11
C12
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.3
C
13
0.25
0.2
0.15
N=5
N=7
N=9
N=11
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.2: NN and NNN concurrence for the XXX model(J1 = 1) with
PBC(odd N )
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
12
C13
0.9
0.8
0.7
C12 / C13
0.6
0.5
0.4
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.3: N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence in
the XXX model(J1 = 1) with PBC
29
Ferromagnetic Nearest-Neighbor Interaction(J1 = −1)
In this section, we discuss the case of ferromagnetic nearest-neighbor interaction. The results for nearest-neighbor concurrence are plotted in the upper
graphs in Fig.3.4(even N ) and Fig.3.5(odd N ). Note that nearest-neighbor
entanglement is absent for all value of J2 . In fact, the absence of nearestneighbor entanglement in the usual periodic Heisenberg XXX model has been
proved analytically [49].
We plot the next-nearest-neighbor concurrence in the lower graphs in Fig.3.4(even
N ) and Fig.3.5(odd N ). We see that next-nearest-neighbor interaction does not
contribute to the enhancement of entanglement between nearest-neighbor pairs.
In addition, we observe that although the qualitataive behavior of entanglement
is similar to the case of J1 = 1, the transition from a non-entangled state to an
entangled state takes place at lower value of J2 for J1 = −1. Also, note that
next-nearest-neighbor entanglement for N = 6 is absent for all value of J2 .
N dependence of concurrence is plotted in Fig.3.6, which shows the similar
oscillation in the next-nearest-neighbor concurrence to the case of the antiferromagnetic nearest-neighbor interaction(Fig.3.3)
30
Plot of C
12
and C
13
Versus J for J = −1, N=4, 6, 8, 10
2
1
1
N=4
N=6
N=8
N=10
C12
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
1
N=4
N=6
N=8
N=10
0.8
C13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.4: NN and NNN concurrence for the XXX model(J1 = −1) with
PBC(even N )
Plot of C
12
and C
13
Versus J for J = −1, N=5, 7, 9, 11.
2
1
1
N=5
N=7
N=9
N=11
C12
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.3
C
13
0.25
0.2
0.15
N=5
N=7
N=9
N=11
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.5: NN and NNN concurrence for the XXX model(J1 = −1) with
PBC(odd N )
31
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= −1
1
C
12
C13
0.9
0.8
0.7
C12 / C13
0.6
0.5
0.4
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.6: N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence in
the XXX model(J1 = −1) with PBC
32
3.1.2
Open Boundary Condition
For open boundary condition, the situation is slightly more complicating as the
concurrences (nearest-neighbor and next-nearest-neighbor) depend on the sites.
Antiferromagnetic Nearest-Neighbor Interaction(J1 = 1)
We denote Cij for the concurrence between the ith and jth site. The results
for C12 and C13 are plotted in Fig.3.7(even N ) and Fig.3.8(odd N ); for C23 and
C24 in Fig.3.9(even N ) and Fig.3.10(odd N ); for C34 and C35 in Fig.3.11(even
N ) and Fig.3.12(odd N ).
Unlike the case for periodic boundary condition, we find that entanglement between certain pairs of nearest-neighbor can be enhanced through the
next-nearest-neighbor interaction for open boundary condition. Like the case
for periodic boundary condition, next-nearest-neighbor entanglement can be
obtained for sufficiently large J2 > 0. Moreover, we observe some interesting peaks in the next-nearest-neighbor entanglement(C35 ) for odd number of
sites(N = 7, 9, 11), as shown in Fig.3.12.
Note that for even number of sites, all the neighboring pairs such as (1, 2), (3, 4), · · · , (N −
1, N ) becomes maximally entangled state at J2 = 1/2, providing a possibility of
quantum teleportation along an arbitrarily long chain by successive Bell state
measurement [63]. This result should be contrasted with the case of periodic
boundary condition(Majumdar-Ghosh model) in which the state is a superposition of two resonance valence bond states.
The variation of the nearest-neighbor and next-nearest-neighbor concurrence at specific value of J2 with N are plotted in Fig.3.13(C12 and C13 ),
33
Fig.3.14(C23 and C24 ), and Fig.3.15(C34 and C35 ). As can be seen these graphs,
it is probably intuitive but interesting to note that C12 tends to have higher
concurrence values compared to C23 and C34 . We could attribute this observation to a unique property of entanglement called monogamy. Since the sites in
the middle of the chain have more sites to interact compared to the sites at the
end, they tend to have stronger correlation with other parties.
Plot of C
12
and C
13
Versus J for J = 1, N=4, 6, 8, 10
2
1
1
N=4
N=6
N=8
N=10
0.8
C
12
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
1
1.5
2
2
1
N=4
N=6
N=8
N=10
0.8
C
13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
2
Figure 3.7: C12 and C13 for the XXX model(J1 = 1) with OBC(even N )
34
Plot of C
12
and C
Versus J for J = 1, N=5, 7, 9, 11.
13
2
1
1
N=5
N=7
N=9
N=11
0.8
C
12
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
1
1.5
2
2
1
N=5
N=7
N=9
N=11
0.8
C
13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
2
Figure 3.8: C12 and C13 in for the XXX model(J1 = 1) with OBC(odd N )
Plot of C
23
and C
24
Versus J for J = 1 , N=4, 6, 8, 10
2
1
2
0.35
N=4
N=6
N=8
N=10
0.3
C
23
0.25
0.2
0.15
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.8
C24
0.6
0.4
N=4
N=6
N=8
N=10
0.2
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.9: C23 and C24 for the XXX model(J1 = 1) with OBC(even N )
35
Plot of C
23
and C
Versus J for J = 1 , N=5, 7, 9, 11.
24
2
1
2
0.35
N=5
N=7
N=9
N=11
0.3
C
23
0.25
0.2
0.15
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
N=5
N=7
N=9
N=11
0.8
C24
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.10: C23 and C24 for the XXX model(J1 = 1) with OBC(odd N )
Plot of C
34
and C
35
Versus J for J = 1 , N=6, 8, 10
2
1
2
1
N=6
N=8
N=10
0.8
C
34
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J
0.5
1
1.5
2
0.5
1
1.5
2
2
N=6
N=8
N=10
0.5
0.4
C
35
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J
2
Figure 3.11: C34 and C35 for the XXX model(J1 = 1) with OBC(even N )
36
Plot of C
34
and C
35
Versus J for J = 1 , N=5, 7, 9, 11.
2
1
2
0.8
N=5
N=7
N=9
N=11
C
34
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J
0.6
0.5
1
1.5
2
0.5
1
1.5
2
2
N=5
N=7
N=9
N=11
0.5
C
35
0.4
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J
2
Figure 3.12: C34 and C35 for the XXX model(J1 = 1) with OBC(odd N )
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
12
C13
0.9
0.8
C12 / C13
0.7
0.6
0.5
0.4
0.3
4
5
6
7
8
9
10
11
N
Figure 3.13: N dependence of C12 (J2 = −1) and C13 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC
37
Plot of C23 and C24 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
23
C24
0.9
0.8
C23 / C24
0.7
0.6
0.5
0.4
0.3
0.2
4
5
6
7
8
9
10
11
N
Figure 3.14: N dependence of C23 (J2 = −1) and C24 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC
Plot of C34 and C35 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
0.7
C
34
C35
0.6
0.5
C34 / C35
0.4
0.3
0.2
0.1
0
5
6
7
8
N
9
10
11
Figure 3.15: N dependence of C34 (J2 = −1) and C35 (J2 = 1.5) in the XXX
model(J1 = 1) with OBC
38
Ferromagnetic Nearest-Neighbor Interaction(J1 = −1)
The results for C12 and C13 are plotted in Fig.3.16(even N ) and Fig.3.17(odd
N ); for C23 and C24 in Fig.3.18(even N ) and Fig.3.19(odd N ); for C34 and C35
in Fig.3.20(even N ) and Fig.3.21(odd N ).
We see that as in the case of periodic boundary condition, entanglement in
nearest-neighbor pairs is absent in the whole region of J2 if nearest-neighbor
interaction is ferromagnetic(J1 < 0). Also, the transition from a non-entangled
state to an entangled state occurs at lower value of J2 . We observe an interasting
shark peak in C35 for N = 7, however peaks are not observed for N = 9 and
N = 11 which are observed in the case of J1 = 1. In addition, absence of
entanglement in the whole region of J2 is observed in C35 for N = 8.
N dependence of concurrence at specific value of J2 are plotted in Fig.3.22(C12
and C13 ), Fig.3.23(C23 and C24 ), and Fig.3.24(C34 and C35 ), which also exhibit
the property of monogamy.
