In Comments on indeterminism and undecidability the abstract reads:

"In a recent paper 1, it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum measurements can be considered as an infinite 1-random or n-random sequence. In this brief note we present our comments on this claim. We have mostly positive but also some negative comments on the arguments of the paper 1. Furthermore, we speculate a logical-axiomatic study of nature which we believe can intrinsically provide quantum mechanical probabilities based on 1(n)- randomness."

What grabbed my attention was something I read towards the end though:

"Finally, we would like to mention an issue that weakens the argument of approach (i). If the locality and experimenters’ free will are assumed, it can be shown that Bell’s theorem rules out any algorithm that predicts experimental results even if algorithm is unknowable."

"Now let’s choose a hidden variable which encodes this algorithm and assume that there are hidden universal Turing machines at points A and B. These Turing machines read the hidden variable when the particles arrive and output the measurement results. These outputs of Turing machines must obey Bell inequalities. This is obvious because such an algorithm and the hidden variable carrying it cannot transmit the spooky action at a distance. Since quantum mechanics can violate Bell’s inequalities, the sequence of outcomes it produces cannot be generated even by an unknowable algorithm. For this reason, the unknowable algorithm cannot save determinism under the assumptions of locality and free will."

But the problem is that I just finished reading a different paper named Quantum Randomness: From Practice to Theory and Back and there I found this:

"The “magic” of the quantum technology capable of producing unbreakable security depends on the possibility of producing true random bits. What does “true ran-domness” 5 mean? The concept is not formally defined, but a common meaning is the lack of any possible correlations. Is this indeed theoretically possible? The answer is negative: there is no true randomness, irrespective of the method used to produce it. The British mathematician and logician Frank P. Ramsey was the first to demonstrate it in his study of conditions under which order must appear (see [24, 32]); other proofs have been given in the framework of algorithmic information theory [9]"

And this:

"Randomness means unpredictability with respect to some fixed theory. The quality of a particular type of randomness depends on the power of the theory to detect correlations, which determines how difficult predictability is (see more in [4, 6]). For example, finite automata detect less correlations than Turing machines. Consequently, finite automata, based unpredictability is weaker than Turing machine, based unpredictability: there are (many) sequences computable by Turing machines (hence, predictable, not random) that are unpredictable, random, for finite automata. In analogy with the notion of incomputability (see [15]), one can prove that there is a never-ending hierarchy of stronger (better quality) and stronger forms of randomness."

So my confusion is this: It is claimed that no algorithm can produce the quantum results because it is ruled out by Bell's theorem. But the other paper states that basically no real number is totally random, and any sequence can be predicted/generated by some unknown algorithm. So this seems like contradiction. I don't know what I'm not seeing/missing.

  • $\begingroup$ Note that Bell's theorem only rules out SOME quantum phenomena from being reproduced using classical physics. $\endgroup$
    – Mauricio
    Commented Apr 4, 2023 at 10:06
  • 1
    $\begingroup$ It's important to distinguish between finite & infinite sequences. It's trivial to generate any finite sequence, so we generally don't consider a finite sequence to be random unless its complexity is high. See en.wikipedia.org/wiki/Kolmogorov_complexity & en.wikipedia.org/wiki/Normal_number $\endgroup$
    – PM 2Ring
    Commented Apr 4, 2023 at 12:29
  • $\begingroup$ There is no such thing as a quantum coin. That's simply a made up system that doesn't exist in nature. Nothing stops you from doing that, of course, but you aren't going to learn anything about reality using such models. $\endgroup$ Commented Apr 5, 2023 at 6:40

1 Answer 1


„It is claimed that no algorithm can produce the quantum results because it is ruled out by Bell's theorem.”

This is not true. Bell’s theorem only rules out some local hidden variables theories that obey a certain assumption (the hidden variables and the settings of the detectors in a Bell test are independent variables). The rejection of that assumption allows you to construct local, hidden variable, deterministic theories such as ’t Hooft’s cellular automaton interpretation.

We also have a deterministic interpretation of QM that does not involve hidden variables (Deutsch-Hayden many worlds interpretation).

Non-local deterministic interpretations also exist (Bohmian mechanics).

It is worth noting that only deterministic theories can be local (EPR argument). This is a very powerful argument in favor of determinism. In conclusion, there is no incompatibility between QM and determinism. In fact, non-deterministic theories are pretty much ruled out by EPR since they have to be non-local and hence, incompatible with Special Relativity.

  • $\begingroup$ Many worlds of any flavor is simply a misunderstanding of quantum mechanics and ruled out both by observation and trivial requirements for the word "science". t'Hofft admits that he can't actually reproduce physics. At least that's a lot more honest than Everett and his followers. Bohmian mechanics is nothing but an extra hard way of calculating the solutions of the Schroedinger equation that requires a non-physical entity to be introduced that does not allow for any backreaction (i.e. none of the usual conservation laws hold). $\endgroup$ Commented Apr 5, 2023 at 6:49
  • $\begingroup$ @FlatterMann, What is your proposal then? $\endgroup$
    – Andrei
    Commented Apr 5, 2023 at 8:24
  • $\begingroup$ My general "proposal" is to take Copenhagen seriously. It is actually trying to tell us something non-trivial about nature. I have been around the quantum mysticism block a couple of times... only to discover that the only real problem we have with quantum mechanics is an educational one: our QM courses and books emphasize the solution theory of the Schroedinger equation so much that all physics is lost. This is especially hard on those theorists who don't like to spend any time in the laboratory. They start looking for explanations in a formalism that was designed to not contain them. $\endgroup$ Commented Apr 5, 2023 at 8:36
  • $\begingroup$ @FlatterMann, Copenhagen is non-local, this is my problem with it. Even QFT is non-local in regards to measurement/collapse. At least, I was not able to find a local/relativistic treatment of collapse. Of course, you can chose Copenhagen over Bohm, but if you want locality you need to go for either many worlds or superdeterminism. $\endgroup$
    – Andrei
    Commented Apr 5, 2023 at 9:13
  • $\begingroup$ Why is Copenhagen non-local (I am not talking about the non-relativistic SE here, merely about the structure of the interpretation)? Or QFT? They are both non-separable, but that's not the same as non-locality. Locality lives in physical space, separability lives in Hilbert space. There is no such thing as collapse. One can't even find a useful definition of that word in the physics literature for all I know. See, this is exactly what I mean by "poor teaching". People are throwing a lot of useless words around instead of giving physical explanations that match the actual observations. $\endgroup$ Commented Apr 5, 2023 at 9:19

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