\n The combination of the polarization and spatial photonic degre
es of freedom open interesting possibilities in the quantum optics and qua
ntum-information domains. The ability to produce and transform beams carry
ing orbital angular momentum &ndash\; the optical vortices - has allowed t
he development of important techniques with potential applications to quan
tum information. In this seminar I will present some optical devices to qu
antum information and I will talk about the topological phase\, associated
with the double connectedness of the *SO* [3] representation in te
rms of maximally entangled states. Under local unitary operations on their
polarization and transverse degrees of freedom\, the vector vortices can
only acquire discrete geometric phase values\, 0 or &pi\;\, associated wit
h closed paths belonging to different homotopy classes on the SO(3) manifo
ld. As the meaning results I will show that the topological phase can be e
videnced through interferometric measurements and that a quantitative rela
tionship between the concurrence and the fringes visibility can be derived
as well.

\n [1] C. E. R. Souza\, J. A. O. Huguenin and A. Z. Khou
ry *Topological phase structure of vector vortex beams*\, JOSA-A ~~31\, No. 5 1007 (2014).~~

\n [2] C. E. R. Souza\, J.
A. O. Huguenin\, P. Milman and A. Z. Khoury *Topological Phase for Spin
-Orbit Transformations on a Laser Beam* PRL** 99**\, 1604
01 (2007).

\n [3] C. E. R. Souza and A. Z. Khoury *A Michelson
controlled-not gate with a **single-lens astigmatic mode converter
*Optics Express **18**\, No. 9 9207 (2010).