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

5. then the state is SEPARABLE.
What is Entanglement?
Consider a state
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. This is a general behavior of Separable States…
What is Entanglement?
Separable State
This is a general behavior of Separable States…
Bari SM&FT 2006

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

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

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

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

15. An example: GHZ [Greenberger, Horne, Zeilinger (1990)]1) 2) 3)
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. Results (3-11 sites) 3 11
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