  We study different aspects of quantum field theory at finite density using
methods from quantum information theory. For simplicity we focus on massive
Dirac fermions with nonzero chemical potential, and work in $1+1$ space-time
dimensions. Using the entanglement entropy on an interval, we construct an
entropic $c$-function that is finite. Unlike what happens in Lorentz-invariant
theories, this $c$-function exhibits a strong violation of monotonicity; it
also encodes the creation of long-range entanglement from the Fermi surface.
Motivated by previous works on lattice models, we next calculate numerically
the Renyi entropies and find Friedel-type oscillations; these are understood in
terms of a defect operator product expansion. Furthermore, we consider the
mutual information as a measure of correlation functions between different
regions. Using a long-distance expansion previously developed by Cardy, we
argue that the mutual information detects Fermi surface correlations already at
leading order in the expansion. We also analyze the relative entropy and its
Renyi generalizations in order to distinguish states with different charge
and/or mass. In particular, we show that states in different superselection
sectors give rise to a super-extensive behavior in the relative entropy.
Finally, we discuss possible extensions to interacting theories, and argue for
the relevance of some of these measures for probing non-Fermi liquids.
