  The detailed characterization of non-trivial coherence properties of
composite quantum systems of increasing size is an indispensable prerequisite
for scalable quantum computation, as well as for understanding of
nonequilibrium many-body physics. Here we show how autocorrelation functions in
an interacting system of phonons as well as the quantum discord between
distinct degrees of freedoms can be extracted from a small controllable part of
the system. As a benchmark, we show this in chains of up to 42 trapped ions, by
tracing a single phonon excitation through interferometric measurements of only
a single ion in the chain. We observe the spreading and partial refocusing of
the excitation in the chain, even on a background of thermal excitations. We
further show how this local observable reflects the dynamical evolution of
quantum discord between the electronic state and the vibrational degrees of
freedom of the probe ion.
