Measures of entanglement at quantum phase transitions
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Measures of Entanglement at Quantum Phase Transitions. G. Morandi F. Ortolani E. Ercolessi C. Degli Esposti Boschi L. Campos Venuti S. Pasini. M. Roncaglia. Condensed Matter Theory Group in Bologna. Spin chains are natural candidates as quantum devices. QUBITS.

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Measures of entanglement at quantum phase transitions
Measures of Entanglement at Quantum Phase Transitions

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

Open Systems & Quantum Information

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A

B

  • Direct product states

  • Nonzero correlations at T=0revealentanglement

  • 2-qubit states

Product states

Maximally entangled

(Bell states)

Open Systems & Quantum Information

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Block entropy Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

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 Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

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

Open Systems & Quantum Information

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  • Local Entropy Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.: 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 Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

  • (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).]

Open Systems & Quantum Information

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Ising model in transverse field Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

  • 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).]

Open Systems & Quantum Information

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Concurrence Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

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

Open Systems & Quantum Information

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Ising model in transverse field Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

2D classical Ising model

CFTwith central chargec=1/2

Critical point

Jordan-Wigner transformation

Exactly solvable fermion model

Open Systems & Quantum Information

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Near the transition ( Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.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 Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

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

Open Systems & Quantum Information

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Quantum phase transitions (QPT’s) Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

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

Open Systems & Quantum Information

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  • The singular term appears in Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.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

Open Systems & Quantum Information

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In this case Hamiltonian. But the GS of strongly correlated quantum systems are generally entangled.

(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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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).]

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[ F.Verstraete, M.Popp, J.I.Cirac, PRL singularity92, 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)

Open Systems & Quantum Information

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[ L. Campos Venuti, M. Roncaglia, PRL singularity94, 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.

Open Systems & Quantum Information

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A spin-1 model: AKLT singularity

=Bell state

Optimal basis:

  • Infinite entanglement length but finite correlation length

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

Typical configurations

Open Systems & Quantum Information

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Conclusions singularity

  • 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

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The end
The End singularity

Open Systems & Quantum Information

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