39
Plot of C
12
and C
13
Versus J for J = −1, N=4, 6, 8, 10
2
1
1
N=4
N=6
N=8
N=10
C12
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
1
N=4
N=6
N=8
N=10
0.8
C13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.16: C12 and C13 for the XXX model(J1 = −1) with OBC(even N )
Plot of C
12
and C
13
Versus J for J = −1, N=5, 7, 9, 11.
2
1
1
N=5
N=7
N=9
N=11
C12
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
1
N=5
N=7
N=9
N=11
0.8
C13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.17: C12 and C13 in for the XXX model(J1 = −1) with OBC(odd N )
40
Plot of C
23
and C
24
Versus J for J = −1, N=4, 6, 8, 10
2
1
1
N=4
N=6
N=8
N=10
C23
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
1
0.8
C24
0.6
0.4
N=4
N=6
N=8
N=10
0.2
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.18: C23 and C24 for the XXX model(J1 = −1) with OBC(even N )
Plot of C
23
and C
24
Versus J for J = −1, N=5, 7, 9, 11.
2
1
1
N=5
N=7
N=9
N=11
C23
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
1
N=5
N=7
N=9
N=11
0.8
C24
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.19: C23 and C24 for the XXX model(J1 = −1) with OBC(odd N )
41
Plot of C
34
and C
35
Versus J for J = −1, N=6, 8, 10
2
1
1
N=6
N=8
N=10
C34
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
0.6
N=6
N=8
N=10
0.5
C35
0.4
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J2
Figure 3.20: C34 and C35 for the XXX model(J1 = −1) with OBC(even N )
Plot of C
34
and C
35
Versus J for J = −1, N=5, 7, 9, 11.
2
1
1
N=5
N=7
N=9
N=11
C34
0.5
0
−0.5
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
0.8
N=5
N=7
N=9
N=11
C35
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.21: C34 and C35 for the XXX model(J1 = −1) with OBC(odd N )
42
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= −1
1
0.9
0.8
0.7
C12 / C13
0.6
0.5
0.4
C12
C13
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.22: N dependence of C12 (J2 = −1) and C13 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC
Plot of C23 and C24 Versus N for J2=−1 J2=1.5 respectively, and J1= −1
1
0.9
0.8
0.7
C23 / C24
0.6
0.5
C23
C24
0.4
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.23: N dependence of C23 (J2 = −1) and C24 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC
43
Plot of C34 and C35 Versus N for J2=−1 J2=1.5 respectively, and J1= −1
0.7
C
34
C35
0.6
0.5
C34 / C35
0.4
0.3
0.2
0.1
0
5
6
7
8
N
9
10
11
Figure 3.24: N dependence of C34 (J2 = −1) and C35 (J2 = 1.5) in the XXX
model(J1 = −1) with OBC
44
3.2
Heisenberg XX chain
The Hamiltonian for the Heisenberg XX model with next-nearest-neighbor interaction is given as
N
y
x
(σix σi+1
+ σiy σi+1
)
H = J1
i=1
N
y
x
(σix σi+2
+ σiy σi+2
)
+ J2
(3.7)
i=1
In this system, the nearest-neighbor interaction as well as the next-nearestneighbor interaction is missing in the z-direction. In this section, we perform
the same numerical analysis performed on the XXX system in the last section.
3.2.1
Periodic Boundary Condition
Antiferromagnetic Nearest-Neighbor Interaction(J1 = 1)
The results for nearest-neighbor concurrence are plotted in the upper graphs in
Fig.3.25(even N ) and Fig.3.26(odd N ); for next-nearest-neighbor in the lower
graphs in Fig.3.25(even N ) and Fig.3.26(odd N ). We find that unlike the XXX
system where next-nearest-neighbor entanglement can be in general induced
only by antiferromagnetic next-nearest-neighbor interation(J2 > 0), whereas
in the case of XX model, next-nearest-neighbor entanglement can be induced
by both antiferromagnetic(J2 > 0) and ferromagnetic(J2 < 0) next-nearestneighbor interaction. Note however that next-nearest-neighbor entanglement
can be induced only when J2 < 0 for N = 6. N dependence of concurrence at
specific value of J2 are plotted in Fig.3.27, which shows a similar oscillation to
45
the XXX system.
Plot of C
12
and C
13
Versus J for J = 1, N=4, 6, 8, 10
2
1
0.5
N=4
N=6
N=8
N=10
0.4
C
12
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J
0.5
1
1.5
2
0.5
1
1.5
2
2
1
N=4
N=6
N=8
N=10
0.8
C
13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J
2
Figure 3.25: NN and NNN concurrence for the XX model(J1 = 1) with
PBC(even N )
46
Plot of C
12
and C
13
Versus J for J = 1, N=5, 7, 9, 11.
2
1
0.35
N=5
N=7
N=9
N=11
0.3
C
12
0.25
0.2
0.15
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.3
0.25
C
13
0.2
0.15
N=5
N=7
N=9
N=11
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.26: NN and NNN concurrence for the XX model(J1 = 1) with
PBC(odd N )
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
12
C13
0.9
0.8
0.7
C12 / C13
0.6
0.5
0.4
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.27: N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XX model(J1 = 1) with PBC
47
Ferromagnetic Nearest-Neighbor Interaction(J1 = −1)
The results for nearest-neighbor and next-nearest-neighbor concurrence are
plotted in Fig.3.28(odd N ). We find that for even number of sites, the results are identical with the case of J1 = 1, which is plotted in Fig.3.25. We plot
the N dependence of concurrence in Fig.3.29.
Plot of C
12
and C
13
Versus J for J = −1, N=5, 7, 9, 11.
2
1
0.4
N=5
N=7
N=9
N=11
C12
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.3
0.25
C
13
0.2
0.15
N=5
N=7
N=9
N=11
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.28: NN and NNN concurrence for the XX model(J1 = −1) with
PBC(odd N )
3.2.2
Open Boundary Condition
The results for C12 and C13 are plotted in Fig.3.30(even N ) and Fig.3.31(odd
N ); for C23 and C24 in Fig.3.32(even N ) and Fig.3.33(odd N ); for C34 and
C35 in Fig.3.34(even N ) and Fig.3.35(odd N ). Note that in the case of XX
model with open boundary condition, the results do not make any difference
48
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= −1
1
C
12
C13
0.9
0.8
0.7
C12 / C13
0.6
0.5
0.4
0.3
0.2
0.1
0
4
5
6
7
8
9
10
11
N
Figure 3.29: N dependence of NN(J2 = −1) and NNN(J2 = 1.5) concurrence
in the XX model(J1 = −1) with PBC
regardless of the sign of the nearest-neighbor interaction. As observed in the
case of periodic boundary condition, it can be seen that next-nearest-neighbor
concurrence could be in general induced by both antiferromagnetic(J2 > 0)
and ferromagnetic(J2 < 0) next-nearest-neighbor interaction. Also, notice that
maximally entangled state of neighboring pairs can also be attained at J2 = 1/2
in the XX model as well. Note that we also observe some interasting peaks in
C35 for odd number of sites which are similar to the XXX case. We plot again
the N dependence of concurrence at specific value of J2 in Fig.3.36(C12 and
C13 ), Fig.3.37(C23 and C24 ), and Fig.3.38(C34 and C35 ) and observe the similar
oscillation although the value of nearest-neighbor concurrence is very stable in
this case. Also, comparing the graphs of N dependence of concurrence in the
XXX model and the XX model(for instance, compare Fig.3.13 and Fig.3.36),
49
it can be seen that nearest-neighbor concurrence in general takes higher value
in the XXX model than the XX model at same value of J2 .
Plot of C
12
and C
13
Versus J for J = 1, N=4, 6, 8, 10
2
1
1
N=4
N=6
N=8
N=10
0.8
C
12
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
1
1.5
2
2
1
N=4
N=6
N=8
N=10
0.8
C
13
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
2
Figure 3.30: C12 and C13 for the XX model(J1 = 1) with OBC(even N )
50
Plot of C
12
and C
Versus J for J = 1, N=5, 7, 9, 11.
13
2
1
1
N=5
N=7
N=9
N=11
0.8
C
12
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
2
0.7
0.6
C
13
0.5
0.4
0.3
N=5
N=7
N=9
N=11
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
2
Figure 3.31: C12 and C13 for the XX model(J1 = 1) with OBC(odd N )
Plot of C
23
and C
24
Versus J for J = 1, N=4, 6, 8, 10
2
1
0.25
N=4
N=6
N=8
N=10
0.2
C
23
0.15
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J
2
0.8
C24
0.6
0.4
N=4
N=6
N=8
N=10
0.2
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
J2
Figure 3.32: C23 and C24 for the XX model(J1 = 1) with OBC(even N )
51
Plot of C
23
and C
24
Versus J for J = 1, N=5, 7, 9, 11.
