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

TL;DR

This paper investigates the properties of quantum state spaces and their implications for constructing entanglement monotones, revealing differences between discrete and continuous convex roof methods in infinite dimensions.

Contribution

It introduces new insights into the convex roof construction of entanglement monotones in infinite-dimensional quantum systems, highlighting when these constructions are valid or not.

Findings

Discrete convex roof may fail to produce valid entanglement monotones.

Continuous convex roof provides a valid generalized construction.

Infinite-dimensional Entanglement of Formation generalization analyzed.

Abstract

We consider two properties of the set of quantum states as a convex topological space and some their implications concerning the notions of a convex hull and of a convex roof of a function defined on a subset of quantum states. By using these results we analyze two infinite-dimensional versions (discrete and continuous) of the convex roof construction of entanglement monotones, which is widely used in finite dimensions. It is shown that the discrete version may be 'false' in the sense that the resulting functions may not possess the main property of entanglement monotones while the continuous version can be considered as a 'true' generalized convex roof construction. We give several examples of entanglement monotones produced by this construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.