A Gramian Approach to Entanglement in Bipartite Finite Dimensional Systems: The case of pure states

TL;DR

This paper introduces a geometric Gram matrix approach to analyze entanglement in bipartite pure states, proposing a new entanglement measure and a non-linear purification method based on the Gram matrix structure.

Contribution

It presents a novel geometrical framework using Gram matrices for quantifying and understanding entanglement in bipartite pure states, including a new measure and purification technique.

Findings

Gram matrix structure characterizes reduced density matrices.

A new geometrical entanglement measure is proposed.

A non-linear purification method based on Gram matrices is developed.

Abstract

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.