In this paper we make a brief review of some basic properties (the matrix elements, the trace, the Glauber formula) of coherent operators and study the corresponding ones for generalized coherent operators based on Lie algebra su(1,1). We also propose some problems.
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}.
The review is devoted to an analysis of definite overcomplete non-orthogonal state systems that are connected with irreducible representations of Lie groups–the so called systems of generalized coherent states. These systems, which the author is the first to propose, are generalizations of Glauber's coherentstate system and arise in natural fashion in physical problems that have dynamic symmetry. They permit a considerable simplification of the solution of the quantum problem by reducing it to a simpler classical problem. The review deals with the properties of generalized-coherent-state systems connected with the simplest Lie groups.
Abstract We provide a formula for computing the overlap between two generalized coherent states of any rank one simple Lie algebra. Then, we apply our formula to spin coherent states (i.e. su ( 2 ) algebra), pseudo-spin coherent states (i.e. su ( 1 , 1 ) algebra), and the sl ( 2 , R ) subalgebras of Virasoro. In all these examples, we show the emergence of a semi-classical behavior from the set of coherent states and verify that it always happens when some parameter, depending on the algebra and its representation, becomes large.