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

This paper clarifies that while Axiom-of-Choice-based deterministic strategies can represent fixed-input no-signalling scenarios, they ultimately fail to constitute valid probabilistic no-signalling strategies because they violate the measurability requirements necessary when inputs are sampled from a probability distribution.

Original authors: Ämin Baumeler, Borivoje Dakić, Flavio Del Santo, Miloš Milovanović

Published 2026-09-07

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CC BY 4.0

Original authors: Ämin Baumeler, Borivoje Dakić, Flavio Del Santo, Miloš Milovanović

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of physics where the rules of cause and effect are tested, scientists often ask a simple question: can information travel faster than light? The answer, according to the fundamental principle known as no-signalling, is a firm no. This rule ensures that if one person makes a choice here, it cannot instantly change the reality of a distant observer, preserving the logical order of the universe. To explore the boundaries of this rule, researchers sometimes imagine games where two players must coordinate their answers without communicating. In these scenarios, they can use strategies that are either fixed and predictable or those that involve a roll of the dice. For decades, the mathematical tools used to describe these games have treated fixed strategies and random ones as distinct categories, with the assumption that a fixed plan is just a special, simple case of a random one.

A recent exchange between physicists has brought a subtle but crucial distinction to the forefront of this discussion. The debate centers on a specific mathematical tool called the Axiom of Choice, a powerful concept that allows mathematicians to construct complex, deterministic functions even when no clear pattern exists. In a previous study, researchers used this axiom to build a strategy that appeared to be a fixed, deterministic plan. However, a group of critics argued that this plan was actually just a probabilistic strategy in disguise, suggesting that the original study had drawn a false line between the two types of plans. The authors of the original study have now replied, clarifying that the critics missed a vital operational detail. They argue that while the critics are mathematically correct in a narrow, point-by-point sense, they fail to capture how the strategy works in a real, physical experiment where inputs are chosen randomly.

The core of the disagreement lies in how one defines a "strategy" when the game begins. In the critics' view, if you look at a single, specific input given to a player, the Axiom of Choice produces a single, definite output. Since a single outcome can be described as a probability of one hundred percent, they conclude the strategy is probabilistic. The authors of the reply agree that this logic holds up if you freeze the game at a single moment. However, they insist that a true strategy must describe the entire flow of the game, from the moment the referee picks a random input to the moment the player gives an answer. In the real world, the referee does not pick a single, fixed number; they pick from a continuous range of possibilities, like selecting a number anywhere between zero and one.

For a strategy to be valid in this full, flowing context, the way it connects inputs to outputs must be measurable. This means that the dependence of the output distribution on the input must satisfy specific mathematical criteria (Borel measurability) that allow the process to be integrated over the random choices of the referee. The authors explain that the strategy built using the Axiom of Choice breaks this rule. While it gives a definite answer for every single number the referee might pick, the collection of all these answers is not measurable, preventing it from being composed with the random selection of inputs to produce a coherent result. It is like trying to calculate the average height of a crowd where every person's height is defined, but the way they are arranged makes it impossible to measure the group as a whole. Because the strategy cannot be combined with the random selection of inputs to produce a coherent result, it fails the test of being a true probabilistic process.

The authors conclude that the distinction they made in their original work remains valid, but it requires a more precise definition of what a probabilistic strategy actually is. They clarify that a probabilistic strategy is not just a list of probabilities for every possible input; it is a complete, measurable process that works when inputs are sampled randomly. The Axiom of Choice strategy, while mathematically consistent in a static sense, is not a valid probabilistic strategy in this operational sense because it lacks the necessary measurability to function in a real experiment. This clarification does not change the laws of physics, but it sharpens the mathematical language used to describe them, ensuring that the difference between a fixed rule and a random process is defined by how they behave in the real world, not just by how they look on paper.

