Quantum mechanics in terms of density operators as the fundamental object? So I've done a two courses in undergrad quantum mechanics, the first began with wave mechanics and then went on to bras and kets, the second course went more into detail regarding bras, kets, hilbert spaces and so on (they retouched on wave mechanics towards the end). But near the end of the second course, they also went into density operators/matrices.
They were presented as another layer of abstraction on top of the more fundamental hilbert space, a convenient tool when dealing with classical statistic ensembles of pure states.
However, reading the textbook and other sources, I've seen it described as more fundamental than pure states, a natural description of subsystems of larger entangled systems, and having the very suggestive and appealing property that there is a one-to-one correspondence between states and operators, unlike in the hilbert space formalism where the correspondence is only up to a phase factor.
So, is there a description of quantum mechanics which takes density operators as the foundational objects, in the same way that the bra-ket formalism begins with rays of a complex space? If so, what does it look like, why isn't QM taught in terms of density operators, and where can I find resources to study it? Also, I have generally heard it is more awkward to work with than bras and kets, why is it?
Finally, if we are representing states by operators, and treating pure states as special cases of density operators, what are they operations on? Or is this just another case of representing abstract mathematical objects as operators for convenience, and there is no meaning to asking what they operate on?
 A: In quantum mechanics, the fundamental objects can never be states (such as density matrices). The fundamental objects are always observables. 
This is due to noncommutativity: contrarily to classical probability theories, where there is an event space (a set together with a $\sigma$-algebra of measurable subsets, the events), also called the universe (in the sense of probability theory), in noncommutative probability theories such as quantum mechanics there is no probabilistic universe, but rather a collection of noncommutative random variables (the observables), and the theory is constructed starting from this collection (it is a C*-algebra, mathematically speaking). 
Therefore any sensible mathematical formulation of a quantum mechanical system as a noncommutative probability theory should take the C*-algebra of observables of the system as a starting point. The corresponding states (noncommutative probabilities), and in particular the normal states w.r.t. a given representation of the algebra (density matrices), are a "derived object": they are obtained by duality on the algebra of observables.
Let me also remark that quantum mechanics can be studied in a very abstract (and in my opinion revealing) way studying the properties of C*-algebras. In fact, many interesting and well-known features of quantum mechanics, such as the fact that observables are operators acting on Hilbert spaces, the difference between pure and mixed states, the existence of inequivalent representations of the algebra of canonical commutation relations in QFT, and many others are all very well understood and explained in the language (and using the abstract properties) of operator algebras.
