When Do Composed Maps Become Entanglement Breaking?
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.
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We thank Daniel Cariello for pointing out how his results [26, 27] together with the techniques from  imply Corollary 3.5. We also thank Ion Nechita for pointing out the different version of Theorem A.1. MC and AMH acknowledge financial support from the European Research Council (ERC Grant Agreement no 337603) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). MW acknowledges the hospitality of the QMATH Centre.
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