Symmetric States and Dynamics of Three Quantum Bits

Abstract

The unitary group acting on the Hilbert space of three quantum bits admits a Lie subgroup, of elements which permute with the symmetric group of permutations. Under the action of such Lie subgroup, the Hilbert space 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 symmetric sector. We provide an analysis of pure states in the symmetric sector of three quantum bits for what concerns their entanglement properties, separability criteria and dynamics. We parametrize all the possible invariant two-dimensional subspaces and extend the previous analysis to these subspaces as well. We propose a physical set up for the states and dynamics we study which consists of a symmetric network of three spin 1/2 particles under a common driving electro-magnetic field. For such set up, we solve a control theoretic problem which consists of driving a separable state to a state with maximal distributed entanglement.