Comment on 'The axiom of choice and the no-signalling principle'

Abstract

The main claim of Baumeler et al. (Proc. R. Soc. A 481, 20240601; arXiv:2206.08467) is that "functional (deterministic) no-signalling resources can be stronger than probabilistic ones" in a certain nonlocal game on a Bell scenario with countably many parties. We disagree and argue that (i) under standard definitions, deterministic no-signalling resources are always probabilistic no-signalling resources; (ii) the deterministic strategy considered in Baumeler et al. can be promoted to a genuinely probabilistic strategy with similar properties; and (iii) a key step in a derivation in Baumeler et al., claimed to hold for all no-signalling strategies, implicitly assumes measurability, leaving a gap in the argument. We propose measurability assumptions which we conjecture would make this derivation rigorous. Taken together, the phenomenon highlighted by Baumeler et al. is best understood as a difference between measurable and non-measurable no-signalling resources.

Accepted for publication in Proc. R. Soc. A