  It is demonstrated that quantum systems classically exhibiting strong and
homogeneous chaos in a bounded region of the phase space can induce a global
quantum diffusion. As an ideal model system, a small quantum chaos with finite
Hilbert space dimension $N$ weakly coupled with $M$ additional degrees of
freedom which is approximated by linear systems is proposed. By twinning the
system, the diffusion process in the additional modes can be numerically
investigated without taking the unbounded diffusion space into account
explicitly. Even though $N$ is not very large, diffusion occurs in the
additional modes as the coupling strength increases if $M \geq 3$. If $N$ is
large enough, a definite quantum transition to diffusion takes place through a
critical sub-diffusion characterized by an anomalous diffusion exponent.
