A comparison of old and new definitions of the geometric measure of entanglement

TL;DR

This paper reviews and compares various definitions of the geometric measure of entanglement, analyzing their properties, equivalences, and differences, and identifies a promising candidate for multipartite entanglement studies.

Contribution

It provides a comprehensive comparison of existing and new geometric entanglement measures, clarifies their relationships, and highlights a particularly suitable measure for future multipartite entanglement research.

Findings

Logarithmic definitions of GM are distinct, unlike linear ones which are equivalent.

A lesser-known GM definition is normalized and weakly monotonous.

Certain graph states have a universal closest separable state for multiple distance measures.

Abstract

Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys 12 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as "maximally correlated states" and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a "universal" closest separable state that minimizes the quantum relative entropy, the Bures distance and the trace distance.