The physical significances and applications of Schmidt decompositions of continuum bipartile entangled state representation

According to the continuum bipartite entangled state representation vertical bar eta > we derive the Schmidt decomposition of vertical bar eta >, respectively in the coordinate representation, momentum representation and the particle number representation, and explain their physical significances. As the applications of the Schmidt decomposition, we directly derive the action of the single-mode squeeze operator on vertical bar eta >, the two-mode squeeze operator's entangled state representation, as well as the matrix element of displacement operator in the Fock space. Generalization of our discussion in this paper to the multipartite entangled state cases is feasible.