  I compute the leading contribution to the ground state Renyi entropy
$S_{\alpha}$ for a region of linear size $L$ in a Fermi liquid. The result
contains a universal boundary law violating term simply related the more
familiar entanglement entropy. I also obtain a universal crossover function
that smoothly interpolates between the zero temperature result and the ordinary
thermal Renyi entropy of a Fermi liquid. Formulas for the entanglement entropy
of more complicated regions, including non-convex and disconnected regions, are
obtained from the conformal field theory formulation of Fermi surface dynamics.
These results permit an evaluation of the quantum mutual information between
different regions in a Fermi liquid. I also study the number fluctuations in a
Fermi liquid. Taken together, these results give a reasonably complete
characterization of the low energy quantum information content of Fermi
liquids.
