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arXiv:1710.10493 · International Journal of Modern Physics A

Bell’s inequality, generalized concurrence and entanglement in qubits

Po-Yao Chang, Su-Kuan Chu, Chen‐Te Ma

Published Mar 10, 2019• 7 citations
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Abstract

It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.

DOI
10.1142/s0217751x19500325

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It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.
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.
Nonlocality and entanglement are not only the fundamental characteristics of quantum mechanics but also important resources for quantum information and computation applications. Exploiting the quantitative relationship between the two different resources is of both theoretical and practical significance. The common choice for quantifying the nonlocality of a two-qubit state is the maximal violation of the Clauser-Horne-Shimony-Holt inequality. That for entanglement is entanglement of formation, which is a function of the concurrence. In this paper, we systematically investigate the quantitative relationship between the entanglement and nonlocality of a general two-qubit system. We rederive a known upper bound on the nonlocality of a general two-qubit state, which depends on the state's entanglement. We investigate the condition that the nonlocality of two different two-qubit states can be optimally stimulated by the same nonlocality test setting and find the class of two-qubit state pairs that have this property. Finally, we obtain the necessary and sufficient condition that the upper bound can be reached.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.
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.
Abstract For a multipartite quantum state, the maximal violation of all Bell inequalities constitutes a measure of its nonlocality (Loubenets 2012 J . Math . Phys . 53 , 022201). In the present article, for the maximal violation of Bell inequalities by a pure bipartite state, possibly infinite-dimensional, we derive a new upper bound expressed in terms of the Schmidt coefficients of this state . This new upper bound allows us also to specify general analytical relations between the maximal violation of Bell inequalities by a bipartite quantum state, pure or mixed, and such entanglement measures for this state as ‘negativity’ and ‘concurrence’. To our knowledge, no any general analytical relations between measures for bipartite nonlocality and entanglement have been reported in the literature though, for a general bipartite state, specifically such relations are important for the entanglement certification and quantification scenarios. As an example, we apply our new results to finding upper bounds on nonlocality of bipartite coherent states intensively discussed last years in the literature in view of their experimental implementations.
Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d − 1)N−1 on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on the maximal violation by an N-qudit state of general Bell inequalities for S settings per site. These new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite Bell inequalities. For some S, d, and N, the new upper bounds are attainable.