There's something subtle about what's map and what's territory in density matrices. I'd like to think to the territory as a pure quantum state and to maps as mixed states, but... If John thinks the electron in the centre of this room is either spin-up or spin-down but he has no idea which (i.e. he assign probability 50% to each), and John thinks the electron in the centre of this room is either spin-east or spin-west but he has no idea which, then for any possible experiment whatsoever, the two of them would assign the same probability distribution to the outcome. There's something that puzzles me about this, but I'm not sure what that is.