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Probability density function characterization of Multipartite Entanglement

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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Probability density function characterization of Multipartite Entanglement

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  1. 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 Bari SM&FT 2006

  2. Objective Explore link between ENTANGLEMENT And Quantum Phase Transitions [see also Vidal et al. PRL (2005); A. Osterloh et al. Nature (2002) ] Bari SM&FT 2006

  3. (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. Bari SM&FT 2006

  4. HOW CAN WE CHARACTERIZE A QPT? Entanglement Energy gap Correlation Length (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: Bari SM&FT 2006

  5. Consider a state What is Entanglement? If one can write then the state is SEPARABLE. Bari SM&FT 2006

  6. What is Entanglement? Bell (or EPR) state Bari SM&FT 2006

  7. What is Entanglement? Separable State This is a general behavior of Separable States… Bari SM&FT 2006

  8. : Eigenvalues of : Dimension of the Hilbert space used to describe the system What is Entanglement? Purity Bari SM&FT 2006

  9. For Separable States… What is Entanglement? Bari SM&FT 2006

  10. B A What is Entanglement? A system of n objects can be partitioned in two subsystems A and B Bari SM&FT 2006

  11. B A Hilbert space of subsystem A Participation Number Bari SM&FT 2006

  12. Objective: evaluate entanglement • Clearly, the quantity will depend on the bipartition, according to the distribution of entanglement among all possible bipartitions Bari SM&FT 2006

  13. B A A B Objective: evaluate entanglement Bari SM&FT 2006

  14. 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) Bari SM&FT 2006

  15. A 1) A 2) A 3) An example: GHZ [Greenberger, Horne, Zeilinger (1990)] For all bipartitions!! Well distributed entanglement Bari SM&FT 2006

  16. An example: GHZ [Greenberger, Horne, Zeilinger (1990)] For all bipartitions!! Well distributed entanglement But low amount of entanglemnt Bari SM&FT 2006

  17. 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 Bari SM&FT 2006

  18. 11 3 Results (3-11 sites) Bari SM&FT 2006

  19. Results (7-11 sites) Bari SM&FT 2006

  20. Results (7-11 sites) This shows that our entanglement characterization “sees” the Quantum Phase Transition! Bari SM&FT 2006

  21. Results (3-11 sites) Bari SM&FT 2006

  22. Results (3-11 sites) Bari SM&FT 2006

  23. 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. Bari SM&FT 2006

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