2
1
0.35
N=5
N=7
N=9
N=11
0.3
C
23
0.25
0.2
0.15
0.1
0.05
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
0.5
1
1.5
2
J
2
N=5
N=7
N=9
N=11
0.8
C24
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J2
Figure 3.33: C23 and C24 for the XX model(J1 = 1) with OBC(odd N )
Plot of C
34
and C
35
Versus J for J = 1, N=6, 8, 10
2
1
1
N=6
N=8
N=10
0.8
C
34
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
J
1
1.5
2
1
1.5
2
2
0.5
N=6
N=8
N=10
0.4
C
35
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
J
2
Figure 3.34: C34 and C35 for the XX model(J1 = 1) with OBC(even N )
52
Plot of C
34
and C
35
Versus J for J = 1, N=5, 7, 9, 11.
2
1
0.8
N=5
N=7
N=9
N=11
C
34
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
J
0.5
1
1.5
2
0.5
1
1.5
2
2
0.7
N=5
N=7
N=9
N=11
0.6
0.5
C
35
0.4
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
J
2
Figure 3.35: C34 and C35 for the XX model(J1 = 1) with OBC(odd N )
Plot of C12 and C13 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
12
C13
0.9
0.8
C12 / C13
0.7
0.6
0.5
0.4
0.3
0.2
4
5
6
7
8
9
10
11
N
Figure 3.36: N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC
53
Plot of C23 and C24 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
1
C
23
C24
0.9
0.8
0.7
C23 / C24
0.6
0.5
0.4
0.3
0.2
0.1
4
5
6
7
8
9
10
11
N
Figure 3.37: N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC
Plot of C34 and C35 Versus N for J2=−1 J2=1.5 respectively, and J1= 1
0.7
C
34
C35
0.6
0.5
C34 / C35
0.4
0.3
0.2
0.1
0
5
6
7
8
N
9
10
11
Figure 3.38: N dependence of C12 (J2 = −1) and C23 (J2 = 1.5) in the XX
model(J1 = 1) with OBC
54
3.3
Summary and Conclusion
We have carried out a comprehensive numerical study on the ground state concurrence of both the Heisenberg XXX model and the Heisenberg XX model with
periodic and open boundary conditions. We have found that although nextnearest-neighbor interactions do not enhance entanglement in nearest-neighbor
pairs in the Heisenberg XXX model with periodic boundary condition, for the
Heisenberg XXX model with open boundary condition, nearest-neighbor entanglement could be enhanced by next-nearest-neighbor interaction. This feature
is also present in the Heisenberg XX model both in periodic and open boundary
condition. Also in the Heisenberg XXX model, next-nearest-neighbor entanglement can be induced only by antiferromagnetic(J2 > 0) next-nearest-neighbor
interaction regardless of periodic or open boundary condition. In addition, we
find that in the XXX system, although the sign of nearest-neighbor interaction
has little effect on the behavior of concurrence, a transition from non-entangled
state to entangled state occurs at a lower value of J2 if the nearest-neighbor
interaction is ferromagnetic(J1 < 0).
In the Heisenberg XX model, we have observed that next-nearest-neighbor
entanglement could be enhanced by both antiferromagnetic (J2 > 0) and ferromagnetic (J2 < 0) next-nearest-neighbor interaction in both periodic and
open boundary condition. This is significantly different from the Heisenberg
XXX model where next-nearest-neighbor entanglement can be induced only by
antiferromagnetic (J2 > 0) next-nearest-neighbor interaction.
55
Chapter 4
Quantum State Transfer
Transmission of a quantum state is an important task in quantum information
processing. One way to transmit a quantum state is via quantum teleportation,
which crucially requires the preparation of an entangled pair and a classical
channel. However, a process called state transfer is a totally different process.
In the state transfer model, it is not necessary to prepare an entangled state
nor a classical channel and it is possible to transfer a quantum state along
a quantum spin chain; the quantum spin chain works as a quantum wire to
transmit a quantum state. The basic idea is that the sender, sitting at one end
of the spin chain first encodes an arbitrary unknown state into the first qubit
of the spin chain. By simply letting the spin chain evolve unitarily in time
under some Hamiltonian, the information encoded by the sender travels along
the chain and the receiver sitting at the other end of the spin chain is able to
recover the quantum state, with high fidelity, at some appropriate time. This
idea was first proposed in Ref.[18], in which the fidelity of this procedure in a
Heisenberg XX chain was calculated up to 80 spins.
56
In this chapter, first we will briefly see how the state transfer scheme works
including the state transfer in the presence of white noise. Subsequently, recent
proposals for the state transfer scheme with better fidelity will be reviewed.
The analysis made in this chapter on the state transfer dynamics will be the
basis for the quantum teleportation scheme along a spin chain, which will be
discussed in the next chapter.
4.1
State Transfer Scheme
Let us first follow Ref.[16] and see how the state transfer actually works in a
spin chain. Consider a spin chain of N sites interacting with Hamiltonian
H=
1
2
y
x
(σix σi+1
+ σiy σi+1
)
(4.1)
i
This is the so-called isotropic XY model or simply Heisenberg XX model and
the strength of interaction is set to 1/2 for convenience. We prepare an arbitrary
unknown state |φ in the spin chain as
|φ = α|0
0
+ β|1
0
(4.2)
where α and β are some unknown parameters. We wish to transfer this quantum
state |φ to the end of the chain. The spin chain is initially prepared in such
a way that all the spins are oriented to the z-direction. Practically, this state
can be achieved by applying an external magnetic field in the z-direction. The
unknown state |φ is encoded into the first qubit. The total system including
57
the unknown state and the rest of the spin chain can be written as
(α|0
0
+ β|1 0 ) ⊗ |00 · · · 0
= α|000 · · · 0 + β|100 · · · 0
≡ α|0 + β|1
(4.3)
We introduce the notation |n , which indicates that the nth site of the spin
chain has been flipped up from an all spin-down state.
Now the total system is subjected to the unitary time evolution under the
Hamiltonian (4.1). Since the state |0 is the zero eigenstate of the Hamiltonian,
it does not evolve and remains to be |0 . On the other hand, |1 evolves into a
superposition of one spin-down state. Therefore, practically speaking, only the
state |1 is relevant to the time evolution. Hence, we define the fidelity of the
state transfer as
F (t) = | N | exp (−iHt)|1 |
(4.4)
This quantity can be computed analytically. To calculate the fidelity, it is
necessary to know the eigenvalues of the Hamiltonian (4.1) as well as its corresponding eigenvectors.
Here, it is worth mentioning that the total spin of z-direction
z
σtotal
≡
σiz
(4.5)
z
[σtotal
, H] = 0
(4.6)
i
commutes with the Hamiltonian, namely
58
holds, which means the total z-component of the spin before and after the time
evolution is conserved.
However, the system is confined to the subspace of one-magnon state, therefore, it is sufficient to know only one-magnon eigenstates. One-magnon eigenstates are known as
|k˜ =
2
N +1
N
sin
n=1
nπk
|n
N +1
(4.7)
and the corresponding eigenvalues are
Ek = −2 cos
kπ
, (k = 1, · · · , N )
N +1
(4.8)
It is clear from Eq.(4.7) that all the one-magnon eigenvalues are superpositions of single-excitation. Note that for periodic boundary system, one-magnon
eigenvectors and its corresponding eigenvalues are [40]
1
|k˜ = √
N
Ek = cos
exp
n
i2πnk
|n
N
(4.9)
2πk
, (k = 1, · · · , N )
N
Now we are ready to calculate fidelity F (t). Using the fact that
(4.10)
k
˜ =
|k˜ k|
1 forms a unit operator in the subspace of one-magnon state, the fidelity F (t)
59
can be computed as
N
F (t) =
k=1
=
=
˜
N | exp (−iHt)|k˜ k|1
2
N +1
2
N +1
N
sin
k=1
N
sin
k=1
πk
N +1
πk
N +1
N | exp (−iHt)|k˜
sin
πkN
N +1
exp (−iEk t)
(4.11)
In general, as can be seen from Eq.(4.11), fidelity is expressed as products of sin
and cos function and the shape of the fidelity is very complicated. To illustrate
how F (t) behaves in time, we have plotted the value of F (t) again time t, for
N = 3, 4 (Fig.4.1), and for N = 9, 10 (Fig.4.2). However, for small N , the
expression for the fidelty takes a relatively simple form, for instance
F (t) = |sin t| ; [N = 2]
√
2
2
F (t) = sin
; [N = 3]
t
(4.12)
(4.13)
It is easy to see that perfect state transfer is possible at time t = π/2 and
√
t = 2π/2 respectively. Actually, it can be proven that perfect state transfer
is possible only when N < 4. The proof can be found in Ref.[16].
