  The 2-receiver broadcast channel is studied: a network with three parties
where the transmitter and one of the receivers are the primarily involved
parties and the other receiver considered as third party. The messages that are
determined to be communicated are classified into public, private and
confidential based on the information they convey. The public message contains
information intended for both parties and is required to be decoded correctly
by both of them, the private message is intended for the primary party only,
however, there is no secrecy requirement imposed upon it meaning that it can
possibly be exposed to the third party and finally the confidential message
containing information intended exclusively for the primary party such that
this information must be kept completely secret from the other receiver. A
trade-off arises between the rates of the three messages, when one of the rates
is high, the other rates may need to be reduced to guarantee the reliable
transmission of all three messages. The encoder performs the necessary
equivocation by virtue of dummy random numbers whose rate is assumed to be
limited and should be considered in the trade-off as well. We study this
trade-off in the one-shot regime of a quantum broadcast channel by providing
achievability and (weak) converse regions. In the achievability, we prove and
use a conditional version of the convex-split lemma as well as position-based
decoding. By studying the asymptotic behaviour of our bounds, we will recover
several well-known asymptotic results in the literature.
