Reply to the Comment on "The Axiom of Choice and the No-Signalling Principle"

Abstract

We clarify the meaning of a probabilistic no-signalling strategy in Ref [Proc. R. Soc. A 481, 20240601]. For every fixed input, the deterministic strategy constructed using the Axiom of Choice can indeed be represented by a Dirac probability distribution over the outputs. The relevant distinction arises when one considers the complete process from input to output, with the input itself sampled according to a probability distribution. In this case, the input and output spaces must be measurable and the dependence of the conditional output distribution on the input must also be measurable. Equivalently, the conditional distributions must form a Markov kernel. The Axiom-of-Choice strategy fails precisely this measurability requirement.