Counterexamples in self-testing

TL;DR

This paper presents explicit counterexamples showing that not all nonlocal games with quantum advantage necessarily certify specific quantum states, challenging assumptions about the universality of self-testing.

Contribution

It introduces the $( ext{G}_1 ext{ or } ext{G}_2)$-game to demonstrate limitations of self-testing, providing the first explicit counterexamples in this area.

Findings

Not all nonlocal games with quantum advantage certify specific states

Some self-testing results cannot be made robust with simple modifications

The introduced game class is of independent theoretical interest

Abstract

In the recent years self-testing has grown into a rich and active area of study with applications ranging from practical verification of quantum devices to deep complexity theoretic results. Self-testing allows a classical verifier to deduce which quantum measurements and on what state are used, for example, by provers Alice and Bob in a nonlocal game. Hence, self-testing as well as its noise-tolerant cousin -- robust self-testing -- are desirable features for a nonlocal game to have. Contrary to what one might expect, we have a rather incomplete understanding of if and how self-testing could fail to hold. In particular, could it be that every 2-party nonlocal game or Bell inequality with a quantum advantage certifies the presence of a specific quantum state? Also, is it the case that every self-testing result can be turned robust with enough ingeniuty and effort? We answer these questions in the negative by providing simple and fully explicit counterexamples. To this end, given two nonlocal games $\mathcal{G}_1$ and $\mathcal{G}_2$, we introduce the $(\mathcal{G}_1 \lor \mathcal{G}_2)$-game, in which the players get pairs of questions and choose which game they want to play. The players win if they choose the same game and win it with the answers they have given. Our counterexamples are based on this game and we believe this class of games to be of independent interest.