Potential capacities of quantum channels

Abstract

We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their "plain" analogues. We thus study upper bounds on some potential capacities: For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first "lift" the channel to a Hadamard channel and then prove that the quantum and private capacity of a Hadamard channel is strongly additive, implying that for these channels, potential and plain capacity are equal. Employing these upper bounds we show that if a channel is noisy, however close it is to the noiseless channel, then it cannot be activated into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the so-called environment-assisted quantum capacity, because we are able to characterize its "potential" version.