In summary: QFT as opposed to QM, bc the Stone-VonNeuman theorem does not hold in that context. Of course no one actually believes that something goes wrong physically, but it is a vexing mathematical problem)
I also think that calling the statistical operator "density matrix" is bad terminology because it is an operator and not a matrix. Unfortunately, it is quite common.
I was curious about the original terminology and just checked Dirac and von Neumann. Dirac starts with the classical case and the phase space density. In analogy to this, he calls the statistical operator the "quantum density" or simply the "density" if it is clear from the context that he talks about the quantum case. Von Neumann doesn't introduce a name right away but later on, he calls it the "statistical operator".
What is the bottom line of this thread? It seems that everybody agrees with @vanhees71 post #32, i.e. the statistical operator itself is time independent in the Heisenberg picture but its matrix elements are time dependent regardless of pictures. Is there still disagreement about what this implies for decoherence? It's hard to tell for me because most of the discussion has been about terminology.
I would say that it ultimately doesn't matter what you call things. The point is that the physics of which picture you are in here differs only by a Unitary transformation. You can go from the Heisenberg picture to the Schroedinger picture, you can then apply a time evolution, and then transform back to the Heisenberg picture. The point is the whole diagram commutes. It's just a case that it's a little more technically challenging, going in one direction in the case of the Decoherence formalism. For the record, I agree with the terminology in post 31 and 32 and is what seems more familiar to me.
But yea, I'm not really sure what the argument is. Are we really arguing that you can't do this?