Quantum Mechanics and Quantum Information

1
Entanglement and Correlation Functions in
Low-Dimensional Systems
M. Roncaglia
Condensed Matter Theory Group in Bologna

• G. Morandi
• F. Ortolani
• E. Ercolessi
• C. Degli Esposti Boschi
• L. Campos Venuti
• S. Pasini

2

• Entanglement is a resource for

teleportation dense coding quantum
cryptography quantum computation

• Strong quantum fluctuations in low-dimensional
  quantum systems at T0

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

3

• 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 T0 reveal entanglement

• 2-qbit states

Product states
Maximally entangled (Bell states)
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Block entropy
B
A

• Reduced density matrix for the subsystem A

• Von Neumann entropy

• For a 11 D critical system

Off-critical
CFT with central charge c
l block size
See P.Calabrese and J.Cardy, JSTAT P06002 (2004)
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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

from S.Gu, S.Deng, Y.Li, H.Lin, PRL 93, 86402
(2004)
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• Bond-charge Hubbard model
• (half-filling, x1)

• Critical points U-4, U0

• Negativity

• Mutual information

• Some indicators show
• discontinuity at transition points, while others
  dont.

from A.Anfossi et al., cond-mat/0502500
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• Ising model in transverse field

• Critical point l1

• 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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Entangled Pair
Entanglement swapping

• With a suitable measure I can concentrate the
  entanglement onto selected qbits.

Bell Measurement
A
D
B
C

• Eventually, the particles A and D are entangled
  even if they have never interacted in the past.

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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 local
  measurements.

j
i
N2 particle state

• Maximum over all local measurement basis

probability of getting
is a measure of entanglement
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Concurrence
For a 2-qbit pure state the concurrence is
(Wootters, 1998)
if

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

Entanglement length
11
Optimal basis
Bell state on i and j
After measure
Calculating the LE requires finding an optimal
basis, which is a formidable task in general
Upper bound entanglement of assistance
Lower bound maximal correlation function
(connected)
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Once an optimal basis is found, the LE becomes
where
is the preconcurrence
The LE is written as an expectation value
with

• Differentiating L w.r.t. all local unitary
  transformations

• Set of extremal equations (very complicate
  solution)

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However, using symmetries some maximal (optimal)
basis are easily found and the LE takes a
manageable form
Spin 1/2
Reality and invariance under one of
Optimal basis is along a-axis
Spin 1
(with two spin ½ at the endpoints)
Reality and invariance under all
Optimal basis is
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Ising model in transverse field
2D classical Ising model CFT with central charge
c1/2
Critical point

• Marshall sign theorem

(analytical result)
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S½ XXZ model
-1
FM
AFM
0
1
BKT
FD
SD
Optimal basis along x
Symmetries U(1)xZ2
Dlt1
Maximal correlations The lower bound is attained
Dgt1
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Spin 1 l-D model
l
D
l Ising-like D single ion

• Invariance under all

• It is possible to calculate the LE without
  making any assumption on ygt

1

• 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

• Infinite entanglement length but finite
  correlation length

• Actually in S1 case LE is related to string
  correlation

Typical configurations
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Open problems

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

• Localizable Entanglement can be distinguished by
  classical correlations?

• Relaxing symmetries
• Excited states (in S1/2)
• Higher-spin systems

• How to generalize the LE for qtrits and so on?

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Conclusions

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

• The complex physics of low-dimensional systems
  can be understood better by studying entanglement
  properties.

• Quantum phase transitions ? No universal recipe.
  Some indicators are analytic at QPT, some others
  not.

• Localizable Entanglement ? We have shown that it
  reduces to some already known correlation
  functions

S1/2 classical correlations
S1 string correlations (maximal LE)
Reference L. Campos Venuti and M. Roncaglia, to
appear in PRL, cond-mat/0503021.