Not a timely comment I know—I was also confused by the power of 2, and I think that simply the correct resolution is that the wave function is a nonlinear simplification of the more fundamental matrix-shaped object, which is the density matrix (explained more here). As to the “what is reality”, I don’t think it’s that much worse than probability theory (you also have to mathematically posit an exponential-dimensional space of states to mathematically formalize the concept of a stochastic process for example, or any BPP algorithm).
I guess we don’t know what’s real but my favorite “sufficient story” for what’s real (and other QM stories are equivalent to it, as I understand) is that the real object is an actual probability distribution on end-of-the universe states, where assuming expansion things can just be modeled as a bunch of elementary particles (probably photons) in e.g. the position basis. The noncommutativity becomes small in the expansion limit, so we get a canonical basis of universe states; this is the ultimate decoherence (where it is rigorous, not an extra assumption), and a (real, not quantum) probability distribution over this basis of “end-of-time states”. This might seem woo-ey, but such a state encodes lots of information; for example, any song on the radio or any light reflected from an object on earth (even very faintly) can be recovered via only small error correction from access to the state of the universe at a later time (just look at frequencies in the shell of photons around earth at a radius corresponding to a particular point in time, adjusting for gravitational lensing and so on).
So a model I like is sort of holographic, where there are two realities: there is the “objective” 3-dimensional reality at time infinity, which is just a probability distribution on states at the end of the universe (nothing quantum, no explicit Born rule) compatible with the big bang. You can imagine some alien race having some supercomputer that models our universe, and it outputs a perfectly reasonable probability distribution on end-of-universe states. But if you sample one of these states, it’s not just a disordered mess—it has things in it like the waveforms of a Miles Davis concert. You can now imagine yourself as that alien trying to interpret it—i.e. trying to explain this particular state/ to find structure in it that you can information-theoretically compress. A natural form of such a structure is to posit an approximate 4-dimensional space-time which can be roughly separated into chaotic microscopic structures (which can be modeled thermodynamically) and irreversible events (like the Miles Davis concert which generates many mutually denoising photons all carrying the same waveform information) which, while not entirely deterministic, are close enough to irreversible to be treated as definite in your compression model. The beings “inside” this universe similarly want to get the best possible compression to understand and interact with their world, so they make a similar set of approximations; we can view “truth” as things where our understanding (insofar as we can write it down by e.g. radioing it out into the universe) agrees with the understanding one would have via access to the end-state.
I don’t think this is likely to be “the answer”—it seems weird to have a theory that requires the heat death of the universe in order to be valid (and I think that other “eventual operator independence” stories can be made). But the piece that’s solid here is that in essentially any model of quantum thermodynamics, entities with different preferred commuting operator bases will tend to have more and more agreement on state as entropy increases, and we can sort of think as consensus reality as the “piece that they will eventually agree on”, perhaps in some not-completely-formal sense.
Note that the eigenvalue story here is incidental: there’s nothing magical here about eigenstates of “measurement operators” (as far as I understand), it is just a nice mathematical model. When an irreversible quantum process occurs (such as a scattered photon causing a phase transition in a magnetic detector system), irreversibility means that we can approximately orthogonally separate end-of-universe states into ones where the detector outputted a zero and ones where it outputted a 1. One nice way to bookkeep this decomposition is to write down an operator (the “measurement operator”) which diagonalizes into these two subspaces (i.e. commutes with their projectors); physics being physics, frequently this is a nice operator (like position, momentum, etc.) which we then say the detector is “measuring”.
Not a timely comment I know—I was also confused by the power of 2, and I think that simply the correct resolution is that the wave function is a nonlinear simplification of the more fundamental matrix-shaped object, which is the density matrix (explained more here). As to the “what is reality”, I don’t think it’s that much worse than probability theory (you also have to mathematically posit an exponential-dimensional space of states to mathematically formalize the concept of a stochastic process for example, or any BPP algorithm).
I guess we don’t know what’s real but my favorite “sufficient story” for what’s real (and other QM stories are equivalent to it, as I understand) is that the real object is an actual probability distribution on end-of-the universe states, where assuming expansion things can just be modeled as a bunch of elementary particles (probably photons) in e.g. the position basis. The noncommutativity becomes small in the expansion limit, so we get a canonical basis of universe states; this is the ultimate decoherence (where it is rigorous, not an extra assumption), and a (real, not quantum) probability distribution over this basis of “end-of-time states”. This might seem woo-ey, but such a state encodes lots of information; for example, any song on the radio or any light reflected from an object on earth (even very faintly) can be recovered via only small error correction from access to the state of the universe at a later time (just look at frequencies in the shell of photons around earth at a radius corresponding to a particular point in time, adjusting for gravitational lensing and so on).
So a model I like is sort of holographic, where there are two realities: there is the “objective” 3-dimensional reality at time infinity, which is just a probability distribution on states at the end of the universe (nothing quantum, no explicit Born rule) compatible with the big bang. You can imagine some alien race having some supercomputer that models our universe, and it outputs a perfectly reasonable probability distribution on end-of-universe states. But if you sample one of these states, it’s not just a disordered mess—it has things in it like the waveforms of a Miles Davis concert. You can now imagine yourself as that alien trying to interpret it—i.e. trying to explain this particular state/ to find structure in it that you can information-theoretically compress. A natural form of such a structure is to posit an approximate 4-dimensional space-time which can be roughly separated into chaotic microscopic structures (which can be modeled thermodynamically) and irreversible events (like the Miles Davis concert which generates many mutually denoising photons all carrying the same waveform information) which, while not entirely deterministic, are close enough to irreversible to be treated as definite in your compression model. The beings “inside” this universe similarly want to get the best possible compression to understand and interact with their world, so they make a similar set of approximations; we can view “truth” as things where our understanding (insofar as we can write it down by e.g. radioing it out into the universe) agrees with the understanding one would have via access to the end-state.
I don’t think this is likely to be “the answer”—it seems weird to have a theory that requires the heat death of the universe in order to be valid (and I think that other “eventual operator independence” stories can be made). But the piece that’s solid here is that in essentially any model of quantum thermodynamics, entities with different preferred commuting operator bases will tend to have more and more agreement on state as entropy increases, and we can sort of think as consensus reality as the “piece that they will eventually agree on”, perhaps in some not-completely-formal sense.
Note that the eigenvalue story here is incidental: there’s nothing magical here about eigenstates of “measurement operators” (as far as I understand), it is just a nice mathematical model. When an irreversible quantum process occurs (such as a scattered photon causing a phase transition in a magnetic detector system), irreversibility means that we can approximately orthogonally separate end-of-universe states into ones where the detector outputted a zero and ones where it outputted a 1. One nice way to bookkeep this decomposition is to write down an operator (the “measurement operator”) which diagonalizes into these two subspaces (i.e. commutes with their projectors); physics being physics, frequently this is a nice operator (like position, momentum, etc.) which we then say the detector is “measuring”.