I am a big fan of E. T. Jaynes. His book Probability Theory: The Logic of Science is the only book on statistics that I ever felt I could understand. Therefore, when he appears to rail against the conclusions of Bell’s theorem in his paper “Clearing up Mysteries—The Original Goal”, I take him seriously. He suggests that perhaps there could be a time-dependent hidden variable theory that could yield the outcomes that quantum mechanics predicts.

However, after reading Richard D. Gill’s paper, “Time, Finite Statistics, and Bell’s Fifth Position” it is very clear that there can be nothing like a classical explanation that yields quantum predictions, time-dependent or otherwise. In this paper Gill reintroduces Steve Gull’s computer network, where a pair of classical computers is tasked to recreate probabilities predicted in a Bell-CHSH delayed choice experiment. The catch is that the challenger gets to choose the stream of bits sent to each of the two spatially separated computers in the network. These bits represent the free choice an experimenter running a Bell-CHSH experiment has to choose which polarization measurements to make. No matter what the classical computer does, no matter how much time-dependent fiddling you want to do, it can never produce correlations that will violate the Bell-CHSH inequality in the long run. This is Gull’s “You can’t program two independently running computers to emulate the EPR experiment” theorem.

Gill presents a nice analogy with playing roulette in the casino. Because of the rules of roulette, there is no computer algorithm can implement a strategy that will beat the house in roulette in the long run. Gill goes on to quantify exactly how long the long run is in order to place a wager against other people who claim they can recreate the probabilities predicted by quantum mechanics using a classical local hidden variable theory. Using the theory of supermartingales, one can bound the likelihood of seeing the Bell-CHSH inequality violated by chance by any classical algorithm in the same way that one can bound the likelihood of long winning streaks in roulette games.

I liked the casino analogy so much that I decided to rephrase Gull’s computer network as a coin guessing casino game I call Bell’s Casino. We can prove that any classical strategy, time-dependent or otherwise, simply cannot beat the house at that particular game in the long run. Yet, there is a strategy where the players employ entangled qubits and beat the house on average. This implies there cannot be any classical phenomena that yields quantum outcomes. Even if one proposes some classical oscollating (time-dependent) hidden variable vibrating at such a high rate that we could never practically measure it, this theory still could not yield quantum probabilities, because such a theory implies we could simulate it with Gull’s computer network. Even if our computer simulation was impractically slow, we could still, in principle, deploy it against Bell’s Casino to beat their coin game. But no such computer algorithm exists, in exactly the same way that there is no computer algorithm that will beat a casino at a fair game of roulette. The fact that we can beat the casino by using qubits clearly proves that qubits and quantum physics is something truly different.

You may have heard the saying that “correlation does not imply causation”.
The idea is that if outcomes `A` and `B` are correlated, the either `A` causes `B`, or `B` causes `A`, or there is some other `C` that causes `A` and `B`.
However, in quantum physics there is a fourth possibilty.
We can have correlation without causation.

In light of Gull and Gill’s iron clad argument, I went back to reread Jaynes’s “Clearing up Mysteries”. I wanted to understand how Jaynes could have been so mistaken. After rereading it I realized that I had misunderstood what he was trying to say about Bell’s theorem. Jaynes just wanted to say two things.

Firstly, Jaynes wanted to say that Bell’s theorem does not necessarily imply action at a distance. This is not actually a controversial statement. The many-worlds interpretation is a local, non-realist (in the sense that experiments do not have unique definite outcomes) interpretation of quantum mechanics. This interpretation does not invoke any action at a distance and is perfectly compatible with Bell’s theorem. Jaynes spends some time noting that correlation does not imply causation in an attempt to clarify this point although he never talks about the many-worlds interpretation.

