Robustness of the fractional topological phase to noise and its experimental demonstration

Abstract

In 1984, Berry discovered the Geometric Phase that became a characterization parameter of a quantum system. According to Berry, the geometric phase carries itself the geometric and the topologic properties of the parameter space. The Fractional Topological Phase (FTP), whose value is 2π/d, where d is the dimension of the bipartite subsystem of a two qudits state, was discovered by Khoury and Oxman in 2011. The FTP 2π/d brings as a signature the dimension d of a subsystem of the Hilbert Space associated with the bipartite state. In this work, we use the Kraus’ Maps formalism to theoretically demonstrate that the FTP is robust to the dephasing and amplitude damping noises. Our calculation uses the theoretical proposal for measuring the FTP discussed in Phys. Rev. A, 87: 042 113 (2013) *. We demonstrate experimentally that the FTP is robust to the dephasing noise. In the theoretical demonstration of the FTP’s robustness that guided our calculation, we based on two experimental proposals. The first one was an interferometer in photon paths variables consisting of two Mach-Zehnder interferometers with PBSs immersed in an interferometer composed of a crystal and a PBS * (this composition can be called a triple composite Mach-Zehnder with PBSs). The second one uses a source of hyperentangled photons pairs, in the polarization and photon paths parameters. For the interferometer in paths variables, we present a simplification that is able to measure the FTP only with the interferometer Crystal-PBS without the Mach-Zehnders. The hyperentangled source is a new technique to measure the FTP with better use of the signal and greater stability of the interferometer. In addition, although there are some losses in relation to the possibilities of SU(d) evolutions of the states, the hyperentangled source is able to eliminate completely the use of interferometry in path variables in the FTP measurement – a procedure not yet used until the publication of our experimental work. We demonstrate experimentally that the hyperentangled source implements the FTP measurement without the need of photon path interferometry. With this setup, we are able to demonstrate the robustness of FTP by the presence of dephasing noise and observe great consistency between the experimental results and the theoretical model predictions. This fact leads us to believe that FTP is a really reliable candidate for the implementation of quantum algorithms based on topological phase. Noise insensitive elements are central to quantum information applications because manipulation of information and its propagation means acting on the quantum state or putting it in contact with environment noise. In general, the state is extremely sensitive to environment noisy operation and interaction, which often destroy the information contained in the state.