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.
