Giuseppe Florio

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Multipartite Entanglement in a Quantum Phase
Transition via Probability Density Function

• Giuseppe Florio
• Dipartimento di Fisica, Università di Bari, Italy

P. Facchi Dipartimento di Matematica, Università
di Bari, Italy S. Pascazio, G. Costantini
Dipartimento di Fisica, Università di Bari, Italy
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Objective

• Explore the link between
• ENTANGLEMENT
• and
• Quantum Phase Transitions (QPT)
• Osterloh, Amici, Falci, Fazio. Nature 2002
• Osborne, Nielsen, PRA 2002
• Calabrese, Cardy, J. Stat. Mech. Theory Expt.
  2004
• Gu, Deng, Li, Ling, PRL 2004
• Vidal, Latorre, Rico, Kitaev, PRL 2005 AND
  MANY MORE...

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An initial remark

• Parisi complex systems
• Manko, Marmo, Sudarshan, Zaccaria multipartite
  entanglement (J. Phys. A 02-03)
• One (or a few) real number(s) is/are not enough
• number of measures (i.e. real numbers) needed
  to quantify multipartite entanglement grows
  exponentially with nnumber of qubits

statistical methods!
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A measure of Entanglement
Purity
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A measure of Entanglement
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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 distribution of characterizes
  multipartite 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.
• P. Facchi, G.F., S. Pascazio, Phys. Rev. A 74,
  042331 (2006)

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An example GHZ Greenberger, Horne, Zeilinger
(1990)
For all bipartitions!!
Well distributed entanglement
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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.
• At T0 the system is frozen in the ground state
  without fluctuations.

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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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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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An example 10 spins Florio et al., J.Phys.
A (in print), quant-ph/0612098
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Results (7-11 spins)
n
GHZ (approximately)
Separable states
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Results (7-11 spins)
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Results (7-11 spins)
This shows that our entanglement characterization
sees the Quantum Phase Transition!
Analogous results for the width of the
distribution
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Results (7-11 spins)
What about the behavior of average and width?
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Results (7-11 spins)
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Results (7-11 spins)

• A consequence

• At the QPT the entanglement of the ground state
  is insensitive to the bipartition. Therefore it
  could be a good tool for generating multipartite
  entanglement.
• Is there a relation between QPT and chaotic
  systems (high value of entanglement, well
  distributed)?

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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.
• From numerical evidences we obtain that the
  amount of entanglement AND the width diverge

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Conclusions and perspectives

• Quantum Phase Transition analytical evaluation
  of entanglement (conformal field theory,
  renormalization group).
• Is a QPT a good tool for distributing
  entanglement? More evidences needed other models
  (XX, XY in progress).
• What is the effect of interactions beyond nearest
  neighbour? (in progress)

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Further details
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Typical states
Independent uniformly distributed random variables
Random point uniformly distributed on the
hypersphere
with distribution
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For it is possible to obtain the
distribution
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Quantum Ising model in 1DPfeuty (1976) Lieb et
al. (1961) Katsura (1962)

• Quantum Ising model in a transverse field

• Coupling limits

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Possible scenarios

• The last conclusion is particularly significant
  the amount of entanglement goes to infinity but
  so does the width of the entanglement
  distribution. Two scenarios are possible

• srel vanishes for larger n. This means that at
  the QPT the entanglement of the GS is
  macroscopically insensitive to the choice of the
  bipartitions.
• srel does not vanishes. In this case the strong
  divergence of s(µmax) would imply that the
  distribution of entanglement is not optimal. This
  means that the amount of entanglement of
  non-contiguous spins partitions macroscopically
  differs from that of contiguous ones