Measures of Entanglement at Quantum Phase Transitions

Open Systems & Quantum Information Milano, 10 Marzo 2006

Measures of Entanglement at Quantum Phase Transitions

Measures of Entanglement at Quantum Phase Transitions

M. Roncaglia

G. Morandi F. OrtolaniE. Ercolessi

C. Degli Esposti BoschiL. Campos Venuti

S. Pasini

Condensed Matter Theory Group in BolognaCondensed Matter Theory Group in Bologna


• Entanglement is a resource for:

teleportation dense coding quantum cryptography quantum computation

QUBITS Spin chains are natural candidates as quantum devices

• The Entanglement can give another perspective for understanding Quantum Phase Transitions

• Strong quantum fluctuations in low-dimensional quantum systems at T=0


• Entanglement is a property of a state, not of an Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

• Direct product states

• 2-qubit states 1001

2

1

11002

1

A BBA

0 jiji BABA

• Nonzero correlations at T=0 reveal entanglement

Maximally entangled(Bell states)

11,00

Product states


Block entropyB A

AAAAS logTr

BA Tr

• Reduced density matrix for the subsystem A

• Von Neumann entropy

• For a 1+1 D critical system

CFT with central charge c

lSA log3

c log

6

cAAS

Off-critical

[ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).]l= block size


RG flow

UVfixed point

IRfixed point

• c-theorem: IRUV cc

(Zamolodchikov, 1986)

Loss of entanglement

Renormalization Group (RG)

Irreversibility of RG trajectories

RG flow

UVfixed point

• Massive theory (off critical) Block entropy saturation


[ S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, 86402 (2004).]

• Local Entropy: when the subsystem A is a single site.

• Applied to the extended Hubbard model

• The local entropy depends only on the average double occupancy

• The entropy is maximal at the phase transition lines (equipartition)


[ A.Anfossi et al., PRL 95, 056402 (2005).]

• Bond-charge Hubbard model (half-filling, x=1)

• Negativity

2/1)(1 AT

ABABN

• Mutual information

)()()( ABBA SSSI

• Critical points: U=-4, U=0

• Some indicators show singularities at transition points, while others don’t.


[ A.Osterloh, et al., Nature 416, 608 (2002).]

Ising model in transverse field

• Critical point: =1

• The concurrence measures the entanglement between two sites after having traced out the remaining sites.

• The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).


11100100 dcba

bcadC 2

yj

yiijC *)(

Concurrence

For a 2-qubit pure state the concurrence is (Wootters, 1998)

if

• Is maximal for the Bell states and zero for product states

For a 2-qubit mixed state in a spin ½ system

22

12

1 zj

zi

zj

zi

yj

yi

xj

xiij σσσσσσσσ=C

;,,0max ijijij CC=C


Ising model in transverse field

][1

zi

xi

xi

hHi

1h

1h

1zP

1h Critical point

2D classical Ising modelCFT with central charge c=1/2

Exactly solvable fermion model

Jordan-Wigner transformation


Local (single site) entropy:

1111 ln2

1ρρTr=Sσσ+I=ρ zz

Near the transition (h=1):

1ln 112

hhσ z

S1 has the same

singularity aszσ

Nearest-neighbour concurrence inherits logarithmic singularity 1ln 1 hhσσ ji

Local measures of entanglement based on the 2-site density matrix depend on 2-point functions

Accidental cancellation of the leading singularity may occur, as for the concurrence at distance 2 sites

1ln112

1 22222, hhσσσσσσC z

izi

yi

yi

xi

xiii


Alternative: FSS of magnetization

100,20,30,=N

2

1

121/8lnln

12

N

π+γ+π+N

π

h+

π=σ C

z

2

2 1

61

~

N

π+=hN

Crossing points:

Exact scaling function in the critical region ξ<N

C. Hamer, M. Barber, J. Phys. A: Math. Gen. (1981) 247.

Standard route: PRG

M,hM=N,hN~~

NENE=Nh, 01

First excited state needed

Shiftterm

Seeking for QPT point


gVHgH 0

)(

cgg

cc

sg

sggggOV )sgn()()(

Letg

cg

Quantum phase transitions (QPT’s)

• First order: discontinuity in

cggge

(level crossing)

• Second order:cgg

n

n

g

e

diverges for some 2n

)()( )()( gegege sr

1)1( d

• GS energy:

• At criticality the correlation length diverges

scaling hypothesis

• Differentiating w.r.t. g


• The singular term appears in every reduced density matrix containing the sites connected by .

