Quantum coherent operators: A generalization of coherent states

Élliott H. Lieb, Jan Philip Solovej

Dušan Popov

In the paper we developed a procedure for constructing generalized coherent states with shifted argument, as a result of the action of the generalized displacement operator. This was based on the action of a pair of nonlinear ladder operators, which generate nonlinear coherent states. To examine the properties of coherent states with shifted argument, the rules of the normal operator ordering technique (DOOT) were used. The results obtained were verified for a series of particular cases, obtaining expressions consistent with those in the literature. The expressions obtained will be, among others, useful in quantum optics, e.g. for calculating the Wigner operator in the representation of coherent states.

Jan Philip Solovej

Zhe Chang

A set of deformed k-boson operators are defined and it is shown that the nonlinear transformations from the usual deformed oscillator operators to the deformed k-boson operators form an Abelian group. It is shown that the multiboson operators satisfy quantum Weyl–Heisenberg group. The generalized Holstein–Primakoff realizations in terms of these multibosons operators are used to define the quantum group-theoretic coherent states of SUq(1,1) and SUq(2). The uncertainties ΔX(k′,q) and ΔP(k′,q) with respect to those group-theoretic and generalized coherent states are computed explicitly and the results show that these coherent states are squeezed.

P. Gluck

Kazuyuki Fujii

The purpose of this paper is to introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1), and to give some applications of them to quantum information theory for graduate students or non--experts who are interested in both Geometry and Quantum Information Theory. In the first half we make a general review of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1) from the geometric point of view and, in particular, prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space. In the latter half we apply a method of generalized coherent states to some important topics in Quantum Information Theory, in particular, swap of coherent states and cloning of coherent ones. We construct the swap operator of coherent states by making use of a generalized coherent operator based on su(2) and show an "imperfect cloning" of coherent states, and moreover present some related problems. In conclusion we state our dream, namely, a construction of {\bf Geometric Quantum Information Theory}.