Quantum entanglement: geometric quantification and applications to multi-partite states and quantum phase transitions

TL;DR

This paper introduces a geometric measure of quantum entanglement applicable to various states, analytically computes it for specific cases, and explores its relation to quantum phase transitions and other entanglement measures.

Contribution

It provides an analytical framework for quantifying entanglement geometrically in multi-partite states and links it to quantum critical phenomena.

Findings

Analytical formulas for two-qubit mixed states

Entanglement behavior near quantum critical points

Connections between geometric entanglement and other measures

Abstract

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.