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).

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