On the one hand, we know that the overall phase of the wave function (of the whole system) is not a measurable quantity, but more an artifact of mathematical description — the physical states are rays in the Hilbert space. The description in terms of density matrix and quantum channels lacks this redundancy (at least, explicitly).
On the other hand, in certain cases it feels like the phase is more physical than it could first seem. But this, of course, is another consequence of axioms of quantum mechanics, and we should be able to describe such situations within any of equivalent formalisms.
How would one, for example, study the Aharonov-Bohm effect using the formalism of density matrix and quantum channels?