Figure 1

The tetrahedron or 3-simplex associated with three-qubit $X$ states. In (a), the 15 operators in Eq. (3) are identified with the points of the simplex (vertices, midpoints of edges, face centers, and body center of the tetrahedron). For clarity, only a few points are labeled. The same tetrahedron is opened out into a planar diagram in (b), resulting in the vertex associated with ${\mathrm{D̂}}_4$ repeated three times. Six lines connecting pairs of face centers and all seven lines through the body center are omitted for clarity. Arrowed lines connecting three operators denote that the product of any two gives the third operator in a cyclic fashion, with a multiplicative $\pm i$. Unarrowed lines denote the product of any two as the third operator, regardless of the order.Reuse & Permissions