  We study the communication capabilities of a quantum channel under the most
general channel model known as the one-shot model. Unlike classical channels
that can only be used to transmit classical information (bits), a quantum
channel can be used for transmission of classical information, quantum
information (qubits) and simultaneous transmission of classical and quantum
information. In this work, we investigate the one-shot capabilities of a
quantum channel for simultaneously transmitting of bits and qubits. This
problem was studied in the asymptotic regime for a memoryless channel and a
regularized characterization of the capacity region was reported. It is known
that the transmission of private classical information is closely related to
the problem of quantum information transmission. We resort to this idea and
find achievable and converse bounds on the simultaneous transmission of the
public and private classical information. then by shifting the classical
private rate to the quantum information rate, the obtained rate regions will be
translated into rate regions of thThis in turn, leads to a rate region for
simulttaneous transmission of classical and quantum information. In the case of
asymptotic i.i.d. setting, our one-shot result is evaluated to the known
results in the literature. Our main tools used in the achievability proofs are
position-based decoding and convex-split lemma.
