Experimental observation of fractional topological phases with photonic qudits

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