  The decoherence of a quantum system $S$ coupled to a quantum environment $E$
is considered. For states chosen uniformly at random from the unit hypersphere
in the Hilbert space of the closed system $S+E$ we derive a scaling
relationship for the sum of the off-diagonal elements of the reduced density
matrix of $S$ as a function of the size $D_E$ of the Hilbert space of $E$. This
sum decreases as $1/\sqrt{D_E}$ as long as $D_E\gg 1$. This scaling prediction
is tested by performing large-scale simulations which solve the time-dependent
Schr{\"o}dinger equation for a ring of spin-1/2 particles, four of them
belonging to $S$ and the others to $E$. Provided that the time evolution drives
the whole system from the initial state toward a state which has similar
properties as states belonging to the class of quantum states for which we
derived the scaling relationship, the scaling prediction holds. For systems
which do not exhibit this feature, it is shown that increasing the complexity
(in terms of connections) of the environment or introducing a small amount of
randomness in the interactions in the environment suffices to observe the
predicted scaling behavior.
