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A game-theoretic probability approach to loopholes in CHSH experiments

Takara Nomura, Koichi Yamagata, Akio Fujiwara

TL;DR

This work recasts the CHSH inequality within game-theoretic probability, avoiding an underlying probability space and focusing on information timing. It formalizes the locality and freedom-of-choice loopholes as structural constraints in sequential hidden-variable games between Scientists and Nature, then presents a loopholes-closed game with two capital processes that monitor (i) convergence to CHSH correlations and (ii) independence-like behavior between settings and hidden variables. A central result shows that these requirements cannot be satisfied simultaneously in the loopholes-closed setting: Nature cannot keep both capital processes bounded, yielding an operational reading of CHSH violations as a consequence of the fundamental informational structure. The framework thus provides a nonprobabilistic, data-driven interpretation of CHSH violations and suggests avenues for extending game-theoretic analyses to other Bell-type scenarios and independence notions.

Abstract

We study the CHSH inequality from an informational, timing-sensitive viewpoint using game-theoretic probability, which avoids assuming an underlying probability space. The locality loophole and the measurement-dependence (``freedom-of-choice'') loophole are reformulated as structural constraints in a sequential hidden-variable game between Scientists and Nature. We construct a loopholes-closed game with capital processes that test (i) convergence of empirical conditional frequencies to the CHSH correlations and (ii) the absence of systematic correlations between measurement settings and Nature's hidden-variable assignments, and prove that Nature cannot satisfy both simultaneously: at least one capital process must diverge. This yields an operational winning strategy for Scientists and a game-theoretic probabilistic interpretation of experimentally observed CHSH violations.

A game-theoretic probability approach to loopholes in CHSH experiments

TL;DR

This work recasts the CHSH inequality within game-theoretic probability, avoiding an underlying probability space and focusing on information timing. It formalizes the locality and freedom-of-choice loopholes as structural constraints in sequential hidden-variable games between Scientists and Nature, then presents a loopholes-closed game with two capital processes that monitor (i) convergence to CHSH correlations and (ii) independence-like behavior between settings and hidden variables. A central result shows that these requirements cannot be satisfied simultaneously in the loopholes-closed setting: Nature cannot keep both capital processes bounded, yielding an operational reading of CHSH violations as a consequence of the fundamental informational structure. The framework thus provides a nonprobabilistic, data-driven interpretation of CHSH violations and suggests avenues for extending game-theoretic analyses to other Bell-type scenarios and independence notions.

Abstract

We study the CHSH inequality from an informational, timing-sensitive viewpoint using game-theoretic probability, which avoids assuming an underlying probability space. The locality loophole and the measurement-dependence (``freedom-of-choice'') loophole are reformulated as structural constraints in a sequential hidden-variable game between Scientists and Nature. We construct a loopholes-closed game with capital processes that test (i) convergence of empirical conditional frequencies to the CHSH correlations and (ii) the absence of systematic correlations between measurement settings and Nature's hidden-variable assignments, and prove that Nature cannot satisfy both simultaneously: at least one capital process must diverge. This yields an operational winning strategy for Scientists and a game-theoretic probabilistic interpretation of experimentally observed CHSH violations.
Paper Structure (11 sections, 4 theorems, 40 equations, 1 table)

This paper contains 11 sections, 4 theorems, 40 equations, 1 table.

Table of Contents

  1. Introduction
  2. Game-theoretic probability theory: a minimal background
  3. From CHSH scenario to a game-theoretic framework
  4. Game-theoretic analysis of CHSH experiments with and without loopholes
  5. A hidden-variable game with the locality loophole
  6. A hidden-variable game with loopholes closed
  7. Concluding remarks
  8. Proofs of Theorems presented in Section 4
  9. Proof of Theorem \ref{['thm:CHSH_LLN_with_commu_loophole']}
  10. Proofs of Theorem \ref{['thm:CHSH_holds']}
  11. CHSH inequality and its assumption

Key Result

Theorem 2.1

In the simple predictive game, Skeptic has a strategy $q : \Omega^\ast \rightarrow \mathcal{P}(\Omega)$ that ensures $\lim_{n \to \infty} K_n = \infty$ unless for all $a \in \Omega$, where $\delta_a$ denotes the Kronecker delta.

Theorems & Definitions (7)

  • Theorem 2.1: Game-theoretic law of large numbers
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem B.1: CHSH inequality
  • proof