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The CHSH Game, Tsirelson's Bound, and Causal Locality

Jacob A. Barandes, Mahmudul Hasan, David Kagan

TL;DR

The paper reframes the CHSH game using indivisible stochastic processes and a causal locality postulate to derive Tsirelson's bound, arguing that no-signaling alone does not constrain quantum correlations. By embedding the dynamics in Barandes's stochastic-quantum correspondence, it shows that causal locality suffices to bound CHSH violations up to the Tsirelson limit while remaining consistent with no-signaling. The approach relies on unistochastic evolution, division events, and a precise causal structure to connect stochastic dynamics with quantum-like correlations, and it suggests that violations beyond Tsirelson's bound would require nonlocal or non-unistochastic dynamics. The work also discusses implications for causality in quantum theory, the role of memory and non-Markovian effects, and possible future formalisms such as quantum combs to broaden the framework.

Abstract

We reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a novel proof of the Tsirelson bound. In particular, we show that unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.

The CHSH Game, Tsirelson's Bound, and Causal Locality

TL;DR

The paper reframes the CHSH game using indivisible stochastic processes and a causal locality postulate to derive Tsirelson's bound, arguing that no-signaling alone does not constrain quantum correlations. By embedding the dynamics in Barandes's stochastic-quantum correspondence, it shows that causal locality suffices to bound CHSH violations up to the Tsirelson limit while remaining consistent with no-signaling. The approach relies on unistochastic evolution, division events, and a precise causal structure to connect stochastic dynamics with quantum-like correlations, and it suggests that violations beyond Tsirelson's bound would require nonlocal or non-unistochastic dynamics. The work also discusses implications for causality in quantum theory, the role of memory and non-Markovian effects, and possible future formalisms such as quantum combs to broaden the framework.

Abstract

We reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a novel proof of the Tsirelson bound. In particular, we show that unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.

Paper Structure

This paper contains 13 sections, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of the CHSH Game
  3. Setup
  4. No-Signaling and NS Boxes
  5. The Stochastic-Quantum Correspondence and Causal Locality
  6. Indivisible Stochastic Systems
  7. Composite Systems
  8. The Stochastic-Quantum Correspondence
  9. Causal Influence and Causal Locality
  10. Stochastic Description of the CHSH Game
  11. The Role of Causal Locality
  12. Proof
  13. Future Directions and Conclusion

Figures (2)

  • Figure 1: A spacetime diagram depicting one round of the CHSH game.
  • Figure 2: At time $t_3$, the configuration $q$ of $Q$ may depend on the random bit $x$ as it lies in the blue-shaded future light cone in the diagram. Similarly, the configuration $r$ of $R$ at time $t_3$ lies in the red-shaded future light cone of random bit $y$, and may therefore depend on $y$. Both configurations $q$ and $r$ will generically depend on the preparation $\Psi$ as it lies in the overlap of their past light cones under the intersection of the dotted lines in the diagram. However at $t_3$, causal locality implies that $r$ cannot depend on $x$ and $q$ cannot depend on $y$ as each lies outside of the future light cones of $x$ and $y$, respectively.