Robust self-testing for nonlocal games with robust game algebras

TL;DR

This paper introduces an operator-algebraic framework for robust self-testing in nonlocal games, linking it to unique states on game algebras and providing new quantitative bounds for robustness.

Contribution

It establishes a novel operator-algebraic characterization of robust self-testing, connecting it to the uniqueness of tracial states on game algebras and extending the Gowers-Hatami theorem to C*-algebras.

Findings

Self-testing corresponds to unique tracial states on game algebras.

Parallel repetition preserves self-testing properties for synchronous games.

Quantitative bounds for robustness are derived using an extended Gowers-Hatami theorem.

Abstract

We give an operator-algebraic formulation of robust self-testing in terms of states on C*-algebras. We show that a quantum correlation p is a robust self-test only if among all (abstract) states, there is a unique one achieving p. We show that the "if" direction of this statement also holds, provided that p is optimal/perfect for a nonlocal game that has a robust game algebra. This last condition applies to many nonlocal games of interest, including all XOR games, synchronous games, and boolean constrained system (BCS) games. For those nonlocal games with robust game algebras, we prove that self-testing is equivalent to the uniqueness of finite-dimensional tracial states on the associated game algebra, and robust self-testing is equivalent to the uniqueness of amenable tracial states. Applying this tracial-state characterization of self-testing to parallel repetition, we show that a synchronous game is a self-test for perfect quantum strategies if and only if its parallel repeated version is a self-test for perfect quantum strategies. As a proof approach, we give the first quantitative Gower-Hatami theorem that is applicable to C*-algebras. Here "quantitative" means there is a constructive bound on the distance between the approximate representations and exact representations. We also demonstrate how this quantitative Gowers-Hatami theorem can be used to calculate the explicit robustness function of a self-test.