When Do Composed Maps Become Entanglement Breaking?

TL;DR

This paper investigates when repeated compositions of quantum maps become entanglement breaking, introducing a Schmidt number-based technique, proving the PPT squared conjecture in dimension three, and extending results to Gaussian channels.

Contribution

It develops a Schmidt number technique to analyze entanglement breaking behavior, proves the PPT squared conjecture in dimension three, and explores its implications for various quantum maps.

Findings

Proves PPT squared conjecture in dimension three.

Introduces a Schmidt number-based method for analyzing entanglement.

Establishes the conjecture for Gaussian quantum channels.

Abstract

For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in dimension three that are also completely copositive we prove the so called PPT squared conjecture in this dimension. We then give more examples of completely positive maps where our technique can be applied, e.g.~maps close to the completely depolarizing map, and maps of low rank. Finally, we study the PPT squared conjecture in more detail, establishing equivalent conjectures related to other parts of quantum information theory, and we prove the conjecture for Gaussian quantum channels.