Non-convexity of private capacity and classical environment-assisted   capacity of a quantum channel
  The capacity of classical channels is convex. This is not the case for the quantum capacity of a channel: the capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities and thus is non-convex. Here we show that this effect goes beyond the quantum capacity and holds for the private and classical environment-assisted capacities of quantum channels. 
David Elkouss and Sergii Strelchuk
quant-ph cs.IT math.IT