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From what I understand of Bell's Theorem, it requires giving up local realism or embracing superdeterminism. I still haven't been able to understand why superdeterminism gets such a bad rap, so I've put together the following toy example to clarify things. If this seems like too much to wade through, my main question is right down at the bottom.

I'll be using the cellular automaton Rule 30. For our purposes, I'm asserting that all of the things which seem almost certainly true about Rule 30 are in fact true, but it shouldn't impact the argument even if they're not.

R30 is typically presented as on the left, below. As with any ECA, you can simply slide the input cells in either direction, relative to the output cell, and while the logic stays the same, any output is skewed horizontally, as on the right. The yellow line intersects all the same cells in each plot, computationally speaking. We'll be using this right-shifted form.

R30 centered and right-skewed

To be explicit, the right-shifted form is given by

$$A_{m+1,n} \leftarrow A_{m,n} \oplus \left(A_{m,n+1} \lor A_{m,n+2}\right)\qquad$$ where $m,n$ are (rows from top, cols from right). Equivalently in algebraic binary form, that's $f(a,b,c):=(a+b+c+bc) \bmod 2$.

One advantage to using the right-shifted version is that it gains the property that every column is periodic in its sequence of white and black cells. In fact, every column will have a period of $2^k$, and $k$ increases monotonically as you travel left.

The assumption is that in the limit, half of all columns will increase $k$ by 1 and thus double the period from there on out, though there is no apparently regularity in the distribution of these columns. This graphic shows where the first several such columns are.

close up showing doubling columns

With that established, we can get to my point: Rule 30 exhibits a form of superdeterminism (as I understand it) where although the correlation between cells in general appears to approach $0$, a significant fraction of every pair of cells has some correlation, however weak, stemming from their single common ancestor and deterministic birth, and other surprising behaviors emerge as well.

So in our example, suppose R30 is the hidden base layer of the universe, and let's say that if a particle is at cell $(y,x)$, we can check its 'spin' in any of the four compass directions, where a black cell as its neighbor in that direction will be an up spin, and a white cell would be down. If you then select any cell at random from an arbitrarily large R30 plot, and any direction you like, you should get effectively random results for spin.

What about two particles? I have almost no understanding of how exactly entanglement can occur, but I did read that electrons in valence shells end up entangled, and I remember high school chem enough to know there are some straightforward mathematics governing that kind of thing. It's not clear to me whether we have any solid understanding of what entanglement physically consists of in some sense, vs. whether we merely have a thorough grasp of the mathematics that describe the resulting system. If local realism isn't a thing, however, it does seem suspicious that particles have to be in close proximity to entangle in the first place.

Suppose two particles have just become entangled:

money shot

...but, suppose that for whatever reason, when this happens, it typically happens such that the particles are occupying a doubling column, and that the turbulent foam of rules controlling them tend to place particles at distances of powers-of-2 apart. (Again, this is very much a toy example to illustrate the spirit of the concept.)

Now we have Alice with her green particle and Bob with his blue particle, and they speed apart in their spaceships to go experiment. We again assume that movement in space only happens along a column in multiples of $4096$ cells or some other contrivance, such that it will have no bearing on their environment with its far smaller period.

As Alice and Bob take measurements, they'll see strange results. To each of them, all of their results will appear random until they compare notes later. On average, this is what they'll see:

  • If Alice and Bob both sample to the west, they will see uncorrelated spins.
  • If they sample to the east, Alice and Bob will have identical spin.
  • If they both sample south, or both sample north, they will have opposing spins.
  • In any case where they sample in different directions, they will have uncorrelated spins.

...to a first approximation. That should be pretty accurate, but with a more careful analysis, you can ferret out more constrained cases that yield increasingly fractional results. For example, if Bob samples the cell to his south, and it's down, and Alice samples the one to her east, instead of being uncorrelated, it will actually have a $\frac{3}{4}$ chance of also being down, which of course becomes a slighter but still significant correlation if you remove some of the constraints.


My overall question here is that given we aren't yet sure if there's some bottom layer or what rules it would follow, are we certain that determinism like this is ruled out? This toy example could in principle be reworked (I suspect) so that it much more closely resembles actual experimental results. I'm wondering at the plausibility of superdeterminism operating on that bottom layer—NOT superdeterminism in the typical sense I read about, where the universe is conspiring to make a run of insanely improbable things happen only to spite you—but the more bland kind that arises in a scenario like this while managing still to create complicated correlations.

Yes, all things being equal, a superdeterministic framework will have negligibly weak correlations in general, but why are we so sure entanglement doesn't entail some process on a base level of reality that slots entangled particles into structures that do have a significant correlation, e.g. being reassigned/nudged into the modulo-$2^k$ relationship that arises in my example?

(I made an honest effort at inquiry here, but I know very little about physics/QM. This is also why I haven't been able to read 't Hooft's book on just this thing.)

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    $\begingroup$ "NOT superdeterminism in the typical sense I read about, where the universe is conspiring to make a run of insanely improbable things happen only to spite you—but the more bland kind that arises in a scenario like this while managing still to create complicated correlations." -- If you could show there was a "bland" type of superdeterminism that reproduced results of quantum mechanics (not a toy example), that might be interesting. But it's hard for me to imagine anything other than the conspiracy you mention in the first part of this statement being strong enough to do this. $\endgroup$
    – Andrew
    Commented Dec 11, 2022 at 17:30
  • $\begingroup$ @Andrew Fair enough, but that also gets me to what I don't understand; what makes the notion of there existing some underlying deterministic model seem so wildly unlikely, to the point where most physicists seem much more ready to throw away locality? It may be tricky to come up with a model that shows QM-like results, but I've never heard anyone explain why it's laughably implausible. $\endgroup$
    – Trev
    Commented Dec 11, 2022 at 17:39
  • $\begingroup$ Superdeterminism is basically saying that your apparatus settings will also be determined by the hidden variables, in addition to determin9ing the state of the system you are observing. It is unclear to me in what way your "model" is superdeterministic. $\endgroup$ Commented Dec 11, 2022 at 19:39

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