Complete Hierarchies for the Geometric Measure of Entanglement

TL;DR

This paper introduces a hierarchical method to compute the geometric measure of entanglement in multiparticle quantum states, with proven convergence and applications in entanglement detection and separability testing.

Contribution

The authors develop a novel hierarchical approach based on multiple copies of quantum states to accurately compute entanglement measures, improving existing methods.

Findings

Hierarchical approximations converge to the true entanglement value.

Method enables computation of geometric entanglement for complex states.

Applications include entanglement witnesses and separability tests.

Abstract

In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.