We present an exact solution for the convex roof of the square root of the three-tangle for rank two-density matrices and for all states within the Bloch sphere. Aside from 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 the optimality of the decompositions under consideration. The footprint of the measure of entanglement consists of a characteristic pattern for the fixed pure states on the surface of the Bloch sphere that constitute the optimal solution. This solution is subject to transformation properties due to the SL invariance of the entanglement measure that extends the optimal decomposition found here to all states within the SL class of GHZ (Greenberger-Horne-Zeilinger) and $W$. The method presented here is directly applicable to the symmetric mixture of generalized GHZ and $W$ states for an arbitrary number of qubits, but the main structure of the three-dimensional tetrahedra is general for all rank-two mixtures of states.

Close

PrevNext

View figure in article

• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum information theory