Parallelism between quantum density matrix and a probability distribution in classical mechanics

In quantum statistical mechanics we often define a density matrix as

$rho = sum_{i} p_{i} | Psi_{i} rangle langle Psi_{i} | $.

Its time volution is determined by the equation:

$rho left(tright) = U left(tright) rho U^{dagger} left( t right) $ and it allows us to determine the expectation value of any observable $O$ of the system at any time as

$langle O (t) rangle = mathrm{Tr} left[ rho(t) O right] $. Defining Heisenberg’s time evolution as $O(t)=U^{dagger} left( t right) O U left( t right) $ it is easy to see the following two things:

1. $langle O rho(t) rangle = mathrm{Tr}left[ rho O(t) right] $

2. $ mathrm{Tr} left[ rho(t) O(t) right] = mathrm{Tr}left[ rho O right] $

Number two is particularly interesting as it means that time evolution of both the probability distribution and the observable leaves measurements unchanged.

I tried to take this as a starting point for a statistical description of classical mechanics. I therefore defined a "classical state" as a probability distribution over the phase space $(textbf{Q}, textbf{P} )$: $phi ( textbf{Q} , textbf{P})$. I know how observables evolve in classical mechanics: $frac{partial}{partial t} f_{t} left( textbf{Q}, textbf{P} right) = left{f_{t} , H right} $. Using relation 2) (which I expect to be true in classical physics as well) I define the classical state at time $t$ such that:

$constant = int dtextbf{Q} dtextbf{P} phi_{t} ( textbf{Q} , textbf{P}) f_{t} left( textbf{Q}, textbf{P} right) $. By deriving both sides of the equation in $t$ and substituting the classical equations of motion for the function $f_{t} $ I get:

$0 = int dtextbf{Q} dtextbf{P} left[ dot{phi}_{t} f_{t} + phi_{t} right{f_{t} , H left} right] $. By expanding Poisson’s brackets, integrating by parts in N dimensions, and equating the integrand to zero I get to the equation:

$dot{phi}_{t} = left{ phi_{t}, H right} $, which kind of looks like Liouville’s theorem but it’s not because it’s missing a $-$ sign in front of the brackets. I expected to get exactly Liouville’s theorem and I don’t understand what went wrong. Maybe relation 2) doesn’t work in classical mechanics, or it needs a little fixing, for example if I claim
$constant = int dtextbf{Q} dtextbf{P} phi_{-t} ( textbf{Q} , textbf{P}) f_{t} left( textbf{Q}, textbf{P} right) $ instead, I should get Liouville’s theorem. Maybe I have to reverse time?