Convex bodies of states and maps (Q2857966)

The authors give a general solution to the question of when convex hulls of orbits of quantum states under unitary actions of compact groups have a nonempty interior in the surrounding space of density matrices. For concrete applications with respect to sets with nonempty interior, quantitative answers depend on a particular choice of volume measure. The main theorem expresses a convex hull of orbits with empty interior in terms of the existence of a proper invariant subspace and the points which are fixed under the compact group. Also, it is given a ``unique characteristic of maximally entangled states in terms of convexed orbits of the local group through them''. In terms of convex bodies, several examples are discussed, e.g. convex body of density operators, separable states, maximally entangled pure states and mixed-unitary channels.