60
1
Fidelity
0.8
0.6
0.4
0.2
0
0
5
10
15
t
20
25
30
Figure 4.1: Time dependence of the state transfer fidelity F (t); N = 3(solid
line) and N = 4(dotted line)
Fidelity
0.8
0.6
0.4
0.2
0
0
5
10
15
t
20
25
30
Figure 4.2: Time dependence of the state transfer fidelity F (t); N = 9(solid
line) and N = 10(dotted line)
61
4.2
State Transfer under White Noise
As discussed in the last section, the state transfer model works quite well,
especially for a small number of sites and its fidelity could be 1 for N = 2
and N = 3. However in practice, a system is often afflicted with noise or
decoherence. In this section, we study the effect of such noise on the fidelity of
state transfer. If the state is exposed to an unpolarized noise, the initial density
matrix ρ0 will be modified to
ρ0 = (1 − f )|1 1| +
f
1
2N
(4.14)
where f denotes the amount of noise which takes on the value 0 ≤ f ≤ 1 and
1 is the 2N × 2N identity matrix. With this initial density matrix, the fidelity
of the state tranfer F ′ (t) can be evaluated as
F ′ (t) =
=
=
(1 − f ) N | exp (−iHt)|1 1| exp (iHt)|N +
1
(1 − f )
N +2
(1 − f )F 2 (t) +
2
N
sin
k=1
f
2N
πk
N +1
f
2N
sin
πkN
N +1
exp (−iEk t) +
f
2N
(4.15)
In this model, the value of f essentially describes the probability that a qubit
will encounter an error due to a bit or phase flip or both. Obviously when
f = 0, Eq.(4.2) reduces to Eq.(4.11). To see the effect of noise clearly, we
have numerically optimized over t and determined the maximum fidelity with
3 different noise levels f = 0, f = 0.01 and f = 0.05, where the range of t is
taken to be 0 ≤ t ≤ N × 100. Results can be found in Fig.4.3. It can be seen
62
that fidelity slowly decreases as the number of sites increases, however, fidelity
remains to be high.
1
Fidelity
0.9
0.8
0.7
0.6
0
5
10
N
15
20
Figure 4.3: Fidelity of the state transfer with different amount of noise levels;
F = 0, F = 0.01, and F = 0.05
4.3
Recent Proposals for High Fidelity State
Transfer
As we have seen above, transfer fidelity decreases as the number of sites increases, and state transfer with perfect fidelity is possible only when N < 4,
even in the absence of noise. Hence, this scheme might be good for short
distance communication but not suitable for long distance communication. Recently, several interesting proposals have been made to achieve high fidelity
state transfer. The reason for decreasing fidelity is mainly due to the dispersion of the excitation along the chain. We can overcome this difficulity either by
63
suitably encoding the initial state with fewer dispersion, or by pre-engineering
the Hamiltonian to somehow re-concentrate the information.
In Ref.[16], it has been shown that transfer of a quantum state in an arbitrarily long chain with perfect fidelity is possible if one is allowed to pre-engineer the
coupling strength between the spin chain sites. Specifically, they have shown
that if the Hamiltonian of the spin chain takes the form of
H=
n
n(N − n) x x
y
[σn σn+1 + σny σn+1
]
2
(4.16)
then it is possible to perform a state transfer with perfect fidelity along an
aribtrarily long chain.
In Ref.[23], the initial state is coded and decoded on multiple-qubit using
gaussian packet, which has minimal dispersion. It has been demonstrated that
this coding gives us near optimal fidelity when the chain is a Heisenberg ring.
This has been shown to be equivalent to the transfer scheme where the sender
and the receiver have access to time-dependent controllable coupling [24]. In
addition, a study has been done on the state transfer scheme with a system
where the sending site and the receiving site are weakly coupled to the chain,
compared to the rest of the chain [25]. Moreover, in very a recent paper [26],
the authors have shown that state transfer with arbitrarily high fidelity can be
achieved only by applying the series of two-qubits gates at the end of the chain.
Particularly interesting one is the scheme in which state transfer is carried
out with parallel spin chains [21, 22]. The system consists of two uncoupled
quantum chains and the sender and the receiver have access to their own qubits.
Each chain is interacting with some Hamiltonian, for instance, the Heisenberg
64
XX Hamiltonian. It is assumed that the Hamiltonian commutes with the total
z-spin component. The receiver is able to check whether or not the transfer
is successful by measuring one of qubits in the receiver’s possesion and if not
successful, the receiver continues to wait until the transfer is successful. In this
model, unlike previous schemes, neither pre-engineering of Hamiltonian nor
suitable coding of state is necessary. It has been extended to the case where
the parallel chain has some asymmetries in the Hamiltonian or the Hamiltonian
has some imperfections.
Besides the state transfer in a Heisenberg XX spin chain, the state transfer
in other kinds of chains has been studied. For instance, M.Avellino et.al. have
studied the state transfer in a spin chain coupled via the magnetic dipole interaction [19]. Their study has shown that with the dipole magnetic interaction, it
is possible to transfer a state in a ring with better fidelity compared to the usual
Heisenberg nearest neighbor interaction, sacrificing the time to transfer. Also,
the state transfer with next-nearest-neighbor interaction has been investigated
[20]. Moreover in Ref.[27], it has been shown that perfect transfer is possible if
one is allowed to perform arbitrary measurements on each qubit in the chain.
65
Chapter 5
Engineering Quantum
Teleportation through State
Transfer
We have already seen that a spin chain can work as a quantum wire to transmit
a quantum state. What else can we do with a spin chain? This basic question
motivates us to consider a spin chain as a resource for quantum teleportation.
First attempts of this kind have been done in Ref.[28, 29], where quantum teleportation via a ring of 3-qubits in the thermal state were studied.
In this chapter, we show that it is possible to teleport an arbitrary unknown
state using a Heisenberg XX spin chain as a resource. For a two-site spin chain,
it is well known that the state of the system becomes a maximally entangled
state through unitary time evolution at some appropriate time, provided that
the spin of one of the sites is initially excited. For a large number of sites,
66
the bipartite state comprising the first and last site can often act as a good
entangled state for teleportation. Using this bipartite state as a resource, high
fidelity quantum teleportation continues to be possible for a large number of
sites. First of all, an explicit scheme for teleportation on a spin chain is presented. Based on this teleportation scheme, we perform a numerical analysis to
analyze the feasibility of the scheme including an analysis where there is substantial noise. Afterwards, extention to the next-nearest-neighbor model will
be studied.
5.1
Teleportation Scheme
Suppose Alice has easy access to the spin at the first site and Bob has easy
access to the spin at the N th site. We consider a scenario in which Alice wishes
to send an arbitrary unknown state |φ to Bob sitting at the N th site. |φ can
be written in general as
|φ = cos (ξ/2)|0 + sin (ξ/2)|1
(5.1)
We also assume that we have an open spin-1/2 chain with N sites interacting
via the Hamiltonian as a resource
1
H=
2
N
y
x
Jx σix σi+1
+ Jy σiy σi+1
(5.2)
i=1
where σx , σy are the Pauli matrices and Jx , Jy are coupling constants between
the neighboring sites. This is the well-known Heisenberg XY open boundary
spin chain model introduced in Chapter 4. For convenience, we assume Jx =
67
Jy = J and we set J = 1 in the following analysis. Remember that the onemagnon eigenvectors of this Hamiltonian (5.2) are
|k˜ =
2
N +1
N
sin
n=1
nπk
|n
N +1
(5.3)
with eigenvalues Ek = −2 cos ( Nkπ
) for k = 1...N . Here, the notation |n has
+1
been defined in Sec.4.1. Note that
N
k=1
˜ forms a unit operator in the
|k˜ k|
subspace of the one-magnon state.
To achieve near perfect teleportation fidelity, it is essential to create a maximally entangled pair at the two ends of the spin chain as a resource for teleportation. It is possible to have a highly entangled pair or even a maximally
entangled pair at the ends of the chain, if the spin chain is initially allowed to
have a pair of maximally entangled state at some properly chosen sites. i.e.
entanglement can eventually percolate through the spin chain. Several studies have been done regarding the transfer and the transport of entanglement
[34, 35, 36].
Let us suppose that Alice and Bob do not in general share an initial entangled state. Using quantum state transfer, we hope to create a highly entangled
pair through unitary time evolution of the entire chain. In Ref.[35, 36], entanglement between nearest-neighbor or next-nearest-neighbor sites is generated in
a Heisenberg XX open spin chain with both infinite and finite number of sites,
starting from an initially unentangled state by unitary time evolution and the
amount of entanglement is studied by employing concurrence. Our approach to
generate entanglement is similar to Ref.[35, 36], except that we are interested
only on the generation of an entangled pair at the two ends of the spin chain
68
rather than the generation of entanglement between nearest or next-nearest
neighboring sites. Intuitively, it is more difficult to have a strong correlation if
the distance of the pair in the chain becomes longer.
