Probability density function characterization of Multipartite Entanglement

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Probability density function characterization of
Multipartite Entanglement
G. Florio Dipartimento di Fisica, Università di
Bari, Italy In collaboration with P.
Facchi Dipartimento di Matematica, Università di
Bari, Italy S. Pascazio Dipartimento di Fisica,
Università di Bari, Italy
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Objective

• Explore link between
• ENTANGLEMENT
• And
• Quantum Phase Transitions
• see also Vidal et al. PRL (2005)
• A. Osterloh et al. Nature (2002)

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(Classical) Phase Transitions

• Discontinuity in one or more physical properties
  due to a change in a thermodynamic variable such
  as the temperature
• Typical example Ferromagnetic system
• Below a critical temperature Tc, it exhibits
  spontaneous magnetization.

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(Quantum) Phase Transitions

• The transition describes a discontinuity in the
  ground state of a many-body system due to its
  quantum fluctuations (at 0 temperature).
• Level crossing between ground state and excited
  states.
• Examples of scaling laws

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What is Entanglement?
If one can write
then the state is SEPARABLE.
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What is Entanglement?
Bell (or EPR) state
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What is Entanglement?
Separable State
This is a general behavior of Separable States
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What is Entanglement?
Purity
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What is Entanglement?
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What is Entanglement?
A system of n objects can be partitioned in two
subsystems A and B
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Participation Number
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Objective evaluate entanglement

• Clearly, the quantity will depend on the
  bipartition, according to the distribution of
  entanglement among all possible bipartitions

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Objective evaluate entanglement
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Objective evaluate entanglement

• The average will be a measure of the amount of
  entanglement in the system, while the variance
  will measure how well such entanglement is
  distributed a smaller variance will correspond
  to a larger insensitivity to the choice of the
  partition.
• The distribution of is a measure of
  entanglement.
• See Facchi P., Florio G., Pascazio S.
• (quant-ph/0603281)

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An example GHZ Greenberger, Horne, Zeilinger
(1990)
For all bipartitions!!
Well distributed entanglement
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An example GHZ Greenberger, Horne, Zeilinger
(1990)
For all bipartitions!!
Well distributed entanglement
But low amount of entanglemnt
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The system Pfeuty (1976) Lieb et al. (1961)
Katsura (1962)

• Quantum Ising model in a transverse field

• It exhibits a QPT for

Energy gap
Correlation Length
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Results (3-11 sites)
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Results (7-11 sites)
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Results (7-11 sites)
This shows that our entanglement characterization
sees the Quantum Phase Transition!
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Results (3-11 sites)
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Results (3-11 sites)
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Conclusions

• Entanglement can be characterized using its
  distribution over all possible bipartitions
  (average AND width).
• This characterization sees the QPT of the Ising
  Model with transverse field.
• Apparently the amount of entanglement AND the
  width diverge.
• Evaluation of analytical expressions in progress.