  We consider entanglement-assisted (EA) private communication over a quantum
broadcast channel, in which there is a single sender and multiple receivers. We
divide the receivers into two sets: the decoding set and the malicious set. The
decoding set and the malicious set can either be disjoint or can have a finite
intersection. For simplicity, we say that a single party Bob has access to the
decoding set and another party Eve has access to the malicious set, and both
Eve and Bob have access to the pre-shared entanglement with Alice. The goal of
the task is for Alice to communicate classical information reliably to Bob and
securely against Eve, and Bob can take advantage of pre-shared entanglement
with Alice. In this framework, we establish a lower bound on the one-shot EA
private capacity. When there exists a quantum channel mapping the state of the
decoding set to the state of the malicious set, such a broadcast channel is
said to be degraded. We establish an upper bound on the one-shot EA private
capacity in terms of smoothed min- and max-entropies for such channels. In the
limit of a large number of independent channel uses, we prove that the EA
private capacity of a degraded quantum broadcast channel is given by a
single-letter formula. Finally, we consider two specific examples of degraded
broadcast channels and find their capacities. In the first example, we consider
the scenario in which one part of Bob's laboratory is compromised by Eve. We
show that the capacity for this protocol is given by the conditional quantum
mutual information of a quantum broadcast channel, and so we thus provide an
operational interpretation to the dynamic counterpart of the conditional
quantum mutual information. In the second example, Eve and Bob have access to
mutually exclusive sets of outputs of a broadcast channel.
