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arXiv:quant-ph/0403099 · Physical Review A

Representation of the SO(3) group by a maximally entangled state

W. LiMing, Zhilie Tang, Cai-Hong Liao

Published Jun 11, 2004• 15 citations
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Abstract

A representation of the $\mathrm{SO}(3)$ 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 $\mathrm{SO}(3)$ sphere breaks an odd or even number of times, corresponding to the double connectedness of the $\mathrm{SO}(3)$ 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.

DOI
10.1103/physreva.69.064301

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ phase.
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.
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.
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.
The structure of the state spaces of bipartite $N\ensuremath{\bigotimes}N$ quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin jjj which transform according to an irreducible representation of the rotation group. A positive map $\ensuremath{\vartheta}$ is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation ${\ensuremath{\vartheta}}_{2}=I\ensuremath{\bigotimes}\ensuremath{\vartheta}$ acting on the composite system can be expressed in terms of the invariant $6\text{\ensuremath{-}}j$ symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of $4\ensuremath{\bigotimes}4$ systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive map is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of $4\ensuremath{\bigotimes}4$ systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.
We investigate the topological phase associated with the double connectedness of the SO(3) representation in terms of maximally entangled states. An experimental demonstration is provided in the context of polarization and spatial mode transformations of a laser beam carrying orbital angular momentum. The topological phase is evidenced through interferometric measurements, and a quantitative relationship between the concurrence and the fringes visibility is derived. Both the quantum and the classical regimes were investigated.
Abstract We present an operational reconstruction of the well-known two-to-one homomorphism between the groups SU(2) and SO(3), grounded in the physical description of quantum state preparation and evolution. Starting from the connection between vectors in three-dimensional physical space and quantum states of two-level systems, we investigate how reversible transformations–modeled as completely positive and trace-preserving maps–give rise to a correspondence between spatial rotations and unitary operations. Our approach reveals how this group-theoretic structure naturally emerges from physical constraints, particularly the preservation of purity and reversibility in quantum processes. Beyond its theoretical relevance, the construction offers a pedagogically accessible framework for introducing core ideas in quantum mechanics and symmetry groups, making the abstract correspondence between SU(2) and SO(3) tangible through experimentally meaningful procedures.