Abstract
This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or ’t Hooft’s Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness (or algorithmic randomness) that in various guises was investigated by Kolmogorov and others and treat Bell’s (Physics 1:195–200, 1964) Theorem and the Free Will Theorem with their implications for randomness.
Highlights
Foundations of Physics (2020) 50:61–104Quantum mechanics commands much respect
One aim of this paper is to provide arguments against this view,3 but even if these turn out to be unsuccessful I hope to contribute to the debate about the issue of determinism versus randomness by providing a broad view of the latter
The antipode defining which particular notion of randomness is meant may vary even within quantum mechanics, and here two main candidates arise (Sect. 3): one is determinism, as emphatically meant by Born [16] and most others who claim that randomness is somehow ‘fundamental’ in quantum theory, but the other is compressibility or any of the other equivalent notions defining what is called 1-randomness in
Summary
The antipode defining which particular notion of randomness is meant may vary even within quantum mechanics, and here two main candidates arise 3): one is determinism, as emphatically meant by Born [16] and most others who claim that randomness is somehow ‘fundamental’ in quantum theory, but the other is compressibility or any of the other equivalent notions defining what is called 1-randomness in. Mathematics as its antipode (see Appendix B for an explanation of this) The interplay between these different notions of randomness is the topic of Sects. 5 I argue that one cannot eat one’s cake and have it, in the sense of having a deterministic hidden variable theory underneath quantum mechanics that is strictly compatible with the Born rule. My analysis relies on some mathematical background presented in (independent) Appendices A–C on the Born rule, Algorithmic (or 1-) randomness, and the Bell and Free Will Theorems
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