  We give a capacity formula for the classical information transmission over a
noisy quantum channel, with separable encoding by the sender and limited
resources provided by the receiver's pre-shared ancilla. Instead of a pure
state, we consider the signal-ancilla pair in a mixed state, purified by a
"witness". Thus, the signal-witness correlation limits the resource available
from the signal-ancilla correlation. Our formula characterizes the utility of
different forms of resources, including noisy or limited entanglement
assistance, for classical communication. With separable encoding, the sender's
signals across multiple channel uses are still allowed to be entangled, yet our
capacity formula is additive. In particular, for generalized covariant channels
our capacity formula has a simple closed-form. Moreover, our additive capacity
formula upper bounds the general coherent attack's information gain in various
two-way quantum key distribution protocols. For Gaussian protocols, the
additivity of the formula indicates that the collective Gaussian attack is the
most powerful.
