Abstract
We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.
Highlights
Enough to describe the tripartite entanglement [12]
The 3-qubit state shows all conceptual issues of many-body Quantum Entanglement that 2-qubit cannot answer
The central question that we would like to address in this letter is the following: What is the quantitative description of the 3-qubit Quantum Entanglement through Quantum Correlator? We first discuss the difficulty of building the relationship of Quantum Correlation and Entanglement from Mermin’s inequality
Summary
Aj ≡ aj · σ; Aj ≡ aj · σ; σ ≡ (σx, σy, σz). The a and a are the unit vectors. For any 3-qubit state, the upper bound of the expectation value of the Mermin’s operator is:. The different choice of Mermin’s operator should provide a different quantification to Quantum Entanglement. The R ≡ (Rx, Ry, Rz) is the generalized R-matrix. Each element of the generalized R-matrix is defined as Rj ≡ (Rjkm). We introduce the orthogonal unit-vectors, c; c , as in the following:. We rewrite the formula as M = 2 cos(θ) a1, Rc +2 sin(θ) a1, Rc. The matrix multiplication of RRT has three possible but not equivalent choices in general: Rj(11J) 1 ≡ Rj1j2j3 |J1=(j2,j3); Rj(22J) 2 ≡ Rj1j2j3 |J2=(j1,j3); Rj(33J) 3 ≡ Rj1j2j3 |J3=(j1,j2),. The inequality saturates the upper bound only when the vectors, x1 and x2, are the corresponding singular vectors for the λ. We will show that γR is not equivalent to γ in general
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