On the other hand, a system in a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:

The density matrix describing a mixed state is defined to be an operator of the form

The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics.

For a discussion of conceptual aspects and a comparison with classical states, see:

For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes for at Caltech.