Self-testing GHZ state via a Hardy-type paradox

Abstract

Self-testing is a correlation-based framework that enables the certification of both the underlying quantum state and the implemented measurements without imposing any assumptions on the internal structure of the devices. In this work, we introduce a self-testing protocol for the Greenberger-Horne-Zeilinger (GHZ) state based on a natural generalization of Hardy's nonlocality argument. Within this framework, we prove that the correlation achieving the maximal Hardy success probability constitutes an extremal point of the quantum correlation set and, moreover, that this point is \emph{exposed}. To address experimentally relevant imperfections, we further develop a robust self-testing analysis tailored to the Hardy construction. Additionally, we show that, in this scenario, the quantum correlation that attains the maximal violation of the Hardy-type paradox coincides with the correlation that yields the maximal violation of the Mermin inequality. This establishes a unified perspective in which the same multipartite correlation admits both a logical-paradox interpretation and a Bell-inequality-based characterization. Collectively, our results pave the way for investigating whether the correlations that maximally violate the generalized $N$-party Hardy paradox remain exposed in higher-party regimes.