We derive a Bell inequality for testing violation of local realism. Quantum nonlocality of several four-qubit states is investigated. These include the Greenberger-Zeilinger-Horne (GHZ) state, $W$ state, linear cluster state, and the state $\ensuremath{\mid}\ensuremath{\chi}⟩$ that has recently been proposed in [Phys. Rev. Lett. 96, 060502 (2006)]. The Bell inequality is optimally violated by $\ensuremath{\mid}\ensuremath{\chi}⟩$ but not violated by the GHZ state. The linear cluster state also violates the Bell inequality though not optimally. The state $\ensuremath{\mid}\ensuremath{\chi}⟩$ can thus be discriminated from the linear cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by considering the usefulness of a family of four-qubit mixed states as resources for two-qubit teleportation. Our results generalize those in [Phys. Rev. Lett. 72, 797 (1994)].We derive a Bell inequality for testing violation of local realism. Quantum nonlocality of several four-qubit states is investigated. These include the Greenberger-Zeilinger-Horne (GHZ) state, $W$ state, linear cluster state, and the state $\ensuremath{\mid}\ensuremath{\chi}⟩$ that has recently been proposed in [Phys. Rev. Lett. 96, 060502 (2006)]. The Bell inequality is optimally violated by $\ensuremath{\mid}\ensuremath{\chi}⟩$ but not violated by the GHZ state. The linear cluster state also violates the Bell inequality though not optimally. The state $\ensuremath{\mid}\ensuremath{\chi}⟩$ can thus be discriminated from the linear cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by considering the usefulness of a family of four-qubit mixed states as resources for two-qubit teleportation. Our results generalize those in [Phys. Rev. Lett. 72, 797 (1994)].

arXiv:quant-ph/0611172Physical Review AMar 26, 200760 citations

Multi-particle Correlations and Characteristic Bell Inequalities

Ch. Schmid, Nikolai Kiesel, Wiesław Laskowski +3 more

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.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.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)Jun 1, 20070 citations

Maximally entangled states and the Bell inequality

Jose Luis Cereceda

arXiv:quant-ph/9812012Physics Letters AMar 1, 19968 citations

Bell inequality, Bell states and maximally entangled states for n qubits

Nicolas Gisin, Helle Bechmann-Pasquinucci

arXiv:quant-ph/9804045Physics Letters ASep 1, 1998270 citations

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

Tomis Prajapati, Harsh Mehta, Shreya Banerjee +2 more

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.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.

Advanced Quantum TechnologiesSep 25, 20252 citations

We derive a Bell inequality for testing violation of local realism. Quantum nonlocality of several four-qubit states is investigated. These include the Greenberger-Zeilinger-Horne (GHZ) state,

W

state, linear cluster state, and the state $\ensuremath{\mid}\ensuremath{\chi}⟩$ that has recently been proposed in [Phys. Rev. Lett. 96, 060502 (2006)]. The Bell inequality is optimally violated by $\ensuremath{\mid}\ensuremath{\chi}⟩$ but not violated by the GHZ state. The linear cluster state also violates the Bell inequality though not optimally. The state $\ensuremath{\mid}\ensuremath{\chi}⟩$ can thus be discriminated from the linear cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by considering the usefulness of a family of four-qubit mixed states as resources for two-qubit teleportation. Our results generalize those in [Phys. Rev. Lett. 72, 797 (1994)].