  We consider classical message transmission under entanglement assistance for
compound memoryless and arbitrarily varying quantum channels. In both cases, we
prove general coding theorems together with corresponding weak converse bounds.
In this way, we obtain single-letter characterizations of the
entanglement-assisted classical capacities for both channel models. Moreover,
we show that the entanglement-assisted classical capacity does exhibit no
strong converse property for some compound quantum channels for the average as
well as the maximal error criterion. A strong converse to the
entanglement-assisted classical capacities does hold for each arbitrarily
varying quantum channel.
