The definition of entropy, be it Boltzmanns ($k_B \text{ln } \Omega $), Gibbs ($ - k_B \sum_i p_i \text{ln } p_i $) or Von-Neumann's ($ - k_B \text{Tr} \hat{\rho} \text{ln } \hat{\rho} $) always relies at least on some probability interpretation, (in the first case, it actually only applies to a system in equilibrium states). In the quantum mechanical case, there isn't an ensemble, but the classical probabilities manifest in the density operator.
Given these formulae, my first impression is that we can only talk about the quantity 'entropy', if we model a system as an ensemble and talk about probabilities.
I'd like to know if there is a definition of entropy that also applies to a micro-state. If that's not the case, what do we actually mean by the statement that 'the entropy of an isolated system increases with time'? What does it mean, especially if we (because we are some kind of omniscient entity) presume to know position and momentum of every particles?
Before I could say something about the entropy of this system, I would have to find an appropriate probability distribution for an ensemble describing the system. My first guess for such a probability distribution would be the one which models the timely behavior of the system. By 'timely behaviour' I mean that the probability density which I choose correctly recreates all the average values that I would also get by measuring the corresponding observables at a selection of different instants in time.
Would that be the right way to go?