The wavefunction in quantum mechanics is not like the probability distribution of (say) where a dart lands when you throw it at a dartboard. (In some but not all imaginable Truly Stochastic worlds, perhaps it’s like the probability distribution of the whole state of the universe, but OP’s intuition-pumping example seems to be imagining a case where A is some small bit of the universe.)
The reason why it’s not like that is that the laws describing the evolution of the system explicitly refer to what’s in the wavefunction. We don’t have any way to understand and describe what a quantum universe does other than in terms of the evolution of the wavefunction or something basically equivalent thereto.
In my view, the big similarity is in principle of superposition. The evolution of the system in a sense may depend on the wavefunction, but it is an extremely rigid sense which requires it to be invariant to chopping up a superposition to a bunch of independent pieces, or chopping up a simple state into an extremely pathological superposition.
I have the impression—which may well be very unfair—that at some early stage OP imbibed the idea that what “quantum” fundamentally means is something very like “random”, so that a system that’s deterministic is ipso facto less “quantum” than a system that’s stochastic. But that seems wrong to me. We don’t presently have any way to distinguish random from deterministic versions of quantum physics; randomness or something very like it shows up in our experience of quantum phenomena, but the fact that a many-worlds interpretation is workable at all means that that doesn’t tell us much about whether randomness is essential to quantum-ness.
It’s worth emphasizing that the OP isn’t really how I originally thought of QM. One of my earliest memories was of my dad explaining quantum collapse to me, and me reinventing decoherence by asking why it couldn’t just be that you got entangled with the thing you were observing. It’s only now, years later, that I’ve come to take issue with QM.
In my mind, there’s four things that strongly distinguish QM systems from ordinary stochastic systems:
Destructive interference
Principle of least action (you could in principle have this and the next in deterministic/stochastic systems, but it doesn’t fall out of the structure the ontology as easily, without additional laws)
Preservation of information (though of course since the universe is actually quantum, this means the universe doesn’t resemble a deterministic or stochastic system at the large scale, because we have thermodynamics and neither deterministic nor stochastic systems need thermodynamics)
Pauli exclusion principle (technically you could have this in a stochastic system too, but it feels quantum-mechanical because it can be derived from fermion products being anti-symmetric, and anti-symmetry only makes sense in quantum systems)
Almost certainly this isn’t complete, since I’m mostly autodidact (got taught a bit by my dad, read standard rationalist intros to quantum, like The Sequences and Scott Aaronson, took a mathematical physics course, and coded a few qubit simulations, binged some Wikipedia and Youtube). Of these, only destructive interference really seems like an obstacle, and only a mild one.
(And, incidentally, if we had a model of Truly Stochastic physics in which the evolution of the system is driven by what’s inside those probability distributions—why, then, I would rather like the idea of claiming that the probability distributions are what’s real, rather than just their outcomes.)
I would say this is cruxy for me, in the sense that if I didn’t believe Truly Stochastic systems were ontologically fine, then I would take similar issue with Truly Quantum systems.
In my view, the big similarity is in principle of superposition. The evolution of the system in a sense may depend on the wavefunction, but it is an extremely rigid sense which requires it to be invariant to chopping up a superposition to a bunch of independent pieces, or chopping up a simple state into an extremely pathological superposition.
It’s worth emphasizing that the OP isn’t really how I originally thought of QM. One of my earliest memories was of my dad explaining quantum collapse to me, and me reinventing decoherence by asking why it couldn’t just be that you got entangled with the thing you were observing. It’s only now, years later, that I’ve come to take issue with QM.
In my mind, there’s four things that strongly distinguish QM systems from ordinary stochastic systems:
Destructive interference
Principle of least action (you could in principle have this and the next in deterministic/stochastic systems, but it doesn’t fall out of the structure the ontology as easily, without additional laws)
Preservation of information (though of course since the universe is actually quantum, this means the universe doesn’t resemble a deterministic or stochastic system at the large scale, because we have thermodynamics and neither deterministic nor stochastic systems need thermodynamics)
Pauli exclusion principle (technically you could have this in a stochastic system too, but it feels quantum-mechanical because it can be derived from fermion products being anti-symmetric, and anti-symmetry only makes sense in quantum systems)
Almost certainly this isn’t complete, since I’m mostly autodidact (got taught a bit by my dad, read standard rationalist intros to quantum, like The Sequences and Scott Aaronson, took a mathematical physics course, and coded a few qubit simulations, binged some Wikipedia and Youtube). Of these, only destructive interference really seems like an obstacle, and only a mild one.
I would say this is cruxy for me, in the sense that if I didn’t believe Truly Stochastic systems were ontologically fine, then I would take similar issue with Truly Quantum systems.