Generalized W-state of four qubits with exclusively threetangle

Abstract: We single out a class of states possessing only threetangle but distributed all over four qubits. This is a three-site analogue of states from the $W$-class, which only possess globally distributed pairwise entanglement as measured by the concurrence. We perform an analysis for four qubits, showing that such a state indeed exists. To this end we analyze specific states of four qubits that are not convexly balanced as for $SL$ invariant families of entanglement, but only affinely balanced. For these states all possible $SL$-invariants vanish, hence they are part of the $SL$ null-cone. Instead, they will possess at least a certain unitary invariant. As an interesting byproduct it is demonstrated that the exact convex roof is reached in the rank-two case of a homogeneous polynomial $SL$-invariant measure of entanglement of degree $2m$, if there is a state which corresponds to a maximally $m$-fold degenerate solution in the zero-polytope that can be combined with the convexified minimal characteristic curve to give a decomposition of $\rho$. If more than one such state does exist in the zero polytope, a minimization must be performed. A better lower bound than the lowest convexified characteristic curve is obtained if no decomposition of $\rho$ is obtained in this way.