Experimental demonstration of the Bell-type inequalities for four qubit Dicke state using IBM Quantum Processing Units

Abstract

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 ($\ddot{I}{ƒ}_x$, $\ddot{I}{ƒ}_y$, and $\ddot{I}{ƒ}_z$). 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 $2.821\pm 0.0019$ using M3 error mitigation, which is within $0.7\ddot{I}ƒ$ of the theoretical maximum $2\sqrt{2}$. In the four-qubit case, we employ a Bell-type inequality tailored for Dicke states and achieve a maximum violation of $2.607\pm 0.029$ 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.