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arXiv:quant-ph/0302202 · Physical Review Letters

Topological Phase for Entangled Two-Qubit States

P. Milman, Rémy Mosseri

Published Jun 12, 2003• 59 citations
arXivHTMLPDF

Abstract

Entangled states play a crucial role in quantum physics, ranging from fundamental aspects to quantum information processing. We show here that entangled two-qubit states can also be used to characterize unambiguously the subtlety of the SO(3) rotation group topology. The well known two distinct families of path in this group are put in one-to-one correspondence with cyclic evolutions of these entangled states, resulting in a pi phase difference. We propose a simple quantum optics interference experiment to demonstrate this topological phase shift.

DOI
10.1103/physrevlett.90.230403

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Entangled states play a crucial role in quantum physics, ranging from fundamental aspects to quantum information processing. We show here that entangled two-qubit states can also be used to characterize unambiguously the subtlety of the SO(3) rotation group topology. The well known two distinct families of path in this group are put in one-to-one correspondence with cyclic evolutions of these entangled states, resulting in a pi phase difference. We propose a simple quantum optics interference experiment to demonstrate this topological phase shift.
We discuss the representation of the SO(3)SO(3)SO(3) group by two-qubit maximally entangled states (MES). We analyze the correspondence between SO(3)SO(3)SO(3) and the set of two-qubit MES which are experimentally realizable. As a result, we offer a new interpretation of some recently proposed experiments based on MES. Employing the tools of quantum optics we treat in terms of two-qubit MES some classical experiments in neutron interferometry, which showed the πππ-phase accrued by a spin-1/21/21/2 particle precessing in a magnetic field. By so doing, we can analyze the extent to which the recently proposed experiments - and future ones of the same sort - would involve essentially new physical aspects as compared with those performed in the past. We argue that the proposed experiments do extend the possibilities for displaying the double connectedness of SO(3)SO(3)SO(3), although for that to be the case it results necessary to map elements of SU(2)SU(2)SU(2) onto physical operations acting on two-level systems.
We propose an experiment to observe the topological phases associated with cyclic evolutions, generated by local SU(2) operations, on three-qubit entangled states prepared on different degrees of freedom of entangled photon pairs. The topological phases reveal the nontrivial topological structure of the local SU(2) orbits. We describe how to prepare states showing different topological phases and discuss their relation to entanglement. In particular, the presence of a $\ensuremath{\pi}/2$ phase shift is a signature of genuine tripartite entanglement in the sense that it does not exist for two-qubit systems.
Quantum mechanical phase factors can be related to dynamical effects or to the geometrical properties of a trajectory in a given space---either parameter space or Hilbert space. Here, we experimentally investigate a quantum mechanical phase factor that reflects the topology of the SO(3) group: since rotations by $\ensuremath{\pi}$ around antiparallel axes are identical, this space is doubly connected. Using a pair of nuclear spins in a maximally entangled state, we subject one of the spins to a cyclic evolution. If the corresponding trajectory in SO(3) can be smoothly deformed to a point, the quantum state at the end of the trajectory is identical to the initial state. For all other trajectories the quantum state changes sign.
We make a geometric study of the phases acquired by a general, pure bipartite two-level system after a cyclic unitary evolution. The geometric representation of the two particle Hilbert space makes use of Hopf fibrations. It allows for a simple description of the dynamics of the entangled state's phase during the whole evolution. The global phase after a cyclic evolution is always an entire multiple of $\ensuremath{\pi}$ for all bipartite states, a result that does not depend on the degree of entanglement. There are three different types of phases combining themselves so as to result in the $n\ensuremath{\pi}$ global phase. They can be identified as dynamical, geometrical, and topological. Each one of them can be easily identified using the presented geometric description. The interplay between them depends on the initial state and on its trajectory, and the results obtained are shown to be in connection to those on mixed state phases.
A representation of the SO(3)\mathrm{SO}(3)SO(3) group is mapped into a maximally entangled two qubit state according to the literature. To show the evolution of the entangled state, a model is set up on an maximally entangled electron pair, two electrons of which pass independently through a rotating magnetic field. It is found that the evolution path of the entangled state in the SO(3)\mathrm{SO}(3)SO(3) sphere breaks an odd or even number of times, corresponding to the double connectedness of the SO(3)\mathrm{SO}(3)SO(3) group. An odd number of breaks leads to an additional $\ensuremath{\pi}$ phase to the entangled state, but an even number of breaks does not. A scheme to trace the evolution of the entangled state is proposed by means of entangled photon pairs and Kerr medium, allowing observation of the additional $\ensuremath{\pi}$ phase.
How to reveal topological phases and their boundaries is an intriguing issue in various systems. Entanglement, which plays a fundamental role in quantum information, has been found profoundly related to topological phases. However, experimentally exploring this relation is precluded by the limited ability to obtain entanglement in many-body systems. In this work, we propose and experimentally demonstrate that the robustness of entanglement, quantified by the von Neumann entropy, can be used to reveal the topological phase with winding number W = 1 and topological phase with W = 0 in quantum walks. With the different robustness of entanglement against perturbations of a parameter, the phase boundaries between the distinct topological phases can be further determined. As a result, our work not only offers a new perspective for quantum walks but also exhibits the deep connection between entanglement and topological physics.