Multipartite Entanglement Measures from Matrix and Tensor Product States

Ching-Yu Huang Feng-Li Lin Department of Physics, National Taiwan Normal University

arXiv:0911.4670

Outline• Quantum Phase Transition• Entanglement measure• Matrix/Tensor Product States• Entanglement scaling• Quantum state RG transformation

Quantum Phase transition

• Phase transition due to tuning of the coupling constant.

• How to characterize quantum phase transition?

a. Order parameters

Frustrated systems

• These systems usually have highly degenerate ground states.

• This implies that the entropy measure can be used to characterize the quantum phase transition.

• Especially, there are topological phases for some frustrated systems (no order parameters)

?

Entanglement measure

• The degeneracy of quantum state manifests as quantum entanglement.

• The entanglement measure can be used to characterize the topological phases.

• However, there is no universal entanglement measure of many body systems.

• Instead, the entanglement entropy like quantities are used.

Global entanglement• Consider pure states of N particles . The

global multipartite entanglement can be quantified by considering the maximum fidelity.

• The larger indicates that the less entangled.

• A well-defined global measure of entanglement is

【 Wei et al. PRA 68, 042307(2003) 】

maxmax

max

2maxlog)( E

Global entanglement density and its h derivative for the ground state of three systems at N. Ising ; anisotropic XY model; XX model.

Our aim

• We are trying to check if the entanglement measure can characterize the quantum phase transition in 2D spin systems.

• Should rely on the numerical calculation• Global entanglement can pass the test but is

complicated for numerical implementation.• One should look for more easier one.

Matrix/Tensor Product States

• The numerical implementation for finding the ground states of 2D spin systems are based on the matrix/tensor product states.

• These states can be understood from a series of Schmidt (bi-partite) decomposition. It is QIS inspired.

• The ground state is approximated by the relevance of entanglement.

1D quantum state representation

1

1

... 11 1

| ... | ...n

n

d d

i i ni i

c i i

31 2

1 1 1 1 2 2 2 3 1

1 1

[3] [ ][1] [2][1] [2]...

...

.... n

n n

n

i n ii ii ic

Γ NΓ1 Γ 2 Γ3

λ1 λ2 λN-1

NdNdDparameters <<» 2# • Representation is efficient• Single qubit gates involve only local update• Two-qubit gates reduces to local updating

Virtual dimension = D

Physical dimension = d

【 Vidal et al. (2003) 】

``Solve” for MPS/TPS

• In old days, the 1D spin system is numerically solved by the DMRG(density matrix RG).

• For translationally invariant state, one can solve the MPS by variational methods.

• More efficiently by infinite time - evolving block decimation (iTEBD) method.

Time evolution• Real time• imaginary time

• Trotter expansion

0 (-iHt) exp| t

( ) / 2[ ] ( )i F G T iF iG Te e e O T

Γ1 Γ 2 Γ3

λ1 λ2

Γ4 Γ5

λ3 λ4

F F

Γ1 Γ 2 Γ3

λ1 λ2

Γ4 Γ5

λ3 λ4

G G

(-Ht) exp (-Ht) exp

|0

0

【 Vidal et al. (2004) 】

Γ (si) Γ (sj)

λi λkλk+1

U

ΘSVD

Γ’(si) Γ’(sj)

λi λ’k λk+1

X (si) Y (sj)

λ'kλ-1iλi

λk+1 λk+1

Evolution quantum state

Global entanglement from matrix product state for 1D system

• Matrix product state (MPS) form

• The fidelity can be written by transfer matrix

where

Nppp

d

ppp

pppAAATr N

N

..., )...( 211...,

21

21

)...( 321 ngggg TTTTTr

is i

ii

iig sBsAT )()( ][][

【 Q.-Q. Shi, R. Orus, J.-O. Fjaerestad, H.-Q. Zhou, arXiv:0901.2863 (2009) 】

ANA1 A2 A3

BNB1 B2 B3

Quantum spin chain and entanglement

• XY spin chain

• Phase diagram: oscillatory (O) ferromagnetic (F) paramagnetic (P)

)2

12

1( 111

zi

yi

yi

xi

xi

N

iXY hH

-0.5 0.5 1.50

0.25

0.5

0.75

1

h

γ

O F pIsing

XY

【 Wei et al. PRA 71, 060305(2003) 】

• A second-order quantum phase transition as h is tuned across a critical value h = 1.

• Magnetization along the x direction . (D= 16)

0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.100.0

0.2

0.4

0.6

0.8

mx

h

D D D D

1)( ** hh

0 2 4 6 8 10 12 14 160

0.20.40.6

0.81

1.2

hc

D

• For a system of N spin 1/2. the separable statesrepresented by a matrix

• The global entanglement has a nonsingular maximum at h=1.1.

• The von Neuman entanglement entropy also has similar feature.

