Let's define this with some history. Circa 1900, it was imagined that electrons orbit the nucleus of an atom much as a planet orbits a star. However, this was soon found to be incorrect, as calculations showed that the electron would fall into the nucleus of the atoms in about 1 / 100 billionth of a second. Clearly this did not happen, so the "planet-like orbit" model of an electron's motion was incorrect. Around this same time, however, it was found that light (traditionally modelled as a wave) displayed particle-like properties referred to as "quanta" or, in modern parlance, "photons". Curiously, something seen as a wave, light, could also behave as a particle. So, scientists thought, what if we went the other way? What if we have something we view like a particle, namely, an electron, that perhaps can also behave like a wave? The electron was then modelled, not as a very tiny point of mass in orbit around a nucleus of an atom, but rather as a sort of standing wave all around the nucleus of the atom. It also turned out that the square root of the height of the wave in a given place gave one the probability of detecting the electron (in the form of a particle) at that particular place. Hence "quantum theory" was born, named from the term "quanta" which described things - like electrons or particles of light - that sometimes were best viewed as waves, and sometimes as particles.
Later refinements, however, led to something quite curious. What if I have 2 electrons around my atom, not just one? Do I now have two waves? (Or "wave functions" which is just a fancier way of saying "waves"). It turned out that, no, one still had only one wave function - the 2 electrons would basically add together and form a composite wave function. The cool thing about this, however, is that - in theory - one could keep going like this, and basically add the wave functions for every single particle of the universe together into one giant composite wave function, known as the wave function of the universe. Of course, this is beyond any computer's ability to do, so this is just a theoretical operation, but the moral of the story is that one can treat even large systems (made up of trillions of particles) as their own composite wave functions. In practicality, for large systems, this is not a convenient or necessary thing to do, but it remains something one can do in theory. Indeed, we could perhaps take the brain (or neocortex) and say there is a composite wave function (built up out of all the wave functions for all the atoms in the neocortex, for example).
The famous Schrodinger's Cat experiment can help take us where we are going here. Imagine a cat in a box, and in that box is a radiocative nucleus with a 50% chance of decaying. If it decays it will trigger a weapon that will kill the cat. If it does not decay, the cat will remain unharmed. So, until one opens the box to check on the cat, the cat is in a "superposition", a mixed state of being if you will 50% alive and 50% dead. We can say the "wave function" of the cat gives us a 50% probability of finding it dead, and 50% probability of finding it alive. This is a well known thought experiment, with apologies to PETA.
Now, what would it mean to say that the entropy of the cat increases? Well, we could perhaps imagine that some of the atoms of the cat get re-arranged in some way (perhaps the cat is scratching itself and disheveling its fur). But this re-arrangement does not in fact trigger the decay the radioactive nucleus - it is in fact irrelevent to the broader setup, that is to say, we can imagine this re-arrangement to not impact the probability of finding the cat either alive or dead. No matter how dishevelled the cat in the box's fur gets, the probability of finding it alive or dead remains the same. It wave function, if you will, from the point of view of somebody outside the box, remains unaffected.
To "increase the entropy" of a wave function of a single particle is a rather simple affair - the wave simply "spreads out" over time, which is to say, as time passes, the probability of finding the particle in one location or another gets smeared out over a larger area. However, the *location* of a particle is only one possible property that a "wave function" can have. A wave function is not a "wave" in "space" - it is basically a mathematical function over Hilbert space, that is to say, over a space of probabilities for some quantity (such as position or momentum). To say a wave function "spreads out" does not mean something in our familiar space - we simply mean this particle gets more chances of have a certain quantity - such as its position say - varied, and the more time that passes, the more varied it gets.
By way of analogy, let's say I am a stock picker trying to create a well balanced portfolio. So "Hilbert Space" in this analogy would be all the possible stocks I could buy. The "wave function" is that subset of stocks I am likely to buy at a particular time. The longer that time goes on, the greater that subset becomes, that is to say, the longer I am in the market, the more likely it is that I will expand the range of stocks I am thinking about buying. So, you could say, the "entropy" of my stock picking operation increases, that is to say, the range of different types of companies, sectors of the market, and so on, increase as time increases.
This same principle applies, albeit in a more complex way, to wave functions for systems built up out of many different parts. It is basically a "state space" - the longer our cat is in the box, the greater its "state space" becomes, though from the outside of the box, one would not know the difference.
A conscious system, then, seems to be a complex system - a nervous system, say - undergoing an increase of its state space, an increase of its internal entropy, and there is some sort of interaction with the environment such that its interaction does not obviously indicate "the state of the state space" if one might clumsily phrase it thus.
As an example, if I am order coffee at a restaurant, I do not give two hoots what the internal monologue of my waitress is, so long as she correctly takes down my order and brings me my coffee - in my case, coffee without cream, and, if they have no cream, then coffee without milk (as the joke from the film "Ninotchka" goes, which of course has now been done to death in nearly everything Slavoj Zizek has ever written, ha ha).
The difference between a conscious waitress and a non-conscious robot, is there exists a wave function of some sort which can describe the "internal state" of the nervous system of the waitress, and there is a well defined way in which this wave function is increasing its "entropy" (or, "state space") wholly unbeknownst to the casual outside observer.
I understand this is a very preliminary and "hand wavy" approach at the moment, and hopefully with increases in neuroscience, physics, and so on, more precision may one day be had, but this picture is I think at least not wholly inaccurate.
To circle back to the cat, at some point, something happens which does impact upon the outside observer - and when she opens the box, finds the cat either alive or dead. But this so-called "collapse of the wave function" is in a sense a separate issue from the issue of the "entropy" of the wave function (and in an Everett Multiverse view the wave function never collapses anyway, but that is another story). Going back to the stock picker analogy, the fact that I buy, say, IBM stock on a Wednesday, as opposed to say Facebook stock, is immaterial to the "range" of stocks I had been considering on that Wednesday. When I say the "entropy of the wave function increased" I mean my "range" of stocks I am considering has increased on that Wednesday as compared to the previous day on Tuesday, and so forth. Somebody observing me purchasing IBM only sees me buying IBM and does not know nor care what particular range of stocks I may have been considering, just prior to purchasing IBM, just as I do not particularly care what my coffee shop waitress' internal monologue might have been, just before I ordered coffee without cream.
A conscious system must, basically, be treated as a unified system described by a single wave function. Just as a single wave function can describe any number of electrons for instance, a single wave function can describe - I would argue - any conscious system. It is the "increasing of the state space" of that wave function of the conscious system (be it a coffee shop waitress, a sponge, or an AI system of some sort) that is the "internal awareness" of the system.
Consciousness is always a wholly private matter, as many philosophers have pointed out. I can only experience "my own" consciousness, not that of someone else. This model would give the reason for that. If concsiousness is really a wave function then it "exists" so to speak only in Hilbert Space, and is thus then really forever inaccesible to the "outside" observer, as much as anything in Hilbert Space.
"...and so forth, and so on" to once again channel Zizek. :)
Unrelated Postcript: Slavoj Zizek discusses the "coffee without cream" joke :)