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arXiv:2410.20241

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

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

Published Oct 26, 2024• 0 citations
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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 ($σ_x$, $σ_y$, and $σ_z$). 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 $0.7σ$ 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.

DOI
10.48550/arxiv.2410.20241

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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.
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.
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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'.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.