Quantum entanglement and the Bell matrix

G. Fano, S. M. Blinder

Liangmeng Xie

Quantum entanglement is a core resource for quantum information processing, and entanglement witnesses are indispensable tools for detection and verification. This paper first clarifies some already-known properties of entanglement witnesses and the Bell State. By applying mathematical tools such as Kronecker product and matrix algebra, this paper analyzes elements in the Hermitian matrix and discusses the conditions for it to form an entanglement witness. This paper also considers both necessary and sufficient conditions of entanglement witness to impose restrictions on elements and investigated several inequality properties of two-qubit entanglement witness to detect Bell states.

Xingyu Guo, Chen‐Te Ma

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.

Shengjun Wu

Weijie Guo, Lian-Fu Wei, Dai-He Fan

Bell’s inequality plays an important role in quantum mechanics. Many experiments have been made to test the Bell’s inequality by using various bipartite entangled sources. However, the maximal violation of the Bell inequality (i.e., the Cirel’son’s limit) has not been reached yet, due to various practically-existing imperfections (e.g., entangled source is not the pure maximally-entangled state, and the measurement settings used are not the best ones). In this paper, we investigate experimentally an approach to enhance the violation degree of Bell’s inequality with the commercialized entangled photons by optimizing the measurement settings for the experimentally-reconstructed density matrix of the entangled photons. It is shown that the measurement setting related to the pure entangled state can not get maximally violation generally. However, upon the density matrix reconstructed by tomographic technique, optimal measurement settings can be found. With these optimal settings, we consequently found that the violation degree of the Bell’s inequality can be increased. This suggests a possible approach to optimize the test of Bell’s inequality.