Well, the probability distribution aspect of a mixed state seems pretty “map-like,” while the quantum superposition aspect seems pretty “territory-like” … but oops! we can decompose the same mixed state into a probability distribution over superpositions in infinitely-many nonequivalent ways, and get exactly the same experimental predictions regardless of what choice we make.
I think the underlying problem here is that we’re using the word “probability” to denote at least two different things, where those things are causally related in ways that keep them almost consistent with each other but not quite. Any system which obeys the axioms of Cox’s theorem can potentially be called probability. The numbers representing subjective judgements of an idealized reasoner satisfy those axioms; call these reasoner subjective probabilities, P_r(event,reasoner). The numbers representing a quantum mixed state do too; call these quantum probabilities, P_q(event,observer).
For an idealized reasoner who knows everything about quantum physics, has unlimited computational power, and has some numbers from the quantum system to start with, these two sets of numbers can be kept consistent: reasoner=observer-->P_r(x,reasoner)=P_q(x,observer). In other words, there is a bit of map and a bit of territory, and these contain the exact same numbers, within the intersection of their domains. The numeric equivalence makes it tempting to merge these into one entity, but that entity can’t be localized to the map or the territory, because it contains one part from each. And the equivalence breaks down, when you step outside of P_q’s domain; if a portion of a quantum system is causally isolated from an observer, then P_q becomes undefined, while P_r does still has a value and still obeys Cox’s axioms.
If the domain of P_r failed to cover all possible events, that would be a huge deal, philosophically. But for P_q to be undefined in some places, that isn’t nearly as interesting.
I think the underlying problem here is that we’re using the word “probability” to denote at least two different things, where those things are causally related in ways that keep them almost consistent with each other but not quite. Any system which obeys the axioms of Cox’s theorem can potentially be called probability. The numbers representing subjective judgements of an idealized reasoner satisfy those axioms; call these reasoner subjective probabilities, P_r(event,reasoner). The numbers representing a quantum mixed state do too; call these quantum probabilities, P_q(event,observer).
For an idealized reasoner who knows everything about quantum physics, has unlimited computational power, and has some numbers from the quantum system to start with, these two sets of numbers can be kept consistent: reasoner=observer-->P_r(x,reasoner)=P_q(x,observer). In other words, there is a bit of map and a bit of territory, and these contain the exact same numbers, within the intersection of their domains. The numeric equivalence makes it tempting to merge these into one entity, but that entity can’t be localized to the map or the territory, because it contains one part from each. And the equivalence breaks down, when you step outside of P_q’s domain; if a portion of a quantum system is causally isolated from an observer, then P_q becomes undefined, while P_r does still has a value and still obeys Cox’s axioms.
If the domain of P_r failed to cover all possible events, that would be a huge deal, philosophically. But for P_q to be undefined in some places, that isn’t nearly as interesting.