Quantum Information Theory

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. We also make a short review of Holonomic Quantum Computation (Computer) and show a geometric construction of the well–known Bell states by making use of generalized coherent states. E-mail address : [email protected] Home-page : http://fujii.sci.yokohama-cu.ac.jp 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. We also present a problem on a possibility of calculation or approximation of coherent state path integrals on Holonomic Quantum Computer. In conclusion we state our dream, namely, a construction of Geometric Quantum Information Theory.