TY - JOUR
T1 - When Do Composed Maps Become Entanglement Breaking?
AU - Christandl, Matthias
AU - Müller-Hermes, Alexander
AU - Wolf, Michael M.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85061649183&partnerID=8YFLogxK
U2 - 10.1007/s00023-019-00774-7
DO - 10.1007/s00023-019-00774-7
M3 - Journal article
AN - SCOPUS:85061649183
VL - 20
SP - 2295
EP - 2322
JO - Annales Henri Poincare
JF - Annales Henri Poincare
SN - 1424-0637
IS - 7
ER -