The Schmidt number and partially entanglement breaking channels in infinite dimensions

TL;DR

This paper extends the concept of the Schmidt number and partially entanglement breaking channels to infinite-dimensional quantum systems, establishing new properties and existence results for states and channels in this setting.

Contribution

It introduces a definition of the Schmidt number for infinite-dimensional states and generalizes properties of entanglement-breaking channels to infinite dimensions.

Findings

Existence of states with a given Schmidt number lacking finite Schmidt rank decompositions.

Generalization of properties of entanglement-breaking channels to infinite-dimensional systems.

Existence of entanglement-breaking channels with Kraus operators all of infinite rank.

Abstract

A definition of the Schmidt number of a state of an infinite dimensional bipartite quantum system is given and properties of the corresponding family of Schmidt classes are considered. The existence of states with a given Schmidt number such that any their countable convex decomposition does not contain pure states with finite Schmidt rank is established. Partially entanglement breaking channels in infinite dimensions are studied. Several properties of these channels well known in finite dimensions are generalized to the infinite dimensional case. At the same time, the existence of partially entanglement breaking channels (in particular, entanglement breaking channels) such that all operators in any their Kraus representations have infinite rank is proved.