1. Topological phase and quantum criptography with spin-orbit entanglement of the photon Universidade Federal Fluminense Instituto de Física - Niterói – RJ - Brasil Antonio Zelaquett Khoury Financial Support: CNPq - CAPES – FAPERJ Financial Support: CNPq - CAPES – FAPERJ INSTITUTO DO MILÊNIO DE INFORMAÇÃO QUÂNTICA

2. Outline Geom. phase for a spin ½ in a magnetic field Geometric quantum computation The Pancharatnam phase Beams carrying OAM Topological phase for entangled states BB84 QKD without a shared reference frame Conclusions

3. Geometric phase of a spin 1/2 in a magnetic field

4. Spin 1/2 in a time dependent magnetic field Spin 1/2 in a time dependent magnetic field BERRY PHASE

5. Geometric quantum computation

6. Geometric conditional phase gate Conditional phase gate J.A. Jones, V. Vedral, A. Ekert, G. Castagnoll, NATURE V.403, 869 (2000) L.-M. Duan, J.I. Cirac, P.Zoller SCIENCE V.292, 1695 (2001)

7. The Pancharatnam phase

8. Pancharatnam phase S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A, V.44, 247 (1956) Collected Works of S. Pancharatnam, Oxford Univ. Press, London (1975).

9. Beams carrying orbital angular momentum

10. Gauss-Laguerre beams carrying OAM (Paraxial Wave Equation) Angular momentum Hermite-Gauss (HG) Rectangular Laguerre-Gauss (LG) Cylindrical

11. Poincaré representation for beams carrying OAM Poincaré representation of first order Gaussian modes Cylindrical lenses at 45 o Astigmatic mode converter

12. Geometric phase from astigmatic mode conversion E.J. Galvez, P.R. Crawford, H.I. Sztul, M.J. Pysher, P.J. Haglin, R.E. Williams, Physical Review Letters V.90, 203901 (2003)

13. Topological phase for entangled states C. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. Khoury IF-UFF P. Milman LMPQ – Jussieu - France

15. Geometric representation for two-qubit states TWO QUBITS  Two Bloch spheres?? Only for product states!!! Bloch sphere (or Poincaré sphere) ONE QUBIT 

16. Geometric representation for two-qubit PURE states Bloch ball SO(3) sphere (opposite points identified) Two-qubit PURE STATES  (Concurrence) Maximally entangled state  Bloch ball colapses to a point!!!! P. Milman and R. Mosseri, Phys. Rev. Lett. 90, 230403 ( 2003 ). P. Milman, Phys. Rev. A 73, 062118 (2006).

17. Topological phase for maximally entangled states Cyclic evolutions preserveing maximal entanglement (“Closed” trajectories)  Two homotopy classes: 0-type trajectories  π-type trajectories  SO(3) sphere

18. Separable polarization-OAM modes

19. Nonseparable polarization-OAM modes Geometric representation on the SO(3) sphere 1 2 3 4

20. Nonseparable mode preparation Holographic preparation of the LG modes PBS

21. Interferometric measurement 1 2 3 4 4’ 1 2 3 4 4  1 (θ = 0 0 ) / 4’  1 (θ = 90 0 ) 4141 CCD θ = 45 0 θ = - 45 0 θ = 0 0 θ = 0 0, 22.5 0, 45 0, 67.5 0, 90 0

22. Experimental results Unseparable mode Separable mode θ=0 0 θ=22.5 0 θ=45 0 θ=67.5 0 θ=90 0 θ=0 0 θ=22.5 0 θ=45 0 θ=67.5 0 θ=90 0

23. Theoretical expressions Unseparable mode Separable mode

24. Calculated images Unseparable mode Separable mode

25. Partial separability and concurrence Partially separable mode Interference pattern (θ=45 0 ) CONCURRENCE

26. BB84 Quantum key distribution without a shared reference frame C. E. Rodrigues de Souza, C. V. S. Borges, J. A. O. Huguenin and A. Z. Khoury IF-UFF L. Aolita and S. P. Walborn IF-UFRJ

27. The BB84 protocol 0 0 1 1 0 0 1 1 ALICE Bennett and Brassard 1984 Polarizers HV +/- HV +/- Polarizers BOB H- 45 o 45 o V Photons

28. 010111100Result HV +/-HV+/- HVBasis 000111101Result HV+/-HV +/-HV+/- Basis     01 1 0 0 Alice and Bob check their basis, but not their results ! ALICE BOB

29. Spin-orbit entanglement Logic basis +/- Logic basis 0/1 Invariant under rotations ! ! ! ! L. Aolita and S. P. Walborn PRL 98, 100501 (2007)

30. BB84 without frame alignment BASIS,,,, Photons,,, Robust against alignment noise ! ! ! ! ALICEBOB

31. Procedure sketch 0 1 + - BOB CNOT X X R(φ) ALICE R(θ)

32. Experimental setup

33. Experimental results Bob’s detector 1 State sent by Alice Bob’s detector 0 Rotation of Alice’s setup Bob’s detector 1 Alice sends 1 Bob’s detector 0, Bob`s detection basis:

34. Conclusions

35. Conclusions Spin-orbit entanglement Topological phase for spin-orbit transformations Potential applications to conditional gates Quantum criptography without frame alignment