FACES FOR TWO-QUBIT SEPARABLE STATES AND THE CONVEX
HULLS OF TRIGONOMETRIC MOMENT CURVES
Abstract: 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.
2000 AMS Mathematics Subject Classification: Primary: 81P16, 52B11, 15A30,
46L30. Keywords and phrases: Separable state, entanglement, trigonometric moment
curve, face, extreme point.