The ability to create an entangled resource between the first and last site
depends crucially on the initial state. As we have mentioned earlier, even if the
initial state is unentangled, it is still possible to create an entangled pair at the
ends of the chain through quantum state evolution. However, this fidelity can
be enhanced if it is possible to prepare a nearest-neighbor maximally entangled
pair somewhere along the chain. Such entanglement could be prepared for
instance by a third party with access to some of the sites along the chain. In
this article, we consider the preparation of high fidelity quantum teleportation
by creating a highly entangled pair at both ends of the chain under two different
scenarios:
• Scenario A: The initial spin chain is a maximally entangled pair at some
suitably chosen nearest neighbor sites;
• Scenario B: The initial spin chain is in a pure separable state with single
excitation.
It is instructive to explore the dynamics of a simple spin chain. Suppose the
spin chain is initially prepared with a one-magnon state, i.e. only one excitation
is allowed in the initial state, for instance, the spin chain could initially be in
the state |l . When the state |l is subjected to unitary time evolution under
the Hamiltonian (5.2), we get using Eq.(5.3) [40]
N
−iHt
e
|l =
bln (t)|n
n=1
69
(5.4)
where the coefficients are given by
N
bln (t)
=
sin
k=1
πkl
N +1
nπk
N +1
sin
exp(−iEk t).
(5.5)
Note that the number of spin-up sites in the chain is conserved during the time
evolution.
Since the bipartite state between the first and the N th site can serve as a
quantum resource, it is instructive to consider the explicit expression for the
entanglement of this bipartite state by tracing out all the sites except the two
ends of the spin chain. In scenario A, denoting the time-dependent reduced
density matrix of the bipartite state as ̺(t), we have
N
̺(t) ≡ Tr2,...,N −1
0
=
0
0
N −1
n=2
N
bln (t)bl∗
m (t)|n m|
n=1 m=1
|bln |2
0
0
|blN |2 bl1 bl∗
N
l
bl∗
1 bN
|bl1 |2
0
0
0
0
0
0
(5.6)
Now we consider the case when we initially have a maximally entangled
√
pair |Φ− = (|01 − |10 )/ 2 at, for instance, lth and kth site in the chain
(where |l − k| ≤ 2). For convenience, we denote the four Bell states as |Ψ± ≡
√
√
(|00 ± |11 / 2, |Φ± ≡ (|01 ± |10 )/ 2. It turns out that |Φ− is the best
state for generating maximal entanglement between the first and last site. Time
70
evolution of the initial state becomes
N
−iHt
e
(|l − |k ) =
n=1
(bln (t) − bkn )|n
(5.7)
Therefore in the same manner, the reduced density matrix for bipartite state
comprising 1st and N th site denoted as ̺˜(t) can be obtained as
N
̺˜(t) ≡ Tr2,...,N −1
1
0
=
2 0
0
N
l,k∗
bl,k
n (t)bm (t)|n m|
n=1 m=1
N −1
n=2
2
|bl,k
n |
0
0
2
|bl,k
N |
l,k∗
bl,k
1 bN
l,k
bl,k∗
1 bN
2
|bl,k
1 |
0
0
0
0
0
0
(5.8)
where the coefficients are
k
l
bl,k
i (t) = bi (t) − bi (t)
5.2
(5.9)
Numerical Analysis
The state given in Eq.(5.19) and Eq.(5.8) acts as a primary resource of the
teleportation scheme. The state ̺(t) in general is not maximally entangled for
arbitrary time. In order to achieve quantum teleportation with high fidelity,
we need to find the optimal time topt at which the fidelity of the state ̺(t) with
the Bell state is maximized.
71
5.2.1
Entangled Pair in the Initial Chain
We find topt numerically so that Tr[˜
̺(topt )|Φ− Φ− |] is a maximum. For numerical simulation, we search for this time topt within the region 0 ≤ t ≤ N × 100,
in which an upper limit for the time used in the optimization is suitably chosen
so that it is a sufficiently large number proportional to the number of sites. The
need for the optimization time to be proportional to the number of sites is since
as the chain becomes longer, it is natural for the system to take longer time
to establish strong correlations between the first and last site. Moreover, the
choice of the initial state considerably affects the maximum possible amount of
entanglement at both ends. If we need to prepare two sites that are entangled,
we envisage a situation in which we are able to prepare an entangled pair only
in the nearest neighbor sites or at most next-nearest neighbor sites. We also
assume that we have full access to the parts of the chain at anytime so that
we can allocate an entangled pair at optimal sites. Among all possibilities of
allocating an entangled pair at site (i, j), we have chosen (N/2, N/2 + 1) for
even N and ( N2+1 − 1, N2+1 + 1) for odd N . The choice is intuitively reasonable
since the information at the chosen sites transport symmetrically under the
time evolution and eventually reach the sites at the both ends of the chain.
5.2.2
Separable Initial Chain
In the same manner as scenario A, we search for the time topt numerically in the
region 0 ≤ t ≤ N ×100. For small N , topt can be found analytically, for instance
√
topt = π/4(N = 2) for N = 2 and topt = 2π/4(N = 3) for N = 3. As in the
previous case, the maximum possible amount of entanglement at both ends is
72
affected significantly by the choice of the initial one-magnon state. Among all
possible one-magnon states, we have chosen the initial state to be |1 for even
N and |(N + 1)/2 for odd N , which is confirmed by numerics to be an optimal
choice.
5.2.3
Local Phase Acquisition
Even if the state between the first and last site, which serves as the entangled
resource for the teleportation protocol, is maximally entangled at some optimal
time, there is in general an arbitrary phase rotation in the final state. As no
state tomography is made to the state at the optimal time, it is important to
estimate and remove the unwanted phase. For instance, for N = 2, state of the
1st and the 2nd site at optimal time becomes
0 0 0
1
0 1 i
̺(topt ) = √
2
0 −i 1
0 0 0
0
0
0
0
(5.10)
Even in this simple case of N = 2, a rotation of α = 3π/2 appears in
the maximally entangled state. Without reversing the phase on the resource,
teleportation fidelity drops down to 2/3, while a properly rotated resource yields
perfect fidelity. For N = 3, α turns out to be zero in general. In order to cancel
the unwanted phase, one has to apply local phase rotation operator U (α) on
the 1st and the N th site of the spin chain, where the operator U (α) is defined
as
73
0
1
U (α) =
0 eiα/2
(5.11)
By operating U (α) on ̺(t), one gets
̺(t, α) = U (α) ⊗ U (α)† ̺(t)U (α)† ⊗ U (α)
0
=
0
0
N −1
n=2
|bln |2
0
0
|blN |2
e−iα bl1 bl∗
N
l
eiα bl∗
1 bN
|bl1 |2
0
0
0
0
,
0
0
(5.12)
which can cancel the phase rotation α. Hence, it is necessary not only to find
an optimal time but also to optimize the phase angle over all possible angle
α. With an estimation of the optimal time and the optimal phase rotation,
̺(topt , αopt ) turns out to be an excellent resource of quantum teleportation via
spin chain.
In the teleportation stage, Alice attaches her qubit |φ to one of the end(0th
site) of the spin chain. Subsequently, she then performs a Bell-state measurement i.e. projecting into one of the Bell states |Ψ+ Ψ+ |, |Ψ− Ψ− |, |Φ+ Φ+ |, |Φ− Φ− |
on her qubits, i.e. Alice’s unknown qubit and the first site of the spin chain. As
in the usual teleportation protocol described in Chapter 1, Bob finally performs
an appropriate unitary transformation on the last qubit of the spin chain, de74
pending upon the measurement result performed by Alice and communicated
to him through classical channels.
5.3
Results
We have evaluated the teleportation fidelity by finding topt and αopt for both
scenarios numerically. Fig.5.1 shows the fidelity of the teleportation with optimal time and appropriate phase rotation for various number of sites from N = 2
up to N = 20 for scenario A(solid line) and scenario B(dotted line). Note that
the fidelity is averaged over all possible parameter ξ of the unknown state |φ
as it generally depends on the parameter ξ.
In scenario A, teleportation continues to be almost perfect for a large number of sites, up to N = 19 for odd number of sites in our simulation. However,
teleportation fidelity is significantly lower when N is even. We observe that
even though we start from a separable initial chain, the perfect quantum teleportation can be achieved for N = 2 and N = 3. This is not a difficult task
to confirm analytically. In addition, the teleportation is also almost perfect for
N = 4(0.999) and N = 6(0.998). In both scenarios, the teleportation fidelity
slowly decreases as the number of sites increases. Intuitively we know that it is
difficult to have strong correlation or entanglement when the distance between
two sites becomes longer. However, the final fidelity appears to be higher than
the fidelity of 2/3 associated with the best classical communication protocol
[30] even for N = 20.
