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International Journal of Quantum Information

A BELL INEQUALITY BASED ON CORRELATION FUNCTIONS FOR THREE QUBITS

Chunfeng Wu, JING-LING CHEN, L. C. Kwek, C. H. Oh

Published Nov 1, 2005• 1 citation
Publisher

Abstract

We discuss a Bell inequality based on correlation function for three qubits. The inequality is violated by any pure entangled state for 3 qubits. The strength of the violation is the same as the result in Ref.13.
DOI
10.1142/s0219749905001249

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Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.
Violation ≠\neq= Quantum
The only entanglement quantity is concurrence in a 2-qubit pure state. The maximum violation of Bell's inequality is monotonically increasing for this quantity. Therefore, people expect that pure state entanglement is relevant to the quantum violation. To understand the relation between violation and entanglement, we extend the study to three qubits. We consider all possible 3-qubit operators with a symmetric permutation. When only turning on one entanglement measure, the numerical result shows a contradiction to the expectation. The maximum violation does not have the same behavior as in 2-qubit pure states. Therefore, we conclude Violation≠\neq=Quantum. In the end, we propose the generalized RRR-matrix or correlation matrix for the new diagnosis of Quantum Entanglement. We demonstrate the evidence by restoring the monotonically increasing result.