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arXiv:math/0301036 · Physics Letters A

Entanglement via Barut–Girardello coherent state for suq(1,1) quantum algebra: bipartite composite system

R. Chakrabarti, S S Vasan

Published May 13, 2003• 14 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(03)00653-4

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Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.
This paper concerns the construction of su(r + 1) Barut–Girardello coherent states in terms of generalized Grassmann variables. We first introduce a generalized Weyl–Heisenberg algebra A(r) (r≥1) generated by r pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit (k-level) systems. It includes the usual Weyl–Heisenberg and su(2) algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock-Bargmann space associated with the algebra A(r). The Barut–Girardello coherent states for su(r + 1) algebras are explicitly derived and their over-completion properties are discussed.