If there is nothing deeper than quantum mechanics, does that mean correlations in entanglement are coincidences?

Question

Again, I am a layman, so please forgive me if I use incorrect terminology.

So according to Bell’s theorem, we cannot have a local theory that determines, say, the spin measurement of either particle in an entangled pair.

Let’s assume that they are anti correlated. This means that the possible outcomes for the entire system are (0,1) or (1,0).

Each particle’s spin can either be measured as 0 or 1. Thus, if the particles are separated in space after being generated, to an experimenter Bob’s perspective, the spin is still equivalent to the flip of a coin.

Now, let’s say Bob measures 0 from his side. This means he also now knows Alice will measure 1. This means the system’s measurement will be (0,1) even though this measurement (instead of (1,0)) was not determined before either Bob or Alice made a measurement.

Now, if there is no local theory that determines the spin, that means there is no local theory that determines that Bob was meant to measure 0. Now let’s also assume there is no non local theory either. So let’s assume that inherently, even in principle, this is unpredictable.

Does this not then become a massive, unexplainable coincidence that the measured outcome on one end, even though it is not previously determined to be 0 or 1 (and thus could be either), still happens to result in a value that ends up being the opposite value on the other end?

What exactly is preventing Bob from measuring 0 and Alice measuring 0 too? The explanation for this from reading on the site is that the system is anti correlated or that angular momentum has to be preserved so the measurements must be anti correlated.

But if the measurement on each side could be 0 or 1, and there are no local or non local influences between the particles and no link, what is ensuring that they remain correlated? To say that they must be correlated because they are correlated due to whatever factor seems like a circular statement.

I have seen the glove analogy presented on this site some times in top upvoted answers here to show why no one should be surprised at this correlation. In this analogy, if I send out two gloves far away in space to different observers and I see a left glove, I know that the other experimenter must have the right glove. However, this assumes the very kind of local hidden variable that Bell sought out to dismiss. It is not that the experimenter is opening a box that has a left glove that from his point of view could have been either the left or the right glove. It is rather that the box in some sense doesn’t have the right or the left glove at all before it’s opened.

Bell of course preferred a non local hidden variable theory to explain this using Bohmian mechanics. But what I’m curious about is how this is made sense of at all if there is no theory at all (local or non local) that explains why a particular measurement is obtained.

Answer 1

For answers, you will need to stop demanding that "a particular measurement is obtained" (which you write at the end). Quantum Mechanics only describes the evolution of the state into a superposition, in which both possible outcomes for Bob are present, but they are "paired" with the consistent outcomes for Alice. That is what we call an entangled state. It follows unambiguously from the math of QM.

So what QM offers you is not one single ultimate result, but it does weed out the inconsistent results! Your question "What exactly is preventing Bob from measuring 0 and Alice measuring 0 too?" is answered: the quantum mechanical time evolution (basically governed by some Schroedinger equation) gives you that restriction automatically.

Other questions you address, that involve "The Measured Outcome", have no meaning if there is no single outcome. The superposition after the interaction only gives you complex amplitudes for the possible outcomes, it does not select one "chosen" result. Now probably you will ask "Why do I experience that there is just one result?"

The reason for that is that you are yourself part of the superposition and in the two (or more) different branches for the results, everything is consistent, so you'll never see inconsistent combinations. We also need to explain why the consistent parts that are there do not interfere with each other, which is by decoherence (very fast fluctuating phase differences).

And finally, since we're at it, you could ask if we can then also guarantee that the Born rule is obeyed for many times repeated interactions. This is often debated. The only thing you can easily prove is that the amplitude in the big superposition, for $n$ times having a certain result in a total of $N$ repetitions, is exactly the square root of the "chance" this would have if the Born rule would govern a collapse into one chosen outcome. Discussing the implications of this is (AFAIK) deprecated on Physics SE...