Symmetric states and dynamics of three quantum bits

Abstract

The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.