OK, my latest research idea that needs some work is something growing out of that paper on the Cerf-Adami inequalities that I'm still tweaking (with Terry Rudolph's help). There's presently a discussion of it, by the way, over at Quantalk.org (who reviewed an earlier version, pre-Terry). I'll post the latest version here in a bit.
My idea is this: all forms of classical entropy essentially count things since, even though it is usually a unitless measure, we conventionally measure it in bits (in information theory). Thermodynamic entropy, when interpreted in the statistical mechanics manner (as a rescaling of the multiplicity), can also be thought of as counting something - and in fact can be converted to the "unitless" bits by dividing by Boltzmann's constant.
So let's take a page from Cerf and Adami and think of entropy as simply counting the bits of information for a state. These bits represent a set. Entropy, then, counts these and is thus the cardinality of the given set. As you'll see in my paper that I'm going to upload in a moment, all the usual definitions of mutual, joint, and conditional entropy arise naturally from this interpretation (since it really is what Cerf and Adami were ultimately doing anyway, so this part, at least, just represents a clarification of their methods).
Now, the interesting thing I was able to do in my paper was to use this to derive the Cerf-Adami inequalities without any reference - explicit or implicit - to Markov processes (which I also show are at least implicitly implied in at least one derivation). I also essentially propose a new way of looking at the second law of thermodynamics.
So my next idea is that, if we want to extend this definition to von Neumann entropies, how do we do it since von Neumann entropies do not necessarily obey the CA inequalities? The thought I had was that classical entropies represent cardinalities of of *well-ordered* sets while von Neumann entropies do not. Hence the latter would simply be a generalization of the former.
The question is: what else can be obtained from this interpretation? Is there anything else we can do that makes this useful?
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