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Advanced Quantum Technologies

Experimental Verification of Bell‐Type Inequalities Using Four‐Qubit Dicke States on Quantum Processors

Tomis Prajapati, Harsh Mehta, Shreya Banerjee, Prasanta Kumar Panigrahi, V. Narayanan

Published Sep 25, 2025• 2 citations
Publisher

Abstract

Abstract Testing for a violation of Bell‐type inequalities provides a standard approach to investigating nonlocal correlations in nonclassical (entangled) states. In this study, a custom measurement operator composed of a linear combination of Pauli matrices (, , and ) is constructed to examine such violations. Both theoretical and experimental analyses of Bell‐type inequality violations using two‐ and four‐qubit Dicke states implemented on quantum computers are presented. Specifically, two methods for preparing four‐qubit Dicke states—gate‐based and statevector‐based—are compared, and their performance is assessed on two IBM superconducting quantum processors, ibm_kyiv and ibm_sherbrooke . In the two‐qubit scenario, a clear violation of the CHSH inequality is observed, with a maximum Bell parameter of achieved using M3 error mitigation, closely approaching the theoretical upper bound of . For the four‐qubit case, a Dicke‐state‐specific Bell‐type inequality is applied and a maximal violation of is reported without additional mitigation using the statevector‐based approach. These findings show that while error mitigation improves outcomes in gate‐based methods, the statevector‐based approach naturally provides higher fidelity with reduced noise. This work underscores the importance of state preparation strategies and noise management in exploring quantum correlations on current quantum computing platforms.

DOI
10.1002/qute.202500366

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Abstract Testing for a violation of Bell‐type inequalities provides a standard approach to investigating nonlocal correlations in nonclassical (entangled) states. In this study, a custom measurement operator composed of a linear combination of Pauli matrices (, , and ) is constructed to examine such violations. Both theoretical and experimental analyses of Bell‐type inequality violations using two‐ and four‐qubit Dicke states implemented on quantum computers are presented. Specifically, two methods for preparing four‐qubit Dicke states—gate‐based and statevector‐based—are compared, and their performance is assessed on two IBM superconducting quantum processors, ibm_kyiv and ibm_sherbrooke . In the two‐qubit scenario, a clear violation of the CHSH inequality is observed, with a maximum Bell parameter of achieved using M3 error mitigation, closely approaching the theoretical upper bound of . For the four‐qubit case, a Dicke‐state‐specific Bell‐type inequality is applied and a maximal violation of is reported without additional mitigation using the statevector‐based approach. These findings show that while error mitigation improves outcomes in gate‐based methods, the statevector‐based approach naturally provides higher fidelity with reduced noise. This work underscores the importance of state preparation strategies and noise management in exploring quantum correlations on current quantum computing platforms.
Violation of the Bell-type inequalities is necessary to confirm the existence of nonlocality in nonclassical (entangled) states. We have designed a customized operator which is made of the sum of the Pauli matrices (σxσ_xσx​, σyσ_yσy​, and σzσ_zσz​). We theoretically and experimentally investigate the violation of Bell-type inequalities using two- and four-qubit Dicke states on IBM Quantum Processing Units (QPUs). We compare two different state preparation methods for the four-qubit Dicke state -- gate-based and statevector-based -- and evaluate their performance on two IBM QPUs, \texttt{ibm\_kyiv} and \texttt{ibm\_sherbrook}. For the two-qubit case, we demonstrate clear violations of the CHSH inequality, with the highest observed Bell parameter reaching 2.821±0.00192.821 \pm 0.00192.821±0.0019 using M3 error mitigation, which is within 0.7σ0.7σ0.7σ of the theoretical maximum 222\sqrt{2}22​. In the four-qubit case, we employ a Bell-type inequality tailored for Dicke states and achieve a maximum violation of 2.607±0.0292.607 \pm 0.0292.607±0.029 without the need for additional mitigation when using the statevector-based method. Our results reveal that advanced error mitigation techniques significantly enhance the observed violations in the gate-based method, while the statevector-based approach inherently yields more robust states with lower noise. This study highlights the critical role of state preparation and mitigation techniques in probing fundamental quantum correlations on near-term quantum hardware.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.
Leggett-Garg's inequalities predict sharp bounds for some classical correlation functions that address the quantum or classical nature of real-time evolutions. We experimentally observe the violations of these bounds on single- and multiqubit systems, in different settings, exploiting the IBM Quantum platform. In the multiqubit case, we introduce the Leggett-Garg-Bell's inequalities as an alternative to the previous ones. Measuring these correlation functions, we find quantum error mitigation to be essential to spot inequalities violations. Accessing only two qubit readouts, we assess Leggett-Garg-Bell's inequalities to emerge as the most efficient quantum coherence witnesses to be used for investigating quantum hardware, among those introduced. Our analysis highlights the limits of current quantum platforms, showing that the above-mentioned correlation functions deviate from theoretical prediction as the number of qubits and the depth of the circuit grow.
Testing and verifying imperfect multi-qubit quantum devices are important as such noisy quantum devices are widely available today. Bell inequalities are known to be useful for testing and verifying the quality of the quantum devices from their nonlocal quantum states and local measurements. There have been many experiments demonstrating the violations of Bell inequalities, but they are limited in the number of qubits and the types of quantum states. We report violations of Bell inequalities on IBM Quantum devices based on the scalable and robust inequalities maximally violated by graph states as proposed by Baccari et al.. The violations are obtained from the quantum states of path graphs up to 57 and 21 qubits on a 65-qubit and two 27-qubit IBM Quantum devices, respectively, and from those of star graphs up to 11 qubits with quantum readout error mitigation (QREM). We are able to show violations of the inequalities on various graph states by constructing low-depth quantum circuits and by applying the QREM technique. We also point out that quantum circuits for star graph states of sizeNNNcan be realized with circuits of depthO(N)O(\sqrt {N})O(N​)on subdivided honeycomb lattices which are the topology of the 65-qubit IBM Quantum device. Our experiments show encouraging results on the ability of existing quantum devices to prepare entangled quantum states and provide experimental evidence on the benefit of scalable Bell inequalities for testing them.
Abstract This laboratory work presents a practical implementation of the CHSH game using the Qiskit framework and IBM quantum hardware. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and aims to provide an experimental demonstration of quantum nonlocality and Bell inequality violation. In Laboratory Work No. 4, participants implement a simplified version of the Clauser–Horne–Shimony–Holt (CHSH) game to test whether IBM quantum processors exhibit genuinely quantum behavior. The CHSH game is a fundamental protocol in quantum information theory that distinguishes classical correlations from quantum correlations through statistical violation of Bell inequalities. Using Qiskit, a two-qubit entangled state is prepared and measurement settings corresponding to different basis choices are applied. By executing the circuit multiple times on a simulator and real quantum hardware, participants collect measurement statistics and compute the CHSH correlation parameter. If the experimentally obtained value exceeds the classical bound, the results provide empirical evidence of quantum entanglement and nonlocal correlations. This laboratory exercise demonstrates how foundational concepts of quantum mechanics—entanglement, superposition, and measurement basis dependence—can be directly tested using cloud-accessible quantum processors. The experiment not only reinforces theoretical understanding of Bell inequalities but also introduces practical skills in circuit construction, parameterized measurements, statistical analysis, and noise considerations in real quantum devices.