1. 1 Separability and entanglement: what symmetries and geometry can say Helena Braga, Simone Souza and Salomon S. Mizrahi Departamento de Física, CCET, Universidade Federal de São Carlos 11 th ICSSUR/ 4 th Feynman Festival, June 22- 26 2009, Palacký University, Olomouc – Czech Republic

2. 2 1.Introduction 2.Polarization vector, correlation matrix 3.Peres-Horodecki criterion 4.New parameters and symmetries 5.Phase space: entangled-like and separable-like regions 6.Distance, concurrence and negativity 7.Examples 8.Conclusions

3. 3 Paradigm of entanglement: the two-qubit system Bohm, two spin-1/2 particles: 1952 Lee and Yang, intrinsic parity Phys. Rev. 104, 822 (1956)

4. 4 Multipartite state up to 2nd order correlation: 4 X 4 matrix (15 parameters) General multipartite state

5. 5 In matrix form

6. 6 Separability

7. 7 Reduction to seven parameters and the X-from Transposition and Peres-Horodecki criterion

8. 8 Additional symmetries related to the eigenvalues of the transposed matrix The eigenvalues of the TM can be obtained from the original state by choosing six among different local reflections

9. 9 Introducing new parameters Eigenvalues of the original matrix Cond. of pos.

10. 10 Do the same with the transposed or locally reflected matrix According to PHC if both ‘distances’ of the locally reflected matrix are positive the state is separable otherwise it is entangled Define ‘phase spaces’: X X = 0 define conic surfaces = 0 and

11. 11 Phase space can be divided in two regions, one for the entangled-like states and the other for the separable-like Systems having the X-form are met in the literarature,

12. 12

13. 13 Werner state Peres-Horod state A.Peres, PRL 77, 1413 (1996) Horodeckis, PLA 223, 1 (1996) R.F. Werner, PRA 40, 4277 (1989)

14. 14 N. Gisin, Phys. Lett. A 210, 151 (1996 )

15. 15 M.P. Almeida et al., Science 316, 579 (2007) Experimental work, entangled photon polarization and simulating dynamical evolution

16. 16 S. Das and G.S. Agarwal, arXiv:0901.2114v2 [quant-ph] 20 May 2009

17. 17 Comparisons between entanglements measures 1. concurrence, 2. Vidal-Werner negativity 3. distance-negativity

18. 18 A.Peres, PRL 77, 1413 (1996) Horodeckis, PLA 223, 1 (1996)

19. 19 R.F. Werner, PRA 40, 4277 (1989)

20. 20 N. Gisin, Phys. Lett. A 210, 151 (1996)

21. 21 M.P.Almeida et al., Science 316, 579 (2007)

22. 22 S. Das and G.S. Agarwal, arXiv:0901.2114v2 [quant-ph] 20 May 2009

23. 23 Conclusions 1.Matrix form symmetries imply physical symmetries when expressed in terms of meaningful physical quantities. 2.Defining new parameters, a phase space and distances, the PHC acquires a geometrical meaning: one can follow the trajectories of states and to discern graphically between entangled and separable states. 3.The negative of squared distance (the distance negativity) can be used as a measure of entanglement, which is comparable to other measures.

24. 24 Thank you!