  Geometrical and topological phases play a fundamental role in quantum theory.
Geometric phases have been proposed as a tool for implementing unitary gates
for quantum computation. A fractional topological phase has been recently
discovered for bipartite systems. The dimension of the Hilbert space determines
the topological phase of entangled qudits under local unitary operations. Here
we investigate fractional topological phases acquired by photonic entangled
qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac
interferometer and the two-photon interference pattern reveals the topological
phase as fringes shifts when local operations are performed. Dimensions $d = 2,
3$ and $4$ were tested, showing the expected theoretical values.
