Quantifying entanglement from the geometric perspective

TL;DR

This review surveys the geometric measure of entanglement, defined by $E(\psi)=1-\Lambda(\psi)^2$ with $\Lambda(\psi)=\max_{\text{separable}}|\langle\text{prod}|\psi\rangle|$, and extends from bipartite to multipartite systems while addressing computational, asymptotic, and mixed-state aspects. It establishes deep connections to the injective tensor norm, tensor eigenvalues, and odeco/fradeco decompositions, highlighting both mathematical structure and algorithmic approaches such as the see-saw method and PPT relaxations. The paper also discusses operational interpretations in LOCC discrimination, links to measurement-based quantum computation and quantum channels, and analyzes typical entanglement scaling in large systems. It concludes with a constructive discussion of how the geometric perspective informs both theory and potential applications, and outlines future directions like higher-order approximations and constrained optimizations that may broaden the utility of entanglement quantification. The work thus provides a rigorous, interdisciplinary framework for understanding and computing entanglement from a geometric viewpoint, with implications for quantum information processing and foundational mathematics alike.

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.