Entropy (usual symbol S) measures the number of specific ways in which a thermodynamic system may be arranged. It measures the number of states. It is understood as a measure of disorder.
Another part of physics which worries about states is Quantum Physics. A Quantum Process is associated to a Quantum Space which turns out to be a Hilbert Space (a complete complex vector space with a metric; basically the nices, simplest high dimensional complex vector space one can conceive of). The measurement is identified with an operator (say A) in said space, which has eigenspaces and eigenvalues (Av = av; where v is a vector called an eigenvector, and a, a complex number, the eigenvalue).
[Haroche, from the ENS lab in Paris which invented optical pumping, thus the laser, 62 years ago, and Wineland, from Boulder, got the Nobel in 2012.]
Both Entropy and the Quantum suffer of the same problem, namely: what is a state? Can state be absolutely defined?
As it is, things have been all too relative.
This is exemplified in Quantum Physics with the Schrodinger Cat Paradox. A cat is put in a box, with an infernal Quantum mechanism that is supposed to gas it (shortly after, the Nazis did for real… Interesting Freudian slip that German and Austrian physicists were involved with the idea of mixtures of dead and live cats).
The question is whether mixing live and dead cat waves is a full description of the system. It obviously stretches credulity. This was the argument of Schrodinger (initiated in exchanges with Einstein).
From the point of view of the cat, inside the box, the waves, states, and chosen Quantum spaces would be quite different
Apply an order on Hilbert spaces, according to fullness of description, and consider only ultrafilters (in the topological sense) as genuinely representative of the best approximation of reality. Hey, nobody said we should not think big… Anyway, that’s my answer to the Multiverse and its multiversists.
Now back to entropy.
As it exists, thermodynamics is about particles. Thus, it infeodated to the problem of states in Quantum Physics. Hence solving the Quantum Cat problem solves the problem of Entropy.
Or does it?
The deepest problem subjacent to Quantum Physics is whether some sort of thermodynamics could be, and thus should be, applied to the isolated particle (I believe it could, and should).
The Haroche and Wineland methods, above, are a step in the right direction, namely measuring what the real states, the ultimate element of reality of the world, are.
So is Entropy useless? Is it physics? Yes, it is physics, just like computer science is science. Both are emergent aspects of the world. Not as fundamental as a future sub-Quantum Physics, but all the fundamentalism, and no more, that we need, much of the time.