Positivity, discontinuity, finite resources, nonzero error for arbitrarily varying quantum channels

We give an explicit example that answers the question whether the transmission of messages over arbitrarily varying quantum channels can benefit from distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the deterministic capacity does have discontinuity points, while that behaviour is, at the same time, not generic: We show that it is in fact continuous around its positivity points. This is in stark contrast to the randomness-assisted capacity, which is continuous in the channel. We then quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) decoding error with respect to the average error criterion that can be achieved over a finite number of channel uses and the number of messages that can be sent. These results also apply to entanglement and strong subspace transmission.