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Robust self-testing with CHSH mod 3

Igor Klep, Nando Leijenhorst, Victor Magron

Abstract

The CHSH mod 3 Bell inequality is a natural testbed for higher-dimensional quantum nonlocality, yet its maximal quantum violation and self-testing properties have remained unresolved. We determine its exact maximal quantum value and show that, up to unitary equivalence and the natural symmetries of the inequality, it admits a unique optimal irreducible strategy; equivalently, there are four symmetry-related optimal irreducible strategies. Each of these strategies uses a maximally entangled two-qutrit state. We further prove that any strategy whose value is within $\varepsilon$ of the optimum is $O(\sqrt{\varepsilon})$-close, up to local isometries, to a direct sum of optimal irreducible strategies.

Robust self-testing with CHSH mod 3

Abstract

The CHSH mod 3 Bell inequality is a natural testbed for higher-dimensional quantum nonlocality, yet its maximal quantum violation and self-testing properties have remained unresolved. We determine its exact maximal quantum value and show that, up to unitary equivalence and the natural symmetries of the inequality, it admits a unique optimal irreducible strategy; equivalently, there are four symmetry-related optimal irreducible strategies. Each of these strategies uses a maximally entangled two-qutrit state. We further prove that any strategy whose value is within of the optimum is -close, up to local isometries, to a direct sum of optimal irreducible strategies.

Paper Structure

This paper contains 22 sections, 18 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Polynomial optimization
  5. CHSH mod $d$
  6. Representation theory
  7. Symmetries of CHSH mod $d$
  8. Upper bound on $\beta_q$ for CHSH mod $3$
  9. Optimizer extraction
  10. Extraction through SOS certificates
  11. Flatness
  12. Robust self-testing with CHSH mod $3$
  13. Discussion
  14. Methods
  15. SOS conditional expectations
  16. ...and 7 more sections

Key Result

Theorem A

The CHSH mod $3$ Bell function has maximal quantum value $\frac{1}{3} + \frac{2\cos(\pi/18)}{3\sqrt{3}}$. Moreover, up to unitary transformations and the natural symmetries of the Bell inequality, there is a unique corresponding irreducible strategy. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Theorem A: Theorem \ref{['thm:chsh_exact_violation']} and Theorem \ref{['thm:unique']}
  • Theorem B: Theorem \ref{['thm:robust']}
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 23 more