Concurrence triangle induced genuine multipartite entanglement measure

TL;DR

This paper introduces a novel geometric approach using concurrence triangles to quantify genuine multipartite entanglement, providing measures that classify entanglement structures and are applicable to mixed states.

Contribution

It develops a new GME measure based on the geometric mean area of concurrence triangles, extending to mixed states via convex roof and purification-based witnesses.

Findings

The GME measure is non-increasing under LOCC.

It classifies separability and entanglement in multipartite pure states.

Effective for detecting GME in mixed states with examples.

Abstract

We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of $N$-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using the geometric mean area of these concurrence triangles, which are non-increasing under local operation and classical communication. The GME measures classify which parts are separable or entangled with the rest ones for non genuine entangled pure states. The GME measures for mixed states are given via the convex roof construction, and a witness to detect the GME of multipartite mixed states is presented by an approach based on state purifications. Detailed examples are given to illustrate the effectiveness of our GME measures.