Eventually Entanglement Breaking Maps

TL;DR

This paper studies linear maps on matrix algebras that become entanglement breaking after repeated composition, showing that unital PPT-channels do so after finite steps and exploring the properties of such maps.

Contribution

It introduces the concept of finite index of separability for linear maps and proves that unital PPT-channels have this property after finite iterations.

Findings

Unital PPT-channels become entanglement breaking after finite iterations.

The class of unital channels with finite index of separability is dense.

Constructs examples of non-PPT maps with finite index of separability.

Abstract

We analyze certain class of linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map is entanglement breaking after finite iterations, we say the map has a finite index of separability. In particular we show that every unital PPT-channel becomes entanglement breaking after a finite number of iterations. It turns out that the class of unital channels that have finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. This analysis is motivated by the PPT-squared conjecture made by M. Christandl that says every PPT channel, when composed with itself, becomes entanglement breaking.