Recently the one-shot quantum no-signalling assisted zero-error classical capacity of a quantum channel has been formulated as a semidefinite programming (SDP) depending only on the Kraus operator space of the channel. In this paper, we study the activated quantum no-signalling assisted zero-error classical capacity by first allowing the assistance from some noiseless forward communication channel and later paying back the cost of the helper. We show that the one-shot activated capacity can also be formulated as a SDP and derive a number of striking properties of this number. In particular, this number is additive under direct sum, and is always greater than or equal to the super-dense coding bound. As a remarkable consequence, we find that one bit noiseless classical communication is able to fully activate any cq-channel to achieve its asymptotic capacity, or the semidefinite fractional packing number. We also show that the asymptotic activated capacity is still equal to the usual no-signalling assisted capacity. Finally, we show that the general asymptotic no-signalling assisted zero-error capacity does not equal to the semidefinite (fractional) packing number by an explicit construction.

Date: 21 October 2015