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 analogously to intersection and union of the typical sets. In this work, we construct a projection operator which behaves very similarly 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 also study the problem of quantum asymmetric composite hypothesis testing.