Post on 17-Mar-2016
description
Multi-qubit Entanglement and Quantum Phase Transition inthe Transverse Ising Model
Afshin Montakhab and
Ali AsadianPhysics Department
Shiraz University, Shiraz, Iran
Overview Multi-qubit entanglement
properties of Ising ground state1. Global entanglement 2. Genuine multi-qubit entanglement3. Entanglement sharing of the ground
state Conclusion and future
development
The Model
N
i
zi
xi
N
i
xi BJH 1
1
We study the transverse Ising model:
and its higher dimensional generalizations
Entanglement properties of the Ising ground state
NxxxN
0;...0;0;21
The ground state of transverse Ising model in zero magnetic field is two-fold degenerate .
NxxxN
1;...1;1;21
Coherent superposition of the above states:
)(2
1 ieGHZ
Previous authors have studied entanglement properties of the ground product states. (e.g. Osborne Nielsen PRA 2002)
We study the entanglement properties of GHZ ground state.
Global Entanglement
Informational approach to multi-qubit entanglement
Brukner-Zeilinger principle: the total information of one qubit is one bit and the total information of N-qubit system is N bit (for pure states)
The total information content of a multi-qubit system can be distributed in local and non-local form
localnonlocaltotal III
ji ii
iiijlocalnon
N
iilocal
N
NINIIandII
......
1 1
1...2
A qubit carries of local information which is one bit for an isolated qubit and N qubit carries of local information
12 2 TrI
)12(1
2
N
iilocal TrI
according to information distribution
localtotallocalnon III
)1(21
2
N
iilocalnon TrI
)1(2 2TrSEN
IE particleoneglPBClocalnon
gl
S linear entropy
0 0.5 1 1.5 2 2.5 3 3.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Glo
bal e
ntan
glem
ent
Global entanglement of 4, 8 and 10 qubit transverse Ising model
4 qubit8 qubit10 qubit
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
d(E
glob
al)/d
First derivatives of global entanglement of transverse Ising model
4 qubit8 qubit10 qubit
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
1st d
eriv
1st derivatives of global entanglement of 2D transverse Ising model
4 qubit9 qubit
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Glo
bal e
ntan
glem
ent
Global entanglement of 2D transverse Ising model
4 qubit9 qubit
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1st d
eriv
1st derivative of global entanglement in 3D (cubic) Ising model
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Global entanglement of 3D (cubic) Ising model
Glo
bal e
ntan
glem
ent
Genuine multi-qubit entanglement
Genuine Entanglement is a form of non-local information which is shared by all the constituents of the system. For a pure state of N qubit, when N is even, there are even and odd bipartitions.
2
21...12 ... NN yyy
evenoddN SS ...12
Information-theoretic measure:
Algebraic measure:
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1st d
eriv
first derivatives of genuine multi-qubit entanglement of 6, 8 and 10 qubit
6 qubit8 qubit10 qubit
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gen
uine
ent
angl
emen
t
Genuine multi-qubit entanglement of transvese Ising model for 6, 8, 10 qubit
6 qubit8 qubit10 qubit
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
1.5
2
2.5
2nd
deriv
2nd derivatives of genuine entanglement of 6, 8, 10 qubit
6 qubit8 qubit10 qubit
Other entanglement which show scaling behavior
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
B/J
entro
py
Entropy of 1, 2, 3 and 4 qubit of 8 qubit Ising chain
1 qubit2 qubit3 qubit4 qubit
0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
B/J
Ent
ropy
Entropy of the half-system
10 qubit sys.8 qubit sys.4 qubit sys.
GHZ and decoherence The GHZ state is often excluded in studies with thermodynamical limit
due to decoherence which can lead to product state
21
21)(
21 GSGSEeGS edecoherenci
The solid line is single-qubit entropy of thermal ground state and
the dashed line is single-qubit entropy (global entanglement) of
Entanglement sharing of the ground state
Genuine three-qubit entanglement of four-qubit system in pure state
klijkjkikijijk SS
We derive this formula from monogamy of entanglement and information distribution in a four-qubit system in pure state. Note that this reduces to KBW tangle if we isolate the fourth qubit i.e,.
jkikijijk S
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
C12
3 an
d C
12
three-qubit entanglement and concurrence of four qubit in pure state
C123C12
Conjecture: all the N-tangle (when N>2 up to N-1) are maximum near the critical point in both scenarios
Conclusion and future development
We study the entanglement properties of the GHZ like state of the Ising ground state.
Thermodynamical behavior of entanglement is observed near critical point for global entanglement, reminiscent of classical phase transition.
Genuine entanglement is more difficult to characterize due to limitation of our system sizes.
We believe all n-tangles are maximized near critical point.
We need to study larger system sizes in order to better study the critical region and extract the critical exponents. (e.g. DMRG, finite-size scaling)
References Caslav Brukner, and Anton Zeilinger. "Operationally Invariant
Quantum Information" Phys. Rev. Lett. 83, 3354 (1999).
Jian-Ming Cai1, Zheng-Wei Zhou1, and Xing-Xiang Zhou2, and Guang-Can, "Information-Theoretic Measure of Genuine Multi-Qubit Entanglement" Phys. Rev.A 74, 042338 (2006).
T. J. Osborne, and M. A. Nielsen, "Entanglement in a simple quantum phase transition" Phys. Rev.A 66, 032110 (2002).
Thank youThank you