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

Problem Statement
This paper addresses a critique raised in a Comment [2] regarding the distinction between deterministic and probabilistic no-signalling strategies established in the authors' previous work [1]. The Comment argues that a deterministic strategy constructed using the Axiom of Choice (AC), as presented in Ref. [1], can be mathematically represented as a probabilistic no-signalling strategy. Specifically, the Commenters contend that since every fixed input $x$ yields a definite output $g(x)$ , the resulting empirical model $e_x = \delta_{g(x)}$ (a Dirac measure) satisfies the definition of a probabilistic strategy. Consequently, they assert that the purported distinction between deterministic and probabilistic strategies in Ref. [1] is invalid.

Methodology and Clarification of Definitions
The authors clarify that the disagreement stems from differing definitions of "probabilistic strategy." While the Comment adopts a pointwise definition, the authors argue that Ref. [1] intended a stronger, operational definition involving the complete stochastic process.

Pointwise vs. Operational View:
- Comment's View: For any fixed input $x$ , the output is a probability measure (specifically a Dirac distribution). Under this definition, the family of distributions $(e_x)$ constitutes a probabilistic empirical model.
- Authors' View (Ref. [1]): A probabilistic strategy must describe a single stochastic experiment where the input itself is sampled from a probability distribution. Therefore, the strategy must define a measurable mapping from the input space to the space of output distributions.
Measurability Requirements:
The authors formalize the operational setting where:
- The input space is $X = [0, 1]$ equipped with the Borel $\sigma$ -algebra $\mathcal{A}$ .
- The output space is $Y = \{0, 1\}$ equipped with the discrete $\sigma$ -algebra $\mathcal{B}$ .
- A strategy is defined by a function $g: X \to Y$ .
- For the strategy to be operational, the map $x \mapsto e_x(B)$ must be measurable for every $B \in \mathcal{B}$ . In measure-theoretic terms, the family $(e_x)$ must form a Markov kernel.
The Failure of the AC-Based Strategy:
The authors demonstrate that while the AC-based strategy assigns a Dirac distribution to every fixed input, the function $g$ constructed via the Axiom of Choice is not measurable. Specifically, the pre-images $g^{-1}(\{0\})$ and $g^{-1}(\{1\})$ do not belong to the Borel $\sigma$ -algebra $\mathcal{A}$ . Consequently, the map $x \mapsto \delta_{g(x)}$ fails to be a Markov kernel.

Key Results

Incompatibility with Random Inputs: Because the AC-based strategy is not measurable, it cannot be composed with a referee's random choice of input (sampled according to a probability measure $\mu$ ). The integral $\int_X e_x(B) d\mu(x)$ , which defines the operational probabilities for the game, is undefined for non-measurable functions.
Distinction Validity: The distinction between "functional no-signalling strategies" (deterministic functions, potentially non-measurable) and "probabilistic no-signalling strategies" (measurable stochastic input-output processes) remains valid. The AC-based strategy belongs to the former category but fails to qualify as the latter in the operational sense intended by Ref. [1].
Context of Finiteness: The authors note that the measurability condition is trivial when the input space is finite and discrete, which explains why the distinction might be overlooked in finite settings but becomes critical in the infinitary setting of Ref. [1].

Significance and Claims
The paper does not claim to introduce new physical phenomena or experimental proposals. Instead, its significance lies in clarifying the mathematical and conceptual framework used in Ref. [1].

Operational Meaning: The authors emphasize that the term "probabilistic strategy" in their original work refers to the complete stochastic process generated when a randomly sampled input is processed. The AC-based strategy, while well-defined in a purely functional framework, lacks this operational meaning because it cannot generate well-defined probabilities when inputs are random.
Resolution of the Comment: The authors conclude that the Comment is formally consistent within its own (weaker) definitions but fails to capture the intended operational meaning of Ref. [1]. The AC-based strategy is a valid deterministic functional strategy but is not a valid probabilistic strategy under the measurability requirements necessary for operational no-signalling games with continuous inputs.

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

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