I am wondering if qualia would lend themselves to a Hilbert space kind of formalism. But one of the main requirements is linearity. In our everyday experience, qualia are not very linear. If a spot of light is kept on made brighter, it would saturate our senses after a certain threshold. In fact, we cannot even imagine a brightness beyond a certain limit.
But I think that a fundamental distinction needs to be made between phenomenal and access consciousnesses. The latter is a result of processes in the brain (pre-Cartesian theater), and it doesn't surprise anyone that there are fundamental non-linearities built in the chain (which can be explained physically).
Imagining (as well as dreaming or hallucinating) anything is also subject to the same access chain, so this is the reason we cannot even imagine a bright light or a loud sound which is 10x what our sense organs are designed to withstand maximally.
But I think that p-consciousness (which is post-Cartesian theater) should be super-linear. This doesn't hold just true of physical qualia but also emotional ones. While emotional qualia (like anger, empathy, or even sex drive) may seem to be purely access driven and not as straightforward as physical qualia (like colors, pitches, and so on), I think they are as elegant and straightforward as the physical ones. A theory of qualia is incomplete if it leaves out emotional qualia solely to the physical workings of the brain. While the hook-ups to the emotional qualia are again access-driven and hence non-linear, the "other side" should be ultra linear. In fact, I think meta ethics should be of much importance as the color inversion problem. Some entheogenic drugs like Ecstasy in fact cause a heightened state of empathy. There is no reason to rule out, in theory, a super-high empathy state (> 1000x), which would solve the world's political problems! And imagine an orgasmic state which is 10000x the regular one!
Of course, one problem is how to quantify emotional qualia to begin with. What does it mean to say some one is having 453x the empathy of another person?
If we assume linearity of the p-qualia space, then the brain maps processes to a certain subspace of this space. The complementary space would have qualia which cannot even be felt or imagined or dreamed of. For example, we do not know what it is to be like a bat, whether they "hear" or "see" or do something else with their echolocation. If they use the Doppler effect to determine the speed of insects, how does this info get mapped? Would it be analogous to color of "our" vision?
I think we should assume linearity of p-consciousness as a first step towards a formal model of qualia.