Tangles of superpositions and the convex-roof extension

TL;DR

This paper explores the convex-roof extension of multipartite entanglement measures, introducing the zero-polytope and convex characteristic curve as tools to analyze entanglement in quantum states, with applications to two- and three-qubit systems.

Contribution

It introduces the zero-polytope and convex characteristic curve concepts for analyzing the convex-roof extension of entanglement measures, enhancing understanding of entanglement in mixed states.

Findings

Zero-polytope identifies states with zero tangle.

Convex characteristic curve provides lower bounds for convex roof.

Applications demonstrated on two- and three-qubit states.

Abstract

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $SL(2,\CC)$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the {\em zero-polytope} is a convex set of density matrices with vanishing tangle whereas the {\em convex characteristic curve} readily provides a non-trivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero-polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.