 This is going in circles. You still need to put in Born's rule by hand somewhere. If you can explain the origin of Born's rule we'll have gotten somewhere. I also don't understand why you call a trace a slice. It's an ensemble average. Try deriving the density matrix formalism by coupling a small system to some collection of ancillae and you'll see that it doesn't really add anything. The trace is just convenient notation for the operator sandwich of closed quantum systems... it's just that your closed system is now an open system tensored with a bath. > You still need to put in Born's rule by hand somewhereI'm not sure what you mean by "putting Born's rule in by hand". We're talking about interpretations here. Born's rule is just an empirical fact. Do you mean that I need to explain why the probabilities are the square of the amplitude? That's kind of like asking me to explain why the speed of light has the value that it has. It doesn't have an explanation. It's just part of the Way Things Are.But I can make the following heuristic arguments:1. Outcomes are probabilistic because this is the only way that classical behavior can emerge from the wave function, and without classical behavior you can't copy information, and without copying information you can't have discussions like the one we're having. It's an anthropomorphic argument (actually it's a info-pomomorphic argument :-)2. The probabilities are the square of the amplitude because that is how you get a useful mathematical model of reality. It is simply an empirical fact that destructive interference happens, so if you want to build a mathematical model of that you need something that can take on negative values. You can't have negative probabilities in classical reality, so the underlying reality must be something other than classical probabilities. The square root of the probability is just the simplest mathematical model that explains the observations.Maybe this will help: the technical paper that my position is based on is here: There are a few ways to derive the Born rule within WMI. None are perfect but their assumptions are fairly reasonable and most importantly the handling of probability is better defined than in classical mechanics or Copenhagen, so it's not a reason to discount the theory. Search: