  In information theory, we often use intersection and union of the typical
sets to analyze various communication problems. However, in the quantum setting
it is not very clear how to construct a measurement which behaves analogous to
intersection and union of the typical sets. In this work, we construct a
projection operator which behaves very similar to intersection and union of the
typical sets. Our construction relies on the Jordan's lemma. Using this
construction we study the problem of communication over authenticated
classical-quantum channels and derive its capacity. As another application of
our construction, we study the problem of quantum asymmetric composite
hypothesis testing. Further, we also prove a converse for the quantum binary
asymmetric hypothesis testing problem which is arguably very similar in spirit
to the converse given in the Thomas and Cover book for the classical version of
this problem.
