Quantifying entanglement from the geometric perspective

TL;DR

This paper reviews the geometric measure of entanglement, a way to quantify quantum entanglement based on the distance to the nearest separable state, highlighting its properties, computation methods, and mathematical connections.

Contribution

It provides a comprehensive overview of the geometric measure of entanglement, including its properties, computational approaches, and links to mathematical problems, advancing understanding in quantum information theory.

Findings

Explains basic properties and operational interpretations of the geometric measure.

Discusses methods for computing the geometric measure of entanglement.

Highlights mathematical relations to eigenvalues and tensor norms.

Abstract

Quantum entanglement between several particles is essential for applications like quantum metrology or quantum cryptography, but it is also central for foundational phenomena like quantum non-locality. This leads to the problem of quantifying the amount of entanglement in a quantum state. We present a review on the geometric measure of entanglement, being a quantifier based on the distance of a state to the nearest separable state. We explain basic properties, existing methods to compute it, its operational interpretations, as well as scaling and complexity issues. We point out intimate relations to fundamental problems in mathematics concerning eigenvalues and norms of tensors. Consequently, the geometric measure of entanglement provides a playground where physical intuition and mathematical rigor benefit from each other.