)1sin0(cos1 ii

iNi e

iii

ii eBB sin)1( , cos)0( ][][

0.0 0.5 1.0 1.5 2.0 2.50.00

0.02

0.04

0.06

0.08

0.10

GE

h

Ising XY

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

M

h

Ising Mx Ising Mz XY Mx XY Mz

Generalization to 2D problem

• The easy part promote 1D MPS to 2D Tensor product state (TPS)

• The different partHow to perform imaginary time evolution a. iTEBDHow to efficiently calculate expectation value a. TRG

【 Levin et al. (2008), Xiang et al. (2008) , Wen et al. (2008) 】

2D Tensor product state (TPS) • Represent wave-function by the tensor

network of T tensors

NNppp

d

ppp

pppTTTTr NN

NN

,1,21,11...,

..., )...( ,1,21,1

,1,21,1

p

l

d

u

r

Virtual dimension = D

Physical dimension = d

Global entanglement from matrix product state for 2D system

• The fidelity can be written by transfer matrix

where

• It is difficult to calculate tensor trace (tTr), so we using the “TRG” method to reduce the exponentially calculation to a polynomial calculation.

)...( 321 ngggg TTTTtTr

A4 A3

A1 A2

B1

B4

B2

B3

Ť4

Ť2

Ť3

Ť1

Double Tensor

is i

ii

iig sBsAT )()( ][][

The TRG method Ⅰ

• First, decomposing the rank-four tensor into two rank-three tensors.

u

S2 d

S4

S3

l

uS1

T T

T T

rd

lγ

γ r

,3

,1

;

lulu

rdrd

Rlurd

VS

US

VUM

,2

,4

,

dldl

urur

Rdlur

VS

US

VUM

The TRG method Ⅱ• The second step is to build a new rank-four

tensor.• This introduces a coarse-grained square lattice.

γ1

γ2

γ3

T’

γ4

γ1 S1

S4 S3

S2

a1

a2

a3

a4

γ2

γ3γ4

432343214121

4321

4321aa

4aa

3aa

2aa

1

,,,,,, SSSS

aaaa

The TRG method Ⅲ• Repeat the above two steps, until there are

only four sites left. One can trace all bond indices to find the fidelity of the wave function.

TA

TC

TB

TD

Global entanglement for 2D transverse Ising

• Ising model in a transverse magnetic field

• The global entanglement density v.s. h, the entanglement has a nonsingular maximum at h=3.25.

• The entanglement entropy

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

h

Mz GE SBP S1

xi

zj

zi

ji

hH ,

Global entanglement for 2D XXZ model

• XXZ model in a uniform z-axis magnetic field

• A first-order spin-flop quantum phase transition from Neel to spin-flipping phase occurs at some critical field hc.

• Another critical value at hs = 2(1 + △), the fully polarized state is reached.

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

<M>

h

<Mx> <Mz> <Mzs>

0 1 2 3 4 5 60.00.10.20.30.40.50.6

h

GE SBP S1

zi

zj

zi

yj

yi

xj

xi

ji

hH )(,

Scaling of entanglement

L spins

ρL SL

LcS LL 2log3

|1|log6 22/ hcS NL

【 Vidal et al. PRL,90,227902 (2003) 】

Entanglement entropy for a reduced block in spin chains

At Quantum Phase Transition Away from Quantum Phase Transition

LLL TrS ln

Block entanglement v.s. Quantum state RG

• Numerically, it is involved to calculate the block entanglement because the block trial state is complicated.

• Entanglement per block of size 2L = entanglement per site of the L-th time quantum state RG(merging of sites).

• We can the use quantum state RG to check the scaling law of entanglement.

1D Quantum state transformationQuantum state RG 【 Verstraete et al. PRL 94, 140601 (2005) 】

• Map two neighboring spins to one new block spin

• Identify states which are equivalent under local unitary operations.

By SVD

RG

qpD

pq AAA

1

)()( :~

p q lRG

α β γ α γ

l)(

)(),min(

1

)()( V ~

22

lpql

Dd

l

pq UA

lllRGp VAA

• For MPS representation for a fixed value of D, this entropy saturates at a distance L ～ Dκ (kind of correlation length)

• Could we attain the entropy scaling at critical point from MPS?

【 Latorre et al. PRB,78,024410 (2008) 】

At quantum point

1 2 5 10 20 30 405060 70 800.30

0.35

0.40

0.45

0.50

1 2 5 10 20 30 405060 70 800.0

0.5

1.0

1.5

2.0

2.5

3.0

S L

L

D D D

D D D

S LL

Near quantum point

• MPS support of entropy obeys scaling law!!

1 2 3 4-0.10.00.10.20.30.40.50.60.7

)ln(182.0

DDS log61)(

Dlog*182.0~

2D Scaling of entanglement

• In 2D system, we want to calculate the block entropy and block global entanglement.

Could we find the scaling behavior like 1D system?

)ln1( LcLS LLxL

Area law

2D Quantum state transformation

• Map one block of four neighboring spins to one new block spin.

• By TRG and SVD method

RG

s1 s2

s4 s3s

RG

The failure

• We find that the resulting block entanglement decreases as the block size L increases.

• This failure may mean that we truncate the bond dimension too much in order to keep D when performing quantum state RG transformation, so that the entanglement is lost even when we increase the block size.

Summary

• Using the iTEBD and TRG algorithms, we find the global entanglement measure.- 1D XY model- 2D Ising model- 2D XXZ model

• The scaling behaviors of entanglement measures near the quantum critical point.

Thank You ForYour Attention!