On properties of the space of quantum states and their application to the construction of entanglement monotones

We consider infinite-dimensional versions of the notions of the convex hull and convex roof of a function defined on the set of quantum states. We obtain sufficient conditions for the coincidence and continuity of restrictions of different convex hulls of a given lower semicontinuous function to the subset of states with bounded mean generalized energy (an affine lower semicontinuous non-negative function). These results are used to justify an infinite-dimensional generalization of the convex roof construction of entanglement monotones that is widely used in finite dimensions. We give several examples of entanglement monotones produced by the generalized convex roof construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.