A general method to estimate the convex roof of an SL-invariant entanglement measure

What is it about?

We describe the method of zero-polytope and (minimal) characteristic curves tailor made for entanglement measures which have the symmetry SL. As example we give a three-qubit mixture of rank twowhere this technique is known to give the exact result and also demonstrate for an example of two qubits where the method fails to give the convex roof but leads a lower bound. In particular it contains a general part on how the zero-polytope changes when one has to do with general mixtures.

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Why is it important?

It is for better describing the method that lead to the exact convex roof for the first time in a non-trivial three qubit systems. It as well reports on the pro's but also where this method fails. The zero-manifolds which emerge for arbitrary rank give an idea of how to proceed in this case.