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Measures of Entanglement at Quantum Phase Transitions

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G. Morandi

F. Ortolani

E. Ercolessi

C. Degli Esposti Boschi

L. Campos Venuti

S. Pasini

M. Roncaglia

Condensed Matter Theory Group in Bologna

Open Systems & Quantum Information

Milano, 10 Marzo 2006

Spin chains are natural candidates

as quantum devices

QUBITS

- Entanglement is a resource for:

teleportation

dense coding

quantum cryptography

quantum computation

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

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

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

A

B

- Direct product states

- Nonzero correlations at T=0revealentanglement

- 2-qubit states

Product states

Maximally entangled

(Bell states)

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Milano, 10 Marzo 2006

Block entropy

B

A

- Reduced density matrix for the subsystem A

- Von Neumann entropy

- For a 1+1 Dcritical system

Off-critical

CFT with central charge c

l= block size

[ See P.Calabrese and J.Cardy, JSTAT P06002 (2004).]

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RG flow

UV

fixed point

IR

fixed point

RG flow

UV

fixed point

Renormalization Group (RG)

- c-theorem:

(Zamolodchikov, 1986)

- Massive theory (off critical)
- Block entropy saturation

Irreversibility of RG trajectories

Loss of entanglement

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- 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)

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

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

- Critical points: U=-4, U=0

- Negativity

- Mutual information

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

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

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Ising model in transverse field

- Critical point: l=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).

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

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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

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Ising model in transverse field

2D classical Ising model

CFTwith central chargec=1/2

Critical point

Jordan-Wigner transformation

Exactly solvable fermion model

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Near the transition (h=1):

S1 has the same

singularity as

Local (single site) entropy:

Local measures of entanglement based on the 2-site density

matrix depend on 2-point functions

Nearest-neighbour concurrence

inherits logarithmic singularity

Accidental cancellation of the leading singularity may occur,

as for the concurrence at distance 2 sites

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Seeking for QPT point

Alternative: FSS of magnetization

Standard route: PRG

First excited state needed

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

Exact scaling function in the

critical region

Crossing points:

Shift

term

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Quantum phase transitions (QPT’s)

Let

- First order: discontinuity in

(level crossing)

- Second order:

diverges for some

- At criticality the correlation length diverges

- GS energy:

scaling hypothesis

- Differentiating w.r.t. g

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

- Local algebrahypothesis: 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.

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

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

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In this case

(sine-Gordon)

Spin 1 l-D model

l

D

l =Ising-like D = single ion

Phase

Diagram

- Symmetries: U(1)xZ2

Around the c=1 line:

Critical

exponents

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- Two-sites density matrix contains the same leading singularity

Derivative

The same for

Crossing effect

- What about local measures
- of entanglement?

Using symmetries:

Single-site entropy

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

Open Systems & Quantum Information

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

Localizable Entanglement

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

j

i

N+2 particle state

- Maximum over alllocal measurement basis

= probability of getting

is a measure of entanglement

(concurrence)

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

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

- Ising model
- Quantum XXZ chain

- MPS (AKLT)

LE = max of correlation

LE = string correlations

1

- :
- The lower bound is attained

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

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A spin-1 model: AKLT

=Bell state

Optimal basis:

- Infinite entanglement length but finite correlation length

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

Typical configurations

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Conclusions

- 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)

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

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

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

Advantages:

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

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

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).

Open Systems & Quantum Information

Milano, 10 Marzo 2006

Open Systems & Quantum Information

Milano, 10 Marzo 2006