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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 resultcontains a universal boundary law violating term simply related the morefamiliar entanglement entropy. I also obtain a universal crossover functionthat smoothly interpolates between the zero temperature result and the ordinarythermal Renyi entropy of a Fermi liquid. Formulas for the entanglement entropyof more complicated regions, including non-convex and disconnected regions, areobtained from the conformal field theory formulation of Fermi surface dynamics.These results permit an evaluation of the quantum mutual information betweendifferent regions in a Fermi liquid. I also study the number fluctuations in aFermi liquid. Taken together, these results give a reasonably completecharacterization of the low energy quantum information content of Fermiliquids.

Renyi entropy, mutual information, and fluctuation properties of Fermi liquids