  In his article "Transcending Classical Invariant Theory" (J.A.M.S., 1989, Vol
2), Roger Howe established a correspondence between representations of a dual
pair of reductive groups. This correspondence is known as Howe's correspondence
or local theta correspondence. In my paper "Unitary Representations and Theta
Correspondence for Classical Group of Type I"(to appear in J. Func. Anal.), we
proved that theta correspondence over the real field preserves unitarity under
certain restrictions, generalizing some results of Jian-Shu Li (Invent. Math.,
Vol 97, 1989). In this paper, we study the compositions of theta
correspondences. Under certain favorable conditions, we prove that the
composition of theta correspodence yields unitary representations. We call this
process quantum induction. Quantum induction resembles parabolic induction in
various ways. Some results in this paper were announced at ICM-2002.
