  The quantum states or Hilbert spaces for the quantum field theory in de
Sitter space-time are studied on ambient space formalism. In this formalism,
the quantum states are only depended $(1)$ on the topological character of the
de Sitter space-time, {\it i.e.} $\R \times S^3$, and $(2)$ on the homogeneous
spaces which are used for construction of the unitary irreducible
representation of de Sitter group. A compact homogeneous space is chosen in
this paper. The unique feature of this homogeneous space is that its total
number of quantum states, ${\cal N}$, is finite although the Hilbert space has
infinite dimensions. It is shown that ${\cal N}$ is a continuous function of
the Hubble constant $H$ and the eigenvalue of the Casimir operators of de
Sitter group. The entropy of the quantum fields on this Hilbert space have been
calculated which is finite and invariant for all inertial observers on the de
Sitter hyperboloid.
