  We measure pressure and entropy of ultracold fermionic atoms in an optical
lattice for a range of interaction strengths, temperatures and fillings. Our
measurements demonstrate that, for low enough temperatures, entropy-rich
regions form locally in the metallic phase which are in contact with a
Mott-insulating phase featuring lower entropy. In addition, we also measure the
reduced density matrix of a single lattice site, and from the comparison
between the local and thermodynamic entropies we determine the mutual
information between a single lattice site and the rest of the system. For low
lattice fillings, we find the mutual information to be independent of
interaction strength, however, for half filling we find that strong
interactions suppress the correlations between a single site and the rest of
the system.
