On Jensen Gap and Capacity of a Shifted Depolarizing Quantum Channel

We study the problem of maximizing the Jensen gap with respect to the probability distribution in a fairly general case, and prove a theorem on the optimal distribution. Using the results obtained, we calculate the one-shot capacity of a certain family of non-unital quantum channels. We show that in sufficiently large dimensions the channel admits one of two modes of an optimal input ensemble depending on the parameters. We also prove that both the fulfillment and the violation of the entanglement-breaking property are possible in any dimension depending on the parameters of the channel.