75
1
Fidelity
0.8
0.6
0.4
0.2
0
5
10
N
15
20
Figure 5.1: Teleportation fidelity via a Heisenberg XX spin chain of various
number of sites from N = 2 to N = 20 in scenario A(solid line) and scenario
B(dotted line) with optimal time and optimal phase angle.
76
5.4
Teleportation under White Noise
We see from the above analysis that the teleportation fidelty via a spin chain
remains high for a sufficiently large number of sites. However in practice,
decoherence effects become significant when the number of coupled sites in
a chain grows. Therefore it is interesting to study the extent of a noisy channel
on the teleportation resource. We apply the same white noise model discussed
in Sec.4.2, which changes the original bipartite state ̺(t) to
̺′ (t) = (1 − F )̺(t) +
F
1
4
(5.13)
where F takes on the value 0 ≤ F ≤ 1 and 1 is the 4 × 4 identity matrix. We
have evaluated the teleportation fidelity in the presence of different amount of
noise, F = 0.001, 0.01 and 0.05. If the noise is tolerable, we see that it is still
possible to teleport through this scheme by finding the optimal time and phase
rotation angle in the same way. Results are plotted for scenario A and scenario
B in Fig.5.2 and Fig.5.3 respectively.
77
1
Fidelity
0.9
0.8
0.7
0.6
0
5
10
N
15
20
Figure 5.2: Teleportation fidelity with 4 different noise levels (from the top,
F = 0, 0.001, 0.01, 0.05) up to N = 20 in scenario A.
1
Fidelity
0.9
0.8
0.7
0.6
0
5
10
N
15
20
Figure 5.3: Teleportation fidelity with 4 different noise levels (from the top,
F = 0, 0.001, 0.01, 0.05) up to N = 20 in scenario B.
78
5.5
Extension to Next-Nearest-Neighbor Model
In this section, we study the effect of next-nearest-neighbor interaction in the
Heisenberg XX model on the generation of entanglement. We consider a system
in which the sites in the chain are mutually interacting with the following
Hamiltonian,
N
N
x
(σix σi+1
HJ =
+
y
σiy σi+1
)
y
x
(σix σi+2
+ σiy σi+2
)
+J
i
(5.14)
i
where J denotes the coupling strength between next-nearest-neighbor sites.
Clearly, when J = 0, the Hamiltonian (5.14) reduces to the usual Heisenberg
XX model. In the following, a system with both open and periodic boundary
condition is studied.
5.5.1
Scheme
Periodic Boundary Condition
Our aim is to generate an entanglement pair between diametrically distant sites
in a ring of N sites(N :even) by unitary time evolution, starting from a state
which is initially in state |1 . Now the initial state |1 is subjected to the unitary
time evolution under the Hamiltonian (5.14). Denoting the time evolved state
as |Ψ(t) ,
|Ψ(t)
= e−iHJ t |1
bn (t)|n
=
n
79
(5.15)
where the coefficients are known to be
1
bn (t) =
N
N
exp
i=1
2k(n − 1)πi
N
× exp −it cos
2kπ
4kπ
+ J cos
N
N
(5.16)
z
Note that the system is confined to one-magnon states since [HJ , σtotal
] = 0
z
holds and the total number of spin-up component is conserved, where σtotal
≡
i
σiz . Given the coefficients bi (t), concurrence between ith and jth site can
be expressed in a rather simple form as
Cij = 2|bi (t)bj (t)|
(5.17)
In our case, i = 1, j = (N/2 + 1).
Open Boundary Condition
In the case of open boundary condition, our aim is to generate a highly entangled pair at both ends of the open chain. However, the analytical expression for
the coefficients of the time evolution bn (t) are not known for the open boundary
case and therefore one generally has to work with the full Hamiltonian explicitly. From the computational point of view, this is indeed hard to carry out
since the dimension of the Hilbert space blows up exponentially as 2N × 2N .
However since the space is confined to the one-magnon subspace, we only need
to consider the one-magnon subspace of the Hamiltonian. In the subspace of
80
one-magnon states, the Hamiltonian can be reduced to a N × N matrix as
˜
HJ =
0
1 J
0
···
···
0
1
0 1
J
···
···
0
J 1 0
1
···
···
0
0 J 1
0
···
...
···
0
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
0
1
J
1
0
1
0
0 0
0
J
1
0
(5.18)
With this Hamiltonian in the subspace, the effort of computation is significantly
reduced.
5.5.2
Results
Periodic Boundary Condition
We have numerically evaluated the concurrence between the 1st site and (N/2+
1)th site, i.e. C1,N/2+1 . To find the maximum achievable concurrence, we have
optimized over the next-nearest-neighbor coupling strength J(0 ≤ J ≤ 2) as
well as the time t(0 ≤ t ≤ 20×N ). Note that our search is restricted to positive
values of J since concurrence (5.17) is invariant under the change of sign for
even N if the boundary condition is periodic. The results are shown in Fig.5.4.
We observe that the presence of next-nearest-neighbor interaction J clearly
enhances the maximum possible entanglement in the diametrically opposite
pair. Particularly, the degree of enhancement is prominent when the number
81
of sites is a multiple of 4, otherwise the presence of next-nearest-neighbor interaction has little effect on the entanglement generation. As N increases, the
concurrence slowly decreases, which is quite intuitive since the longer the chain
becomes, the more difficult it is to have strong correlation.
1
Concurrence
0.8
0.6
0.4
0.2
6
8
10
12
N
14
16
18
20
Figure 5.4: Maximum concurrence between the 1st and (N/2 + 1)th site,
C1,N/2+1 up to N = 20 with HJ (thick line) and H0
We note that this can be used as a resource to teleport an unknown quantum
state. It is easy to see that the bipartite density matrix comprising the 1st and
(N/2 + 1)th site, defined as B(t) ≡ Tr [|Ψ(t) Ψ(t)|] gives
0
B(t) =
0
0
N
n=2,n= N
+1
2
2
|bn |
0
0
|bN/2+1 |2 b1 b∗N/2+1
b∗1 bN/2+1
|b1 |2
0
0
0
0
0
0
(5.19)
where the partial trace has been taken for all the sites except the 1st and (N/2+
82
1)th site. By finding the optimal time topt at which the state Tr [|Φ± Φ± |B(topt )]
yields maximum, the state B(topt ) works as a good resource for quantum teleportation from the 1st site to (N/2+1)th site.
Open Boundary Condition
In a same manner, we have numerically evaluated the concurrence at the both
ends of the open chain, now for both even and odd N . Time t and the nextnearest-neighbor coupling strength J are optimized in the same way to give
the maximum concurrence. The results are presented in Fig.5.5. Note that in
the open boundary case, concurrence is invariant under the change of the sign
of J for both even and odd N . It can be seen that as in the periodic case,
the concurrence is significantly enhanced by the presence of the next-nearestneighbor interaction. In fact, we have seen in the previous section that it is
possible to generate a highly entangled pair without the next-nearest-neighbor
interaction if one has a full access to any site in the chain. In this case, we
can choose the state |1 and |N/2 + 1 for even and odd N as the initial state
of the spin chain. In Fig.5.5, we have also plotted the maximum concurrence
C1,N (optimized over the time t, 0 ≤ t ≤ 20 × N ) with this choice of the initial
state under the usual Heisenberg XX Hamiltonian H0 (dotted line).
5.6
Summary and Discussion
In conclusion, we have shown numerically that teleportation with fidelity better
than any classical protocol is possible using a Heisenberg XX spin chain with up
to 20 spins. We have also evaluated the teleportation fidelity in the presence of
83
1
Concurrence
0.8
0.6
0.4
0.2
4
6
8
10
12
N
Figure 5.5: Maximum concurrence between the 1st and N th site(C1,N ) up to
˜ J (thick line) and H0 together with the concurrence C1,N with
N = 12 with H
optimally chosen initial state under the usual Heisenberg XX Hamiltonian H0
(dotted line)
noise. However, in this study, our consideration is limited to a one-magnon state
as the initial state of the spin chain. Teleportation with better fidelity than the
above result might be possible if one optimizes over all possible configurations
of the initial spin chain. In fact, numerical analysis shows that it is possible
to obtain a state with higher entanglement if one starts with an initial state
with more than two excitations. In addition, we have studied the effect of nextnearest-neighbor interaction in the Heisenberg XX model on the generation of
entanglement and found that the presence of next-nearest-neighbor interaction
enhances the possible amount of entanglement in both open and periodic chains.
84
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90
Appendix A
Exactly Solvable Models
In this appendix, I will briefly review some exactly solvable models; the Heisenberg XXX model and Heisenberg XY model.
