Optical transformation from the coherent state to the quantum entangled state of continuous variables

Optical Fresnel diffraction is described by the Fresnel integration transformation. In this paper, starting from the coherent state and its function space we find a direct route to obtaining the quantum entangled state of continuous variables and the corresponding function space. The mutual integration transformation between these two function spaces are revealed. This transformation is helpful to introduce the Hermite-polynomial excitation state in quantum optics.