Convex bodies of states and maps

TL;DR

This paper provides a general framework for determining when the convex hulls of quantum states and channels under group actions have non-empty interiors, with applications to entanglement and state classification.

Contribution

It introduces a unified approach to analyze convex hulls of quantum states and channels, characterizing maximally entangled states and comparing geometric properties of convex bodies.

Findings

Convex hulls of orbits can have non-empty interiors under certain conditions.

Maximally entangled states are characterized by properties of convex hulls of their orbits.

Largest inscribed balls and maximum volume ellipsoids in convex bodies generally differ.

Abstract

We give a general solution to the question when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density states. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.