The idea that entropy represents the uncertainty in our description of a system works perfectly well in quantum statistical mechanics (actually better than in classical statistical mechanics where the entropy of a system diverges to minus infinity as the precision of the description increases). The entropy is zero when we have a pure state, a state perfectly defined by a wave function, and greater than zero when we have a mixed state.

Regarding your second point, how does the practical impossibility of measuring the initial state invalidate the idea that there is uncertainty about the state?

The post I was replying to claims that probability is not a physical property of an observed system, that it is a property of an observer trying to observe a system. The examples given in the quoted link all talk about experiments like rolling dice or tossing coins, and explain that knowledge of mechanics shows that these things are perfectly predictable mechanical processes, and so any "probability" we assign to them is only a measure of our own lack of knowledge of the result, which is ultimately a measure of our lack of knowledge of the initial state.

So, the link says, there's no such thing as a "fair coin" or a "fair coin toss", only questions of whether observers can predict the state or not (this is mostly used to argue for Bayesian statistics as the correct way to view statistics, while frequentist statistics is considered ultimately incoherent if looked at in enough detail).

I was pointing out however that much of this uncertainty in actual physics is in fact fundamental, not observer dependent. Of course, an observer may have much less information than physically possible, but it can't have more information about some system than a physical limit.

So, even an observer that has the most possible information about the initial state of a system, and who knows the relevant laws of physics perfectly, and has enough compute power to compute the output state in a reasonable amount of time, can still only express that state as a probability. This probability is what I would consider a physical property of that system, and not observer-dependent. It is also clearly measurable by such an observer, using simple frequentist techniques, assuming the observer is able to prepare the same initial state with the required level of precision.

> an observer may have much less information than physically possible, but it can't have more information about some system than a physical limit.

Still the probability represents the uncertainty of the observer. You say that "the most possible information" is still not enough because "measurement is a time-consuming process" and it's not "possible to measure" with infinite precision. I'd say that you're just confirming that "the lack of knowledge" happens but that doesn't mean the physical state is undefined.

You call that uncertainty a property of the system but that doesn't seem right. The evolution of the system will happen according to what the initial state was - not according to what we thought it could have been. Maybe we don't know if A or B will happen because we don't know if the initial state is a or b. But if later we observe A we will know that the initial state was a. (Maybe you would say that at t=0 the physical state is not well-defined but at t=1 the physical state at t=0 becomes well-defined retrospectively?)

I would say that any state that we can't measure is not a physical state. So if we can only measure, even in theory, a system up to precision dx, then the system being in state a means that some quantity is x±dx. If the rules say that the system evolves from states where y>=x to state A, and from states where y<x to state B, then whether we see it in state A or state B, we still only know that it was in state a when it started. We don't get any extra retroactive precision.

This is similar to quantum measurement: when we see that the particle was here and not there, we don't learn anything new about its quantum state before the measurement. We already knew everything there was to know about the particle, but that didn't help us predict more than the probabilities of where it might be.

> I would say that any state that we can't measure is not a physical state.

Ok, so if I understand correctly for you microstates are not physical states and it doesn't make sense to even consider that at any given moment the system may be in a particular microstate. That's one way to look at things but the usual starting point for statistical mechanics is quite different.

That would be a bit too strong a wording. I would just say that certain theoretical microstates (ones that assume that position and velocity and so on are real properties that particles possess and can be specified by infinitely precise numbers) are not in fact physically distinguishable. Physical microstates do exist, but they are subject to the Heisenberg limit, and possibly other, less well defined/determined precision limits.

Beyond some level of precision, the world becomes fundamentally non-deterministic again, just like it appears in the macro state description.