One success of the probabilistic method in quantum information theory is Hastings’s example of quantum channels, for which the capacity to transmit information is non-additive. The exegesis of that example has generated lot of activity in the community.
Here we consider a related question: for , can we find two quantum channels and such that
This question is well-known to be much simpler than the one solved by Hastings (which corresponds to ). However, one can make it a bit harder by demanding some extra properties from the channels , for example of being PPT-inducing (meaning that applying the channel on one part on a bipartite system always outputs PPT states). Existence of such channels was proved by Moto Fukuda and Ion Nechita for by using tools from free probability.
Updated January 29, 2018: In a previous version of this entry we claimed to be able to push down to . The argument turned out to be erroneous, as pointed to us by Motohisa Fukuda. Therefore the problem remains open, and the case is probably the most important one!