Measuring entropy and mutual information in the two-dimensional Hubbard model

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.