Effect of Phase Factor in the Geometric Entanglement Measure of Three-Qubit States

TL;DR

This paper investigates how the phase factor influences the geometric entanglement measure in three-qubit states, revealing domain structures and the special role of the W-state in entanglement properties.

Contribution

It provides a detailed analysis of the applicable domains for the geometric entanglement measure in symmetric three-qubit states with a phase factor, highlighting the emergence of three domains at phase π/2.

Findings

Three applicable domains identified for phase π/2 due to W-state emergence.

Highly entangled states are near domain boundaries, less entangled states are farther away.

Neighbors of W-state are more entangled than GHZ neighbors, but GHZ neighbors have a wider range.

Abstract

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. Each of expressions has its own applicable domain and thus the whole state parameter space is divided into subspaces that are ranges of definition for corresponding expressions. The purpose of this paper is to examine 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, deduce the boundaries of all subspaces, and find allocations of highly and slightly entangled states. It is shown that there are three applicable domains when the phase factor is $\pi/2$ while other cases have only two domains. The emergence of the three domains is due to the appearance of the additional W-state. We show that most of highly entangled states reside near the boundaries of the domains and states located far from the boundaries become less-entangled and eventually go to the product states. The neighbors of W-state are generally more entangled than the neighbors of Greenberger-Horne-Zeilinger(GHZ) state from the aspect of the geometric measure. However, the range of the GHZ-neighbors is much more wider than the range of the W-neighbors.