Geometric Measure of Entanglement and Shared Quantum States

TL;DR

This paper derives explicit formulas for the geometric measure of entanglement in three-qubit states composed of four orthogonal product states, revealing geometric interpretations and special shared states with unique properties.

Contribution

It provides explicit expressions for the geometric entanglement measure for specific three-qubit states and explores the properties of shared states at parameter boundaries.

Findings

Three different formulas for the entanglement measure depending on parameter ranges

Shared states on boundary surfaces have unique entanglement features

Geometric interpretations aid understanding of entanglement properties

Abstract

We give an explicit expression for the geometric measure of entanglement for three qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space. Each expression of the measure has its own geometrically meaningful interpretation. Such an interpretation allows oneself to take one step toward a complete understanding for the general properties of the entanglement measure. The states that lie on joint surfaces separating different ranges of definition, designated as shared states, seem to have particularly interesting features. The properties of the shared states are fully discussed.