)(sg

OV

• Local algebra hypothesis: every local quantity can be expanded in terms of the scaling fields permitted by the symmetries.

• Any local measure of entanglement contains the singularity of the most relevant term.

• The best suited operator for detecting and classifying QPT’s is V , that naturally contains . Moreover, FSS at criticality

/)sgn( /)()(LLggLOLV

cs

g

s

• Warning: accidental cancellations may occur depending on the specific functional form next to leading singularity

)(sg

O )( L

00


N

i

z

i

z

i

z

i

y

i

y

i

x

i

x

i SSSSSSSJH1

2

111 ][ )(

Spin 1 D model

D

=Ising-like D = single ion

2ziS=

D

e

In this case

Phase Diagram

• Symmetries: U(1)xZ2

Around the c=1 line:

KH x 4cos2

1 22(sine-Gordon)

cDD Criticalexponents

)2/( KK )2/(1 K


[ L.Campos Venuti, et. al., PRA 73, 010303(R) (2006).]

Crossing effect

Derivative ccz DDDD~S sgn

2

0.64=

Single-site entropy

2.59=λ 42.29=Dc0.82=

cczi

zi ~SS sgn1

The same for

• What about local measures of entanglement?

zzzz=S 1ln12/ln1 2ziS=z

Using symmetries:

• Two-sites density matrix contains the same leading singularity)2(

ij


Localizable Entanglement

• LE is the maximum amount of entanglement that can be localized on two q-bits by local measurements.

i j

s

sss

ij EpL )(|max}{

N+2 particle state

Nsss ,,1 • Maximum over all local measurement basis

ssp = probability of getting

sE | is a measure of entanglement

[ F.Verstraete, M.Popp, J.I.Cirac, PRL 92, 27901 (2004).]

(concurrence)


LE = max of correlation LE = string correlations

[ L. Campos Venuti, M. Roncaglia, PRL 94, 207207 (2005).]

Ising model Quantum XXZ chain

MPS (AKLT)

Dλ

Calculating the LE requires finding an optimal basis, which is a formidable task in general

However, using symmetries some maximal (optimal) basis are easily found and the LE takes a manageable form

Spin 1/2 Spin 1

totN SiL exp1 1

• The LE shows that spin 1 are perfect quantum channels but is insensitive to phase transitions.

jiijL max

• :The lower bound is attained

CE


A spin-1 model: AKLT

N

ijiji SSSSJH

1

2 ])(3

1[

• Infinite entanglement length but finite correlation length

• Actually in S=1 case LE is related to string correlation

0explim1

1

k

k

jlljString

SSiSO

|| jk

=Bell state

000000Typical configurations

Optimal basis: 2/11,0


Conclusions

References:L.Campos Venuti, C.Degli Esposti Boschi, M.Roncaglia, A.Scaramucci, PRA 73, 010303(R) (2006).L.Campos Venuti and M. Roncaglia, PRL 94, 207207 (2005).

• Localizable Entanglement It is related to some already known correlation functions. It promotes S=1 chains as perfect quantum channels.

• The most natural local quantity is , where g is the driving parameter across the QPT.

ge /

• it shows a crossing effect• it is unique and generally applicable

Advantages:

• Low-dimensional systems are good candidates for Quantum Information devices.

• Several local measures of entanglement have been proposed recently for the detection and classification of QPT. (nonsystematic approach)

• Open problem: Hard to define entanglement for multipartite systems, separating genuine quantum correlations and classical ones.

• Apart from accidental cancellations all the scaling properties of local entanglement come from the most relevant (RG) scaling operator.


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Measures of Entanglement at Quantum Phase Transitions