Secondly, Jaynes wanted to say that Bell’s theorem does not imply that quantum mechanics is the best possible physical theory that explains quantum outcomes. Here his argument is half-right and half-wrong. He spends some time suggesting that maybe there is a time-dependent hidden variable theory that could give more refined predictions than predicted by quantum theory. However, the suggestion that any classical theory, time-dependent or otherwise, could underlie quantum mechanics is refuted by Bell’s theorem and this is clearly illustrated by Gull’s computer network or by Bell’s casino. Jaynes learned about Gull’s computer network argument at the same conference that he presented “Clearing Up Mysteries”. His writing suggests that he was surprised by the argument, but he did not want to rush to draw any conclusions to from it without time to get a deeper understanding of it. Nevertheless, Jaynes larger point was still correct. Bell’s theorem does not imply that there is not some, non-classical, refinement of quantum mechanics that might yield more informative predictions than quantum mechanics does and Jaynes was worried that people would not look for such a refinement.

Jaynes spent a lot of effort trying to separate epistemology, where probability theory rules how we reason in the face of imperfect knowledge, from ontology, which describes what happens in reality if we had perfect information. Jaynes thought that quantum mechanics was mixing these two branches together into one theory and worried that if people were mistaking quantum mechanics for an ontological theory then they would never seek a more refined theory.

While Bell’s theorem does not rule out that there may be a non-classical hidden variable theory, Colbeck and Renner’s paper “No extension of quantum theory can have improved predictive power” all but eliminates that possibility by proving that there is no quantum hidden variable theory. This can be seen as a strengthening of Bell’s theorem, and they even address some of the same concerns that Jaynes had about Bell’s theorem.

To quote Bell [2], locality is the requirement that “…the result of a measurement on one system [is] unaffected by operations on a distant system with which it has interacted in the past…” Indeed, our non-signalling conditions reflect this requirement and, in our language, the statement that P

_{XYZ|ABC}is non-signalling is equivalent to a statement that the model is local (see also the discussion in [28]). (We remind the reader that we do not assume the non-signalling conditions, but instead derive them from the free choice assumption.) In spite of the above quote, Bell’s formal definition of locality is slightly more restrictive than these non-signalling conditions. Bell considers extending the theory using hidden variables, here denoted by the variable Z. He requires P_{XY|ABZ}= P_{X|AZ}× P_{Y|BZ}(see e.g. [13]), which corresponds to assuming not only P_{X|ABZ}= P_{X|AZ}and P_{Y|ABZ}= P_{Y|BZ}(the non-signalling constraints, also called parameter-independence in this context), but also P_{X|ABYZ}= P_{X|ABZ}and P_{Y|ABXZ}= P_{Y|ABZ}(also called outcome-independence). These additional constraints do not follow from our assumptions and are not used in this work.

The probabilistic assumptions are weaker in Colbeck and Renner’s work than in Bell’s theorem, because they want to exclude quantum hidden variable theories in addition to classical hidden variable theories. Today, if one wants to advance a local hidden variable theory, it would have to be a theory that is even weirder than quantum mechanics, if such a thing is even logically possible. It seems that quantum mechanics’s wave function is an ontological description after all.

I wonder what Jaynes would have thought about this result. I suspect he would still be looking for an exotic hidden variable theory. He seemed so convinced that probability theory was solely in the realm of epistemology and not ontology that he would not accept any probabilistic ontology at all.

I think Jaynes was wrong when he suggested that quantum mechanics was necessarily mixing up epistemology and ontology.
I believe the many-worlds interpretation is trying to make that distinction clear.
In this interpretation the wave-function and Schrödinger’s equation is ontology, but the Born rule that relates the norm-squared amplitude to probability ought to be epistemological.
However, there does remain an important mystery here:
Why do the observers within the many-worlds observe quantum probabilities that satisfy the Born rule?
I like to imagine Jaynes could solve this problem if he were still around.
I imagine that would say that something like, “Due to phase invariance of the wave-function … *something something* … transformation group … *something something* … thus the distribution must be in accordance with the Born rule.”
After all, Jaynes did manage to use transformation groups to solve the Bertrand paradox, a problem widely regarded as being unsolvable due to being underspecified.