Tangles of superpositions and the convex-roof extension
Andreas Osterloh, Jens Siewert, and Armin Uhlmann

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, 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.
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