Maxim Evgenievich Shirokov

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.

On characterization of positive maps preserving continuity of the von Neumann entropy

We show that a positive linear map preserves local continuity (convergence) of the entropy if and only if it preserves finiteness of the entropy, i.e. transforms operators with finite entropy to operators with finite entropy. The last property is equivalent to the boundedness of the output entropy of a map on the set of pure states.