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A Study of Entanglement and Squeezing of Multiqubit Cluster Systems

Mojtaba Jafarpour, Ahmad Akhound

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

We study entanglement and squeezing of a cluster of spin systems under the influence of the two-axis countertwisting Hamiltonian. The squeezing parameters given by Wineland et al and also by Kitagawa et al. are chosen as the criteria of spin squeezing. The criterion of pairwise entanglement is chosen to be the concurrence and that of the bipartite entanglement the linear entropy. We also define a new squeezing parameter η, which plays a direct role in the investigation of the relationship between squeezing and entanglement. We observe that if the system is squeezed according to the Wineland’s criterion, it is squeezed according to the Kitagawa’s also, but the reverse is not always true. Moreover, if the system is squeezed according to Kitagawa’s criterion, it is pairwise entangled simultaneously and vice versa. It is also observed that the entropy is a linear function of the parameter η2.

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We study entanglement and squeezing of a cluster of spin systems under the influence of the two-axis countertwisting Hamiltonian. The squeezing parameters given by Wineland et al and also by Kitagawa et al. are chosen as the criteria of spin squeezing. The criterion of pairwise entanglement is chosen to be the concurrence and that of the bipartite entanglement the linear entropy. We also define a new squeezing parameter η, which plays a direct role in the investigation of the relationship between squeezing and entanglement. We observe that if the system is squeezed according to the Wineland’s criterion, it is squeezed according to the Kitagawa’s also, but the reverse is not always true. Moreover, if the system is squeezed according to Kitagawa’s criterion, it is pairwise entangled simultaneously and vice versa. It is also observed that the entropy is a linear function of the parameter η2.
We show that spin squeezing implies pairwise entanglement for arbitrary symmetric multiqubit states. If the squeezing parameter is less than or equal to 1, we demonstrate a quantitative relation between the squeezing parameter and the concurrence for the even and odd states. We prove that the even states generated from the initial state with all qubits being spin down, via the one-axis twisting Hamiltonian, are spin squeezed if and only if they are pairwise entangled. For the states generated via the one-axis twisting Hamiltonian with an external transverse field for any number of qubits greater than 1 or via the two-axis countertwisting Hamiltonian for any even number of qubits, the numerical results suggest that such states are spin squeezed if and only if they are pairwise entangled.
We show that the two definitions of spin squeezing extensively used in the literature [M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993) and D.J. Wineland et al., Phys. Rev. A 50, 67 (1994)] give different predictions of entanglement in the two-atom Dicke system. We analyze differences between the definitions and show that the spin squeezing parameter of Kitagawa and Ueda is a better measure of entanglement than the commonly used spectroscopic spin squeezing parameter. We illustrate this relation by examining different examples of a driven two-atom Dicke system in which spin squeezing and entanglement arise dynamically. We give an explanation of the source of the difference using the negativity criterion for entanglement.
Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classification scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett.95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
The quantum information is a new interdisciplinary science in the 21st century, and it has become a research field which attracts strongly much attention in the world. Recently, the study of the quantum computer is in the ascendant. In the process of finding a scheme for quantum computation, quantum chaos aroused much attention. It has been shown that it is possible to induce quantum chaos when running quantum computer and consequently it may destroy the operational condition of the quantum computer. The relationship between quantum entanglement and quantum chaos has already been investigated, and the spin squeezing is regarded as another classical quantum property indeed, then is there a natural relationship between quantum chaos and spin squeezing? In this paper a very typical QKT model in the quantum chaos has been discussed,and the properties of spin squeezing in the quantum chaotic system has been studied. The relation between spin squeezing parameters and chaotic parameter in the two different definitions is worked out by numerical calculation. The stronger that the regular regions are than chaotic ones, which are chosen firstly, in the classical phase space, the stronger the spin system which the initial states locate at the centre of elliptical regions as a result. If the initial states are chosen another condition, the contrary result is obtained naturally. It indicates the spin squeezing is very sensitive to the classical chaos and the classical chaos suppresses the spin squeezing. It is interesting that the definition by Wineland is more sensitive than the one by Kitagawa and Ueda, consequently a conclusion, which the quantum chaos can be described by spin squeezing, is acquired.