  We examine distinct measures of fermionic entanglement in the exact ground
state of a finite superconducting system. It is first shown that global
measures such as the one-body entanglement entropy, which represents the
minimum relative entropy between the exact ground state and the set of
fermionic gaussian states, exhibit a close correlation with the BCS gap,
saturating in the strong superconducting regime. The same behavior is displayed
by the bipartite entanglement between the set of all single particle states $k$
of positive quasimomenta and their time reversed partners $\bar{k}$. In
contrast, the entanglement associated with the reduced density matrix of four
single particle modes $k,\bar{k}$, $k',\bar{k}'$, which can be measured through
a properly defined fermionic concurrence, exhibits a different behavior,
showing a peak in the vicinity of the superconducting transition for states
$k,k'$ close to the fermi level and becoming small in the strong coupling
regime. In the latter such reduced state exhibits, instead, a finite mutual
information and quantum discord. And while the first measures can be correctly
estimated with the BCS approximation, the previous four-level concurrence lies
strictly beyond the latter, requiring at least a particle number projected BCS
treatment for its description. Formal properties of all previous entanglement
measures are as well discussed.
