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

TL;DR

The paper critiques a recent claim about the strength of deterministic versus probabilistic no-signalling resources, emphasizing the importance of measurability assumptions in the analysis.

Contribution

It refutes the claim that deterministic no-signalling resources are stronger than probabilistic ones and highlights the role of measurability assumptions in the derivation.

Findings

Deterministic no-signalling resources are equivalent to probabilistic ones under standard definitions.

A deterministic strategy can be transformed into a probabilistic one with similar properties.

Measurability assumptions are crucial for the validity of the derivation in the referenced work.

Abstract

The main claim of arXiv:2206.08467 is that "functional (deterministic) no-signalling resources can be stronger than probabilistic ones" 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 arXiv:2206.08467 can be promoted to a genuinely probabilistic strategy with similar properties and (iii) a key step in the derivation in arXiv:2206.08467, 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 in arXiv:2206.08467 is best understood as a difference between measurable and non-measurable no-signalling resources.