Indeed, when you consider more complicated setups like GP, or Continuous Measurements, you get equations that are nonlinear in the state. What puzzles me is that you can get nonlinear equations for operators even in standard QM

I'm thinking in standard QM in the Heisenberg picture where an observable O evolves with Heisenberg's equation dO/dt = i [H, O] which can be nonlinear in the observable (e.g. if H has terms like σzσz the eq for σx will have a term σzσy and so on)

I think the reference where I learned about this was the original paper by Haake et al link.springer.com/article/10.1... in there the quantum equations of motion for observables are nonlinear, and so is the map they find in the classical limit.

I always found this argument a bit confusing. QM is linear in the state but you can also write CM as a linear equation on the phase space density using Poisson brackets. On the contrary, if you write the Heisenberg eq. for the evolution of an operator in QM you can get a nonlinear equation, as in CM

Thank you for doing this! I'd like to be included in the list✨