I believe you have made a mistake in the second equation. There $\rho(p,q)$ should be the hamiltonian $H(p,q)$ right?

How are you sure that that is the only theoretical constraint? It seems to me that it is impossible to gather all the information to the limit given by heisenberg's uncertainty principle, but I cannot articulate why. This is my central question

Thank you for taking the time to answer. I see, it would take an enormous amount of data to register the measurements. But I am guessing that it would even be theoretically impossible, because any measurement would basically make the information you already gathered useless right? Even if you would have a huge memory disk

I adjusted it, thank you for your comment

Thank you for your answer. What do you exactly mean with substituting time averages with ensemble averages?