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Advanced Quantum Technologies

Violation of Qudit Mermin's Inequality and Its Application to Quantum Computing

Jinwon Yoo, Stefania Sciara, Nicola Montaut, Agnes George, Roberto Morandotti

Published Mar 31, 2026• 0 citations

Abstract

ABSTRACT Mermin's inequality provides a fundamental test to verify the quantum nonlocality of multipartite Greenberger–Horne–Zeilinger (GHZ) states, by also serving as a key resource for measurement‐based quantum computing. While the theoretical foundation of Mermin's inequality test is well established for both qubit and qudit systems, the experimental validation for qudits remains elusive. In this work, we perform qudit Mermin's inequality test implemented on a qutrit (i.e., three‐level system) GHZ state, where the multipartition is achieved by deterministically entangling the time and frequency bins of a biphoton state via standard fiber‐optic components. The direct measurement of Mermin's operator yields an expectation value of 8.612(44), surpassing the bound of 6 for local realism by 59 standard deviations. We demonstrate the potential of the verified GHZ state for high‐dimensional quantum computing by utilizing the nonlocality results to compute a ternary logic function through non‐adaptive high‐dimensional measurement‐based quantum computation. Our scheme is readily scalable to higher dimensions without additional loss or modifications to the experimental setup.

DOI: 10.1002/qute.202500925

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We investigate experimentally the violation of the Bell, Mermin and Svetlichny inequalities to local realism prediction. Using the maximum-likelihood technique to construct the density matrix and calculate the fidelity of the Greenberger-Horne-Zeilinger (GHZ) state. The values of the three inequalities are measured. The experimental resuls show that the three inequalities have different violation degrees to local realism prediction. So we can choose a suitable method of nonlocality to estimate the security of quantum channel.

Abstract Non-locality is not only one of the most prominent quantum features but can also serve as a resource for various information-theoretical tasks. Analysing it from an information-theoretical perspective has linked it to applications such as non-adaptive measurement-based quantum computing (NMQC). In this type of quantum computing the goal is to output a multivariate function. The success of such a computation can be related to the violation of a generalised Bell inequality. So far, the investigation of binary NMQC with qubits has shown that quantum correlations can compute all Boolean functions using at most 2 n − 1 qubits, whereas local hidden variables (LHVs) are restricted to linear functions. Here, we extend these results to NMQC with qutrits and prove that quantum correlations enable the computation of all three-valued logic functions using the generalised qutrit Greenberger–Horne–Zeilinger (GHZ) state as a resource and at most 3 n − 1 qutrits. This yields a corresponding generalised GHZ type paradox for any three-valued logic function that LHVs cannot compute. We give an example for an n-variate function that can be computed with only n + 1 qutrits, which leads to convenient generalised qutrit Bell inequalities whose quantum bound is maximal. Finally, we prove that not all functions can be computed efficiently with qutrit NMQC by presenting a counterexample.

In this paper, we theoretically study the relation between concurrence and nonlocality depicted by Bell-type inequality violation of quantum mechanics prediction versus local realism prediction for the GHZ (Greenberger -Horne-Zeilinger) class states. Analytical expressions of concurrence, violations of the Mermin inequality and the Svetlichny inequality are obtained. Through numerical calculations, the relationship between entanglement and nonlocality of GHZ-class states is discussed. Our results show that the concurrence is consistent with the degree of nonlocality described by violations of the two Bell-type inequalities of GHZ-class states. The Bell operator and its parameters can obviously reveal the nonlocal features of quantum states.

Multipartite Bell tests provide a correlation-only route to benchmarking quantum processors, but their application at large scales is hindered by the rapid decay of many-body correlators under noise and exponentially many terms in conventional Bell expressions. Here we address these scalability obstacles by introducing a finite-setting generalized Mermin family of state-tailored Bell inequalities with analytic certification bounds, in which the measurement-setting number

m

provides an additional certification dimension complementary to the system size

n

. We show that, for the powers-of-two setting choices considered here, increasing

m

leaves the ideal normalized multipartite quantum value unchanged while lowering the relevant classical bounds, thereby strengthening the Bell-violation ratios and yielding an improved noise-robustness scaling compared to the standard Mermin inequality. We test this construction experimentally on a programmable superconducting processor by preparing Greenberger-Horne-Zeilinger (GHZ) states of up to 80 qubits. Using randomized sampling for direct Bell-operator estimation, we observe Bell ratios that grow exponentially with system size, certify a nonlocality depth of 14, and show that increasing

m

strengthens both the Bell ratio and depth certification. All results are obtained solely from measured correlators and analytical bounds, without readout correction, tomography, or model-based mitigation. Generalized Mermin inequalities therefore provide a sharper Bell benchmark for noisy large-scale GHZ states.

Using spectral joint measurements of the qubits, we propose a scheme to test the tripartite Mermin inequality with three qubits dispersively coupled to a driven cavity. First, we show how to generate a three-qubit Greenberger-Horne-Zeilinger (GHZ) state by only one-step quantum operation. Then spectral joint measurements are introduced to directly confirm such tripartite entanglement. Assisted by a series of single-qubit operations, these measurements are further utilized to test the Mermin inequality. The feasibility of the proposal is robustly demonstrated by the present numerical experiments.