A.1
Heisenberg XXXs=1/2 -Algebraic Bethe Ansatz
The analytical solution was first given by Bethe in 1931 by means of Bethe
ansatz for antiferromagnetic Heisenberg model. Later on, an algebraic approach to this analytical solution was developed, called the Algebraic Bethe
Ansatz(ABA). I will briefly overview ABA by closely following Faddeev’s lecture note [31, 32]. Also, a very elementary but clear introduction to spin chain
models can be found in Ref.[33].
The Hamiltonian of the Heisenberg XXX model reads
H=
α,n
sαn sαn+1 −
91
1
4
(A.1)
where α takes x, y, z and the constant 1/4 is taken for later convenience. We
define an operator called the ’Lax operator’ Ln,a as follows.
Ln,a (λ) = λIn ⊗ Ia +
a
sαn ⊗ σ α
z
is−
n
λ + isn
=
is+
λ − iszn
n
i
= (λ − )In,a + iPn,a
2
(A.2)
(A.3)
(A.4)
where Ia denotes the identity matrix acting on the auxiliary space and λ is
called the spectrum parameter. Historically, the Lax operator originates from
Inverse Scattering Method(ISM) and it carries explicit information about the
specific model. Also, we define the permutation operator P as
1
P = (I ⊗ I +
2
α
σα ⊗ σα)
(A.5)
which acts in C 2 ⊗ C 2 as P (a ⊗ b) = b ⊗ a. Now we want to find a commutation
relation between Lax operators. The following relation holds for Lax operators.
Ra1 ,a2 (λ − µ)Ln,a1 (λ)Ln,a2 (µ) = Ln,a2 (µ)Ln,a1 (λ)Ra1 ,a2 (λ − µ)
(A.6)
where the operator called R-matrix is defined as
Ra1 ,a2 = λIa1 ,a2 + Pa1 ,a2
Note that Eq.(A.6) is one kind of Quantum Yang-Baxter Equation.
92
(A.7)
Furthermore, we define the monodromy matrix Ta (λ) as the product of Lax
operators
TN,a (λ) = LN,a (λ) · · · L1,a (λ)
(A.8)
It turns out that the commutation relation for the monodromy matrix Ta (λ) is
identical with the commutation relation for the Lax operator, that is
Ra1 ,a2 (λ − µ)Tn,a1 (λ)Tn,a2 (µ) = Tn,a2 (µ)Tn,a1 (λ)Ra1 ,a2 (λ − µ)
(A.9)
We write the matrix element of the monodromy matrix in the auxiliary space
as
A(λ) B(λ)
TN,a =
C(λ) D(λ)
(A.10)
Also, we define a trace of T (λ)
F (λ) = Tra Ta (λ) = A(λ) + D(λ)
(A.11)
It follows from the commutation relation (A.9) that
[F (λ), F (µ)] = 0
(A.12)
This relation essentially corresponds to the integrability of the system. Now our
aim is to express the Hamiltonian in terms of F (λ). To see this, first note that
one can rewrite the Hamiltonian (A.1) in terms of the permutation operator as
H=
1
2
n
Pn,n+1 −
93
N
2
(A.13)
Hence, our aim of expressing the Hamiltonian in terms of F (λ) will be completed
if we could find a direct relation between P and F (λ). Note that
Ln,a (i/2) = iPn,a
(A.14)
d
Ln,a (λ) = In,a
dλ
(A.15)
TN,a (λ)(i/2) = iN PN,a PN −1,a · · · P1,a
(A.16)
It follows that
d
Ta (λ)|λ=i/2 = iN −1
dλ
d
F (λ)|λ=i/2 = iN −1
dλ
n
PN,a · · · Pˆn,a · · · P1,a
(A.17)
P1,2 · · · Pn−1,n+1 · · · PN −1,N
(A.18)
n
where the hat on P denotes its absence. Also,
d
d
1
F (λ)F −1 (λ)|λ=i/2 =
ln F (λ)|λ=i/2 =
dλ
dλ
i
Pn,n+1
(A.19)
n
Therefore, from the comparison of Eq.(A.13) and Eq.(A.19), the Hamiltonian can be written as
H=
N
i d
ln F (λ)|λ=i/2 −
2 dλ
2
(A.20)
So the problem of diagonalizing the original Hamiltonian (A.1) is now equivalent to diagonalizing F (λ).
94
Now we define the vacuum state Ω as
Ω=
n
⊗| ↑
n
(A.21)
From Eq.(A.3), it is easy to show that
C(λ)Ω = 0
(A.22)
A(λ)Ω = α(λ)N Ω
(A.23)
D(λ)Ω = δ(λ)N Ω
(A.24)
where α(λ) = (λ + i/2) and δ(λ) = (λ − i/2). It can be seen from Eq.(A.22)
that C(λ) works as an annihilation operator. Morever, it is clear that the state
Ω is the eigenstate of the operator F (λ), namely
F (λ)Ω = (λ + i/2)N + (λ − i/2)N Ω
(A.25)
Now let us study the effect of the operator B(λ) on the vacuum state Ω.
It turns out that B(λ) acts as a rising operator. To see this, we define the
so-called Bethe vector Φ({λ}) as
Φ({λ}) = B(λ1 )B(λ2 ) · · · B(λl )Ω
(A.26)
We want to show that the Bethe vector is actually the eigenstate of F (λ) for
some set of λs.
Now we want to find out the commutation relations between matrix elements
of the monodromy matrix T (λ), namely A(λ), B(λ), C(λ) and D(λ). Explicitly
95
working from Eq.(A.9), we get
[B(λ), B(µ)] = 0
(A.27)
A(λ)B(µ) = f (λ − µ)B(µ)A(λ) + g(λ − µ)B(λ)A(µ)
(A.28)
D(λ)B(µ) = h(λ − µ)B(µ)D(λ) + k(λ − µ)B(λ)D(µ)
(A.29)
with coefficients
f (λ) =
i
λ+i
i
λ−i
, g(λ) = , h(λ) =
, k(λ) = −
λ
λ
λ
λ
(A.30)
By using the commutation relations above,
l
l
λm − λ + i
i
i
λm − λ − i
F (λ)Φ({λ}) =
(λ + )N +
(λ − )N Φ({λ})
λm − λ
2
λm − λ
2
m=1
m=1
(A.31)
if
λj + i/2
λj − i/2
l
N
=
k=j
λj − λk + i
λj − λk − i
(A.32)
holds. This is the so-called Bethe ansatz equation and it has to be solved for
λ.
Lastly, let us derive some important observables; Momentum and Energy.
To begin with, let us define a operator U as
U = i−N Tra TN,a (i/2) = P1,2 P2,3 · · · PN −1,N
(A.33)
Here, we used Eq.(A.16) and the relation for the permutation operator P,
96
Pn,a1 Pn,a2 = Pa1 ,a2 Pn,a1 = Pn,a2 Pa2 a1 . We note that
U Xn = Xn+1 U
(A.34)
holds for any local operator Xn . By the definition of the momentum operator,
it follows that U can be written as the exponential of momentum operator Π,
U = eiΠ
(A.35)
Therefore, from (A.31)
U Φ({λ}) = iN F (i/2)Φ({λ}) =
j
λj + i/2
Φ({λ})
λj − i/2
(A.36)
taking the logarithm on both sides,
ΠΦ({λ}) =
p(j)Φ({λ})
(A.37)
j
where
p(λ) =
1 λ + i/2
ln
i λ − i/2
(A.38)
Similarly, the energy spectrum can be obtained as
HΦ({λ}) =
E(λj )Φ({λ})
(A.39)
j
where
E(λ) = −
1
1
2
2 λ + 1/4
97
(A.40)
Notice that both momentum and energy are additive.
The discussionn given here can naturally be extended to more general systems, including spin-s, Heisenberg XXZ model and excellent discussions regarding those models can be found in [31, 32].
