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We analyze the facial structures of the convex set consisting of all twoqubit separable states. One of faces is a four dimensional convex bodygenerated by the trigonometric moment curve arising from polyhedralcombinatorics. Another one is an eight dimensional convex body, which is theconvex hull of a homeomorphic image of the two dimensional sphere. Extremepoints consist of points on the surface, and any two of them determines anedge. We also reconstruct the trigonometric moment curve in any evendimensional affine space using the qubit-qudit systems, and characterize thefacial structures of the convex hull.

Faces for two qubit separable states and the convex hulls of trigonometric moment curves