Self-testing GHZ state via a Hardy-type paradox

TL;DR

This work develops a device-independent self-testing protocol for the tripartite GHZ state based on a generalized Hardy paradox. It proves that the maximal tripartite Hardy probability is achieved by GHZ correlations that form an extremal, exposed point of the quantum set, and it shows these correlations coincide with maximal Mermin inequality violations, unifying paradox-based and inequality-based views. A robust, swap-based analysis supplemented by the NPA hierarchy yields quantitative fidelity bounds for both the state and measurements under realistic noise. The results illuminate the geometric structure of quantum correlations and open avenues for exploring exposed extremality in higher-party Hardy scenarios and other paradoxes.

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.

Figures (1)

• Figure 1: Plots illustrate the robustness of the self-testing result corresponding to the correlation that maximally violates the tripartite Hardy paradox. To quantify the quality of self-testing of the state, we adopt the fidelity with respect to the chosen reference state \ref{['figmeritstate']} as the figure of merit. For the self-testing of the measurements, the relevant figure of merit is defined in \ref{['figmeritmeasure']}. Because the reference measurements of $A_1$, $A_2$, and $A_3$ are similar, it is sufficient to display the plot for $A_1$ only. Throughout our analysis, the parameter $\varepsilon_2$ denotes the admissible deviation from the ideal zero-probability condition. All numerical results reported here are obtained using the level-6 outer approximation of the quantum set $\mathcal{Q}$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof