A Sharp Geometric Measure of Entanglement

TL;DR

The paper introduces the Sharp Geometric Measure (SGM), a new entanglement measure that overcomes limitations of existing geometric measures by effectively distinguishing LU inequivalent states and quantifying genuine multipartite entanglement.

Contribution

It proposes the SGM with a closed-form expression based on Riemannian geometry and defines GMS, a measure of GME that addresses key limitations of previous measures.

Findings

SGM provides a closed-form expression using Riemannian geometry.

GMS effectively quantifies genuine multipartite entanglement.

SGM overcomes limitations of existing geometric measures in distinguishing states.

Abstract

Despite their elegance and widespread use, the current Geometric Measures (GMs) of entanglement exhibit a significant limitation: they fail to effectively distinguish Local Unitary (LU) inequivalent states due to the inherent nature of their definition. We illustrate the impact of this limitation using the fidelity of the teleportation protocol as an example. To address this issue, we introduce the Sharp Geometric Measure (SGM) by modifying the standard definition of the Geometric Measure. We show that the closed-form expression of the SGM can be equivalently derived using the Riemannian structure of both the composite state space and the reduced density operator space. Furthermore, we define a measure of Genuine Multipartite Entanglement (GME) derived from the SGM, which we term GMS. We demonstrate that GMS resolves two key limitations of some existing GME measures, thereby establishing its utility and effectiveness in quantifying GME.