Hierarchies of Geometric Entanglement

TL;DR

This paper introduces a hierarchical set of geometric measures to quantify and distinguish different types of entanglement in multipartite quantum states, providing new insights into the structure and ordering of entanglement in various quantum states.

Contribution

It develops a comprehensive framework of generalized geometric entanglement measures that hierarchically quantify bipartite and multipartite entanglement in pure quantum states.

Findings

Establishes a hierarchy between GHZ and W states based on geometric entanglement.

Provides a method for explicit evaluation of multipartite geometric entanglement components.

Shows scale invariance and self-similarity of entanglement in symmetric states.

Abstract

We introduce a class of generalized geometric measures of entanglement. For pure quantum states of $N$ elementary subsystems, they are defined as the distances from the sets of $K$-separable states ($K=2,...,N$). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows to discriminate among the different contributions. The extended measures are applied to the study of entanglement in different classes of $N$-qubit pure states. These classes include $W$ and $GHZ$ states, and their symmetric superpositions; symmetric multi-magnon states; cluster states; and, finally, asymmetric generalized $W$-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between $GHZ$ and $W$ states, clarifying the original result of Wei and Goldbart that $W$ states possess a larger global entanglement than $GHZ$ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.