Geometric measures of entanglement and the Schmidt decomposition

TL;DR

This paper generalizes the geometric measure of entanglement for pure states by relating it to the distance to the closest separable state and connecting it with the Schmidt decomposition's eigenstructure.

Contribution

It extends the geometric entanglement measure to unnormalized states and links it to the Schmidt decomposition, providing new interpretations of the closest separable state.

Findings

Entanglement measure $\, ext{sin}^2 heta$ equals the distance to the closest separable state.

Components of the closest separable state relate to eigenvectors of reduced density matrices.

Norm of the closest separable state relates to eigenvalues in Schmidt decomposition.

Abstract

In the standard geometric approach to a measure of entanglement of a pure state, $\sin^2\theta$ is used, where $\theta$ is the angle between the state to the closest separable state of products of normalized qubit states. We consider here a generalization of this notion to separable states consisting of products of unnormalized states of different dimension. In so doing, the entanglement measure $\sin^2\theta$ is found to have an interpretation as the distance between the state to the closest separable state. We also find the components of the closest separable state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector.