  The transmission of classical information over a classical channel gave rise
to the classical capacity theorem with the optimal rate in terms of the
classical mutual information. Despite classical information being a subset of
quantum information, the rate of the quantum capacity problem is expressed in
terms of the coherent information, which does not mathematically generalize the
classical mutual information. Additionally, there are multiple capacity
theorems with distinct formulas when dealing with transmitting information over
a noisy quantum channel. This leads to the question of what constitutes a
mathematically accurate quantum generalization of classical mutual information
and whether there exists a quantum task that directly extends the classical
capacity problem. In this paper, we address these inquiries by introducing a
quantity called the generalized information, which serves as a mathematical
extension encompassing both classical mutual information and coherent
information. We define a transmission task, which includes as specific
instances both classical information and quantum information capacity problems,
and show that the transmission capacity of this task is characterized by the
generalized information.
