Formalizing CHSH Rigidity in Lean 4

2026-04-04

AI Summary

This work presents the first formal verification of the CHSH rigidity theorem in Lean 4, rigorously proving that any strategy achieving a near-optimal violation of the CHSH inequality must be locally isometric to the canonical two-qubit strategy. By integrating techniques from linear algebra, quantum information theory, and interactive theorem proving, the authors not only provide a complete machine-checked verification of this foundational result in quantum information but also uncover a logical gap in the original proof by McKague et al. This effort substantially enhances the mathematical rigor and reliability of the CHSH rigidity theorem and establishes a crucial foundation for the formal verification of quantum cryptographic protocols.

Abstract

Violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality certifies genuine quantum correlations. In this work, we formalize in Lean 4 the rigidity theorem -- any strategy achieving near-optimal CHSH value must be locally isometric to the canonical qubit strategy. In the course of formalization, we identified a gap in the argument of McKague, Yang, and Scarani (arXiv:1203.2976).

Problem

• CHSH inequality
• rigidity theorem
• quantum correlations
• formal verification
• Lean 4

Innovation

• CHSH rigidity
• formal verification
• Lean 4
• quantum correlations
• local isometry