Phase-factor Dependence of the Geometric Entanglement Measure

Any pure three-qubit state is uniquely characterized by one phase and four positive parameters. The geometric measure of entanglement as a function of state parameters can have different expressions that are eigenvalues of the stationarity equations. Each expression has its own applicable domain; thus, the whole state parameter space is divided into subspaces that are ranges of definition for corresponding eigenvalues. These subspaces are invariant under parametrization of states and show the geometry of entangled regions of Hilbert space. The purpose of this paper is to examine the phase (γ)-dependence of the entanglement and the applicable domains for the most general qubit-interchange-symmetric three-qubit states. First, we compute the eigenvalues of the non-linear eigenvalue equations and the nearest separable states for the permutation-invariant three-qubit states with a fixed phase. Next, we compute the geometric entanglement measure, find allocations of highly- and slightly-entangled states and deduce the boundaries of all subspaces. Given a fixed γ, the boundary is the set of states for which eigenvalues of the stationarity equations are degenerate. Thus, the boundary separates subspaces with different eigenvalues, and these eigenvalues coincide on the boundary. When γ≠π/2, there are two invariant subspaces, and boundary states are double degenerate. The entanglement of quantum states is phase independent in the first subspace and increases with γ in the second subspace. However, there are three invariant subspaces at γ≠π/2, and boundary states are either doubly or triply (among them infinitely) degenerate. KCI Citation Count: 0