That's right, except that the opposite of "local" in this case is not "global". (This is physics, not software engineering.) The opposite of "local" in this case is "non-local", which refers to state that propagates faster than the speed of light. Bell's theorem shows that describing reality requires some kind of non-local state. In the case of both Copenhagen and Bohm, that non-local state resides in the wave function. But here's the thing: you cannot know the state of the wave function because of the no-cloning theorem. The best you can do is prepare sub-systems in known states. So QM really is different. You cannot predict the outcome of a quantum experiment even in principle and even with arbitrarily advanced technology.
What matters is the reason that it's not knowable. Some things are knowable in principle but not in practice because of technological or economic constraints, like whether or not there are life-supporting planets in the Andromeda galaxy. But in principle, if you could build a big enough telescope, you could know. Quantum experiments are not predictable even in principle. Even with arbitrarily advanced technology and unlimited resources, you can never know the outcome of a quantum experiment (assuming QM is correct, of course). That is an operational definition of the idea that the information required to predict a quantum experiment does not exist in our universe.
This is the reason I hedged with "assuming QM is correct, of course". You are recapitulating the EPR argument. The reason the Bell inequalities are a thing is that they refute the EPR argument. It is not possible for QM to be completed as a local hidden-variable theory. If it turns out that the information required to predict the result of a QM experiment actually exists, then QM is not merely incomplete, but actually wrong. That is possible, of course. But I'll give long odds against.
Heisenberg says that we cannot know both position and velocity with arbitrary precision.
This is inherent to any wave-based system. BUT, this only means we cannot know (as in theoretically prevented) all of the variables to arbitrary enough precision to accurately predict the outcome.
It doesn’t mean, however, that there aren’t any initial conditions even prior to measurement.
Nor does Bell’s inequality doesn’t negate this. Note Lso that non-local does not imply that causality is broken (you cannot transmit information FTL via decoherence).
In fact, one of the more interesting (and unexplored) possibilities is that the boolean-logic law of excluded middle is wrong.
This is because Bell’s derivation is pure arithmetic and logic. It’s the one bit of QM that any student can follow.
Lest this is handwaved away, know that there are entire branches of constructivist mathematics that do just this.
> It doesn’t mean, however, that there aren’t any initial conditions even prior to measurement.
This is the EPR argument.
> Bell’s inequality doesn’t negate this.
Bell shows that there is no measurement you can make, even in principle, that will give you the information you need to predict the outcome of a quantum experiment.
You can, if you wish, insist that those initial conditions exist notwithstanding our inability to measure them even in principle. But you could equally well insist that the outcomes of quantum experiments are determined by an invisible pink unicorn. Both hypotheses are equally unfalsifiable (if QM is correct).
I have actually coined the term IPU (Invisible Pink Unicorn) as an intentionally derisive description of hypothetical constructs that cannot be measured even in principle. Many QM interpretations contain IPUs. Bohmiam particle positions, for example, are an IPU.
One thing to look at is how is the theory formulated. Standard QM seems to build its Hilbert space of wave functions from functions over R^3N. It has a Hamiltonian built out of the notions of that space. So configuration space seems pretty crucial. But configuration of what? If you say particles with positions do not exist, then what exactly is the relevance of this space? What is the primitive stuff whose behavior can be right or wrong from our perspective?
It is also odd to say that position cannot be measured. We can tell in an experiment whether something ended up over there or over here. It would be reasonable to then try to have a theory that correlates the position measurements with something that has a position. Now it is not necessarily the case that there has to be such a thing, but it seems like a reasonable first step.
We can even see trails of particles in cloud chambers and the like. Why is that an IPU?
I will grant that it does not have to be the case that the only possible explanation is that of particles with position. But it certainly seems like if there is such a theory (and, of course, there is), then it would seem reasonable to consider it as quite plausible.
It also helps to ask you what is real in your theory. Are wave functions real? They certainly can't be measured in their entirety. Are operators the real thing? We don't measure them, but rather get something close to their eigenvectors/eigenvalues. Are those real?
Many worlds is the closest version with nothing added, but even that requires some kind of mass density function to make explicit connection with our lived experience. While it doesn't add too much in the way of extra mathematical structure in the theory (integrate over the wave function in a certain way: https://arxiv.org/abs/0903.2211 ), the implication in terms of what it says reality is actually like certainly involves a heck of a lot of IPUs.
> If you say particles with positions do not exist, then what exactly is the relevance of this space?
That's a good question. The real answer is that no one actually knows. I think this is actually the biggest mystery in QM. But let me start with this, because I didn't make myself clear:
> It is also odd to say that position cannot be measured.
When I said that Bohmian positions are an IPU I did not intend that to mean that particle positions can't be measured. Obviously they can. The IPU-ness of Bohmian positions has to do with their ontological status, not their epistemic status. On Bohm's theory, a particle position considered along some axis is a real (in the mathematical sense) value, which is to say, it contains an infinite amount of information. But this information cannot be accessed in the same way that information stored in (say) a book can. I can open a book, even a book with an infinite number of pages, to any page and start reading it, and having read any page, I can go back and read that same page again. The information stored in Bohmian positions doesn't work that way. The laws of physics somehow conspire to hide all that information so that it can only be accessed serially and non-repeatably. The first time you measure a particle's position you get the most significant bits of its position. Those are then lost forever. You can never measure them again. The next time you measure a particle's position you get the next most significant bits of what that particle's position originally was, and so on. But you can never go back and do a second experiment to verify that the result you got for any of your measurements was actually correct and not a result of experimental error.
So the much-vaunted determinacy of Bohmian mechanics is not a reflection of the determinacy of the underlying metaphysical reality. It is really nothing more than a rhetorical trick. All the randomness is still there, it's just "pre-computed" and stored in particle positions in a way that it can only be accessed so that the world behaves exactly as if it were "really random" (whatever that means).
This same kind of trick is made manifest in a thought experiment [https://www.mathpages.com/rr/s9-07/9-07.htm] proposed by Kevin Brown. He points out that, if pi is normal (which is almost certainly is) then all of the results of all experiments ever conducted could be produced by a "cosmic Turing machine" computing the digits of pi. (See the two paragraphs beginning with "Even worse, there need be no simple rule of any kind relating the events of a deterministic universe.") Bohmian positions have exactly the same ontological status as the cosmic Turing machine. Only the window-dressing is different.
> It also helps to ask you what is real in your theory. Are wave functions real?
