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Communications in Theoretical Physics

Degree of Entanglement and Violation of Bell Inequality by Two-Spin-1/2 States

K. Berrada, Y. Hassouni, Hichem Eleuch

Published Oct 1, 2011• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/56/4/15

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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.
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In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.
In the present paper, we examine the quantum entanglement for more general states of two-qubit system in the context of spin coherent states (SCSs). We consider the concurrence as a quantifier of entanglement and express it in terms of SCSs. We determine new set of maximally entangled conditions that provide the maximal amount of entanglement for certain values of the amplitudes of SCSs for the case of pure states. Finally, we examine the entanglement of a class of mixed states of the two qubits and provide the range in which the entanglement value is maximal with respect to the values of the amplitudes of SCSs.
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.