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Here I present a method how intersections of a certain density matrix of ranktwo with the zero-polytope can be calculated exactly. This is a purelygeometrical procedure which thereby is applicable to obtaining the zeros of SL-and SU-invariant entanglement measures of arbitrary polynomial degree. Iexplain this method in detail for a recently unsolved problem. In particular, Ishow how a three-dimensional view, namely in terms of the Boch-sphere analogy,solves this problem immediately. To this end, I determine the zero-polytope ofthe three-tangle, which is an exact result up to computer accuracy, andcalculate upper bounds to its convex roof which are below the linearized upperbound. The zeros of the three-tangle (in this case) induced by thezero-polytope (zero-simplex) are exact values. I apply this procedure to asuperposition of the four qubit GHZand W-state. It can however be applied toevery case one has under consideration, including an arbitrary polynomialconvex-roof measure of entanglement and for arbitrary local dimension.

Exact zeros of entanglement for arbitrary rank-two mixtures: how a geometric view of the zero-polytope makes life more easy