Bell’s inequality and entanglement in qubits

Abstract

Abstract:We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell's inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell's inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell's inequality to the generalized concurrence of a pure state with respect to different bipartitions.

Tripartite entanglement and quantum correlation

“…Hence Quantum Entanglement cannot be a source of violation for Mermin's inequality. Choosing other operators can show the monotonically increasing behavior in the GHZ class [7,8]. However, the proposal is also not general.…”

“…In other words, the 2-qubit state is too unusual. It is hard to extend the relationship to a general n-qubit state [7,8]. Indeed, various difficulties of many-body Quantum Entanglement already appear in the 3-qubit state.…”

Maximally entangled state and Bell’s inequality in qubits

“…We use seven-qubit quantum states as an example to estimate intensity of quantum entanglement in the maximally entangled state through Bell's inequality [11,18]. A ground state of a toric code model [3,4] on a disk manifold can be shown to have local maximum quantum entanglement in our quantum states through an upper bound of Bell's inequality [9,14,17]. We also discuss relations between an upper bound of Bell's inequality and concurrences of pure states of a region [13].…”

“…A ), and the entanglement entropy of the region A, S A , through the two maximal eigenvalues of the 3 × 3 Rmatrix R ij ≡ Tr(ρσ i ⊗ σ j ) [14,15], where i = 1, 2, 3 and j = 1, 2, 3, and σ x , σ y , σ z are the Pauli matrices defined in [16]. For some n-qubit quantum states, an upper bound of Bell's inequality is also a monotonic function with respect to the generalized concurrence of the pure state of a region A [9,17], C A (m, ψ) ≡ 2(1 − 2 m−1 Tr A ρ 2 A ), and the entanglement entropy of the region A through the generalized R-matrix [9,17], R i 1 i 2 ···in ≡ Tr(ρσ i 1 ⊗ σ i 2 ⊗ · · · ⊗ σ in ) ≡ R Iin , where i α = x, y, z and α = 1, 2, · · · , n are the site indices. The generalized R-matrix can be rewritten as a 3 n−1 × 3 matrix R Iin with the first index being a multi-index I = i 1 i 2 · · · i n−1 and the second index being i n .…”

Computing the Maximal Violation of Bell Inequalities for Multipartite Qubit via Partially Symmetric Tensor

“…In [20], Horodecki et al showed that for arbitrary mixed two-qubit state the maximal violation of CHSH inequalities is related to the eigenvalues of correlation matrix, and derived a necessary and sufficient condition for violating the inequalities accordingly. Based on this idea, the authors [21] generalize the the the maximal violation of the Bell's inequality in a nqubit system by using the unfolded matrix of correlation tensor. It should be noted that this generalization is direct.…”