Towards correlation self-testing of quantum theory in the adaptive Clauser-Horne-Shimony-Holt game

TL;DR

This paper investigates the adaptive CHSH game as a method for correlation self-testing in quantum theory, analyzing its effectiveness across various generalized probabilistic theories to distinguish quantum correlations.

Contribution

It provides a comprehensive analysis of the adaptive CHSH game across different theories, showing quantum theory's unique optimal performance and advancing the understanding of correlation self-testing.

Findings

Quantum theory outperforms theories with minimal or maximal tensor products.

No alternative theories outperform quantum in the adaptive CHSH game.

Quantum performance cannot be recovered in certain generalized probabilistic theories.

Abstract

Correlation self-testing of a theory addresses the question of whether we can identify the set of correlations realisable in a theory from its performance in a particular information processing task. Applied to quantum theory it aims to identify an information processing task whose optimal performance is achieved only by theories realising the same correlations as quantum theory in any causal structure. In [Phys. Rev. Lett. 125 060406 (2020)] we introduced a candidate task for this, the adaptive CHSH game. Here, we analyse the maximum probability of winning this game in different generalised probabilistic theories. We show that theories with a joint state space given by the minimal or the maximal tensor product are inferior to quantum theory, before considering other tensor products in theories whose elementary systems have various two-dimensional state spaces. For these, we find no theories that outperform quantum theory in the adaptive CHSH game and prove that it is impossible to recover the quantum performance in various cases. This is the first step towards a general solution that, if successful, will have wide-ranging consequences, in particular, enabling an experiment that could rule out all theories in which the set of realisable correlations does not coincide with the quantum set.