  The methods of Information geometry have been glowing up to develop various
subjects of theoretical physics, including quantum information systems. The
present article has two purposes. The first one is to develop general theory of
homogeneous statistical manifolds. In particular we construct explicit examples
of homogeneous statistical manifolds of low dimension. The second purpose is to
classify $3$-dimensional Lie groups admitting non-trivial conjugate symmetric
left invariant statistical structure.
