Separability and Entanglement-Breaking in Infinite Dimensions

TL;DR

This paper characterizes separable states and entanglement-breaking channels in infinite-dimensional quantum systems, providing new integral representations and examples that extend finite-dimensional results.

Contribution

It introduces a general integral representation for separable states in infinite dimensions and presents the first example of non-countably decomposable separable states, also generalizing entanglement-breaking channel structure.

Findings

Existence of separable states not countably decomposable

Infinite-dimensional entanglement-breaking channels without Kraus rank-1 operators

Generalized structure theorem for entanglement-breaking channels

Abstract

In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional entanglement-breaking channels having no such representation.