Completely positive classical structures and sequentializable quantum   protocols
  We study classical structures in various categories of completely positive morphisms: on sets and relations, on cobordisms, on a free dagger compact category, and on Hilbert spaces. As an application, we prove that quantum maps with commuting Kraus operators can be sequentialized. Hence such protocols are precisely as robust under general dephasing noise when entangled as when sequential. 
Chris Heunen (University of Oxford), Sergio Boixo (USC and Harvard   University)
quant-ph