Introduction to Coherent States and Quantum Information Theory

TL;DR

This paper introduces theorems on coherent states and their applications in quantum information, focusing on geometric aspects and demonstrating swap and cloning of coherent states.

Contribution

It provides a geometric framework for coherent states and applies it to quantum information tasks like state swapping and cloning, introducing new methods based on Lie algebra representations.

Findings

Resolution of unity via curvature forms on parameter space

Construction of swap operator for coherent states

Demonstration of imperfect cloning of coherent states

Abstract

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}.