  A key open issue in condensed matter physics is how quantum and classical
correlations emerge in an unconventional superconductor from the underlying
normal state. We study this problem in a doped Mott insulator with information
theory tools on the two-dimensional Hubbard model at finite temperature with
cluster dynamical mean-field theory. We find that the local entropy detects the
superconducting state and that the difference in the local entropy between the
superconducting and normal states follows the same difference in the potential
energy. We find that the thermodynamic entropy is suppressed in the
superconducting state and monotonically decreases with decreasing doping. The
maximum in entropy found in the normal state above the overdoped region of the
superconducting dome is obliterated by superconductivity. The total mutual
information, which quantifies quantum and classical correlations, is amplified
in the superconducting state of the doped Mott insulator for all doping levels,
and shows a broad peak versus doping, as a result of competing quantum and
classical effects.
