Gull's theorem revisited

TL;DR

This paper revisits Gull's Fourier-based proof of Bell's theorem, framing it as a no-go theorem for classical distributed computing with shared randomness, and clarifies the proof by filling gaps and correcting errors.

Contribution

It provides a corrected and complete version of Gull's Fourier-theoretic proof of Bell's theorem, emphasizing the role of shared randomness in classical distributed systems.

Findings

Bell's theorem can be viewed as a no-go result for classical distributed computing.

Shared i.i.d. randomness is essential for the proof.

The proof is clarified and gaps are filled with a third computer or pseudo-random generator.

Abstract

Steve Gull, in unpublished work available on his Cambridge University homepage, has outlined a proof of Bell's theorem using Fourier theory. Gull's philosophy is that Bell's theorem (or perhaps a key lemma in its proof) can be seen as a no-go theorem for a project in distributed computing with classical, not quantum, computers. We present his argument, correcting misprints and filling gaps. In his argument, there were two completely separated computers in the network. We need three in order to fill all the gaps in his proof: a third computer supplies a stream of random numbers to the two computers representing the two measurement stations in Bell's work. One could also imagine that computer replaced by a cloned, virtual computer, generating the same pseudo-random numbers within each of Alice and Bob's computers. Either way, we need an assumption of the presence of shared i.i.d. randomness in the form of a synchronised sequence of realisations of i.i.d. hidden variables underlying the otherwise deterministic physics of the sequence of trials. Gull's proof then just needs a third step: rewriting an expectation as the expectation of a conditional expectation given the hidden variables.