Entanglement and Topological order in self-dual cluster states

Vlatko Vedral

University of Oxford, UK &National University of Singapore

ContentsTopological order and Entanglement.

XX model.

Cluster states.

Dual transformation.

Boundary effects, Phase transition and criticality.

Entanglement as an order parameter.W. Son, L. Amico, S. Saverio, R. Fazio, V. V., arXiv:1001.2565

Topological order A phase which cannot be described by the Landau

framework of symmetry breaking. Three different characterization of the topological order.

◦ Insensitivity to local perturbation. ◦ Ground state degeneracy to the boundary condition.◦ Topological entropy.

Relationship between the topological order and fault tolerance.

Conceptual relationship between topological order and entanglement.◦ Entanglement is global properties in the system.◦ Entanglement is sensitive to degeneracy (Pure vs Mixed )

Criticality indicatorLong range order

Off-diagonal LRO

Even more creative : Two dimensional phase transitions.

Entanglement order? (c.f. Wen) Fractional Quantum Hall effects.

| |† ( ) ( ) x yx y const

| |† †( ) ( ) ( ) ( ) x yx x y y const

Order treeDifferent Orders

Long range order(e.g. 2D Ising)

Short range order(e.g. KT)

Off-diagonal LRO(e.g. BCS)

Quantum – ground state –Topological(e.g. FQHE)

Topological, finiteT order ?

Symmetry breaking

Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)

Entanglement (Block ent. & Geometric Ent.)Separability

Block entanglement (Entropy)

Geometric entanglement

)]([)( AALogTrS

2||min)( sep

QPT in XX model

What is quantum phase (transition) in many-body system? (XX model)

1

23

12

3

Thermal state and purity (XX model)

Cluster statesConstruction of the cluster state.

Hamiltonian for cluster state.

Usefulness of cluster states for measurement based quantum computation.

CP CP CP CP CP

)(iNk

kii XZSi

iSH

)(iNk

kii XZS

CZCi

nin

i

N

i

CPiUC

Full Spectrums of Cluster HamitonianFull Spectrums

For the case of N=4

CZCi

nin

i

C

CZi

CZZ ji

CZZZ kji

CZZZZ 4321

40 E

21 E

02 E

23 E

44 E

}1,0{in

Geometric entanglementPhysical meaning;

Mean field correspondence.

Numerical evaluation.

Symmetries can be applied for closest separable state. (XX model with perturbation.) Can entanglement b

e a

topological order parameter?

Entanglement as Energy

Think of phase transition as tradeoff between energy and entropy:

TdSdUdF

Quantum phase transitions: tradeoff between entanglement and entropy:

)ln(~ GpSE

Clusters:

12ln

2

1

1ln

2 /2

JT

e

NC

N

kTJ

Diagonalising ClusterJordan Wigner transformation leads

to free fermions (“hopping” between next to nearest neighbours)

Probability looks like N independent fermions

Then do the FT and Bogoliubov…

nnnnn

l

l

k

zkl

ccccH

c

22

1

1

Dual transformation (Fradkin-Susskind).

Definition.

Duality ◦ Emergence of qusi-particles (discuss XX).◦ Identification of critical point. ◦ Change of state and entanglement.

Sensitivity to the boundary condition in the dual transformation.

Mapping of Cluster into Ising 1D Cluster Hamiltonian.

State transformation.

Hamiltonian without boundary term.

Ising state.

i

iiiC XZXH 11

Cluster

Self-dual Cluster Hamiltonian

Model

Solution

Geometric entanglement and criticality

Topological order in Cluster stateInsensitivity to local perturbation.

No degeneracy in the ground state.

String order

Highly entangled state (E~N/2).

CNCH c

1)1(1

21

N

k

yN

zk

yNO

DiscussionApplied standard methods of statistical

physics and solid state to computing; Can think of entanglement as

equivalent to energy (free energy)Should do the same analysis in 2D (JW

ambiguity)Can all topological phases support

computing?Could we map between circuits and

clusters?

ReferencesL. Amico, R. Fazio, A. Osterloh, V. V, Rev.

Mod.Phys. 80 (2008)

Xiao-Gang Wen, Quantum Field theory of Many-body systems (2004)

W. Son, L. Amico, F. Plastina, V. V Phys. Rev. A 79(2009)

W. Son, V. V., OSID volume 2-3:16 (2009) Michal Hajdušek and V. V. New J. Phys. 12 (2010) A. Kitaev, Chris Laumann, arXiv:0904.2771 A. Kitaev, J. Preskill, Phys. Rev. Lett. 96 (2006) R. Raussendorf, D.E. Browne, H.J. Briegel, Phys. Rev.

A 68 (2003)