A.2
XY Model
The XY model can be solved exactly by means of Jordan-Wigner transformation
which in essence maps the Hamiltonian into free fermion model. The model was
first investigated in detail by Lieb, Shultz, and Mattis [64]. In this section, I
will briefly review the eigenvalue problem of Heisenberg XY model by following
Ref.[2] and Ref.[50]. The model Hamiltonian is given as
H=−
J
2
N −1
j=0
y
x
[(1 + γ)σjx σj+1
+ (1 − γ)σjy σj+1
] − Γσjz
(A.41)
where γ denotes the degree of anisotropy and Γ and J is coupling strength and
an external magnetic field respectively. To solve this Hamiltonian, we carry out
the following transformation, called Jordan-Wigner transformation [65].
i−1
ci =
[−σjz ]σi−
(A.42)
[−σjz ]σi+
(A.43)
j=0
i−1
c†i =
j=0
98
Note that we have
i−1
i−1
c†i ci
[−σjz ]σi− = σi+ σi−
(A.44)
1
1
σi− σi+ = (1 − σiz ); σi+ σi− = (1 + σiz )
2
2
(A.45)
[−σjz ]σi+
=
j=0
j=0
Using the above relations, it is easy to check that ci satisfies the following
fermionic anticommutation relation.
ci , c†j
= δij
(A.46)
{ci , cj } = 0
(A.47)
ci c†i + c†i ci = σi− σi+ + σi+ σi− = 1
(A.48)
For i = j
For i = j, assuming that i < j and using σl− σlz = −σlz σl−
j−1
j−1
ci c†j
+
c†j ci
[−σkz ]σj+
= σi
[−σkz ]σi− = 0
+ σj
(A.49)
k=i
k=i
Now we want to rewrite the Hamiltonian in terms of the fermionic variable ci .
+
−
x
σix σi+1
= (σi+ + σi− )(σi+1
+ σi+1
)
+
−
+
−
= σi+ σi+1
+ σi+ σi+1
+ σi− σi+1
+ σi− σi+1
= c†i c†i+1 + c†i ci+1 − ci c†i+1 − ci ci+1
99
(A.50)
Similarly,
σiz = 2σi+ σi− − 1 = (2c†i ci − 1)
(A.51)
Collecting all these terms, the Hamiltonian is
H = −2
1
1
(c†i ci − ) + λ
2
2
i
(c†i ci+1 + ci c†i+1 + γc†i c†i+1 + γci ci+1 )
i
(A.52)
where λ = J/Γ. Thus, the original Hamiltonian has been mapped into noninteracting fermion system. In rewriting the Hamiltonian in terms of fermionic
variable, we neglect terms coming from the boundary condition. These terms
can actually be taken into account, however, we will omit them here for simplicity of discussion.
It is worth mentioning that if we have Heisenberg XXX interaction, that is,
if we have some extra terms in the z-direction
z
σiz σi+1
=
i
i
+
−
(2σi+ σi− − 1)(2σi+1
σi+1
− 1)
= 2
i
1
1
(c†i ci − )(c†i ci+1 − )
2
2
(A.53)
then the resulting system is no longer a free fermion system but an interacting
fermion system because of the presence of the cross term c†i ci c†i+1 ci+1 .
The Hamiltonian (A.52) can be further transformed into the quadratic form
using matrices A and B as
N −1
N −1
c†i Aij cj
H=
i,j=0
1
+
c† Bij c†j + h.c. + N
2 i,j=0 i
100
(A.54)
where Aii = −1, Aii+1 = − 21 γλ = Aii+1 , Bii+1 = − 12 γλ, Bi+1i = 21 γλ
A general method to diagonalize a Hamiltonian having the above quadratic
form has been developed. First, we demand that Hamiltonian takes the following form with η.
ωq ηq† ηq + const
H=
(A.55)
q
Let us consider the linear transformation of ci (Bogoliubov transformation) to
express the Hamiltonian in terms of variable η. The transformation reads
N −1
ηq =
gqi ci + hqi c†i
(A.56)
gqi c†i + hqi ci
(A.57)
i=0
N −1
ηq† =
i=0
where coefficients gqi and hqi can be taken to be real. From the condition that
ηq must also satisfy fermionic anticommutation relations, we get
(gqi gq′ i + hqi hq′ i ) = δqq′
(A.58)
(gqi hq′ i − gq′ i hqi ) = 0
(A.59)
i
i
If the Hamiltonian has the form of (A.55), we also have
{ηq , H} = ωq ηq
101
(A.60)
It follows that
ωq gqi =
j
ωq hqi =
j
(gqj Aji − hqj Bji )
(A.61)
(gqj Bji − hqj Aji )
(A.62)
which can be written using simple notations as
(A − B)Φq = ωq Ψq
(A.63)
(A + B)Ψq = ωq Φq
(A.64)
where the component of the vector Φ and Ψ are
[Φq ]i = gqi + hqi
(A.65)
[Ψq ]i = gqi − hqi
(A.66)
(A + B)(A − B)Φq = ωq2 Φq
(A.67)
(A + B)(A − B)Ψq = ωq2 Ψq
(A.68)
combining together
This is a N × N eigenvalue problem. Originally, the problem was a 2N × 2N
eigenvalue problem, however by the method given above, the size of the space
which we have to diagonalize has been significantly reduced. The eigenvalue
equation can now be solved and, at the same time, the constant term and ωq
102
can be obtained. The Hamiltonian can be finally written as
H=2
q
ωq ηq† ηq −
ωq
(A.69)
q
where
ωq =
(γλ sin φq )2 + (1 + λ cos φq )2
(A.70)
Note that the Hamiltonian (A.2) is written in a general form and some other
models can be derived easily as the limit of Hamiltonian (A.2). For instance,
γ = 0 corresponds to the isotropic Heisenberg XY model and γ = 1 corresponds
to the Ising model in a transverse magnetic field.
103
[...]... teleportation and superdense coding in Chapter 1 In Chapter 2, some results regarding the properties of entanglement in spin chain models will be reviewed In Chapter 3, we extend the study of spin chains to systems possessing next-nearest-neighbor interactions in addition to the usual nearest-neighbor interactions and study the effect of next-nearest-neighbor interactions on the entanglement properties of... states associated with quantum spin chain systems[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] In particular, it has been shown that one-dimensional Heisenberg spin chains can act as a quantum wire for the transmission of an unknown quantum state from one site to another [18] In this thesis, we study one-dimensional quantum spin chain models in the context of quantum information science Specifically,... properties of entanglement in the ground state of the Heisenberg XXX and XX models with next-nearest-neighbor interaction In addition, we will also propose a new scheme of quantum teleportation in a quantum spin chain via measurement process This thesis is structured as follows Firstly, we review some key ideas in quantum information technology including simple quantum information processes like quantum. .. analytical results obtained so far in quantum spin chain systems such as the Heisenberg XXX and XY model 2.1 Measure of Entanglement Entanglement is a crucial component of quantum information processing Primitives for quantum information processing such as quantum teleportation and superdense conding can be carried out with the help of entangled states It is then natural to ask about the entanglement of an... then moves on to a study of the dynamics of the spin chain models for the rest of 3 the chapter In Chapter 4, we review quantum state transfer scheme along a spin chain and we propose a quantum teleportation scheme using a Heisenberg spin chain as a resource in Chapter 5 1.2 Notations In this section, we summarize some frequently used notations used in this thesis • Qubit 0 1 |0 = |... quantum computation - quantum teleportation can act as a universal computational primitive [14] In addition to quantum teleportation with qubits, quantum teleportation with continuous variables has also been studied and demonstrated via squeezed states [15] 7 Chapter 2 Entanglement in Quantum Spin Chain In this chapter, we begin a brief review some quantitative measures of entanglement Using these measures,... demonstrated that quantum spin 2 chains are potentially useful resources for quantum information processing Moreover, the effective Hamiltonian for many realistic systems such as quantum dots and cavity QED systems can be treated as simple one-dimensional or two dimensional quantum spin chains [54, 55, 56] These studies have given rise to a vast amount of literature devoted to the study of entanglement. .. states such as spin glass and spin liquid In fact, a simple and natural way to incorporate frustration into a spin chain is to consider next-nearest-neighbor interaction of the form N H= J1 Hi,i+1 + J2 Hi,i+2 (3.1) i=1 where H could be a Heisenberg XXX model or an XY model Recently, entanglement properties in these next-nearest-neighbor models have been studied in the ground state as well as in the thermal... state devices An initial effort in this direction is the study of quantum spin chains Quantum spin chain models, such as the Heisenberg XXX model, Heisenberg XXZ model, and Heisenberg XY model, have always been useful as natural theoretical models for studying magnetism Surprisingly, these one-dimensional models are exactly solvable, that is, the full spectrum of the Hamiltonian can be obtained, by means...the existence a purely quantum correlation called entanglement , a notion that was first discussed by Einstein, Podolsky, and Rosen and now commonly known as the EPR paradox Entanglement is perhaps one of the most striking and peculiar property of quantum mechanics Entanglement is not the only property that has been exploited in quantum information processing Another important quantum mechanical property ... Summary Entanglement plays a central role in quantum information processing and its usefulness as a resource for quantum information processing in quantum spin chain models has been vastly studied In. .. including simple quantum information processes like quantum teleportation and superdense coding in Chapter In Chapter 2, some results regarding the properties of entanglement in spin chain models will... Quantum Spin Chain In this chapter, we begin a brief review some quantitative measures of entanglement Using these measures, we will look at some analytical results obtained so far in quantum spin
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