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Laser Physics

Entanglement of bipartite partly non-orthogonal 1 2 -spin coherent states

Mostafa Mansour, Zakaria Dahbi

Published Jun 29, 2020• 21 citations
Publisher

Abstract

Abstract We investigate the entanglement of a bipartite partly non-orthogonal state of the form | ϕ ⟩ = u | ξ ⟩ 1 ⊗ | 1 ⟩ 2 + v | ξ ⟩ 1 ⊗ | 0 ⟩ 2 + w | ξ ′ ⟩ 1 ⊗ | 1 ⟩ 2 + z | ξ ′ ⟩ 1 ⊗ | 0 ⟩ 2 through 1 2 -spin coherent states by adopting the concurrence as a measure to quantify the degree of entanglement of the two-qubit system. After, we study the amount of entanglement of mixed states represented by a statistical mixture of partly non-orthogonal bipartite pure states. We also express the lower and upper bounds of the squared concurrence of mixed states in terms of the amplitudes of spin coherent states.

DOI
10.1088/1555-6611/ab935e

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Abstract We investigate the entanglement of a bipartite partly non-orthogonal state of the form | ϕ ⟩ = u | ξ ⟩ 1 ⊗ | 1 ⟩ 2 + v | ξ ⟩ 1 ⊗ | 0 ⟩ 2 + w | ξ ′ ⟩ 1 ⊗ | 1 ⟩ 2 + z | ξ ′ ⟩ 1 ⊗ | 0 ⟩ 2 through 1 2 -spin coherent states by adopting the concurrence as a measure to quantify the degree of entanglement of the two-qubit system. After, we study the amount of entanglement of mixed states represented by a statistical mixture of partly non-orthogonal bipartite pure states. We also express the lower and upper bounds of the squared concurrence of mixed states in terms of the amplitudes of spin coherent states.
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.
In this paper, we use the concurrence vector as a measure of entanglement, and investigate lower and upper bounds on the concurrence of a superposition of bipartite states as a function of the concurrence of the superposed states. We show that the amount of entanglement quantified by the concurrence vector is exactly the same as that quantified by III concurrence, so that our results can be compared to those given in Phys. Rev. A 76, 042328 (2007). We obtain a tighter lower bound in the case in which the two superposed states are orthogonal. We also show that when the two superposed states are not necessarily orthogonal, both lower and upper bounds are, in general, tighter than the bounds given in terms of the III concurrence. An extension of the results to the case with more than two states in the superpositions is also given.
In this paper, we investigate two different entanglement measures, the negativity and concurrence, in the case of pure and mixed states of two-qubit system basing on the spin coherent states. For two-qubit pure states, the negativity is the same as the concurrence. For mixed states, using a simplified expression of concurrence in Wootters' measure of entanglement, we write the bounds of Verstraete et al.1as a function of some new parameters and we compare the both measures for a class of mixed states.