FACES FOR TWO-QUBIT SEPARABLE STATES AND THE CONVEX HULLS OF TRIGONOMETRIC MOMENT CURVES

We analyze the facial structures of the convex set consisting of all two-qubit separable states. One of the faces is a four-dimensional convex body generated by the trigonometric moment curve arising from polyhedral combinatorics. Another one is an eight-dimensional convex body, which is the convex hull of a homeomorphic image of the two-dimensional sphere. Extreme points consist of points on the surface, and any two of them determine an edge. We also reconstruct the trigonometric moment curve in any even-dimensional affine space using the qubit-qudit systems, and characterize the facial structures of the convex hull.