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

I present an exact solution for the convex roof of the square root of the threetangle for rank two density matrices and for all states within the Bloch sphere.

Aside the formerly known two tetrahedra it contains three additional optimal tetrahedra and connecting triangular optimal decompositions.

The remaining optimal decompositions are one-dimensional.

Optimal decompositions are proved to contain as many states from the zero-polytope as possible, a property that is called zero-state locking; it will be the working horse throughout this work.

In addition, an inequality is derived which decides about the optimality of the decompositions under consideration.

The footprint of the measure of entanglement consists in a characteristic pattern for the fixed pure states on the Bloch sphere surface which constitute the optimal solution.

This solution is subject to transformation properties due to the SL-invariance of the entanglement measure which renders the optimal decomposition found here to all the states within the SL-class of GHZ and W.

The method presented here is directly applicable to the symmetric mixture of generalized GHZ and W states for arbitrary number of qubits but the main structure of the 3-dimensional tetrahedra is general for all rank-two mixtures of states.