In quantum mechanics, pure states may be represented by (subspaces spanned by) vectors in a Hilbert space, which may be added. This is physically meaningful, and in wave mechanics leads to visible interference patterns.
But we can also discuss mixed states. They are usually introduced as a book-keeping device for classical uncertainty about the quantum state of a system, but it is also possible to have a formalism where they are foundational and kets and bras are the derived objects.
If we take such a perspective and think of density operators as our fundamental unit, then how would we "add" two pure state density operators in the sense of some operation that would map the density operators associated with two kets to the density operator of the sum of those kets?