The exact convex roof for GHZ-W mixtures for three qubits and beyond

TL;DR

This paper provides an exact analytical solution for the convex roof of the square root of the threetangle in three-qubit states, revealing optimal decompositions and transformation properties related to entanglement measures.

Contribution

It introduces a novel exact solution for the convex roof of the threetangle, including a proof of zero-state locking and an inequality for optimal decompositions.

Findings

Exact convex roof solution for three-qubit states

Optimal decompositions involve zero-polytope states

Transformation properties due to SL-invariance

Abstract

I present an exact solution for the convex roof of the square root of the threetangle for all states within the Bloch sphere. The working horse that optimal decompositions contain as many states from the zero-polytope as possible which can be called zero-state locking is proved and an inequality is derived which decides about the optimality of the decompositions under consideration here. The footprint of the measure of entanglement consists in a characteristic pattern for the fixed pure states on the surface which form the optimal solution. The solution is subject to transformation properties due to the SL-invariance of the entanglement measure.