  The environment of an open quantum system is usually modelled as a large
many-body quantum system. However, when an isolated quantum system itself is a
many-body quantum system, the question of how large and complex it must be in
order to generate internal equilibration is an open key-point in the
literature. In this work, by monitoring the degree of equilibration of a single
spin through its purity degradation, we are able to sense the chaotic behaviour
of the generic spin chain to which it is coupled. Quite remarkably, this holds
even in the case of extremely short spin chains composed of three spins, where
we can also reproduce the whole integrable to chaos transition. Finally, we
discuss implications on quantum control experiments and show that quantum chaos
reigns over the best degree of control achieved, even in small chains.
