Open Systems

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Measures of Entanglement at Quantum Phase
Transitions
M. Roncaglia
Condensed Matter Theory Group in Bologna

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

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• 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

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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 T0 reveal entanglement

• 2-qubit 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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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, x1)

• Critical points U-4, U0

• Negativity

• Mutual information

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

A.Anfossi et al., PRL 95, 056402 (2005).
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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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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 CFT with central charge
c1/2
Critical point
Jordan-Wigner transformation
Exactly solvable fermion model
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Near the transition (h1)
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 (QPTs)
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 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.

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

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

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Spin 1 l-D model
l
D
l Ising-like D single ion
Phase Diagram

• Symmetries U(1)xZ2

Around the c1 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 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
(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 LE shows that spin 1 are
• perfect quantum channels but is insensitive to
  phase transitions.

• The lower bound is attained

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A spin-1 model AKLT
Bell state
Optimal basis

• Infinite entanglement length but finite
  correlation length

• Actually in S1 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
  S1 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).
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The End