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Physical Review A

Bell’s inequality for an entanglement of nonorthogonal states

A. Mann, Barry C. Sanders, William J. Munro

Published Feb 1, 1995• 108 citations
Publisher

Abstract

Given an entanglement of two systems involving nonorthogonal states, we find the Schmidt decomposition for the state. The relation between the Schmidt representation and an ideal measurement of the degree of entanglement of the states is discussed, and a Bell inequality is shown to be violated. The maximal violation of the Bell inequality provides a measurement of the degree of entanglement. The entangled coherent states are provided as a concrete example of the Bell inequality for entangled nonorthogonal states.

DOI
10.1103/physreva.51.989

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Given an entanglement of two systems involving nonorthogonal states, we find the Schmidt decomposition for the state. The relation between the Schmidt representation and an ideal measurement of the degree of entanglement of the states is discussed, and a Bell inequality is shown to be violated. The maximal violation of the Bell inequality provides a measurement of the degree of entanglement. The entangled coherent states are provided as a concrete example of the Bell inequality for entangled nonorthogonal states.
We show that bipartite entanglements involving non-orthogonal states are {\it necessarily nonmaximally} entangled, however {\it small} the non-orthogonality may be. How the deviation from maximal entanglement is related to nonorthogonality is quantified by using two independent measures of entanglement corresponding to the violation of Bell's inequality and the entropy measure respectively. This result is true even if one of the subsystems has orthogonal states. An application of this is that a maximal violation of Bell's inequality in entangled neutral kaons is not possible in the presence of CP violation.
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 violated by the quantum mechanical predictions made from an entangled state of the composite system. In this paper we examine this inequality and entanglement measures in the construction of the coherent states for two-qubit pure and mixed states. we find a link to some entanglement measures through some new parameters (amplitudes of coherent states). Conditions for maximal entanglement and separability are then established for both pure and mixed states. Finally, we analyze and compare the violation of Bell inequality for a class of mixed states with the degree of entanglement by applying the formalism of Horodecki et al.
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.
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.
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.