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Physical Review A

Robustness of the fractional topological phase to dephasing

R. A. Ribeiro, A. A. Matoso, Luis E. Oxman, Antonio Zelaquett Khoury, Sebastião Pádua

Published Apr 1, 2019• 1 citation
Publisher

Abstract

In this work, we demonstrate theoretically via Kraus maps that the fractional topological phase in qudits is robust to phase noise. In our proposal, dephasing noise is inserted in an optical setup designed for measuring the fractional topological phase on photonic qudits encoded in path variables after local unitary operations are applied. Qudit states can be prepared with photon pairs generated by spontaneous parametric down-conversion crossing a multiple-slit array. Polarization as an additional degree of freedom and a two-photon longitudinal interferometer are the basic ingredients of the proposed setup. Although the visibility of the interference pattern decreases when noise is added, our calculation shows that the topological phase is preserved. This result stimulates further studies for the use of fractional topological phase in quantum gates due to its robustness in environments where phase noise is present.

DOI
10.1103/physreva.99.042101

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Scientific Reports•Jan 24, 2019•4 citations
In this work, we demonstrate theoretically via Kraus maps that the fractional topological phase in qudits is robust to phase noise. In our proposal, dephasing noise is inserted in an optical setup designed for measuring the fractional topological phase on photonic qudits encoded in path variables after local unitary operations are applied. Qudit states can be prepared with photon pairs generated by spontaneous parametric down-conversion crossing a multiple-slit array. Polarization as an additional degree of freedom and a two-photon longitudinal interferometer are the basic ingredients of the proposed setup. Although the visibility of the interference pattern decreases when noise is added, our calculation shows that the topological phase is preserved. This result stimulates further studies for the use of fractional topological phase in quantum gates due to its robustness in environments where phase noise is present.
In this work we performed two experiments to observe the fractional topological phases in two qudits photonic systems. The qudits are encoded in the transverse position degree of freedom of photon pairs generated by the spontaneous parametric down conversion after being transmitted through multiple slits. The photons in this phenomenon are quantum correlated in momentum and energy. Initially we review the theory about the origin of the phases acquired by the evolving quantum states of one and two qubits, where we identify the dynamic, the geometric and the topological phases. Then we examine the generalization of topological phases for two qudits, and discuss its relation with the concept of connectedness of a space. Next we show a way to measure the topological phase, considering that it must appear as a relative phase between two interfering quantum states. To realize the measurement we build an optical interferometer with a spatial light modulator inside it to implement the necessary unitary operations. The polarization is used as an auxiliary degree of freedom for the operations to act conditionally. The results are close to the theoretical predictions, showing the dependence of the phase with the Hilbert space dimension of the state. In a second setup we measure the topological phase for qudits without using a longitudinal interferometer. A new source of polarization entangled photon pairs is used in this new setup. The entanglement in the auxiliary degree of freedom, polarization, is used as a resource to eliminate the interferometer, propitiating a significant improvement in the visibility of the measured interference curves, and a large increase in the signal to noise ratio. In this new setup, we are also capable of realize the unitary operations in both parts of the two qubit system.
We observe experimentally the fractional topological phase (FTP) robustness to the dephasing noise in two-qudit photonic systems. We use a source of photon pairs generated by spontaneous parametric down-conversion, hyperentangled in the polarization and photon path degrees of freedom. The dephasing noise is described theoretically via Kraus maps and experimentally implemented via the spatial light modulator. The FTP is a great candidate to explore in the implementation of quantum processing and communication due to its robustness to dephasing noise.
In this work we propose two experimental setups to observe fractional topological phases in qudits. The degree of freedom in which the qudits are encoded is the transverse position of photon pairs generated by spontaneous parametric down-conversion after being transmitted through multiple slits. The photons generated in this phenomenonare quantum correlated in energy and momenta. Initially we discuss the basic theory about topological phases, the context in which they are fractional, and its relation with entanglement. Then, we see how local unitary operations can be applied in this degree of freedom with the aid of spatial light modulators that introduce a programmable phase prole on the wavefront of the photons. Thus, the desired fractional phases canbe observed through polarization-controlled two-photon interference. The roles played by entanglement and Hilbert space dimension are revealed by the behavior of the interference fringes as the unitary operations are continuously changed. These ideas are supported by analytical and numerical calculations of the expected interference fringes. We also show how the so called Hong-Ou-Mandel eect is present in the proposed interferometer, which requires simultaneity on the recombination of the photons generated to allow for the interference
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.
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=2d=2d=2, 3, and 4 were tested, showing the expected theoretical values.
Pairs of photons simultaneously entangled in their path and polarization degrees of freedom are used to measure the topological phase acquired by bipartite entangled states. Conditional phase local unitary operations having the polarization degree of freedom as the control variable are applied. Qudits of arbitrary dimensions are encoded on the photons transverse positions while polarization entanglement is used as an auxiliary resource for quantum interference measurements. With this scheme the fractional phases predicted for dimensions d = 2, 3 and 4 could be measured with visibilities for the interference curves beyond the limit allowed for classical sources, which is expected for a source of quantum correlated photons. The strategy of perform a quantum interferometry experiment with photons entangled in an auxiliary degree of freedom and apply unitary local operations conditioned to this auxiliary variable shows an increase to the signal to noise ratio, simplifies alignment and can be used in different applications. This offers an interesting perspective for the efficient implementation of phase gates in quantum computing with hyperentangled photon sources in polarization and path degrees of freedom. Furthermore, one can conjecture whether the measured phase can serve as a dimensionality identifier of the Hilbert space dimension for an unknown state preparation.